بسم الله الرحمن الرحيم

رب علما ِوقل ًزدني

العظيم الله صدق

الباحثة رحاب نوري

On the dynamics of some Prey-predator

Models with Holling type-IV functional response

AbstractThis thesis deals with the dynamicalbehavior of prey-predator models with nonmonotonic functional response. Two

types of prey-predator models are proposed and studied.

The first proposed models consisting of Holling type-IV prey-predator model

involving intra-specific competition

However, the second proposed models consisting of Holling type-IV prey-predator model involving intra-specific competition in prey population with stage structure in

predator. Both the models are represented mathematically by nonlinear

differential equations. The existence, uniqueness and boundedness of the

solution of these two models are investigated. The local and global stability

conditions of all possible equilibrium points are established.

The occurrence of local bifurcation (such as saddle-node, transcritical and pitchfork) and Hopf bifurcation near each of the equilibrium points are discussed. Finally, numerical simulation is used to study the

global dynamics of both the models.

Chapter one

Preliminary

Mathematical model; is mathematical construct, often an

equation or a relation ship, designed to study the behavior of

a real phenomena.

Definition: Any group of individuals, usually of a single

species, occupying a given area at the same time is known as

population.

The application of mathematical method to problem in

ecology has resulted in a branch of ecology know as

mathematical ecology,.

There are three different kinds of interaction between any

given pair of species, these are described as bellow.

mutualisms or symbiosis (+,+), in which the existence of

each species effects positively on the growth of the other

species (i.e. increases in the first species causes

increases in the growth of the second species )

Competition (-,-), in which the existence of each species

effect negatively on the growth of the other species.

prey-predator (-,+) in which the existence of the first species

(prey) effects positively (increases) on the growth of the

second species (predator), while the existence of the

second species

(predator) effects negatively (decreases) on the growth

of the first species (prey).

Functional response; is that function, which refers to the

change in the density of prey attacked per unit at time per

predator as the prey density changes .the functional response

can be classified into three classes with respect to its

dependence on the prey or predator ,those are :

prey-dependent functions : the term prey-dependent means

that the consumption rate by each predator is a function of

prey density only, that is . In the following, the most

commonly used prey-dependant types functional reasons are

presented

)(),( xfyxf

(i) lotka-volterra type: in which the consumption rat of

individual predator increases linearly with prey. Hence the lotka

- volterra type of functional response can be written as follows:

Where is the consumption rate of prey by p rate of prey

by predator.

0;)( xaxxf0a (1)

(1)

2 4 6 8 10x value

20

40

60

80

100y

(ii) Holling type-I

In which the consumption rate of individual predator increases

lineally with prey density and when the predator satiate then it

becomes constant. Hence the Holling type-I functional

response can be written as follows:

Where is the value of prey density at which the predator

satiate at constant

x x0

)(

xxf

10 20 30 40x value

20

40

60

80

100y

(iii) Holling type-II: in which the consumption rate of individual

predator increases at a decreasing rate with prey density until it

becomes constant at satiation level. It is a hyperbola the

maximal value asymptotically, and defined by

Here is a search rate, h is the time spent on the

handling of one prey, is the maximum attack rate and b is

the half saturation level

2)( abf

ha 1

A

a

(3)xbax

AhxAxxf

1)(

2 4 6 8 10x value

0.5

1

1.5

2y

(iv) Holling type-III:

In this type of functional response the consumption rate of

individual predator accelerates at first and then decelerates

towards satiation level. It is defined as:

2

2

2

2

1)(

xbax

AhxAxxf

(4)

2 4 6 8 10x value

0.5

1

1.5

2

2.5

y

(V) Holling type-IV:

The type of functional response suggested by Andrews. The

Holling type IV functional response is of the form.

(5)

axixMxxf

2)(

2. Ratio-dependent function :

The prey-dependent functional responses do not

incorporate the predator abundance in it forms. Therefore,

Aridity and Ginsberg suggested using new type of functional

response called ratio-dependent, in which the rate of prey

consumption per predator depends not only on prey density

but instead depends on the ratio between the prey and

predator density. The Aridity -Ginsberg ratio dependent

response can be written as:

xbyax

AhxyAx

yxAh

yxA

yxf

1

),( (6)

3.predator-dependent functions;

In this type, the functional responses are depending on both

prey and predator densities. It is observed that high predator

density leads to more frequent encounters between

predators.

