Learning predator promotes coexistence of prey species in host–parasitoid systems
DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY …
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DYNAMICS OF A MODEL THREE SPECIESPREDATOR-PREY SYSTEM WITH CHOICEDouglas MagomoUniversity of Southern Mississippi
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The University of Southern Mississippi
DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM
WITH CHOICE
by
Douglas Magomo
A Dissertation Submitted to the Graduate Studies Office of The University of Southern Mississippi in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
Approved:
August 2007
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C o p y r ig h t b y
D o u g l a s M a g o m o
2007
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The University of Southern Mississippi
DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM
WITH CHOICE
by
Douglas Magomo
Abstract of a Dissertation Submitted to the Graduate Studies Office of The University of Southern Mississippi in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
August 2007
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ABSTRACT
DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM
WITH CHOICE
by Douglas Magomo
August 2007
Studies of predator-prey systems vary from simple Lotka-Volterra type to nonlinear
systems involving the Holling Type II or Holling Type III functional response functions.
Some systems are modeled to represent a simple food chain, while others involve mutual
ism, competition and even switching of predator-prey roles. In this study, we investigate
the dynamics of a three species system in which the principle predator has a choice of two
prey, while the prey species change their behavior from being prey to predator and vice
versa.
Biological and mathematical conditions for the existence of equilibria and local stabil
ity are given. A proof to show the nonexistence of periodic solutions in the corresponding
two species system is also given. Global stability of the coexistence equilibrium in the
top predator-prey system is demonstrated through numerical simulations. The resulting
quintic polynomial through full symmetry analysis provide for conditions for a cusp bifur
cation using resultants theory. Various numerical simulations to illustrate the population
dynamics of the corresponding two species system as well as the three species system
model are given.
ii
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ACKNOWLEDGMENTS
I w ould like to express my sincere thanks to the professors who have agreed to be part
o f this research struggle, in their capacities as advisors and com m ittee mem bers.
M y special acknow ledgem ents are reserved for my supervisors D r Sherry H erron, Dr.
Louis Rom ero and Dr. Joseph Kolibal.
D r Herron allow ed m e to conduct such research in the m athem atics education program
dom ain and I thank her for her encouragem ent.
Dr. Louis Rom ero is a researcher with Sandia N ational Laboratories, New M exico,
A lberqueque. His contribution is very unique in that w e never met physically and no one
had refered me to him for supervision. Here is a selfless professor who was w illing to
be bothered in the m iddle o f the night through phone calls of academ ic enquiry from a
stranger. He displayed som e strange patience with m e and I can only thank God who
opened his heart towards the success o f this research endeavor.
Dr. Joseph Kolibal is well known am ong students as a strict and detailed professor.
He would sit down with me to shape the research and correct m ost o f the write-up.
Thank you professors and I hope that in som e way, I would becom e the person you
collectively produced and be a representation o f each one o f you through your respective
contribution.
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TABLE OF CONTENTS
A B ST R A C T ................................................................................................................ ii
ACKNOWLEDGEMENTS.............................................................................................iii
LIST OF ILLUSTRATIONS.................................................................................... vi
LIST OF T A B L E S .................................................................................................. viii
NOTATION AND GLOSSARY................................................................................ ix
1 Background........................................................................................................... 1
1.1 Predator-prey Models 1
2 The Model Three Species S y s te m ..................................................................... 10
2.1 Mathematical Model 10
3 Two-Species S y ste m ............................................................................................. 14
3.1 Invariant Set 14
3.2 Local Stability Analysis 18
3.3 Symmetry and bifurcation in 2D 29
3.4 Other Corresponding Two Species Systems 35
4 THREE SPECIES S Y S T E M ........................................................................... 42
4.1 Invariant Set 42
4.2 Symmetry-Breaking Bifurcation 57
5 Discussions and Conclusions...................................................................................59
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5.1 General observations 59
A Computational Systems A n a ly s is ...................................................................... 63
A.l Matlab Code for Sensitivity of One Parameter 63
A.2 Matlab Code for Time Series 63
A.3 Matlab Code for Quintic Eigenvalue Problem 65
BIBLIOGRAPHY..................................................................................................... 68
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LIST OF ILLUSTRATIONS
Figure
2.1 Three sp e c ie s ........................................................................................................ 10
2.2 An increase of bij in the response function a ,j means increased interaction. . 12
2.3 An increase of c,y in the response function a i;- means a decreased but pro
longed interaction................................................................................................... 12
3.1 F l o w i n g ........................................................................................................... 17
3.2 Phase flow near zero ........................................................................................... 22
3.3 Time evolution in 2 D ........................................................................................... 25
3.4 Stability regions in the corresponding two species system.................................. 25
3.5 Transcritical........................................................................................................... 27
3.6 Single species stabilization.................................................................................. 28
3.7 Spiral and a sa d d le .............................................................................................. 29
3.8 Two spirals and a saddle ..................................................................................... 30
3.9 Low p r e y .............................................................................................................. 30
3.10 High predator........................................................................................................ 31
3.11 Saddle form ation................................................................................................. 31
3.12 Single coexistence .............................................................................................. 32
3.13 Environmental carrying capacity l im it ............................................................... 32
3.14 Branching and Limit points.................................................................................. 35
3.15 Cubic r o o t s ........................................................................................................... 38
3.16 Large d ................................................................................................................. 40
3.17 Small d ................................................................................................................. 40
3.18 Spiral and a sa d d le .............................................................................................. 41
3.19 Zoom near z e r o .................................................................................................... 41
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4.1 Pitchfork .............................................................................................................. 49
4.2 Projection of cusp ................................................................................................. ^8
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LIST OF TABLES
Table
1.1 Variables descriptions and response functions...................................................... 3
1.2 S ign ificance o f the re sp o n se fu n c tio n ................................................................................... 4
1.3 Parameter meaning for Ruxon’s system................................................................ 6
3.1 2D equilibria........................................................................................................ 21
3.2 Parameters and variables describing the predator-prey system (3.28)................ 37
3.3 Equilibrium solutions for the two species system................................................ 39
4.1 Assigning values to certain parameters, k and d vary........................................... 46
4.2 Block matrices ..................................................................................................... 55
4.3 Critical population density at critical parameter values....................................... 57
viii
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NOTATION AND GLOSSARY
G e n era l U sage a n d T erm inology
The notation used in this text represents fairly standard m athem atical and com putational
usage as used in M athem atical B iosciences publications. In m any cases these fields tend
to use different preferred notation to indicate the same concept, and these have been
reconciled to the extent possible, given the interdisciplinary nature o f the m aterial. In
particular, the notation for partial derivatives varies extensively, and the notation used is
chosen for stylistic convenience based on the application. W hile it w ould be convenient to
utilize a standard nom enclature for this im portant sym bol, the m any alternatives currently
in the published literature will continue to be utilized.
The rate o f change o f a population, that is, the rate o f increase or decrease o f pop
ulation size, where the population species are distinguished by the variables x ,y and z
is denoted by and ^ or x ,y and z respectively. The blackboard fonts are usedat a t at
to denote standard sets o f numbers: R for the field o f real num bers, C for the com plex
field, Z for the integers, and Q for the rational num bers. The capital letters, A , B , • • • are
used to denote m atrices, including capital G reek letters, e.g., A for a diagonal matrix.
The Jacobian matrix is denoted by the letter J. Functions which are denoted in bold
face type typically represent vector valued functions, and real valued functions usually
are set in lower case Rom an or G reek letters. Caligraphic letters, e.g., 7 , are used to
denote param eter spaces, w hile lower case letters such as i , j , k , l , m ,n . r , som etim es with
subscripts, r \ , r 2 ,ri,,bj,kj represent rates, environm ental carrying capacities and other pa
ram eters w hich influence the dynam ics o f the three species system. M atrices are typeset
in square brackets or parenthesis. In general the absolute value o f num bers is denoted
using a single pairs o f lines, e.g., | • |. Single pairs o f lines around m atrices indicates the
determ inant o f the matrix.
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Chapter 1
Background
1.1 P re d a to r-p re y M odels
The study o f predator-prey system s dates back to the 1950s with attem pts to m odel real
life system interactions through discrete and continuous m athem atics. The fam ous Lotka-
Volterra equations (1.1) assum ed that in a sim ple predator-prey system, predation would
linearly depend on the availability o f the prey species. The growth rate o f each species
was considered to be a function o f a single param eter. The rate at which predation oc
curred was also a linear com bination o f the interacting variables, in this case, the predator
variable and the prey variable. It was then within the limits o f available technology and
m athem atical analysis skills that such a system gained infirmity. Thus, for an n-species
system we w ould have the system
^
- ± = Xj (b /+ Y , au xj ) , 1, ; '= ,n , (1.1)
d x mwhere for each i, —2 denotes the rate o f growth o f species x,, w hile /;, represented the
atintrinsic population growth or decline in the absence o f o ther species; a,j is the predation
rate, positive if it is for the predator, and negative if the equation is representing the prey
population, [6],
A num ber o f criticism s arose out o f system (1.1), [11], [15] [25], [4], [6], [1] . First,
the linear functional response o f predation was not realistic. Predators kill w hen there is
need and not because o f abundance and availability o f prey population, and prey species
m ortality occurs for reasons other than predation only. Factors such as com petition for
food am ong prey and diseases definitely affect the rate o f increase o f prey population.
C onstant rates expressed as param eters, (1.1), and b, are not representative o f a m ore
realistic situation as these could be functions that express environm ental changes which
affect the growth o f any population, such as the hunting strategies and hiding strategies
o f both predator species and prey species respectively, [11]. Yet, som e m odification on
these param eters define sym biosis and m utualism in population dynam ics [29], [34], [41 ].
Therefore m odifications are being m ade such that these param eters are actually functions
which respond to environm ental changes.
1
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CHAPTER 1. BACKGROUND 2
By considering response functions such as the Holling Type II function, Holling
[1965], [27], allowed for the tim e needed by a predator to kill or consum e each prey,
as well as the lapse in tim e before it is hungry again. Therefore, the response function
1 + m x
expresses the lim iting factor in the predation term o f the predator species where m > 0
is the environm ental carrying capacity o f the predator population x, [25]. O thers, [4],
regard the Holling Type II function as one that describes consum ption by predation as an
increasing function o f prey, which saturates at higher prey densities.
Therefore a simple two species system taking into account the above considerations
would be represented as follows:
d x ,~jt = f { t ) x - g ( t ) x y ,
di = -* « )> • +dt 1 + my
where the functions f ( t ) , h(t) and g(t) are positive definite. Unfortunately, system s that
are explicit as well as non-linear are not easily analyzed. Equations in explicit form
are often expressed im plicity even though the resultant system becom es more non-linear.
Also, unfortunately, the Holling Type II function does not describe decreasing predation
at lower prey densities, [4], First introduced by H olling (1959), and subsequently used by
other researchers, [3], [26], [27], the H olling Type III sigm oidal-shaped function
2ax(1.2)
1 + bx2
not only describes decreasing predation at lower prey densities, but also describes predator
switching to different prey as well as accounting for predator hunting experience and
gam ing strategies. The three species m odel w e will consider in this research utilizes this
type o f response function.
Studies o f predator-prey system s with ’’sw itching” were made after it was discovered
that in certain circum stances species which initially act as predators becom e prey under
alm ost sim ilar environm ental conditions. W hether this switch is due to population size
fluctuations or slight environm ental changes, rem ains an open question. H ernandez and
Barradas [11], provided a two species predator-prey system with a switching effect. They
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CHAPTER 1. BACKGROUND 3
determ ined that the equations representing this sw itching phenom enon be m odeled as:
d x ( x \ x y_ = n , ^ - _ j + a , 2_
d y (■> y \_ =r 2 > , h _ _ j + a 2 , _ .
where
« i 2 (*,y) = y2 and g 2i M = ^ X2 - (1.4)\ + c \ y l 1 + C 2XZ
Table 1.1 describes the param eters as well as the trophic functions that govern the system
(1.3).
Table 1.1: Variables descriptions and response functions.
Symbol Description
X population density for species x
y population density for species y
r\ intrinsic rate o f increase o f species x
r 2 intrinsic rate o f increase o f species y
k \ environm ental carrying capacity o f species x
k-2 environm ental carrying capacity o f species y
a.\2 interaction coefficient, expressing effect o f interaction between species x and y
«21 interaction coefficient, expressing effect o f interaction between species y and x
In their paper, H ernandez and Barradas [11] desired to model this sw itching behavior
by considering the quadratic-ratio functional response form to be as described in (1.4).
Their m athem atical choice was designed to account for saturation effects, that is, that
predation cannot go beyond what the environm ent can sustain and that food availability
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CHAPTER 1. BACKGROUND 4
is limited. Also by saturation, we m ean to express that population densities do not grow
without bound. These assum ptions are m ore realistic biologically. Viewed differently, the
functional response function oc;y in (1.4) is a m odification o f the H olling Type III function.
We sum m arize the significance and properties o f the trophic function , /, j = 1,2, / / j
as described in [11]. As a result o f this choice, the m odel two dim ensional system is such
Table 1.2: Significance o f the response function.
Sign or m agnitude o f a Significance
a X2 > 0, ( a 2\ < 0) a positive contribution to the density-dependent factor in the per capita growth rate o f species x
a 2\ > 0, {a,\2 < 0) a positive contribution to the density-dependent factor in the per capita growth rate o f species x
a ij m easure o f intensity o f contribution
that there w ould be no periodic solutions expected. Therefore, the only possible solutions
are equilibrium solutions, no limit cycles, since the only unstable steady state w ould be a
saddle. By the Poincare-Bendixon theorem , limit cycles are possible if there are unstable
foci or nodes. Variable outcom es were noted due to the use o f the a -function and variation
o f certain sensitive param eters. The association or interaction o f one species with another
prom oted an increase in the growth rate and equilibrium density o f the other, so m uch that
when a critical size o f the form er is reached a reverse effect is observed.
W hat we seek to accom plish by this study is to understand the dynam ics o f a sim ilar
system but one which includes a third force, a predator that influences the dynam ics of
these variable outcom es. The top predator is assum ed to have a choice o f food, hence the
use o f the response function defined by (1.2). This understanding would include know l
edge o f w hether in the presence o f the top predator, the ’’sw itching” behavior betw een the
prey species would continue, and in the event o f predation by the top predator, w hether
the lower predator continues to prey on the bottom species independent o f the population
sizes.
