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# Systems Modelling and Simulation - Prey predator model. The goal of this experiment is to model the population dynamics of animals both predator and prey when they are present in an environment.

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Introduction

Systems Modelling and Simulation

Assignment No 2

Predator Prey Model

By

Vishal Vora (roll no 32)

Submitted in partial fulfilment of the course requirement to

Prof. Girja Sharan & Mr. Bhavesh Dharmani

Predator Prey Model

Goal:

The goal of this experiment is to model the population dynamics of animals both predator and prey when they are present in an environment.

Description:

1. Initially at time t=0, the population of prey is some value say x0 and that of predator is y0
2. Prey eats vegetarian food and predator eats prey
3. If there are no predators, the population of preys will grow exponentially over time and will be growing at the rate of the difference of the birth and death rate.
4. If there are predators, the population of prey will decrease to the extent they interact and the predator is successful in hunting the prey (Alpha)
5. The population of the predator is dependent on the growth rate of the predator due to prey availability (Beta) and the death rate of the predator.

Assumptions:

1. The forest is large and the preys are born and die continuously.

Middle

→ starvation due to death due to non-availability of prey

Thus it can be written as

Y= [µ- αy; -λ+ βx][x;y]’

If µ=1, λ=1, α=.01, β=.02

Then we can plot the graph for different initial values of prey and predator and find the population model.

Let the initial values of prey and predator be [20 20].

Matlab Code:

Create a file lotka.m with the following code. This will help us use the lotka model with different values of alpha and beta.

MATLAB Code:

function yp = lotka(t,y)

%LOTKA   Lotka-Volterra predator-prey model.

global ALPHA BETA

yp = [y(1) - ALPHA*y(1)*y(2); -y(2) + BETA*y(1)*y(2)];

type lotka

function yp = lotka(t,y)

%LOTKA   Lotka-Volterra predator-prey model.

global ALPHA BETA

yp = [y(1) - ALPHA*y(1)*y(2); -y(2) + BETA*y(1)*y(2)];

Following code is then used to get the output

t0=0; tfinal =15;

y0 =[20 20]’

global ALPHA BETA

ALPHA =0.01; BETA=0.02;

[t,y] = ode23('lotka',[t0 tfinal],y0);

subplot(1,2,1)

plot(t,y)

title('Time history')

subplot(1,2,2)

plot(y(:,1),y(:,2))

title('Phase plane plot') This shows that population of the predator will initially fall and that of prey will fall.

Conclusion

Now if we make the interaction rate and prey growth rate due to predation equal, the population of both the predator and the prey will reach a high of 400 before decreasing. Thus the interaction rate and predation rate decide the behaviour of the system.

ALPHA =0.01;

BETA=0.01;

[t,y] = ode23('lotka',[t0 tfinal],y0);

plot(t,y)

subplot(1,2,1)

title('Time history')

subplot(1,2,2)

plot(y(:,1),y(:,2))

title('Phase plane plot') Results:

The predator prey model shows that:

1. The values of predator and prey depend on the interaction rate and the growth rate of predator.
2. The model shows that with proper interaction both predator and prey are able to survive. The equilibrium will keep shifting but the process of population increasing and decreasing will keep on repeating. Thus equilibrium in this case will be the repeat of the cycle.

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