The most commonly used predator dependent functional

response is that proposed by DeAngalies and independently

by Beddingnon, which is now know as DeAnglis-Beddington

functional response:cxby

axAhxByAxyxf

1),(

(7)

The dynamics of Holling type IV prey predator model

with intra- specific competition

Chapter two

Mathematical models

Let x be the density of prey species at time ,y be the density

of predator species at time that consumes the prey species

according to Holling type IV functional response then the

dynamics of a prey–predator model can be represented by the

following system of ordinary differential equation.

t ttt

(8)

),(

),(

22

12

yxyfyxxxedy

dtdy

yxxfxxybxax

dtdx

with and Note that all the parameters of system

(8) are assumed to be positive constants and can be described

as follow:

is the intrinsic growth rate of the prey population; is the

intrinsic death rate of the predator population; the parameter

is the strength of intra-specific competition among the prey

species; the parameter can be interpreted as the half-

saturation constant in the absence of any inhibitory effect; the

parameter is a direct measure of the predator immunity from

the prey; is the maximum attack rate of the prey by a

predator; represents the conversion rate; and finally is the

strength of intra-specific competition among the predator

species .

0)0( x

aa

d

0)0( y

b

e

e

Now I worked on my research and I finished to finding and

proofing the following

1. Existence and Local Stability analysis of system (8) with

persistence

(1) The trivial equilibrium point always exists.

(2) The equilibrium point always

exists, as the prey population grows to the carrying capacity in

the absence of predation.

)0,0(0 E

0,1 baE

(3) There is no equilibrium point on as the predator

population dies in the absence of its prey.

(4) The positive equilibrium point exists in

the interior of the first quadrant if and only if there is a positive

solution to the following algebraic nonlinear equations:

We have the polynomial from five degrees

axisy

),( **2 yxE

0)( 012

23

34

45

5 AxAxAxAxAxAxf

And its exists under the condition, and when I

investigated varitional matrix for my system and

substitute we get the eigenvalues are

and Therefore, is locally

asymptotically stable if and only if

0,1 baE 01 ax

1Ebaba

abedy

21

dbaba

abe

22

While is saddle point provided that

And when I investigated varitional matrix for my system

and substitute we get is locally

asymptotically stable if and only if

and

)0,(1 baE

dbaba

abe

22

),( **2 yxE 2E

2*)*()*2(** Rybxxyx

RxyxeRb )2()( **2*223

And the persistence is proof by using the Gard and Hallam

method.

2.Global dynamical behavior of the system (8)

I proof the global dynamical behavior of system (8) and

investigated by using the Lyapunov method

3.The System(8) has a Hopf –bifurcation.

Near the positive equilibrium point at the parameter value

)*2(***)( 2*

*

xyxRbybx

4.The system (8) is uniformly bounded

In this section the global dynamics of system (8) is studied

numerically. The system (8) is solved numerically for different sets

of parameters and for different sets of initial condition, by using

six order Runge-Kutta method

5. Numerical analysis

0.01 0.75e 0.01d 1 0.75 0.50 0.2b 1

a

(8a)

2.5 3.5 4.5 5.5 6.51.5

2.5

3.5

4.5

(a)

x

y

initial point (3.0,3.0)

initial point (4.0,2.0)

initial point (6.0,3.0)

Stable point (4.71,3.99)

0 1 2 3 4 5

x 104

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8b

Time

popu

latio

n

Fig1: (a) Globally asymptotically stable positive point of

system (8) for the data starting from (8a) with

different initial values. (b) Time series of the attractor

given by Fig 1(a)

3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

a

Prey

Pre

dato

r

0 2 4 6 8 10

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5b

Time

Pop

ulat

ion

Fig. 2: (a) The system (8) approaches asymptotically to

stable point 0,51 E with the data given in (8a)

05.0dwith starting from (3, 3), (b) Time series of the

attractor given by Fig1 (a).