The system (1.3) is also considered to be one that models mutual dynam ics, [16],
These functions = 1,2 instead are regarded as the com petition coefficient for
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CHAPTER 1. BACKGROUND 5
which different qualitative long term dynam ic behavior m ay occur depending on whether
the interaction is considered weak or strong. M utual benefit and strength according to
Kooi, et. al, [16] was dependent on w hether the product a \ 2 • « 2 i was less than one,
for weak interaction, or greater than one, for strong interaction. The m agnitude o f this
product expressed the strength o f m utualism between the species. The w ork by Barradas,
et. al, [11] considered the sam e system w here these com petition coefficients were in fact
response functions (1.4) that provided the sw itching o f prey-predator roles.
Studies o f chaos in population system s date back to the 1970’s w here difference-
equation m odels o f a single population exhibited chaotic behavior, M ay (1995). Rai and
Kumar, [27] m ade som e extensive analysis o f why chaos is rarely observed in population
dynam ics and G ilpin [7], and Iyengar et. al. [37] observed quasi-cyclical behavior o f tra
jectories in ecological system s and defined this behavior as spiral chaos . They concluded
that ecological system s with high levels o f nonlinearities exhibited chaotic behavior and
notable tim e delays. As described above, nonlinearities are necessary to account for sat
uration effects. Tim e delays are due to m aturation o f organism s or the flow o f nutrients
through ecosystem s. All these are characteristic ingredients of chaotic dynam ics in eco
logical m odels.
Ruxon [30] modified a model by M cCann and Yodzis, [25] by introducing a term that
defines the im m igration o f a constant rate o f individuals into the resource population as
a fraction o f the corresponding environm ental carrying capacity. His m odel included a
population floor effect on a continuous-tim e population model. The m odel
described the rates o f change o f the predator P, the consum er C, and the resource popu
lation R over time. Table 1.3 describe the param eters that influence the dynam ics o f this
system. To include a situation where there w ould always be some resource that is im mune
to exploitation, Rs, exploitable resource H was defined as a piece-w ise function
With a 5% size o f irremovable resource, R uxon’s model exhibits chaotic behavior under
certain param eter values. Ruxon concluded that with im position o f population floors,
xpy pPC
ct{C + Co)x pypPC x cy cCH
a ( C + C0) a ( H + R 0yx cy cC H
a ( H + Ro) a ’
R - R s R > R S.
0 otherwise.(1.5)
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CHAPTER 1. BACKGROUND
Table 1.3: Param eter m eaning for R uxon’s system.
6
Param eter Significance
x p m ass specific m etabolic rate o f species P m easuredrelative to production-to-biom ass ratio o f resource population R
x c mass specific m etabolic rate o f species C m easuredrelative to production-to-biom ass ratio of resource population R
y P m easure o f ingestion rate per unit m etabolic o f species P o r C
P(). Ro and C0 are initial population sizes
a resource population carrying capacity
H exploitable resource relative to production-to-biom ass ratio o f resource population x
Rs resource individuals im m une to exploitation
continuous-tim e predator-prey m odels show chaotic behavior. An overall look at this
three species system leaves us wondering w hat dynam ics can com e out o f a system where
the resource R and the consum er C som etim es interchange roles, while the predator P is
free to choose on which to feed, given the abundance or availability o f either o f the two.
W ith this in mind, one three species system worth noting concerns the predator-prey
relationship betw een lion, zebra and w ildebeest in which the lion had a food choice be
tween the two, depending on the population densities. Fay and Greeff, [6] produced a
model that tries to fit the census data o f these anim als at the Kruger National Park, South
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CHAPTER 1. BACKGROUND 7
Africa as reported during the 1970s. Their system,
dxdt
dzdt
0.125(xy + yz)
= —g(t) ( l - -0 .8 1 x y + 0.015xz,
= 0.34z (1 - - 0 .15yz + 0.02xz,
contains the functions g(t) and v(t) , m odifications that account for seasonal calving of
w ildebeest and cropping o f lion respectively. The m ixed quadratic term s in the system
define the interaction term s where the corresponding coefficients are the predation rates.
A positive sign on the m ixed term m eans a positive contribution towards the per capita
growth rate o f the corresponding species, w hile a negative sign produces the opposite
effect.
The overall system resem bles the general three species system w here the lion species
equation incorporates a H olling Type II functional response function, the other two prey
species having the logistic part and the predation contribution terms. W hile this model
system is explicitly responding to the available data, the authors did not explore the gen
eral dynam ics or variable interactions between the species. Very specific param eter val
ues such as predation rates, growth rates and environm ental carrying capacity values were
found, values that provide the best fit to available data. Also in this m odel system , it is
assum ed that the lion population had equal chances o f m aking a kill on either the zebra
or the wildebeest. Positive term s reflecting m utual benefit between zebra and w ildebeest
characterize this model system. A particular case sim ilar to this system could provide
extensions o f analysis o f our system where instead o f fixed param eters, we w ould wish to
consider a num ber o f possible scenarios.
Predator-prey interactions determ ine m any aspects o f population dynam ics and com
m unity structure o f estuarine and m arine ecosystem s. In a M aster’s thesis on a simple
three species food chain, M agom o [22] m odeled the dynam ical system along the Leslie-
Gower, [17] schem e and the resulting behavior in certain param eter regions was chaos
through period doubling bifurcations. S im ilar system s that were o f Kolm ogorov type also
produced interesting dynam ics such as H opf-bifurcation and chaos, [5], [35], [9]. Letel-
lier and A ziz-A laoui, [17] considered a sim ilar three species system but regarded the last
equation o f the above system to take the form
( 1.6)
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CHAPTER 1. BACKGROUND 8
Their original predator equation was o f the logistic form, but they considered instead a
is directly proportional to the num ber o f m ales as well as females o f the z species. M any
researchers consider that the growth rate takes into account this proportionality, [17], [5],
[11], In their study, bifurcation param eters w ere defined and chaos was observed through
variation o f the intrinsic growth rate o f the prey species x. We are therefore m otivated
into considering the top predation equation either to take this form (1.6) or the one (1.7)
in which predator dynam ics is represented by a logistic model with carrying capacity
proportional to the num ber o f prey species x and y, in this case, carrying capacity for the
predator species z , k = k^xy:
The study by Song and X iang, [42] revealed that variation o f sensitive param eters resulted
in period doubling cascades leading to chaos. How ever their m odels on pest control
produced periodic solutions where the interaction term s are m ostly linear. This study is
com plicated in that we include these response functions a,y that give rise to predator-prey
role interchange under certain param eter conditions, while the top predator has a choice
o f the seem ingly com peting species.
The organization o f our research is as follows: In C hapter 2 we define our model sys
tem in relation to the biological hypothesis provided for by Barradas, [11], We explain
why it is im portant to analyze this three species predator-prey system and how we de
sire to analyze this system. The equations that model this system are explained in their
relationship to biological realities.
In C hapter 3 we analyze the corresponding two species system. We prove that given
the choice o f the trophic functions em ployed in this m odel, response functions that m an
age the sw itching phenom enon, there are no periodic solutions in the positive half plane.
We obtain equilibrium solutions and analyze local stability o f these equilibria. In trying
to sim plify the highly nonlinear system, we analyze the system under symmetry. U nder
certain param eters, we observe som e exchange o f stability between certain critical points
and show that some equilibrium solutions undergoes a transcritical bifurcation. We fur
ther show phase portraits o f the two species interactions. Phase diagram s are shown to
reveal bifurcation o f equilibria as som e sensitive param eter, nam ely the environm ental
carrying capacity, is varied.
In C hapter 4 we further break down the three species system into other two species
systems. Phase portraits o f the resulting two species system reveal the nature o f stability
squared term a ^ z 2 to justify the fact that the m ating frequency o f the predator population
(1.7)
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CHAPTER 1. BACKGROUND 9
o f the resulting equilibrium solutions. Analysis o f the two species system through sym
m etry produce a pitchfork bifurcation. Analysis o f the three species system, again by
considering symmetry, lead towards obtaining a cusp bifurcation using resultant theory.
C hapter 5 provides a discussions and conclusion. We briefly review the results o f the
model analysis in relation to possible ecological situations.
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Chapter 2
The Model Three Species System
2.1 M ath em a tica l M odel
The snail-lobster model that was studied by B arradas et. al, [11] is the ecological basis for
the three species predator-prey system under this study. Both lobsters and snails are prey
to dogfish. This principal predator preys on either o f the two depending on the availability
or abundance o f the prey species.
We therefore gain the m otivation to study the three species system in which lobsters
and snails have this sw itching effect while the two rem ain prey to dogfish.
Figure 2.1: Relationship betw een food chain species. Reverse arrow on the bottom two species predicts switching.
It is this unique setup that m otivates us into the need to analyze the dynam ics o f this
system,
predator
lobster
(2 .1)
— = r3z - z f i i 23 (x ,y ,z), at
10
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CHAPTER 2. THE MODEL THREE SPECIES SYSTEM 11
where the loss in the predator population z is proportional to the reciprocal o f per capita
availability o f the two prey species x and y. This loss, represented by the function /3i 23
could be represented by either o f the two forms:
ZP m = — , m ultiplicative saturation form.
(2 .2)P \23 = ----------------- additive DeAngelis interference form , [24].
q + m{x + y)
The environm ental carrying capacity for the top predator z is therefore represented by
some prey ratio-dependent expression o f the m ultiplicative form or the D eAngeles inter
ference form (2 .2 ).
The response functions a \ 2 and a 2\ are as described by the equation (1.4) and the
system param eters are as described in Table 1.1, [11]. The param eters r(, i — 1,2,3
are the intrinsic growth rates o f species i. The carrying capacity o f the environm ent for
species are represented by the param eter kj, / = 1,2. In the absence o f predation, each
individual species grows to its corresponding carrying capacity. The sign o f a ,7, see Table
1 .2 , determ ines which species, betw een species x and y, assum es the role o f predator or
prey. If a \ 2 > 0 then the species x preys on species y. If a 2 \ > 0 then the species y preys
on species x.
Increasing b,j, the param eter which can be described as the intensity interaction pa
rameter, m eans an increase in reaction o f the species interaction, see Fig. 2.2 on page
12. Increasing the param eter c (/ produces a negative response, it prolongs the interac
tion, hence a dum ping effect, see Fig. 2.3 on page 12. The H olling Type III functional
response function that describe the predation on species x and y by the top predator z is
used;
P n { x ,z ) = J 0'* Z2 and /323(>’, z) = \ • (2.3)a 1 + x z d 2 + y L
We observe that the param eters £0;, i = 1,2 are the predation rates; d 2 is the value o f y at
which the per capita removal rate o f y becom es half the predation rate (0 2 . The same can
be said o f the param eter d\ in relation to the prey species x.
Com bining these term s, the system (2.1), under consideration takes the form
dx / x \ ( b \ y — y2\ x y C0\x2z = r\x 1 +
d t \ k\ J \ \ + c \ y 2 J k \ \ + d \ x 2 '
d y ( . y \ ( b 2x - x 2 \ x y o>2 y 2z— = r2y 1 - 7 - +d t \ k2 J \ 1 + C2X 2 J k2 1 + d 2y 2 ’
dz ( 1 z= Oz[ 1 -
dt ' \ k2xy
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CHAPTER 2. THE MODEL THREE SPECIES SYSTEM 12
a
0 02
0 1 0 2 0 3 0 4
b0.5
Figure 2.2: An increase o f bjj in the response function a tj means increased interaction.
a
0 2 0 3
c0 5
Figure 2.3: An increase o f c,7 in the response function (Xjj m eans a decreased but p ro longed interaction.
ordxdt
dydt
dzdt
= r xx |^1
= r2y 1
x \ ( b \ y — y 2
k\
y_k2
+
+b 2 X — x 2
q + m (x + y )
4y _ (0{X2Zk\ d\ + x 2
xy a>iy2z
h d2 + y 2
(2.4a)
(2.4b)
(2.4c)
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CHAPTER 2. THE MODEL THREE SPECIES SYSTEM 13
depending on whether w e require a quadratic-ratio dependent predator environm ental car
rying capacity proportional to the product o f the prey density, K = k^xy or linear com bina
tion o f the prey densities equal to the carrying capacity o f the predator, K = q + m {x + y ) .
We concentrate on the system defined in equations (2.4a)-(2.4c) in which we consider
an additive saturation term for both species x and y. The third equation o f the above sys
tem contains a m odified Holling Type II functional response function which characterize
interference between the two prey species w hile com petition for food am ong the predator
species experience this saturation effect o f the prey species. The param eters d\ and di
quantify the extent to which the environm ent provides protection to prey species x and
y respectively. We note that the predator equation, (2.4c) is the logistic type equation
whereby the carrying capacity is proportional to the additive com bined prey abundance,
m (x + y) , [36], The param eter q, norm alizes the residual reduction in the predator popu
lation z because o f severe scarcity o f prefered food.
The last equation, (2.4c), also shows that in the absence of both prey species x and y,
then the predator species goes to extinction as long as q < r$, otherw ise this w ould grow
unboundedly to infinity if this inequality is reversed, which is not acceptable biologically.
2.1.1 D iscussion
O ur model (2.1) represents a generalized three species predator-prey system. The spe
cific m odel represented by the equations (2.4a)-(2.4c) is a build-up from the two species
system that exhibited a sw itching phenom enon and which was presented by H ernandez
and Barradas, [11], We have chosen to consider the response function with the functional
response characteristics o f not only providing a choice o f food by the top predator, but
also giving the predator the trait o f not killing prey even when it exists in abundance. In
the same way, the prey species attains the ability to hide and evade being killed.
The nature o f the logistic function for the top predator is that saturation is density-
dependent. This m eans that the characteristic sigm oidal shape levels off with increasing
time to the additive linear sum o f the prey species. Com petition am ong predators controls
an otherw ise exponential population rise and we assum ed that the environm ental carrying
capacity was expressed as a prey-density-dependent linear form. In the proceeding chap
ters, w e analyze this system in stages o f corresponding two species to the three species
system.
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Chapter 3
Two-Species System
3.1 In v a r ia n t Set
The system defined by equations (2.4a)-(2.4c) reduces to the corresponding two species
system in the absence o f the top predator species z. In that case, the equations governing
the corresponding two species system are described in [11], Barradas et. al determ ined
the dynam ics o f this m odel through com putational num erical sim ulations. In this chapter
we reconsider the corresponding two species system w hose equations model the switching
phenom enon that was docum ented in [11],
The variables x(t) and y( t) represent population densities. It is essential that solutions
with x ( t) > 0 and y ( t) > 0 initially, rem ain in the positive quadrant for all time. The
param eters involved are as defined in Table 1.1.
