Consider the predator-prey model from page 36 of the book: D_n + 1 = R(1 - D_n/K

Question

Consider the predator-prey model from page 36 of the book: D_n + 1 = R(1 - D_n/K)D_n + D_n - P_n D_n P_n + 1 = Q D_n P_n. Here, D_n is the density of prey and P_n is the population of predator at time n. (a) Using Matlab, plot the first 70 generations of D_n and P_n with the following parameter values: K = 100, R = 1.5, Q = 0.023 with D_0 = 50. P_0 = 0.2. Hand in your code and the graph. You should observe oscillations of both prey and predator for these parameter values. (b) Try decreasing the parameter Q (the measure of the efficiency of utilization of prey for reproduction by predators), while keeping all other parameters as in part (a). Investigate what happens to the oscillatory solution as you do this. You will find that there exists a critical threshold Q_c which separates the oscillatory from non-oscillatory regime. Estimate the value of Q_c numerically using the matlab code you wrote in (a). Hand in the appropriate graphs for various values of Q showing your estimates. (c) Determine the non-zero steady state of this system, for general K, R, Q. (d) For general K, R, Q, determine a 2 times 2 matrix whose eigenvalues determine the stability of the system. (e) Compute the eigenvalues of this matrix for parameter values in (1) (give then numerical values). Comment on whether these eigenvalues agree with the observed behavior of the system. (f) When K = 100, R = 1.5 and Q = Q_c (the value you obtained numerically in (b)), what are the corresponding eigenvalues? How does it explain the threshold value of Q_c? (g) For general parameter values of K, R, determine Q_c as a function of K and R.

Explanation / Answer

(a) % MATLAB code

D(1)=50; P(1)=0.2

K=100; R=1.5; Q=0.023

for indx=2:72

D(indx)= R*(1-D(indx-1)/K)*D(indx-1) + D(indx-1) - P(indx-1)*D(indx-1)

P(indx)= Q*D(indx-1)*P(indx-1)

end

figure(1)

plot(D)

title('Density of Prey')

figure (2)

plot(P)

title('Population of Predator')

%End of program

(b) change the value of Q in above program and run. Suggestion to run it easily

write above program as function.

%name of function is "preyPredator" it takes input the value of 'Q' and return 'D' and 'P'

function [D,P] = preyPredator(Q)

%paste above program exactly just remove Q=0.023 line.

end

Now from the main program call it for decreasing value of Q and then plot D and P as above to get Qc.

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