Natural Systems Modeling

Midterm Assignment

lorenzo rossato

lrossato@islander.tamucc.edu

March 7, 2014

Academic year 2013/2014

lrossato@islander.tamucc.edu

CONTENTS CONTENTS

Contents

1 Two spatial dimensions environment modeling 2

2 Terms for the preyed species 2

2.1 A limit to herbivores growth . . . . . . . . . . . . . . . . . . 22.2 Advective ow for preys . . . . . . . . . . . . . . . . . . . . 3

3 Terms for the predator species 4

3.1 Advective ow for preys . . . . . . . . . . . . . . . . . . . . 43.2 The density-dependent dispersal . . . . . . . . . . . . . . . . 4

4 Results and conclusions 5

Bibliography 8

1

2. Terms for the preyed species

1 Two spatial dimensions environment model-

ing

When species interact the population dynamics of each species is aectedin a way that can be predicted fairly well by complex mathematical expres-sions. We consider here a system involving two species interacting in a 2Denvironment, introducing every term of a certain importance for reachingsome kind of verisimilitude with a basic real system. We shall start fromscratch with the simple Lotka-Volterra1 model [1]

S1t

= D1

(2S1x2

+2S1y2

)+ a1S1 b1S1S2

S2t

= D2

(2S2x2

+2S2y2

) a2S2 + b2S1S2

where we already introduced a diusive term to allow these species to spreadacross the environment. From this moment we will add as many terms aswe think it is useful since the simulator will take all the computational eortto solve the system.

Yet the complications and the number of terms and coecients couldgrow so large that for the purpose of this work it is better to immediatelydene a circumstantial goal and nature of the model. Ideally we will besimulating the interaction between the many-time-studied wolves and deerssystem in a square region (of dimension L L) which presents a mountainin the centre so that the diusion is constrained by the terrain steepness

Di(x, y) = D0

1 11 + exp

[ci,2

((x L

2

)2+(y L

2

)2)]ci,1 (1)

where the Roman index i denotes the species and ci, n are quasi-arbitrarilydecided coecients. Such these random coecients will be denoted with cthroughout the entire report.

2 Terms for the preyed species

2.1 A limit to herbivores growth

For what concerns the preyed species, we might like to introduce a limit onits growth which could be an Ilev type function [2] for an herbivore grazing

Rm(1 e(S1S1 )

)forS1 S1 (2)

where Rm is the maximum grazing rate of the herbivore, is the Ilev con-stant and S1 denotes the threshold for food availability.

1The predator-prey model was rst investigated and theorised independently by Alfred

Lotka in 1925 and by Vito Volterra in 1926.

2

2.2 Advective ow for preys 2. Terms for the preyed species

Another possible limit in a population growth is the already studiedlogistic function which for simplicity will be preferred as more handy thanthe Ilev function. Once again the carrying capacity should be somehowaltitude-dependent so that

K(x, y) = K0

1 11 + exp

[c((x L

2

)2+(y L

2

)2)] (3)

where K0 denotes the original carrying capacity.

2.2 Advective ow for preys

Habitat heterogeneity inuences parameters such as the birthrate, speciesinteraction and above all the dispersal. This eect provides a biased motionof animals toward regions with more favourable conditions and it is givenby the expression [2]

u1

(

x(fxS1) +

y(fyS1)

)(4)

where u1 is the magnitude of the advective ow and f and is a term thatgives the preferred direction of this ow. One could parametrises in fboth the tendency of avoiding concentrations of predators (f1) and the owtoward favourable regions (f2). The rst statement could be easily turn tomath with

f1,x(x, y) S2x

f1,y(x, y) S2y

making the prey spreading against the gradient of the predators distribution,but the second term is implicitly a bit more complicated. The decision couldbe either letting them spread towards regions in which they diuses faster(ideally meaning that they can get access to more food) so that

f2,x(x, y) D1x

f2,y(x, y) D1y

or letting them move in protected places not easily accessible to predatorsso that

f2,x(x, y) D2x

f2,y(x, y) D2y

3

3. Terms for the predator species

which is a term that has not to be confused with f1. All these assumptionsare made to relate a biased spread to environmental heterogeneity. If diu-sion describes random dispersal what can make a species move preferablytowards some favourable regions is the condition of the habitat itself. Inother words, an animal should be capable of grasp towards which directionto move to nd better living conditions just from the changes in the en-vironment he nds as he moves. On the contrary, if the habitat worsenseventually the animal will turn back. If there are no changes in the environ-ment there's no biased motion and thus the advective term should vanishand we should have just diusion. This could be a rough explanation ofwhy I chose to insert the gradient of some function for advection. Maybean even more complicated version of this term would see u1 dependent onspace [2].

3 Terms for the predator species

3.1 Advective ow for preys

Environmental heterogeneity aects predators as much as preys. This timefor simplicity f = f1, meaning that predators are attracted by aggregationsof preys so that

f1,x(x, y) S1x

f1,y(x, y) S1y

So, once again, predators are deemed capable of tracking preys by thisbiased random walk toward the highest concentration of food.

3.2 The density-dependent dispersal

Predators behaviours have been studied since prehistoric times (then for ourdefence and now for their protection) and the main feature of their natureis either the ability to gather in packs to enhance the chances of successor the solitary life of some predators at the very top of the food chain.Introducing a density-dependent term means forcing the spreading of S2species and thus giving it, if not solitary behaviour, at least the preferenceto hunt in smaller groups. In formulas this term is

[

x

(S2S2x

)+

y

(S2S2y

)](5)

which is a concept rather similar to advective ow.