D efin ition 3.1.1. In v a r ia n t set: An invariant set for a system o f coupled differential
equations is a region D , o f the phase space with the property that any solution starting
within T> rem ains within it for all time. Note that the phase space o f a dynam ical system
is the space o f species interaction w here all the possible states are represented.
(3.1)
T h eo rem 3.1.1. (Invariant Set) The set
D = {(x,>’) e R |0 < x < k\ , 0 < y < k f ) ,
is an invariant set where any solution o f (3.1) starting inside T) will remain inside D f o r
all time.
Proof: The system (3.1) is equivalent to
x (t) = ;c(to)exp' _ r \x{s) + y ( s ) a i 2{s)
(3.2a)
14
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CHAPTER 3. TWO-SPECIES SYSTEM 15
y( t) = y(f0)expr2y ( s ) x ( s ) a 2i( s ) l
/"2------ ;-------1-------- :-------- dS (3.2b)
If species x is the prey species, then it is because the response function a \ 2 is negative
while OC21 > 0. For this sign definite o f the response functions we require that
• b\ < y ( t ) < k2,
• 0 < x(t) > b2 < k\.
Substituting the m axim um possible values o f x(t) and y( t) into equations (3.2a)-(3.2b),
we observe that
• j ( 0 0 as t —> °° and
• x( t) —> 0 as t —>• °°.
The assertion o f the theorem follows im m ediately for all t > to. A sim ilar approach is
arrived at if the species x becom es the predator w hile y species is the prey in the two
dim ensional system (3.1). □
Definition 3.1.2. The solution o f (3.1) is said to be ultim ately bounded if there exists
B > 0 such that, for every solution (x ( t ) ,y ( t ) ) o f (3.1), there exists T > 0 such that
||x ( t) ,y (r) || < B for all t > to + T, where B is independent o f the particular solution,
while T may depend on the solution.
Theorem 3.1.2. Let e > 0. I f max {M f } < /?, < k ^ , i = 1,2, then the set Te defined by
is positively invariant with respect to the system, (3.1) where
M f = k\ T £, M f = k2 ~\~ £•
and £ > 0 is sufficiently small.
Proof: The condition b\ < k \r \ ensures that k\r \ term is much larger than a \ 2 for any
r e = j(x,y) £ M |0 < x < M f , 0 < y < Mf j,
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CHAPTER 3. TWO-SPECIES SYSTEM 16
forward tim e t > to. Therefore
x(t ) < x(t) n - ^ ( 0
< rM > )k\k\ + e —x(t)
= M f - x ( r )
A standard com parison argum ent shows that
0 < x(to) < M f ==> x( t ) < M f , t > to.
The p roof is com plete if we apply the sam e reasoning for the solution y(t ) . □
The choice o f (Xjj as explained in C hapter 1 provides that the dynam ics o f the system
for w hatever choice o f param eters w ould only yield sinks, saddles and spiral solutions
with no periodic solution in the positive quadrant as alluded to by the following lem m a
3.1.5. However before we give this lem m a, we m ention without proof, B endixson’s N eg
ative Criterion, (see p roof in [14]) in the form o f a theorem . Re-define system (3.1) with
equationsx = f { x , y ) ,
y = g(x,y) .
T h eo rem 3.1.3. Let the two species system be defined by equations (3.3). There are no
closed paths in a sim ply-connected dom ain o f the phase plane on which d f / d x + d g / d y
is o f one sign.
(3.3)
The B endixson’s Negative Criterion is extended to give rise to the so-called D ulac’s test:
T h eo rem 3.1.4. Let the two species system be defined by equations (3.3) and let the
function p ( x , y ) be one that adm its continuous firs t partia l derivatives. Then there are no
closed paths in a sim ply-connected dom ain o f the phase p lane on which
d (p (x, y ) f ( x , y )) j d x + d (p (x, y )g (x, y )) / d y is o f one sign.
The two theorem s 3.1.3 and 3.1.4 for the B endixson’s test lead us to our lemma.
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CHAPTER 3. TWO-SPECIES SYSTEM 17
— i---------- 1---------- 1---------- 1---------- 1---------- 1---------- 1---------r i i
le- ; j j ; j
i i k ]
= ! \ i s \ \
i *
i I i ! i i ! i^ 0 5 10 15 20 25 30 35 40 45 50
XFigure 3 .1 : Phase portrait o f the flow in the half plane. Notice the direction o f arrows towards equilibrium solutions w hich are not periodic, in the positive half plane, the invariant region.
L em m a 3.1.5. There are no periodic solutions to the system (3.1) in the positive h a lf
plane R 2 .
Proof: Since R 2 is sim ply connected, we define the C 1 function p ( x , y ) = — whichxy
adm its continuous first partial derivatives in R 2 .
A pplying D ulac’s Test [14], [40], [32], to system (3.1), then
P ( x .y ) f { l ,y ) = i ( t - f ) ^ b ' y - y 2y \ k \ ) \ 1 + c i y 2/ k\ ’
n( X , 1 o f , y \ , ( b2 x - x 2 \ 1p( x , y ) g ( x , y ) = — 1 - —
Therefore
x \ k2 ) \ 1 + c2x J k2
d ( p ( x , y ) f { x , y ) ) = _ n _ d x k \ y '
d ( p{ x , y ) g ( x , y ) ) r2
c)y k2x
d { p { x , y ) f ( x , y ) ) + d(p ( x , y ) g{ x , y ) ) = _ / _n_ + < Qso that
d (P (*d x ' d y V^O7 b2x /
for all G R + , r \ , r 2 G R +. Therefore there are no periodic solutions in the positive
half plane R “L see Fig. 3.1. □
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CHAPTER 3. TWO-SPECIES SYSTEM 18
3.2 L ocal S tab ility A nalysis
The corresponding two species system, although studied by H ernandez and B arradas [ 11 J,
is here re-visited for som e thorough m athem atical calculations that include stability of
equilibria. The following theorem s and definitions are necessary to understand the subse
quent m athem atical analysis.
D efin ition 3.2.1. E q u ilib r iu m so lu tion C onsider a general autonom ous vector field,
that is, a vector field associated with the first order autonom ous differential equation,
x = f ( x ) , x G R ” . (3.4)
An equilibrium solution o f (3.4) is a point x e R" such that
/ ( * ) = 0,
that is, the solution o f (3.4) does not change in time.
In ecology, the balance o f nature can be divided into three categories:
1. the claim that natural populations have a m ore or less constant num ber or individu
als,
2 . the claim that natural system s have a more or less constant num ber o f species,
3. and the claim that com m unities o f species m aintain a delicate balance o f relation
ships, w here the removal o f one species could cause the collapse o f the w hole com
munity.
In this study, an equilibrium solution is that state in a dynam ical system where the net rate
o f change in population density is zero.
D efinition 3.2.2. Stability Let x( t ) be any solution o f the system (3.4). This solution is
stable if solutions starting close to x(t ) at a given tim e remain close to x(t ) for all later
times. It is asym ptotically stable if nearby solutions actually converge to x(t ) as t —> °°.
In our study o f species interaction, stability is understood to m ean that the interacting
species m aintain relatively steady population sizes within a given param etric region, for
increasing time.
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CHAPTER 3. TWO-SPECIES SYSTEM 19
In order to understand the stability o f solutions to our system (3.1), w e seek to under
stand the nature o f solutions near an equilibrium solution. A perturbation o f the equilib
rium solution jc(?) so that
x = x ( t ) + y ,
and substituting into (3.4), then Taylor expanding leads to
x = x ( t ) + y = f { x ( t ) ) + D f ( x ( t ) ) y + Q(\y\2), (3.5)
where D f is the derivative o f / and | • | denotes the norm on M". Since x( t ) = f ( x ( t ) )
equation (3.5) simplifies to
y = D f ( x ( t ) ) y + 0( \ y \ 2). (3.6)
Equation (3.6) describes the evolution o f orbits near the equilibrium solution x(t) . U n
derstanding the dynam ics o f the system (3.4) require understanding the dynam ics near
the equilibrium solution. Locally, the analysis o f the system (3.4) can be com plete if we
consider the linear part as defined by (3.6) w ithout the higher order terms. Therefore
y = D f ( x ( t ) ) y (3.7)
is the corresponding linearization to the system (3.4). In general, D f ( x ( t ) ) defines the
linearized matrix, called the Jacobian o f the system (3.4). In species interactions, the
eigenvalues o f this Jacobian carry im portant inform ation about the stability nature o f equi
librium solutions. A few o f the theorem s here will clarify local dynam ics and hence global
dynam ics o f the two-species system (3.1).
For stability o f equilibrium solutions o f (3.1), the following steps are necessary:
1. determ ine if trivial solution y = 0 o f (3.7) is stable.
2. show that the stability or instability o f the solution y = 0 o f (3.7) im plies the stability
or instability o f (3.4).
To determ ine the stability o f y = 0, the following theorem is necessary:
Theorem 3.2.1. Suppose all the eigenvalues o f D f ( x ) have negative real parts. Then the
equilibrium solution x = x o f the nonlinear vector fie ld (3.4) is asym ptotically stable .
Definition 3.2.3. The phase portrait o f a dynam ical system (3.4) is a partitioning o f the
state space into orbits.
By looking at the phase portrait, one can determ ine the num ber and types o f asym p
totic states to which the system tends as t —> °° (and as t —> — if the system is invertible).
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CHAPTER 3. TWO-SPECIES SYSTEM 20
Theorem 3.2.2. I f the linearization m atrix 3 has no zero or purely imaginary eigenval
ues, then the phase portra it fo r the nonlinear system near an equilibrium po in t is sim ilar
to the phase portra it o f its linearization.
This theorem , cited as the H artm an-G robm an Theorem in W iggins [40], provides
a m icrocosm ic understanding o f the nonlinear dynam ics of the system. In particular,
coupled with the following definitions, one gets a clearer sense o f the justification behind
analysis o f corresponding linear system.
Definition 3.2.4. Guckenheimer and Holmes [8] Two C vector fields, /'. g are said to
be Gk equivalent (k < r) if there exists a Qk diffeom orphism h which takes orbits <j>f (x) of
/ to orbits (jc) o f g, preserving senses but not necessarily param etrization by time. If h
does preserve param etrization by time, then it is called a conjugacy.
Definition 3.2.5. Guckenheimer and Holmes [8]: Structural Stability A C vector
field / is structurally stable if there is an e > 0 such that all C1, £ pertubations o f / are
topologically equivalent to / .
The need to establish topological equivalence betw een vector fields stem s from the
fact that the m odel system under scrutiny has nonlinearities which m ake it hard to analyze.
However if there is topological equivalence or structural stability betw een the vector field
and its corresponding linear field, then the dynam ics o f one infers the dynam ics o f the
other, at least locally.
Definition 3.2.6. A n equilibrium solution is called hyperbolic if there are no eigenvalues
with zero real parts.
Theorem 3.2.3. The phase portraits o f system (3.1) near two hyperbolic equilibria ( x q .v o )
and (xi ,} '|), are locally topologically equivalent i f and only i f these equilibria have the
same num ber n_ and n + o f eigenvalues with Re(X < Oj and with Re(X > 0), respectively.
Therefore, if the eigenvalues o f the Jacobian matrix at the equilibrium solution have
nonzero real parts, then the solution o f this system will not only yield asym ptotic behavior,
but, by H artm an’s theorem and the stable m anifold theorem , it will also provide the local
topological structure o f the nonlinear system.
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CHAPTER 3. TWO-SPECIES SYSTEM
Table 3.1: Equilibrium solutions for the two species system.
21
The two species will die out with increased time and is not interesting.
The x species persists w hile the y species eventually disappear due to predation.
The y species persists w hile the x species eventually disappear due to predation.
The nonzero coexistence equilibrium solution.
By equating the right hand side o f (3.1) to zero, and solving the system
b \ y - y 2 'r \ x ^1 -
r2y ( 1 -
+
+
xy+ c \ y 2 ) k\
b2 X — x 2 \ x y
k 21 + c2 x 2
w e obtain the equilibrium solutions shown in Table 3.1.
= 0 ,
= 0 .
L em m a 3.2.4. The equilibrium solution (0 ,0 ) o f (3.1) is unstable.
Proof: Let
f = «x[
g = n y
which are the right hand side o f (3.1).
Since
2k2
b \ y - y 2
1 + c \ y 2
b2x — x 2
1 + c2x 2
xy
xy
~ki
d_d x
d_dy
d_
d x
d_dy
r\x
r \x
1 - *
k\+
- i - ) +
n y i - +
r2y 1 - Lki
+
b \ y - y 2
1 + C[_y2
b \ y - y 2
1 + c i y 2
b2x — x 2
1 + c2 x 2
b2x — x 2
1 + c2 x 2
xy_
k\
xyk\
xy_
kixy
k2
= r l -
-
k\
- ^ k 2
= r2 -
2 r \ x y a 12
k\ k\ '
2b\ - 3y 2 c, (b iy 2 - y 3)1 + c i y 2
2b2 — 3x
1 + c2x 2
2 r2y x a 2\ k 2 k2
(1 + c \ y
2 c 2 (b2 x 2
2 \ 2
(1 +C2X:2 \ 2
(3.8)
(3.9)
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CHAPTER 3. TWO-SPECIES SYSTEM 22
2
y
0
Figure 3.2: Phase portrait o f solutions starting near the origin. N otice that the flow is away from the origin, the zero solution is unstable (k = 3, r\ = r2 = 0.3, b\ = b 2 = 5, c\ = c2 = 0.5).
0.2 0.4 0.6 1.2 1.4 1.6
then let
2 r xx y c tn r xx y a n r \xJ\ \ = r \ ----------- = r i ----------------------------- ,
ki k\ ^ ^ ki
h i =xy ( 2bi — 3y 2c 1 ( b ty 2 - y 3)ki V 1 + c i y 2 (1 + c i >’2 ) 2
xy f 2b2 — 3x 2 c2 (b2 X2 — x3'•^21 = T
k 2 V 1 + C2X
2 r2y x a 2Xh i = r2 - — h ——
K2 K2
(1 + c 2 x 2 ) 2
r2y x a 2ir 2 ~ T ~ + ~ T ~ k i k 2
r iyk i
d =
/ i f d p dx ~5y
1\ dx d y ,
J\\ h i h \ h i
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
From equations (3.10) to (3.13) we observe that at the equilibrium solution (0 ,0 ), 7] 1 =
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CHAPTER 3. TWO-SPECIES SYSTEM 23
r \ , J 12 = 0, J 2 \ = 0 and J 2 2 = r2. Therefore
J \\ h i h \ -h i (0 ,0 )
n 0
0 r2(3.15)
The eigenvalues o f the Jacobian m atrix (3.15) are r\ and r2 . Since biologically these
param eters are the growth rates o f the two respective populations, these are always posi
tive param eters. It follows that the equilibrium solution (0 ,0) is always unstable. □
Define two param eters
&2X = r 1k\
P = r2 + ^ - a 2 \{k\ ) . k i
Lemma 3.2.5. The equilibrium solution ( k \ .0) is stable asym ptotically i f X < 0 and un
stable i f X > 0 .