0 1 2 3 4 50

2

4

6

8

10

12a

Prey

Pre

dato

r

initial point(3.0,3.0)

0 1 2 3 4 5 6 70

2

4

6

8

10

12b

Prey

Pre

dato

r

initial point(6.0,4.0)

0 2 4 6 8 10

x 104

0

2

4

6

8

10

12c

Time

popu

latio

n

0 2 4 6 8 10

x 104

0

2

4

6

8

10

12d

Time

Pop

ulat

ion

Fig2: Globally asymptotically stable limit cycle of system

(8) for the data given by Eq. (8a) with 75.0

(a) The solution approaches to limit cycle from inside

(b) The solution approaches to limit cycle from outside (c) Time series for Fig (1a) (d) Time series for Fig (1b).

Chapter three

Stability Analysis of a stage

structure

prey-predator model with

Holling type IV functional response

1.The mathematical models),,( 2112

2 yyxfxxybxax

dtdx

),,()( 21222

111 yyxf

xxxyeyDd

dtdy

),,( 2132212 yyxfydDydtdy

)9(

1.All the solutions of system (9) which initiate in 3

are uniformly bounded2. Existence and Local Stability analysis of system (9)

are investigate

(2a) The trivial equilibrium point always exists.(2b) It is well known that, the prey population grows

to the carrying capacity in the absence of ba

predator, while the predator population dies in the absence of the prey, then the axial

equilibrium point

0,0,1 baE always exists.

(3) There is no equilibrium point in the as the predator population dies in the

absence of its prey.(4) The positive equilibrium point ),,( *

2*1

*2 yyxE

exists in the 3. Int

Now we obtain the following third order polynomial

equation.

0012

23

3 BxBxBxB

4.The Bifurcation of system (9)the occurrence of a simple Hopf bifurcation and local

bifurcation (such as saddle node,pitck fork and a

transcritical bifurcation) near the equilibrium points of

system (9) are investigated.

3. Global dynamical behavior of system (9) are

investigatewith the help of Lyapunov function

5.Numerical analysis

In this section the global dynamics of system (9) is studied

numerically. The system (9) is solved numerically for

different sets of parameters and for different sets of initial

condition, by using six order Runge-Kutta method

0.05 0.35,e 0.25,D 0.01,d 2, 0.75, 1, 0.2,b ,25.0

21

da

(9a)

00.5

11.5

00.2

0.40.6

0.80

0.5

1

1.5

Prey

(a)

Immature Predator

Mat

ure

Pred

ator

Initial point(0.85,0.75,0.65)

Initial point(0.65,0.55,0.45)

Initial point(0.45,0.35,0.25)

Stable point(0.38,0.08,0.44)

0 0.5 1 1.5 2

x 105

0

0.4

0.8

1.2(b)

Time

Popu

latio

ns

Fig. (1): (a) The solution of system (9) approaches

asymptotically to the positive equilibrium point starting

from different initial values for the data given by Eq.

(9a). (b) Time series of the attractor in (a) starting at

(0.85, 0.75, 0.65).

01

23

0

0.5

10

1

2

Prey

(a)

Immature Predator

Mat

ure

Pred

ator

Initial point(0.85,0.75,0.65)

Initial point(0.85,0.75,0.65)

0 2.5 5 7.5

x 104

0

0.5

1

1.5

2

2.5(b)

Time

Popu

latio

ns

0 2.5 5 7.5

x 104

0

0.5

1

1.5

2

2.5(c)

Time

Popu

latio

ns

Fig. (2): (a) Globally asymptotically stable limit cycle

of system (9) starting from different initial values for

the data given by Eq. (9a) with

(c) Time series of the attractor in (a)

5.0a (b) Time series of the attractor in (a) starting at

(0.85, 0.75, 0.65).

(1.25, 0.25, 0.75).starting at

0 2 4 6 8

x 104

0

1

2

3

4

5

6

Time

Popu

latio

ns

Fig. (3): The trajectory of system (9) approaches

asymptotically to stable point 0,0,51 E for 1awith the rest of parameter as in Eq. (9a).