4

4. Results and conclusions

4 Results and conclusions

Starting from the primitive system (just Lotka-Volterra's with diusion)and adding one new term at a time we can get a sense of the entity ofthis change. Referring to Fig. 1, we note a major change when addingthe advective term and an almost insignicant variation with the density-dependent dispersal term meaning that with those coecient we observea kind of 'pack-eect'. In other words, population S2 grows faster if theyhunt together, at least this could be an interpretation of the earlier burstin their number occurring at S1 = 1000. This is conrmed by Fig. 2 thatshows minimal changes for various values of when only its relative termis added to the basic system. If we take away the mountain and leave onlythe diusion the equilibrium will have a dierent location but the systemwill head directly for it (light line in Fig. 1). Thus the kind of 'U-turn'we see at S1 v 2500 is due to the heterogeneity of the habitat, which slowsdown the convergence to the equilibrium.

0 500 1000 1500 2000 2500 3000Prey

0

100

200

300

400

500

600

Pre

dato

r

= 0, u1 = 0, u2 = 0

= 0, u1 = 0.8, u2 = 0.3

= 0.4, u1 = 0.8, u2 = 0.3

Figure 1: Phase space of the system with various sets of parameters. The light line is the system with

all the terms but without the mountain.

There are varieties and varieties of possible congurations for this systembut a single appealing video can completely explain what words struggle for.Video 3 show how could ideally seem two species of animals introduced ina new habitat that can somehow host both; in fact these two species startwith a very peaked and restricted population but growing fast in numberand dispersing swiftly in the surroundings. We clearly see how S1 spreadsover the mount shying away from S2 which strives in vain to get to the topof the mountain and instead its spreading occurs by means of a high-densitywave brushing the sides of the rise.

5

4. Results and conclusions

0 500 1000 1500 2000 2500 3000Prey

0

100

200

300

400

500

600

Pre

dato

r

= 0

= 5

= 20

Figure 2: Dependence of the system on .

The major diculty in this project has been to translate all the aboveconcepts in 2D formulas, which also means an implicitly increased compu-tational time. Another remark could regard the multiplicity of coecients.Their number makes the simulation rather complex so that one should playwith it many times to try dierent combinations. With such complicatedformulas it's impossible to nd theoretical relations between parameters;hence the only way to test them is just playing with the system. Disregard-ing computational time, one could at this point move on adding a spatialdependence to a1 , b1 and b2 an maybe even some time-dependent trigono-metric functions to simulate seasons.

In any case, even without trying many combinations of coecients wewere able to get some interesting insights. Diusion itself was alreadyenough to reach the equilibrium and to avoid long oscillations around it.Anyway, the most important feature the system has shown is that environ-mental heterogeneity plays a very important role in the system's dynamics,slowing down the spreading of species and creating a natural separationbetween dierent species depending on their characteristics.

6

4. Results and conclusions

Figure 3: Evolution of the system, here the parameters are u1 = 0.8 , u2 = 0.3 , = 0.4.

7

REFERENCES REFERENCES

References

[1] J. D. Murray.Mathematical Biology, An Introduction. Third Ed. Springer,2002. isbn: 0-387-95223-3.

[2] Akira Okubo and Simon A. Levin. Diusion and Ecological Problems.Second Ed. Springer, 2001. isbn: 0-387-98676-6.

8

Mathematica Codes

This is the Mathematica code used for this work.

Environment_6b.nb

L = 80;

Tmax = 200;

H*diffusion *LD0 = 0.6;

D3@x_, y_D := D0 1 - 11 + ExpA 1

400IIx - L

2M2 + Iy - L

2M2ME

12

;

D3 b@x_, y_D := D0 1 - 11 + ExpA 1

400IIx - L

2M2 + Iy - L

2M2ME

14

;

H*Logistic growth*Lr = 0.4;

K0 = 1;

K3@x_, y_D := 3 K05

1 -1

1 + ExpA 1700

IIx - L2

M2 + Iy - L2

M2ME ;

H*Lotka-Volterra*La2 = 0.4;

b1 = 1.5;

b2 = 1.3;

H*advective flow *Lu1 = 0; H*0.8*Lu2 = 0; H*0.3*LH*Density-dependent dispersal *L = 0; H*0.4*L

system =

:t u@t, x, yD D0 Hx,x u@t, x, yD + y,y u@t, x, yDL +u1 Hx H H-x D3@x, yD - x v@t, x, yDL u@t, x, yDL + y H H-y D3@x, yD - y v@t, x, yDL u@t, x, yDL L +r u@t, x, yD 1 - u@t, x, yD

K0- b1 u@t, x, yD v@t, x, yD,

t v@t, x, yD D0 Hx,x v@t, x, yD + y,y v@t, x, yDL +u2 Hx H Hx u@t, x, yDL v@t, x, yDL + y H Hy u@t, x, yDL v@t, x, yDLL + Hx Hv@t, x, yD x v@t, x, yDL + y Hv@t, x, yD y v@t, x, yDLL - a2 v@t, x, yD +b2 u@t, x, yD v@t, x, yD, u@0, x, yD == K0

2ExpB-0.05 Kx - L

2O2 + Ky - L

2O2OF,

v@0, x, yD == K05

ExpB-0.05 Kx - L2

O2 + Ky - L2

O2OF, uH0,1,0L@t, 0, yD uH0,1,0L@t, L, yD 0,vH0,1,0L@t, 0, yD vH0,1,0L@t, L, yD 0, uH0,0,1L@t, x, 0D uH0,0,1L@t, x, LD 0,vH0,0,1L@t, x, 0D vH0,0,1L@t, x, LD 0>;

sol = NDSolve@system , 8u, v

U@t_, x_, y_D := u@t, x , yD . sol;V@t_, x_, y_D := v@t, x , yD . sol;

H*--------------------VISUALISE ------------------------------------------------*L

Manipulate @Row@8ContourPlot @U@t, x, yD, 8x, 0, L