Proof: Equations (3.8) can be re-w ritten as
n 1 - H +
r2 1 — f +
b i y - y 2
1 + c i y 2
b 2 X — x 2
1 + c2x2
k\x
ki
= 0 ,
= 0,
so that for nonzero x and y then
r 1 -r \ x , y a 2 1
k\ k\a,2 \xn y .
k 2
= 0 ,
= 0 .(3.16)
W hile J \ 2 = J21 = 0 at the equilibrium solution (&i, 0), the other Jacobian entries J\ \ and
J2 2 are sim plified by (3.16) to get
•̂ 11 J 12J 21 J 2 2 (*1,0)
- n 0
0 X(3.17)
Since r\ is a strictly positive value that represents the growth rate o f the x population, we
observe that the eigenvalues o f the Jacobian (3.17) have negative real parts if X < 0. Under
these conditions the equilibrium solution (&i,0) is asym ptotically stable. O therw ise if
X > 0 then (fci, 0) is unstable. □
Lemma 3.2.6. The equilibrium solution (0. A2) is stable asym ptotically i f p < 0 and un
stable i f p > 0.
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CHAPTER 3. TWO-SPECIES SYSTEM 24
Proof: Sim ilar to Lem m a 3.2.5 on page 23, the entries J \ 2 = 72i = 0- By (3.16), J 2 2
simplifies to —r2 while the entry J\ \ simplifies to p . Thus
J 11 J \ 2
h \ J22 (*1,0)P 00 — r2
(3.18)
The eigenvalues o f the Jacobian m atrix in this case are fJ. and —r2 with strictly positive
param eters. The equilibrium solution (0, k f) is asym ptotically stable if ju < 0 and unstable
if ju > 0. □
The non-trivial equilibrium solution (x*,y*), (x* f 0 and y* 0) as obtained from
(3.8) is im portant for coexistence o f the two sw itching species. S tability analysis o f this
coexistence equilibrium is im portant. Equations (3.16) will help sim plify the entries J\ \
a n d 7 22 in the Jacobian matrix, such that (3.10) and (3.13) become
r \ xJ \ 1 =
k\, r2y
J22 = — k-2
respectively. Substituting (3.19) into (3.14), we obtain
(3.19)
V ,v * ) -
/ n x*k\
\ h \
J 12
r y f k2 /
(3.20)
L em m a 3.2.7. The coexistence equilibrium solution (x*.y*) is asym ptotically stable i f the
inequality$ ̂r \r 2 X y
J \ 2h \ <k \k 2
(3.21)
is satisfied.
Proof: The Jacobian matrix in (3.20) has trace t and determ inant d given by
t = -r \x * r2 y*k\ k2
, , r \x r2yand d = ------ J n J2\.
k\ x2
Since trace is always negative, for asym ptotic stability we require that the determ inantf\x* r2y*
be positive. This is possible i f --------------> J \ 2J 2 \ - It m ust be m entioned that for two-k\ k 2
dim ensional system s, the necessary and sufficient condition for local stability o f a fixed
point is that the trace o f the Jacobian m ust be strictly negative and the determ inant o f the
Jacobian m ust be strictly positive. □
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CHAPTER 3. TWO-SPECIES SYSTEM 25
B
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
x stabilizes at about 5.5
1,5
).5
—r ~ r t i l l
>• stabilizes at about 0.4
0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80
1 t
Figure 3.3: Time series plots o f populations x( t ) and y( t ) stabilizing to the coexistence equilibrium solution with increasing time.
Since the trace o f the Jacobian m atrix is always negative, we then present a schem atic
view o f all the possible equilibria and the nature o f stability in the trace-determ inant plane,
(Fig. 3.4).
D e t .Stablespiral
Stable node4Det
T r a c e J
Saddle
Figure 3.4: Stability regions in the corresponding two species system.
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CHAPTER 3. TWO-SPECIES SYSTEM 26
A pplying the H artm an-Grobm an theorem for each equilibrium point and observing
the hyperbolicity o f the particular point, one can make conclusions about the long term
behavior o f the system near the equilibrium point. The problem arising from applying this
tool is w hen the equilibrium solution is nonhyperbolic, which m eans that the real part of
the eigenvalue corresponding to the m atrix jacobian at that equilibrium solution is zero.
In this case we w ould consider higher order term s’ contribution to the system behavior.
In some cases norm al fo rm analysis will determ ine which higher order term s contribute
m ore towards system behavior about the equilibrium point.
A bifurcation is a qualitative change in the behavior o f solutions as one or m ore param
eters are varied. The param eter values at which these changes occur are called bifurcation
points. If the qualitative change occurs in the neighborhood o f a fixed point or periodic
solution, then it is called a local bifurcation, otherw ise it is global. For m ore explanations
and descriptions about bifurcations, see [31], [40], [32], [8].
We include diagram s o f bifurcation analysis o f equilibria as we vary sensitive param
eters, in this case the the carrying capacity k o f the species environm ent. We note that the
equilibrium density o f the species x change, (bifurcate) as we vary this parameter.
Lemma 3.2.8. I f X — 0 then the equilibrium solution to the system (3.1) undergoes a
transcritical bifurcation.
Proof: The coordinate transform ations k \ , y ^ y , X ^ X, preserve the dynam ics
o f the original system , which we write as
x — F ( x , X ) , x €
where x = (x.y).
For this we verify that
• F(0,0) =0.
# c>F(0 , 0 ) Qd x
The two fixed points of
are given by
and
x — F ( x , X ) , x e l 2.
x = 0
x = X.
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CHAPTER 3. TWO-SPECIES SYSTEM 27
Hence for A < 0 there are two fixed points; x = 0 which is unstable and x = A which is
stable. These two coalesce at A = 0. For A > 0,x = 0 is stable and x = A is unstable, see
fig. 3.5. This m eans we have a transcritical bifurcation. □
Figure 3.5\ Exchange o f stability occurs at A = 0, a bifurcation known as transcritical bifurcation. The dotted line is the unstable manifold.
A plot o f one species density against its corresponding carrying capacity provides
some graphical way o f observing hysteresis as the carrying capacity param eter k is in
creased. Hysteresis is the lack o f reversibility o f system structure or com position as a
param eter is varied.
The disappearance o f the top nodal sink as it fuses with the non-trivial saddle equi
librium, further confirming another bifurcation o f equilibria, provided som e favorable
predation conditions at the expense o f a friendly environm ental co-existence habitat. If
unchecked, drastic changes occur leading to a rapid loss o f predator population followed
by a subsequent rise in prey population, possibly explaining the reverse effect that was
w itnessed in the South A frican Islands, [11], By further increasing the sensitive param eter
m, the predator population disappears. N otice the characteristic role switch exhibited by
the exchange o f species interaction in the form ation o f saddle points between Fig. 3.7 and
Fig. 3.11. The transition o f exchange o f equilibria from a low coexistence equilibrium
o f one species, through a saddle and two foci, to a low coexistence, then disappearance
o f coexistence equilibria is sim ulated num erically through phase plane diagram s o f the
phase portraits o f the two dim ensional system, (Fig. 3.6 to Fig. 3.13). These variations
are due to changes in the carrying capacity o f one o f the species. This approach not only
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CHAPTER 3. TWO-SPECIES SYSTEM 28
7
6
S
4
3
2
1
0
0.5 2.5 3 3.5 40 1 1.5 2X fcj
Figure 3 .6 : U nder certain conditions, the single species system stabilizes at the respective environm ental carrying capacity.
shows the trajectory o f the system , but also indicates the type o f equilibrium in term s of
its stability state.
It can observed that depending on choices o f param eters, one obtains som e variations
o f population interactions. Hysteresis is biologically im portant in that som etim es contin
ued increase o f a sensitive param eter m ay result in som e abrupt or sharp changes in the
dynam ics o f the w hole system. It w ould require a lot o f energy to force the system back
to its original form. Hysteresis m ay as well lead to extinction o f som e species. The upside
down s-shaped form o f the solution curve is typical o f hysteresis, with upper and lower
branches com prising stable equilibria w hile the m iddle branch is unstable, Fig. 3.8. At
the turning points o f this curve, small changes in k m ay provoke som e catastrophic jum p
between stable branches.
We have discussed the stability conditions o f solutions o f the corresponding two
species system. We have observed that for single species equilibria, there exists a critical
param eter value at which the solution becom es nonhyperbolic, that is, that eigenvalues of
the Jacobian for the linearized system have zero real parts. In this case we conclude that
the system becom es structurally unstable so m uch that the dynam ics near the equilibrium
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CHAPTER 3. TWO-SPECIES SYSTEM 29
9
a
7
e
4
3
2
1
0
0 ’ 2 3 4 5 6 7X
Figure 3.7: There is a critical /: value at which a saddle point is born while the stable spiral persists.
solution cannot be the sam e as those o f the corresponding nonlinear system when the
sensitive param eter is zero. A sum m ary o f these observations is thus, as follows:
1. Orbits and therefore vector fields are structurally stable in the vicinity o f hyperbolic
fixed points, Thm. 3.2.2.
2. In the neighborhood o f bifurcation points, the orbits and therefore the associated
vector field is structurally unstable.
3.3 Symmetry and bifurcation in 2D
The two species system (3.1) can be described from a sym m etric view point.
1. Full sym m etry is defined when all the corresponding param eters are the same, that
is, when we haver\ = n = r,
k\ = k 2 = k,
b\ = b 2 = b,
Cl = c2 = c,
i i i ................................ 1 ------------------- 1
i
/
i 1 /
T T ' 7 ................i, i
i * i1
T - i - 7 ...................
f i r
V. i ________ . .
5
■
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CHAPTER 3. TWO-SPECIES SYSTEM 30
9
7
6
5
4
3
2
00 1 2 5 6 74
X
Figure 3 .8 : Increasing & from 2 to 3 leads to three equilibrium solutions, two stable spirals with a saddle between them.
7
6
5
4
3
2
1
00 0. 5 1 15 2 2. 5 3.5 4. 5 54
X
Figure 3 .9 : For certain param eter choices, a spiral focus with low x-species density and high y-species density em erges, (k\ = 2,&2 = 5 ,c i = 0.5, C2 = 3).
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CHAPTER 3. TWO-SPECIES SYSTE M 31
7
6
5
4
3X
2
0~ 1- —
3.50.50 1.5 2 2.5 3 4.5 54
Figure 3 JO: Phase portraits showing a stable spiral focus with low ^-species density and high y-species density and a saddle.
3.5
2.5
0.5
0 0. 5 1.5 2 2. 5 3
Figure 3 .11 : Increasing k\ from k\ — 3 to k\ = 5 leads to the disappearance o f one spiral solution through a saddle form ation.
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CHAPTER 3. TWO-SPECIES SYSTEM 32
4
5
>5
2
1.5
1
).5
01.5 2 2.5 S0 0.5 1
X
Figure 3 .12 : A single coexistence equilibrium solution exists for k\ > 4.5.
2
1,8
15
y1.4
1.2
^2 1
08
0.8
0.4
0.2
0
Figure 3 .13 : No coexistence in certain param eter space, each species exists independent o f the other and is lim ited by the environm ental carrying capacity.
o 0.2 0.4 0.6 1 1.4 15 2
k\ x
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CHAPTER 3. TWO-SPECIES SYSTEM 33
so that (3.1) becom es
d x ( x \ f b y — y 2 \ x y
* =rxV-l) + \ T T ^ ) l d?=r>,(i-r) + (bx-xl'\v
(3.22)
dt y V k J ' \ 1 + c x 2 J k
2. Efficient asymmetry, [38] is when the predation rate differs from the assim ilation
rate. Excessive efficient asym m etry is understood to mean that the predator is hav
ing problem s in capturing its prey. Therefore in our model
b\ ^ b2 either b\ > £>2 or b 2 > b\ ,
c \ 7̂ c 2 either c\ > c2 or C2 > c\.
3. Driving asym m etry is when the environm ental carrying capacities and the growth
rates are different so m uch that one o f the species dom inate the other. This usually
results in extinction o f one with the persistence o f the other.
Given these points, a perfectly sym m etric system (3.22) would be ideal for consideration
given the sw itching behavior as described in [11]. It is through variation o f a sensitive
param eter such as the environm ental carrying capacity that one obtains different equilibria
with different stability conditions.
From system (3.1) we already observed that (3.8) lead to equilibria such that there is
an equilibrium at x = 0 for all param eter values. Now if we consider the point (k\ ,0 ) as
the bifurcating point, then, near this point
t ( \ r,X l y ( X l 2 nfi(x,y) = n ~ — + — = 0,(3 23)r2y x a 2i
h ( x , y ) = r2 ~ — + = 0.ki k 2
Equilibrium solutions are obtained by solving the system (3.24) simultaneously,
x f \ ( x , y ) = 0 ,(3.24)
y f 2 ( x , y ) = 0 .
N on-zero solutions o f equation (3.24) had to be solved numerically, m ore so, under sym
m etry considerations. Let r\ — r2 — r, k \ — k 2 — k, and note that the sensitive param eter
is k.
As the param eter k is varied, two fixed points approach each other, collide, and are
destroyed. U sually this phenom enon occurs when we have what is referred to as a saddle-
node bifurcation or fo ld . The following conditions are necessary, but not sufficient for a
fold bifurcation to occur:
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CHAPTER 3. TWO-SPECIES SYSTEM 34
• F ( x c, kc) — 0.
• D XF m ust have precisely one zero eigenvalue, w hile the other eigenvalue has a
nonzero real part at (xc, kc).
In our case, the function F ( x , k ) is the two dim ensional right hand side o f (3.24)
for w hich D XF defines the Jacobian o f the linearized system at the critical equilibrium
(xc,yc) and the critical sensitive param eter kc. However because o f symmetry, we can
only expect either a transcritical bifurcation or a pitchfork bifurcation. To distinguish
between the two, we w ould require that d 3 F / d x 3 / 0 for a pitchfork bifurcation. In
this case, the sign o f this third partial derivative determ ines w hether the branching is
subcritical, (d 3 F / d x 3 > 0) or supercritical, (d 3 F / d x 3 < 0 ) .
Figure 3.14 shows num erical results at the bifurcation point, i.e. we have for b\ =
= 3, r\ = r2 — 0 .3, then
xc = 3.0737, y c = 3.0737, kc = 3.0000,
correct to four decim al places. We verify the num erical results by substituting x = y in
(3.24), and substituting param eter values into the resulting polynom ial
rx bx2 — jc3r 1 = 0
k k( 1 + cx2)
and sim plifying we get
x3 — 3x2 + 0.05x — 0.85 = 0.
The general solution o f a cubic polynom ial equation o f the form
ax 3 + bx2 + cx + d = 0
is given by
x = \/ —ft3 be y 2 7 a 3 ^ 6 a 2
be27a3 6 a 2
d 2 a
c ft2 3 a 9a 2
+ \/ —ft3 be \ 2 7 a 3 6 a 2
d_ 2 a
3 ( ~ b 3 beV V 27a3 6 a2d_
2 a+
c3 a
iF_ 9a2
b_ 3 a
(3.25)
By substituting a = l ,f t = —3 ,c = 0 .0 5 , d = —0.85, we obtain that x = 3.0737. This
m eans that the equilibrium solution at the branching point is the coexistence equilibrium
(3 .0737,3 .0737).
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CHAPTER 3. TWO-SPECIES SYSTEM 35
6
5
4
BPx 3
2
1
BE0 L-10 •5 0 5 10 15 20 25 X 35 4C
k
Figure 3.14: Variation o f param eter k with x showing BP-branching point, LP-fold bifurcation and H P-H opf bifurcation.
3.4 Other Corresponding Two Species Systems
In the previous section we analyzed the two species system that exhibited sw itching be
havior. We observed that variation o f either the interaction param eter b or the environ
m ental carrying capacity under symmetry, produced dynam ics that include bifurcation
o f equilibria in certain param eter regions. This m odel (3.1) was betw een the two prey
species in the absence o f the top predator. There is a critical carrying capacity value kc
where we observed that beyond this value, one o f the species disappeared. This m ay or
m ay not be the reality should there be the existence o f the top predator, which during
the period o f low density o f one due to predation, the top predator would prey on the
dom inant prey species, thus reversing the gain.
W hen one species is absent or occurs at low density due to predation we look at
the dynam ics o f the rem aining system. In that case we have either o f the following two
species system s, whose interaction can be understood by analyzing one o f the two system s
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CHAPTER 3. TWO-SPECIES SYSTEM 36
(3.26) and (3.27),
(3.26)
or(oy2z
d + y 2 ’(3.27)
Equations (3.26) and (3.27) are equivalent in that the reduced system is one o f a
predator-prey relationship.
We define the equivalent system to be
and observe that this system is a sim ple two species predator prey system. The predation
term is one incorporating the Holling Type III response function. In Table 3.2 we ex
plain the m eaning o f the term s and the corresponding param eters that are involved in this
system. The predator species equation is logistic. The positive contribution in the growth
rate o f the predator is achieved if there is m inim um com petition am ong predators given
the food abundance and choices that they enjoy.
For equilibrium solutions we equate to zero the right hand side o f (3.28) and solve the
resulting system.
(3.28)
(3.29)
s \U• If u — 0 then u s \ —
iOuv= 0 and v = 0 or v = q.
k d + u2
If v = 0 then for u ^ 0 we have that u = k.
• If both m / 0 and v ^ 0 then v — q + mu and
2 oCl .su + u ( q ( 0 + m(D + — ) — sd = 0.
K(3.30)
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CHAPTER 3. TWO-SPECIES SYSTEM 37
Table 3 .2 : Param eters and variables describing the predator-prey system (3.28).
Symbol Description
u population density for the prey species u
V population density for the predator species v
Si intrinsic rate o f increase o f prey species u
S2 intrinsic rate o f increase o f predator species v
k environm ental carrying capacity o f the prey species u
q and m constants defining environm ental carrying capacity o f species v
d predation saturation term
Equation (3.30) is solved again using the form ula developed by Cardano [1545], for
cubic polynom ial equations w here, in this case, a = s / k , b = —s , c = q ( 0 + m ( 0 + s d / k , d =
—s d .
Before solving this system, we w ish to recall D escartes’ Rule o f Sign for polynom ials:
L em m a 3.4.1. The num ber o f positive roots o f a polynomial, counting multiplicity, is
equal to the num ber o f sign changes o f the polynom ial or that num ber decreased by an
even integer.
Since all param eters are positive, the polynom ial equation (3.30) alternates in sign.
Therefore we expect either three positive roots, or one positive root depending on the
choice o f param eters.
Substituting the coefficients into (3.25) we obtain the coexistence equilibrium solution
o f the form (u*,q + mu*). Specific param eter values w ould result in num erical solutions
to the cubic polynom ial. We here choose
• an environm ental carrying capacity, k = 5,
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CHAPTER 3. TWO-SPECIES SYSTEM 38
/ ( « ) 60 f ( u ) = m3 — 5 u2 +5u — 5 >—J
40
I///
/
20
//
u = 4.0739/7
4 /-2 -1 0 _____________2 3 5 6
^ ---- ft-2 0
/ - 4 0 -
Figure 3.15: D epending on param eter choices, we get either three or one real roots, in this case for q = 1 — d .m — (O = 0 .2,5] — S2 = 0.3, £ = 5 we have a single coexistence equilibrium solution.
• the growth rate o f the prey species, 5] = 0.3,
• the predation saturation factor, d = q = 1,
• the predation rate, ( 0 = 0.2,
• and the predator com petition control factor, m — 0.2.
These values lead to the cubic polynom ial 0.06w3 — 0.3w2 + 0 .3u — 0.3 = 0, so that
dividing throughout by 0.06 the equation reduces to
m3 — 5m2 + 5m — 5 = 0. (3.31)
The graph o f the cubic function f ( u ) = u 3 — 5u2 + 5u — 5 has a single positive zero
which occurs at u = 4.0739, using N ew ton’s M ethod. So, the corresponding value for
v that results in the coexistence equilibrium is v = 1 + 0 .2 x 4.0739. The coexistence
equilibrium solution is (4 .0739 ,1 .8148), given the selected choice o f param eter values.
O ther param eter values choice could result in three coexistence equilibrium solution. This
m eans that by keeping all the param eters but one constant and varying that param eter,
equilibrium solutions bifurcate from one to three.
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CHAPTER 3. TWO-SPECIES SYSTEM
Table 3.3: Equilibrium solutions for the two species system.
39
The two species will die out over time.
The u species persists while the v species eventually disappear due to predation.
The v species persists while the u species eventually disappear due to predation.
The nonzero coexistence equilibrium solution.
In summary, Table 3.3 shows the type o f equilibrium solutions that is obtained under
this model.
We state w ithout p roof the following lem m a showing through phase portraits, the
nature o f stability o f equilibria.
L em m a 3.4.2. The coexistence equilibria i f existing, com prise globally asym ptotically
stable spirals and unstable saddle points.
The coexistence equilibrium solution for this two species system explains the fact
that with low density o f one prey source for the top predator, the predator is now finding
it hard to search for food and m ake a kill. In the process, this leads to a decrease in
population size o f the predator population. M eanw hile the prey species struggles to regain
as there also is less food source for survival. In this case, the other prey species begins to
grow in size because o f heavy presence o f the top predator against its im m ediate threat.
The follow ing figures, (Fig. 3.16 to Fig. 3.19) show phase portraits o f som e coexistence
equilibrium solutions, (saddles and spirals), depending on certain param eter variations.
These phase portraits are obtained as the carrying capacity of the prey species is varied
while other param eters rem ain fixed.
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CHAPTER 3. TWO-SPECIES SYSTEM 40
v
40
20
5 62 4 6 7O t
U
Figure 3 .16 : For large carrying capacity d — 5 o f the predator population, the only coexistence eauilibrium is the stable SDiral.
6
S
4
2
0
0 0.2 0.4 0& f .2 14 21 I d
U
Figure 3.17: There is a critical d (d=5.45) value at w hich a saddle point betw een two spirals is bom .
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CHAPTER 3. TWO-SPECIES SYSTEM 41
6
5
4
3
2
1
0
0 6 1.2 1.4 1.6 20 0 4
Figure 3 .18 : There is a critical (= 3.25) value at w hich a saddle point is born w hile the stable spiral persists.
“ I------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1------------1T
' I f f I I I................ i.............. V0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Figure 3 .19 : By zoom ing near the origin, the two spirals and a saddle persists for very small d (= 0 .15).
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Chapter 4
THREE SPECIES SYSTEM
4.1 Invariant Set
The system given by equations (2.4a)-(2.4c) provides meaningful solutions for all
(x, y,z) G R 3 such th a tx > 0 ,y > 0 ,z > 0. The variables x ,y and z are population den
sities o f the interacting species, therefore it is essential that solutions o f (2.4a)-(2.4c),
initially positive, will remain positive for all time.
Lemma 4.1.1. The non-negative region
^ 3 = {(*,;y,z) £ R + : x < k \ , y < k 2 , z < q + m(k\ + k2)}
is positively invariant fo r system (2.4a)-(2.4c).
Proof: In TIt, we have the following inequalities
x = 0 > 0 when x = 0 or Jc > 0 when 0 < x < k \ ,
y — 0 > 0 when y = 0 or y > 0 when 0 < y < k 2 ,
z > r^z for z < q + m(k\ + k 2).
Thus the nonnegative region Q. is positively invariant for the system (2.4a)-(2.4c). □
Lemma 4.1.2. A ll nonnegative solutions o f the system (2.4a)-(2.4c) are ultim ately uni
fo rm ly bounded in fo rw ard time, and thus they exist fo r all positive time. Moreover, the
system is dissipative.
Ideas o f the p roof o f this lem m a require that under certain environm ental conditions
where the predator-prey relationship is well defined, the response function that should
be em ployed m ust definitely determ ine which species is prey and which is predator for
the corresponding two species system. However we note that all the three species are
governed by the logistic part o f the equations, thus in forw ard time
x < n * ( l —x / k \ ) ,
y < r2y(\ - y / k 2),z < r 3z (l - z / h ) w here £3 > q + m{k\ + k 2).
42
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CHAPTER 4. THREE SPECIES SYSTEM 43
This leads to
x ( t ) < xoexp(C t) for som e constant C,
y( t) < yo exp (D t ) for som e constant D, £>2 < y < ^2 ,
z ( t ) < zoexp(r3 t) w here £3 > q + m(k\ + £2 )-
4.1.1 Symmetry
Equations (2.4a)-(2.4c) are to be considered for determ ination o f equilibria. Equilibrium
solutions and their stability analysis are necessary to understand the dynam ics o f the sys
tem under consideration. We observe that little w ould be obtained from this nonlinear
system w ithout considering special structural conditions that m ay influence the dynam
ics o f the system. We therefore begin by considering the system to have a sym m etric
structure.
Definition 4.1.1. A system is said to be sym m etric in the biological sense o f interactions
if all the interacting species have the same environm ental conditions and their growth
rates are the same.
For equilibria, the right hand side o f each o f the equations (2.4a)-(2.4c) is equated to
zero. To determ ine equilibrium solutions under symmetry, that is, by consideration o f the
transform ation (x , y , z ) — *• (y ,x,z) , we first equate the right hand side o f (2.4a)-(2.4c) to
zero. This m eans that the third equation (2.4c) simplifies to
w = q + 2 mx, (4.1)
while the first equation, under symmetry, becom es
/ x \ f b \ x — x 2\ x 2 (Ox2(q + 2 mx)
<42)
Sim plifying (4.2), by substituting (4.1) into (4.2), we obtain a quintic polynom ial in
the variable x and param eters b , c , d , k , m , q , r and ( 0 (ignoring subscripts):
dkr — x(rd + k(Oq) + x2 (kr + krcd + bd — 2 km(o)
— x 2(r + d + kcq ( 0 + cdr) + x 4(ckr + b — 2ckm(o) — x5(l + cr), (4.3)
whose solutions will give rise to the x and y com ponents o f equilibrium solutions o f the
form (x*,y*,z*). Stability o f equilibria under sym m etry is an im portant analysis.
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CHAPTER 4. THREE SPECIES SYSTEM 44
4.1.2 Real Roots under Symmetry
From (4.3) let r — 2m ( 0 = T. Then the quintic polynom ial can be w ritten as
—do u5 + a\U4 — (32 M3 + — CI4 U + £?5 (4.4)
where
ao = 1 + rc ,
d\ = k c z + b,
(32 — r + d + red + kcqco,
(33 = k t + fcrcd + bd,
d4 — rd + kq(0 ,
«5 = krd.
Since all the param eters are positive, we recall that
• k- is the param eter defining the environm ental carrying capacity o f the prey species.
• c- is the param eter that provides the dum ping effect o f the predation reaction.
• d- is the param eter that saturates the prey population due to predation.
• b- and ( 0 are the param eters that increase or decrease predation.
• q- is the param eter that norm alizes the residual reduction in the predator population
due to scarcity o f favorite food.
• m- is the param eter that saturates the predator population with food abundance.
• r- is the prey population growth rate.
It is therefore expected that for coexistence or persistence o f the three species we
require that r > 0 .
Lemma 4.1.3. The condition T > 0 is d necessdry but not sufficient condition f o r the
quintic polynomiol to hdve 5 ,3 or 1 red I roots.
Proof: By D escartes’s sign rule on (4.4) we observe that all the a,-, i — 0 , . . . 5 are
positive, provided T > 0, that is m ( 0 < § . There are therefore five sign changes, m eaning
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CHAPTER 4. THREE SPECIES SYSTEM 45
that we w ould get a m axim um of 5 positive rational roots, 3 possible rational roots or 1
rational root. □
Param eter variations will always be key to the determ ination o f the num ber o f positive
real roots o f the quintic polynom ial (4.3). The choice to fix some o f these param eters is
often guided by such pre-determ ined analysis o f possible outcom es through sensitivity
analysis. Obviously, there are so m any param eters in the polynom ial and some have to
be fixed w hile the overall dynam ics o f the system under sym m etry depend on certain
particularly sensitive param eters.
4.1.3 Tinning Points
Given the conditions for existence o f positive rational roots, we choose some param eter
values that m eet these conditions. Table 4.1 lists the param eter values for the model. To
decrease the com plexity o f this model, all param eters except those that directly increase or
decrease population interaction or predation, were held constant. It should be m entioned
that the param eters k and d are not necessarily unique in generating dynam ics that lead to
chaos in som e param etrization o f the model. B arradas and H ernandez [11] in their two
species system analysis that resulted in hysteresis, considered the environm ental carrying
capacity as one o f the sensitive param eters. E isenberg and M aszle [4] regarded their
sensitive param eter to be the saturation param eter which tends to stabilize interaction in
predation.
The corresponding quintic polynom ial (4.3) that results due to sym m etry w ould there
fore be w ritten as
p(x) = —1.2x5 + (0.1£ + 5)x4 + ( - 0 . 4 - \ . 2 d - 0 A k ) x 3
+ (0.2 k + 5 d + 0.2 kd ) x 2 + ( - 0 . 4 d - 0.2k)x + OAkd. (4.5)
The turning points for this equation are determ ined by solving the system
p{x) = 0 ,
p'{x) = 0,
wherep{x) — aox5 T a \ x 4 + a2 x 2 + a^x2 + a^x + a$ ,
p' (x ) = 5<2o* 4 + 4a i x 3 + 3a2 x 2 + 2 a^x + <34 ,
p' (x) denote the derivative with respect to x o f the polynom ial p(x ) , and the a j , i = 0 . . . 5
are as in Table 4.1. Using the resultant o f the quintic equation (see [33], [39], [2], [15],
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CHAPTER 4. THREE SPECIES SYSTEM
Table 4.1: A ssigning values to certain param eters, k and d vary.
46
Param eter Value
b 5
c 0.5
m 0.5
q 1
r 0.4
( 0 0 .2
T 0 .2 (calculated, t = r — 2 mat)
ao 1 .2
a\ 0 A k + 5
(32 0 .4 + \ .2d + 0 . \ k
(33 0.2k + 5d + 0 .2kd
(34 0 A d + 0.2k
(35 OAkd
[19]), we obtain that the discrim inant with respect to the two polynom ial equations (4.6)
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CHAPTER 4. THREE SPECIES SYSTEM 47
for the quintic polynom ial (4.5) is
ao a\ a 2 <23 (24 <35 0 0 0
0 ao a\ a 2 (23 (24 a 5 0 0
0 0 ao a\ (22 a-i (24 a5 0
0 0 0 ao a\ a 2 <33 (24 <35
bo b\ bi h b4 0 0 0 0
0 bo b\ bi h b4 0 0 0
0 0 bo b\ b2 bj. b4 0 0
0 0 0 bo b\ b2 b~i b4 0
0 0 0 0 bo b\ b 2 b3 b4
We observe that in this case the i = 0 . . . 4 are m ultiples o f the a,- i — 0 . . . 4 such
that bo = 5ao,b{ — 4 a \ , b 2 — 3a 2 , b 3 = 2a3 and b4 = <34. We require that for a non-trivial
solution to be obtained, this determ inant m ust be zero;
^■(p ,p') = 0-
Ideally, the resultant condition w ould determ ine param etric conditions for the sensitive param eters k , d that guarantees the existence o f turning points. They could also provide the param eter region where there are these turning points. However the determ inant o f this m atrix is an odd m ixture o f sum m ation o f term s involving these two param eters:
rR(/,y ) = 10252.08253d5 + 79.76225893Ar5d4 - 0.15704064? - U 6d 6
+ 0.1849344? — 2k1 + 136.3582648d4 +415.1209820W 3 - 17329.86828£2d3
+ 185.6106455fc3d + .817275648fc6d4 + 3022.876625k4d4 - 182.5052844ArV
+ .340328448(fc2d - 1385.453962d4£ + 265880.1060d4fc2 + 50222.36202ArV
— 0.26515968? - I k ’d 2 +0.2013696? - I k ’d4 - 14.69000908ksd 5 + 1227.224383d 6
- .361961472&6d5 - 504461.2110d5k - 113330.1009d5k2 - 8459.885367k3d 5
+ .152262912£6 - 0 . 3772416c - I k ’d 5 + ,680656896/W2 + 0.6144? - 5ks
- 1,721032704fc5d 6 - 1.782616320fc6d - 2.403071424£6d2 - 3.381784320fc6d3
+ .453771264d3 +437.7058714£2d 2 - 0.89088? - 4£8d2 - 0.4608? - 5k%d4
-0 .12288? - 4k*d5 + 36.75547238d 1 - 0.31544832? - 1 k1 d 5 - 0.26781696? - 1 k7d
- . 127401984fc4d7 - 1815Al%212d1k - 293.852676Id7*2 - 9.55514880Ar3d7
+ 0.56721408?- U 3 + 27.51232512>t4 - 6195.408007Ar3d2 - 229.4885099£4d
- 1342.401032/t4d 2 - 35696.95305Ar3d 3 -4 8 0 2 .1 29833fc4d 3 + 4.0042629\2k5
-355.1150589k4d 5 -39 .18777139A:5d - 94.45107687k5d2 - 0.98304? - 4k8 d 3
- 0.98304? - 4k*d - 81.93927172k4d6 - 60428.31572d 6k - 17214.25804d6fc2
- 1789.955851A:3d6.
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CHAPTER 4. THREE SPECIES SYSTEM 48
The critical values o f k and d at w hich this resultant is zero have to be obtained.
However given the two param eter polynom ial it is w ise to fix one o f these param eters and
then solve the resultant equation to obtain the values o f the other such that the com bination
would result in the resultant to be zero, [19]. For these critical values, the fact that the
resultant equals zero is a necessary and sufficient condition guaranteeing the existence o f
the turning points.
4.1.4 Bifurcations
We have tried to determ ine the dynam ics o f the three species system through sym m et
ric analysis where w e considered the transform ation (x , y , z ) — > (y , x , z )• The resulting
sim plifications led to the one-dim ensional vector field
dx— = p ( x ,d , k ) , i g R , G R, (4.7)dt
where p is a C r function on som e set R x R . This general vector (4.7) field can further
be reduced to a one param eter, one-dim ensional vector field such that for som e critical
param etric value Ao the point (jco, Ao) is a fixed point. In that case we have that
p (x 0,Ao) = 0 .
Stability o f the fixed point (jco, Ao) is im portant, and so is the need to determ ine the
dynam ics o f the system locally about this fixed point as A, the sensitive param eter, is
varied.
The flow o f the system described by (3.1) is said to be qualitatively the same as the
flow o f its linearization if, by the H artm an-G robm an theorem , the eigenvalues o f the Ja-
cobian o f the linearized system are hyperbolic, that is, the eigenvalues o f the linearization
has no eigenvalues on the im aginary axis.
Definition 4.1.2. A fixed point ( jc, A) = (0 ,0 ) o f a one-param eter fam ily o f one-dim ensional
vector fields is said to undergo bifurcation at A = 0 if the flow for A near zero and jc near
zero is not qualitatively the same as the flow near jc = 0 and A = 0 .
In the situation where these eigenvalues are non-hyperbolic, qualitative resem blance
o f the flow near the fixed point is not guaranteed. Norm al form analysis procedure deter
m ines the contributing terms that play m ajor roles in determ ining system behavior near
the fixed point.
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CHAPTER 4. THREE SPECIES SYSTEM 49
Generically, the one-param eter fam ily o f vector fields defined as
— = p ( x ,X ) = Xx — x3 + 0 (x:4) at
(4.8)
which provides the pitchfork bifurcation shown in Fig. 4.1 can further be perturbed to
Figure 4 .1 : G eneric diagram o f the pitchfork bifurcation resulting from equation (4.8) ignoring higher order term s. The dotted line is the unstable equilibrium .
give the form
Equation (4.9) is a degenerate case o f a fc-codimension bifurcation where k = 2. Theoreti
cally, w hat this says is that the system (3.1) is not structurally stable and is o f codim ension
The study o f behavior o f fixed points is ideally a m uch sim pler approach tow ards anal
ysis o f structural stability o f (3.1). If it were easy to determ ine the equilibrium solutions
o f the system (2.4a)-(2.4c), we w ould sim ply look for those fixed points which are non-
hyperbolic. However, we w ent around this problem because o f the m any non-linear and
rational term s in system (2.4a)-(2.4c).
We wish to determ ine the conditions that give rise to a cusp bifurcation [31] through
the use o f the Theory o f Resultants.
A cusp bifurcation occurs when the follow ing conditions are satisfied:
x .2
(4.9)
k = 2 .
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CHAPTER 4. THREE SPECIES SYSTEM 50
1. A fixed point with a zero eigenvalue.
2. A second derivative o f the vector field vanishes in the direction o f the eigenvector
corresponding to the zero eigenvalue.
The sym m etric transform ation (x , y , z ) > (y ,x,z) reduced the three species system into
a one-dim ensional system
x — p ( x , n ) , x £ M, p € M2. (4.10)
We seek conditions which give rise to a cusp bifurcation. We note that the function p ( x , p )
is smooth since it is a polynom ial function in both x and A, hence adm itting continuous
partial derivatives. In that case w e are led into determ ining the resultant o f the system
p{x) = 0
p'[x) = 0
p"(x) = 0 ;
this last system being a m ultivariate resultant. Therefore
pi (u) = - 1 .2jc5 + (0 .Ik + 5)x4 + ( - 0 .4 + \ . 2d + 0 . \ k ) x 3 + (0.2k + 5d + 0.2k d )x 2
+ (—0.4c? - 0.2k)x + OAkd = 0
p 2 („) = - 6 x 4 + (0.4k + 20 )* 3 + ( - 1 .2 + 3.6 d + 0.3 k)x 2 + (0.4k + 1 0 d + 0 A k d ) x
- 0 .4 c /-0 .2 k = 0
p 3 (n) = - 2 4 x 3 + (1 .2k + 60)x2 + ( - 2 .4 - 7.2c/ + 0.6k)x + 0.4k + 1 Od + OAkd = 0.(4.11)
The process to determ ine w hich param eters influence m ost o f the system behavior is done
through sensitivity analysis. The nonlinear term s in the model system im ply that there are
m any param eters w hose variation will lead to com plicated dynam ics. We assign values
to some o f the param eters as a way to non-dim ensionalize the system. Appendix A .l p ro
vided a num erical approach to determ ine w hich param eter was giving different dynam ics
as that param eter was veing varied. We assigned values to some o f the param eters, leav
ing two free param eters. We observed that the use o f num erical techniques to find the
critical values o f these param eters, though still involving the use o f the theory or resul
tants, w ould be em ployed, only this tim e under a polynom ial system with two quadratic
and one linear equation in these sensitive param eters. Such a problem is often referred to
as a codim ension 2 problem.
The quintic polynom ial (4.5) is reconsidered to determ ine conditions that give rise to
cusp bifurcation. To determ ine these conditions, the system of polynom ial equations
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CHAPTER 4. THREE SPECIES SYSTEM 51
p(x) = 0
p'{x) = 0
p"{x) = o,
where (4.5) is derived from the general quintic equation (4.3) by considering the values
in Table 4.1 is considered. The transform ation
P2 ^ P2~XP3
transform s the polynom ial p 2 in (4.11) into a linear equation in k and d so that the resultant
is then applied to the system o f polynom ial equations
p i ( jc ) = - 1.2x5 + (0. I k + 5)x4 + ( - 0 .4 + 1 .2d + 0 A k ) x 3 + (0.2k + 5d + 0.2kd)x 2
+ (—0.4d - 0 .2k)x + OAkd = 0,
p 2 (x) = — 24jc3 + (1.2 k + 60)x2 + ( - 2 .4 - 1.2d + 0.6k)x + 0.4A: + 10rf + OAkd = 0,
P 3 ( jc ) = 18x4 + ( - 0M - 40)x3 + ( 1.2 + 3.6d + 0.3k )x 2 - OAd - 0.2k.(4.12)
Treating x as a constant and introducing a variable y, we hom ogenize the system to obtain:
p\ = (—1.2x5 + 5x4 — 0.4x3 )y2 + (O.lx4 + 0.1x3 — 0 .2 x ) k y + (1.2x3
+ 5x2 - 0 A x ) d y + (0.2x2 + 0 .4 )kd
P2 = (—24x3 + 60x2 — 2.4x)y2 + (1.2x2 + 0.6x + 0.4)A:y+ (—7 .2 x + \ 0 ) d y (4.13)
+ (0.4)1W
p 3 = (18x4 — 40x3 + 1,2x2 ) y + (—0.8x3 + 0.3x2 - 0 .2 )£ + (3.6x2 - 0 . 4 )d.
The first two equations o f (4.13) are degree two hom ogeneous polynom ials in the vari
ables k , d , y . The last equation o f (4.13) is a linear hom ogeneous equation in k , d and
r-*
The eigenvalue problem p ( X ) x — 0, w here p ( A) = ^ A 'Aj, was extensively studiesi=0
in [33], [19], [18], [15], [2], We are m ainly concerned with regular m atrix polynom ials,
that is, those polynom ials p ( A) such that d etp (A ) is not identically zero for all A 6 C. In
such polynom ials, the eigenvalues are precisely the roots o f the scalar polynom ial defined
by detp (A ).
D efinition 4.1.3. Let p ( A) be an n x n m atrix polynom ial o f degree k > 1. A pencil
L(A) = AX + y with X , Y belonging to a field o f real or com plex num bers, is called a
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CHAPTER 4. THREE SPECIES SYSTEM 52
linearization o f p (X ) if there exist unim odular m atrix polynom ials E ( X ) , F ( A) such that
E ( X ) L ( X ) F ( X ) = ’ p(A) 00 I ( k - \ ) n
The spaces o f pencils provide a convenient platform to look for linearizations o f struc
tured polynom ials, such as those presented by sym m etric or skew -sym m etric coefficient
m atrices or those defining palindrom ic structures, [21], [12] . Unfortunately, a num ber
o f contributions towards analysis o f system s that are not necessarily structured in any
way have mainly been those w hose coefficient m atrices are square. The problem under
scrutiny has resulted in rectangular coefficient m atrices and the num erical approach to de
term ine the needed eigenvalues and eigenvectors requires that these m atrices be square.
This process requires that the rectangular m atrices that define the eigenvalue problem be
m ultiplied by som e random ly generated m atrices o f the same dim ension. W hile this pro
cedure fails to work for m any num erical analysis problem s due to generation o f m any
useless extrem e eigenvalues, this procedure always w orks for the problem under study in
that the dynam ics are localized and infinite eigenvalues or zero eigenvalues are not worth
considering for the practical problem o f predator-prey dynamics.
A nother observation is that while the theory that include °° and 0 as eigenvalues re
quires that A 5 be nonsingular (for °° eigenvalue) and Ao be nonsingular for 0 eigenvalue,
these requirem ents w ould be necessary if we desire to determ ine lower and upper bounds
for the resulting eigenvalues, [13], [20], It should be m entioned that the values o f the
determ ined eigenvalues are in fact the population sizes o f the prey species as well as
the critical param eter values which give rise to the cusp bifurcation. Therefore we are
interested in finite nonzero positive eigenvalues only.
Com panion m atrices are often considered useful form s o f linearizations o f p ( X ) , [21],
[12].
D efin ition 4.1.4. A com panion m atrix [21] C(X) = X X + Y from the linearization o f p (X )
is called the first com panion form if
X = diag(A *,/(*_!)„) and Y =
Ak - 1 A * _ 2- I n 0
Ao0
- I n 0
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CHAPTER 4. THREE SPECIES SYSTEM 53
It is called the second com panion form if
X = diag(A *,/(*_!)„) and Y =
Ak - 1
Ak - 2
Ao
-In
0
0
0
- I n0
/ kI
0=0
In general, for an eigenvalue problem o f the form p ( X ) y / = j ^ A'A, J y/ = 0, the
linearization
L(X) — X
will reduce the £th order eigenvalue problem into a first order equation. In our situa
tion, k — 5 so that the quintic eigenvalue problem is written in block m atrix form and
represented by its corresponding linearization:
Ak 0 . . . 0 ' Ak - l A k- 2 • • • ^ 0
0 In +~In 0 . . . 0
0
. 0 0 In . 0 - I n 0 _
' a 5 0 . . . 0 ' A 4 A 3 ■■■ A o
0 h
••• 0+
- 1 5 0 . . . 0
0 0 h .0 - I s 0 _
L(X ) = X
It should be m entioned that linearization o f this quintic eigenvalue problem is not unique.
Equivalent transform ations o f the block matrices
'A 5 0 . . . 0 ' ' a 4 A 3 ■■■ Ao0 I5 and
- I s 0 . . . 0
0
0 1
0
0 - I s 0
yield the same results. This is consistent with the standard linearization procedure o f
introducing the variables x\ = X k 'x , X2 = X k l x X3 = Xk - 2 , k - 3. ■Xk- 1 = X x ,X k = X.
For writing convenience and easy notation let k = x , d = y, y = z and x — X, and
consider the lexicographical arrangem ent o f the linear and quadratic term s in x . y and
z to be
and
{x2 , xy ,x z ,y 2 , yz , z2}
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CHAPTER 4. THREE SPECIES SYSTEM 54
respectively.
We therefore rewrite the hom ogeneous polynom ials (4.13) as
pi (x.y.z, A) = —0 .2x —0.4y + (0.3x + 3.6y + 1.2 z)A 2 + (—0 .8x — 40z)A 3 + 18zA4,
P2 ( x , y , z , k ) = 0 .4ry + ( -0 .2 x z -0 .4 y z )A + (0.2xy + 5yz)A2
+ (0 . Ixz 4- 1.2 yz — 0.4z2 )A 3 + (0 . \xz + 5z2 )A4 + (—1.2 z2 )A^,
/?3 (x.y,z, A) = 0 .4 x y + 10yz + (0.6xz — l . l y z — 2.4z2)A + ( 1 .2xz + 60z2)A2 + ( - 2 4 z 2)A3.(4.14)
This arrangem ent leads us to solve a quintic eigenvalue problem o f the form
+ Ag, + A2g 2 + A3g 3 + A4g4 + A5g5 1 ^ = 0 ;
with f . g j , i = 1,5 functions o f x .y and z; and
f ( x , y , z ) = { —0.2x,0.4xy,0.4xy + 10yz},
g i (x , y , z ) — {0, - 0 .2 x z - 0 .4 y z ,0 .6 x z - 7 .2 y z - 2 .4 z 2},
g 2 (x,y,z) = {0.3x + 3 .6 y + 1 .2 z ,0 .2 x y + 5yz, 1.2xz + 60z2},t o (4.15)
g 3 (x,y,z) = { - 0 .8 x - 4 0 z ,0 .1 x z + 1 .2 y z - 0 .4 z , - 2 4 z },
g4 (x,y,z) = {18z.0.1xz + 5z2 ,0},
g5(*,:y,z) = {0, — 1 .2z2,0}.
The coefficients in the quadratic system o f equations, com bined with those o f the origi
nal system o f hom ogenized polynom ials will form a 6 x 6 matrix whose determ inant is
the resultant o f the three polynom ials. A ppendix A.3 provides the num erical approach
through a M atlab program to solve the resultant o f three polynom ials. Exact values o f the
critical param eters k and d, as well as the prey species density x w here a cusp bifurcation
occurs are generated.
We let yq = A4 t/r, i//2 = A3 y/, y/3 = A2 t/r, 1//4 = A yf and y/5 = yr the quintic eigen
value problem p ( X ) y / = I ^ ^ 'A j I yt = 0 is transform ed into the equationV«=o )
A 5 ( X y / i ) + A 4 yf\ + A 3 y/ 2 + A 2 y/3 + A 1 \f/ 4 + Ao y/ 5 = 0 . (4.16)
From (4.14) and (4.15) we form block m atrices that define the m atrix polynom ial that
in turn defines the quintic eigenvalue problem . We define block m atrix entries o f (4.16),
using coefficients o f polynom ials (4.14) as shown in Table (4.2). These block m atrices
are not square m atrices. A num erical technique to m ake them square before one could
linearize the quintic eigenvalue problem requires that each o f these block m atrices is
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CHAPTER 4. THREE SPECIES SYSTEM
Table 4.2: Form ation o f block m atrix entries for the quintic eigenvalue problem .
/ , = [ -0 .2 , -0 .4 ,0 ]
h = [0 ,0 .4 ,0 ,0 ,0 ,0 ]
/ 3 = [0 ,0 .4 ,0 ,0 ,1 0 ,0 ]
= [Aoi, Ao2 ,Ao3];
A01 = oneBlock{3, dout, 1, / I }
Aq2 — oneB lock{3,dout, 2 , /2 }
A03 = o n eB lo ck { 3 ,d o u t,2 ,/3 }
g\ = [0 , 0 , 0 ]
g 2 = [ 0 ,0 ,- 0 .2 ,0 ,- 0 .4 ,0 ]
g 3 = [0 ,0 ,0 .6 ,0 , - 7 .2 , -2 .4 ]
M = [^ 10, ^ 11, ^ 12];
A 10 = oneB lock{3,dout, 1 , g l }
A n = oneB lock{3,dout, 2 ,g2}
A 12 = oneB lock{3 ,dou t,2 ,g3}
hi = [0 .3,3.6,1.2]
h 2 = [0 , 0 .2 ,0 , 0 ,5 ,0]
/i3 = [0 ,0 ,1 .2 ,0 ,0 ,6 0 ]
A2 = [^21 , ^ 22 , ^ 23];
A21 = oneB lock{3,dout, 1,/?1}
A 2 2 = oneB lock{3 ,dout,2 ,/i2}
A23 = oneB lock{3,dout,2 ,/?3}
ii = [ - 0 .8 ,0 ,- 4 0 ]
h = [0 ,0 ,0 .1 ,0 ,1 .2 , —0.4]
I3 = [0 ,0 ,0 ,0 ,0 ,-2 4 ]
A 3 = [^31 , ^ 32 , ^ 33];
A31 = oneBlock{3, dout, 1, i \ }
A32 = oneB lock{3 ,dou t,2 ,/2}
A33 = oneBlock{3, dout, 2, /3}
j \ = [0,0,18]
j i — [0 , 0 , 0 . 1 , 0 , 0 ,5]
h = [0 , 0 , 0 , 0 , 0 , 0 ]
A4 — [A41 ,A42,A43];
A41 = oneB lock{3,dout, 1, j 1}
A42 = oneB lock{3,dout, 2, y'2}
A43 = oneB lock{3,dout,2 ,y '3}
h = [0 , 0 , 0 ]
k2 = [0 , 0 , 0 , 0 , 0 , - 1 .2 ]
k3 = [0 , 0 , 0 , 0 , 0 , 0 ]
^ 5 = [^51 , ^ 52 , ^ 53]-
A51 = oneB lock{3,dout, 1 , k \ }
A52 = oneB lock{3 ,dou t,2 ,£2}
A53 = oneB lock{3,dout,2 ,£3}
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CHAPTER 4. THREE SPECIES SYSTEM 56
m ultiplied by the transpose o f a random ly generated m atrix of the same dim ension. The
technique in solving the quintic eigenvalue problem , as said earlier, requires linearization
o f the system through com panion m atrices so that we would eventually solve the generic
eigenvalue problem
A x = Xx.
The m atrix A ideally should be square,even though this problem is actually one o f solving
n polynom ials in n + 1 variables X , x \ , . . . , x n. To solve this system, the system o f po ly
nom ials is augm ented with an equation o f a linear com bination o f the variables jc,- using
some random ly chosen coefficients, [18]. Problem s m ay arise where the system m ay have
n isolated solutions instead o f n + 1 solutions. Corrections are often m ade by considering
the solution o f the system
q\ = (A + <?n)(xi + e \ 2)
q 2 = (X + e2 l )(x2 + e22)
q n = (A + €n \) (xn -r s n2)
q n + 1 = c \ x \ H H c nx n + Cn+ 1 •
It is observed that the resulting eigenvalues and eigenvectors that are obtained must
be selected carefully as som e are extrem e cases which do not respond to the biological
system requirem ents. The approach that was em ployed for this contains a section for
checking the eigenvalues through solution identity o f the corresponding polynom ials, (see
A ppendix A.3). A m ore detailed explanation about solutions o f hom ogeneous system s
through resultant theory, w here the Bezout resultant approach is discussed, can be found
in [33] and [18].
Results o f the M atlab code that num erically solved the quintic eigenvalue problem
using block m atrices defined in Table (4.2) are given in Table (4.3). By considering the
first set o f values from table 4.3, as well as recalling the param eter values that provided for
the quintic polynom ial (4.3), we observe that the corresponding coexistence equilibrium
under the sym m etric transform ation
is num erically determ ined to be
(x,y,z) = (0 .1561 ,0 .1561,2 .0000).
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CHAPTER 4. THREE SPECIES SYSTEM 57
Table 4.3: Critical population density at critical param eter values.
X k d
0.1561
0.1377
0 .1 0 0 2
0.0905
-0 .0 5 0 1
-0 .0 4 5 2
This critical equilibrium is the cusp equilibrium . The Jacobian m atrix at the coexistence
equilibrium is
J n h i J\ 3 \ / 8.8826 0.1335 1050.7264
h \ h i J 1 3 = 0.1335 8.8826 1050.7264
h \ Jyi J 3 3 J \ 0 .1894 0.1894 -6 4 .6 2 9 8
The eigenvalues A/, i = 1,2,3, o f the Jacobian m atrix evaluated at this equilibrium are
A, = 14.0729, A2 = 8.7491, A3 = -6 9 .6 8 6 7 ,
which shows stability in the direction o f the top predator species and instability in the
direction o f the two sw itching prey system.
We obtained critical values for x and p defined in Table 4.3 for the system (4.10),
where p = (k .d) . By a param eter-dependent shift o f the coordinate E, = x + 8 ( p ) and
expanding p ( x ) ( = p(E, — 8 ( p) ) in Taylor series, and taking into account that p ( 0 ,0 ) =
p ' ( 0 , 0 ) = p " ( 0 , 0 ) = 0 but p"'(x) ^ 0 , we obtain the corresponding normal form for the
cusp bifurcation that assum es the generic equation
f\ = j 3 i + / 3 2 t ? - t 7 3-
A plot o f the equilibrium m anifold
M = {(i7,j3i,02) : 0 i + 021] — T73 = 0}
has its projection onto the ( 0 i ,0 2)-plane as shown in Fig. 4.2.
4.2 Symmetry-Breaking Bifurcation
W hile analysis o f the system under sym m etry assum es that individuals in the same species
are identical, obviously they are not. We however continue to study sym m etry by con
sidering that alm ost sym m etric system s are regarded as small perturbations o f sym m etric
systems.
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CHAPTER 4. THREE SPECIES SYSTEM 58
0f t
Figure 4.2\ O ne-dim ensional cusp projection o f the equilibrium m anifold. A nondegenerate fold bifurcation occurs upon crossing either branch (7j and Ti) at any point other than the origin.
Under sym m etry
In the case o f sym m etric eigenvectors, we have a turning point type o f bifurcation, whereas
in the antisym m etric case we obtain a pitchfork bifurcation. We also observe that if the
full system is stable about a fixed point, the same dynam ics are observed in the sym m etric
system, that is, there is no loss o f stability. H owever this is not the case in the antisym
m etric system, w here change o f stability is observed through a pitchfork bifurcation.
any any eigenvector o f the three species system is either symmetric
or antisym m etric, represented by
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Chapter 5
Discussions and Conclusions
5.1 General observations
M ost published m odels o f predator-prey system s that exhibit com plicated dynam ics such
as chaos involve term s that result in oscillations. Indeed an ideal predator-prey system
should show cyclical behavior owing to the fact that when prey density is low, predator
density, if solely dependent on this prey for food, w ould also decrease due to scarcity
o f food. The result is that a critical population size for the predator species is reached,
beyond which extinction occurs. D uring this period o f low predation, prey density rises
again, w hich in turn increases predator density. In reality the predator population does not
depend solely on one food source. This has led to m odel systems exhibiting either lim it
cycles or stable foci.
We briefly explain the assum ptions we considered for the model. M uch o f this re
search was centered on understanding the specific system interaction that exhibits the
switching behavior in the special tw o-species system. Instead o f approaching the m odel
system through the standard cyclical behavior o f predator-prey dynam ics, the interact
ing term s rule out the possibility o f lim it cycles. The effect o f the interaction between
the corresponding two species was expressed through the response a,y, i , j = 1 , 2 , / ^ j .
The predation intensity was m easured by the absolute value o f this coefficient Ofy. The
positive or negative sign o f a ,7 reflected a positive or negative contribution to the density-
dependent factor in the per capita growth rate o f the species. This way, one w ould be able
to determ ine which species is the predator and which is the prey.
The resulting equations could have been re-scaled by introducing new variables for
which the carrying capacity o f the environm ent disappeared, but in an effort to reflect
ecological reality, w e chose to keep the original m odel. Indeed the carrying capacity be
came the sensitive param eter we chose to vary while other param eters rem ained constant.
The work by B arradas et. al. showed m ore variations o f param eters. They tried to tie
the resulting phase portrait diagram s to environm ental changes. In our study, only the
carrying capacity param eter becam e the sensitive param eter which varied. It is difficult
to vary two or m ore param eters sim ultaneously given the nonlinearities involved in the
model. We instead chose to approach the sim plifications by considering the system under
59
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CHAPTER 5. DISCUSSIONS AND CONCLUSIONS 60
symmetry. The results o f this approach lead to the dynam ics o f a pitchfork bifurcation
outcome.
We also observed that even when the dynam ics involve the top predator and one prey
species, through response functions o f the H olling Type III where there rem ains the choice
factor, the overall result is that we have only stable foci in the event there is the existence
o f the coexistence equilibrium solution.
The standard approach to obtaining equilibria and determ ining the nature o f stabil
ity could not be pursued due to nonlinearities in the system. Considering the system
under sym m etry w e required the use o f the theory o f resultants to determ ine conditions
necessary for turning points. In this case, sym m etry m eant trying to provide the sam e en
vironm ental conditions for the three species. W hile this m ay not be realistic biologically,
a perturbation o f the original system under sym m etry lead to sym m etry-breaking analy
sis. The change o f stability through sym m etry-breaking again gave rise to characteristic
sym m etry-breaking bifurcation o f the pitchfork type. Conditions for a cusp bifurcation
further sim plified the analysis, but we observed that w ithout com putational num erical
analysis, a cusp bifurcation could only be observed through simulations.
It is im portant to find conditions giving rise to cusp bifurcation. The dynam ics of
populations leading to a cusp bifurcation need careful param eter variations because the
existence o f small perturbations near the cusp peek could result in some catastrophic jum p
between stable branches. Ecologically, this could lead to catastrophic regim es where one
species m ay go extinct.
O ur results show that these bifurcation diagram s allowed the visualization o f different
types o f interactions and the dynam ics o f variability o f certain param eters. Specifically,
an increase in the environm ental carrying capacity £,, led to different equilibria and their
stability conditions. We noted that the increase in carrying capacity o f one species resulted
in a direct decrease in the abundance o f the other species.
The results o f this m odel show that in the presence o f the third top predator, the system
stabilizes and the three species is observed to persist. The conditions for coexistence in the
species are a requirem ent for persistence. The top predator is understood to play a m ajor
ecological role in controlling population abundances in a given ecosystem . The fact that
under very low density o f one prey species, the two species system continues to persist,
m eans that the top predator acts as a regulatory m echanism for continued existence o f the
three species, albeit under set environm ental conditions.
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CHAPTER 5. DISCUSSIONS AND CONCLUSIONS 61
5.1.1 Extensions
The dynam ics o f the three species system was considered as a w hole only under condi
tions o f sym m etry and hence perturbations leading to sym m etry-breaking. Suggestions
for analysis o f this system included considering one o f the species as a linear m ultiple of
the other. On the analysis o f the corresponding two species system, we realized that the
top predator w ould continue to m ake food choices, despite the absence o f the other food
source. We note the deficiency in this regard but in reality, alternative food sources are
usually provided in the environm ent. The dogfish-lobster-snail relationship is one where
the dogfish does not solely depend on lobsters and snails, but also feed on other m arine
life. Despite this observation one m ight w ish to change the m odel system o f the corre
sponding two species system to accom m odate the fact that the top predator no longer has a
choice in the absence o f low abundance o f one prey system due to predation. A nother pos
sibility w ould be to change the density-dependent carrying capacity o f the environm ent
for the predator population in favor o f a response function of the H olling Type II which
w ould benefit the growth rate o f the predator. These changes w ould perhaps provide a
model that is m ore biologically realistic.
We wished that through sensitivity analysis o f param eters that influence the dynam ics
o f the system, we would identified values that gave rise to interesting dynam ics. It is true
that given the m any nonlinearities in the m odel system, a num ber o f param eters could
have been chosen for sensitivity analysis w hose change produced som e drastic system
behavior. We therefore wish to further this study by doing more sensitivity analysis o f the
param eters involved.
The approach we chose o f analyzing the system under full sym m etry [38] is ju s t one
approach to analyze the system, especially given lim ited com puter resources. However
the algebraic com putations involved through the use o f Resultants Theory necessitated
com putational num erical analysis code to produce num erical results needed for a cusp
bifurcation, (see Appendix A.3). The com putational techniques em ployed are not yet
com pletely understood. We believe that through the use o f DSTool or AUTO, software
packages designed to solve dynam ical system s with non-linear term s like this m odel, one
can obtain param etrizations where interesting dynam ics such as chaos is obtained.
The sw itching o f predator-prey role is a rare phenom enon in ecological system s. This
switching provides an intrinsic m echanism for which persistence o f species could be ex
plained, for such a switch excludes the predator as sole resource beneficiary. These inter
actions occur in the presence o f o ther species w hich benefit from the resulting outcom es.
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CHAPTER 5. DISCUSSIONS AND CONCLUSIONS 62
That benefit either leads to extinction o f one or m ore species while certain conditions or
to continued existence o f all species. We believe that m ore work is required for systems
exhibiting such properties.
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Appendix A
Computational Systems Analysis
A.l Matlab Code for Sensitivity of One Parameter
•/, INITIAL CONDITIONS
’/, speciel specie2p . xO = [3 3] ;
[t x] = odel5s(Spred, p.tt, p.xO, p .options,AO, p);
figure(l), elf, plot(t,x(:,1),’r ’), legend(’x ’),
figure(2), elf, plot(t,x(:,2),’r ’), legend(’y ’),m m m m n m m n m nfunction xp = pred(t,x,AO,p)
xp = x;
f = (5*x(l) - x(l)*x(l))/(l + 0.5*x(l));
g = (5*x(2) - x(2)*x(2))/(1 + 0.5*x(2));
m m m m m m m m m m m m m m m m m m m m m m n
xp(1) = p.R0*x(l)-p.R0*x(l)*x(l)/A0 + g*x(l)*x(2)/A0;
xp(2) = p.R0*x(2)-p.R0*x(2)*x(2)/A0 + f*x(l)*x(2)/A0;
A.2 Matlab Code for Time Series
function threespecies ’/, Paper scaling
63
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APPENDIX A. COMPUTATIONAL SYSTEM S ANALYSIS
p.dt = 1; p.tf = 1500.0; p.tt = 0:p.dt:p.tf;
*/.p.A0=.55;
’/, options for ode solverp.options = odeset('abstol’,le-8,’reltol', le-8, ’maxstep’,p.dt);
'/. INITIAL CONDITIONS
'/, preyl prey2 predatorp.xO = [0.1265 0.1265 2*0.1265]; X under symmetry
x = p.xO; p .limit=[100:1:500]; for bl=2:.l:5 [t x] = ode23(@pprey, p.tt, p.x0,p.options,bl, p);
yl=x(p.limit,1): y2=x(p.limit,2) y3=x(p.limit,3)°/.ymaxl=max(yl) ;’/,ymax2=max(y2) ; ymin3=min(y3); ymax3=max(y3); plot(bl,ymax3,’r ’); hold onplot(bl,ymin3,’*g’) hold on'/.axis ([0 1 0 2]) ; legend(’umax’,’umin’)'/.legend (’ umax ’)
end
"/.figure (1), elf, plot (t ,x(: , 1), ’r ’), legend(’A ’),
function xp = pprey(t,x,bl,p)
xp = x;
fl = ( (3*x(2)-x(2)*x(2))/(l+0.2*x(2)*x(2))); f2 = ( (3*x(l)-x(l)*x(l))/(l+0.2*x(l)*x(l))); f3 = x(3)/(l+0.2*x(l)+0.2*x(2))-l;
xp(l) =
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APPENDIX A. COMPUTATIONAL SYSTEM S ANALYSIS
0.3*x(l)-(0.3*x(l)*x(l))/bl+(f2*x(l)*x(l))/bl-0.2*x(l)*x(l)*x(3)/(l+0.2*x(l)*x(l));
xp(2) =0.3*x(2)-(0.3*x(2)*x(l))/bl+(fl*x(2)*x(2))/bl-0.2*x(2)*x(2)*x(3) /(1+0.2*x(2)*x(2));
xp(3) = x(3)*f3 ;
global Alin
dout=3; M=12; N=10;
’/, k=l we have a linear polynomial, and for "/. k=2,3 we have quadratic polynomials
'/iP_l (x, y , z , \lambda) =-0. 2x-0. 4y+ (0. 3x+3. 6y+l. 2z)\lambda‘2+(-0. 8x-40z)\lambda~3 ’/, +18z\lambda~4
*/.p_2 (x, y , z , \lambda) = 0 . 4xy+ (-0. 2xz-0. 4yz) \lambda+ (0. 2xy+5yz) \lambda~2’/, + (0. lxz+1. 2yz-0. 4z~2) \lambda~3"/, + (0. lxz+5z~2) \lambda”4+(-1. 2z~2)\lambda~5
A.3 Matlab Code for Quintic Eigenvalue Problem
y."/.p_3 (x, y , z , \lambda) 0.4xy+10yz+(0.6xz-7.2yz-2.4z‘2)\lambda
+(1.2xz+60z~2)\lambda~2+(-24z~2)\lambda~3•/.*/."/, set the polynomials fl = [-0.2,-0.4,0]; f2
gl = [0,0,0]; g2 = [0,0,-0.2,0,-0.4,0] ; g3 = [0,0,0.6,0,-7.2,-2.4] ;
hi = [0.3,3.6,1.2]; h2 = [0,0.2,0,0,5,0]; h3 = [0,0,1.2,0,0,60]
il = [-0.8,0,-40]; i2 = [0,0,0.1,0,1.2,-0.4]; i3 = [0,0,0,0,0,-24];
jl = [0,0,18]; j2 = [0,0,0.1,0,0,5] ; j3 = [0,0,0,0,0,0] ;
kl = [0,0,0]; k2 = [0,0,0,0,0,-1.2] ; k3 = [0,0,0,0,0,0] ;
"/, We will now make the rectangular resultant matrices °/, A_k, k=l,5, such that we want'/t(\lambda"5 A5+\lambda~4 A4+\lambda~3 A3+\lambda~2 A2
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APPENDIX A. COMPUTATIONAL SYSTEM S ANALYSIS
‘/,+\lmabda A1 + AO) to have a'/, nontrivial null vector. These matrices will be '/. 12 x 10
A01 = oneBlock(3,dout,l,fl); A02 = oneBlock(3,dout,2,f2); A03 oneBlock(3,dout,2,f3);
AO = [A01; A02; A03] ;
A10 = oneBlock(3,dout,l,gl); A12 = oneBlock(3,dout,2,g2); A13 oneBlock(3,dout,2,g3);
A1 = [A10;A12;A13];
A21 = oneBlock(3,dout,1,hl); A22 = oneBlock(3,dout,2,h2); A23 oneBlock(3,dout,2,h3);
A2 = [A21;A22;A23];
A31 = oneBlock(3,dout,l,il); A32 = oneBlock(3,dout,2,i2); A33 oneBlock(3,dout,2,i3);
A3 = [A31;A32;A33];
A41 = oneBlock(3,dout,1,jl); A42 = oneBlock(3,dout,2,j2); A43 oneBlock(3,dout,2,j3);
A4 = [A41;A42;A43];
A51 = oneBlock(3,dout,l,kl); A52 = oneBlock(3,dout,2,k2); A53 oneBlock(3,dout,2,k3);
A5 = [A51; A52; A53]
Q=rand(M,N) ; L5 = Q ’*A5 ; L4= Q ’*A4 ; L3=Q’*A3;L2=Q’*A2; L1=Q’*A1; L0=Q’*A0 '/. We'/.get square matrices F and G by multiplying
[M,N] = size(L5) ; I = eye(M); Z= zeros(M,M) ;/.Here I considered the linearization suggested by Mackey et.al.'/, in their paper "Vector Spaces of Linearization for Matrix ‘/.Polynomials" page 1 equation (1.1)F= [ Z,I,Z)Z,Z;Z,Z,I,Z,Z;Z,Z,Z,I,Z;Z,Z,Z,Z,I;L0,L1,L2,L3,L4]; G = [ -I,Z,Z,Z,Z;Z,-I,Z,Z,Z;Z,Z,-I,Z,Z;Z,Z,Z,-I,Z;Z,Z,Z,Z,L5];
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APPENDIX A. COMPUTATIONAL SYSTEM S ANALYSIS
[V,E]=eig(F,-G) D= diag(E) ;
pp = isfinite(D) ; D=D(pp); U=V(l:15,pp) ;
'/, In the following loop, type in j=l,2,3,..10 % you will see that some answers are good, others are not.’/, Here we are getting the solution by the technique of '/» normaliing the eigenvector so its last component is one
for k =1:N for j=l:37 ’/,j=input (’ j ’)phi=U(:,j); phi = phi/phi(N) ; lam=D(j); x=phi(6) ; y=phi(9) ; ans= [x,y,lam]checkl= -0.2*x-0.4*y+(0.3*x+3.6*y+l.2)*lam~2
+(-0.8*x-40)*lanr3+18*lam"4 '/, check’/,1st equation
check2= 0. 4*x*y+ (-0. 2*x-0. 4*y) *lcim+ (0. 2*x*y+5*y) *lam*2+(0.l*x+l.2*y-0.4)*lam*3+(0.l*x+5)*lam“4-l.2*lam~5 '/, check 2nd equation
check3= 0.4*x*y+10*y+(-7.2*y+0.6*x-2.4)*lam+(1.2*x+60)*lam"2 + (-24)*lam~3 "/, check 3rd eqn.
endend
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INDEX
asym ptotically stable, 19
B endixson’s Negative Criterion, 16 bifurcation, 26 block m atrix, 53
codim ension 2, 50 coexistence equilibrium , 24 com panion m atrix, 52 com petition, ii conjugacy, 2 0
cusp bifurcation, 49
difference-equation, 5 Driving symmetry, 33 D ulac’s Test, 17
Efficient symmetry, 33 equilibrium solution, 18 excessive asym m etry, 33
full symmetry, 29
H olling Type II, ii H olling Type III, ii hom ogeneous polynom ial, 54 hysteresis, 27
Jacobian, 20
lexicography, 53 Lotka-Volterra, ii
m utualism , ii
non-hyperbolic, 49 nonhyperbolic, 26 normal form, 26
pencil, 51 phase portrait, 19 pitchfork bifurcation, 49 Poincare-Bendixon, 4
71
population floors, 5
quasi-cyclical, 5quintic eigenvalue problem , 53
spiral chaos, 5 stability, 18 structurally stable, 2 0
sw itching, ii sym m etry-breaking, 57
topological equivalence, 2 0
topologically equivalent, 2 0
transcritical bifurcation, 26
unim odal matrix polynom ial, 52
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