Question

% 1) Find the solution to the discrete model
% X_{t+1} = X_t +4Y_t
% Y_{t+1} = .5 X_t
% if X_0 = 6 and Y_0 = 0
% Plot time series of the solution, and plot the solution in a phase plane.
% If you use 35 timesteps the numbers will be enormous; try it with timesteps=8.

Answer 1

My attempt at the matlab code

>> clear all;
>> timesteps=8;
>> x=[6; 0]; xhistory=x;
>> A=[1 4; 0.5 0];
>> for t=1:timesteps;
x=A*x;
xhistory=[xhistory x];
end
>> time=0:timesteps;
>> subplot(2,1,1)
>> plot(time,xhistory(1,:),'-'), xlabel('time'), ylabel('populations'), hold on
>> plot(time, xhistory(2,:))
>> subplot(2,1,2)
>> plot(xhistory(1,:), xhistory(2,:)), xlabel('juveniles'), ylabel('adults'), hold on
>> [V,D]=eig(A)

V =

0.9701 -0.8944
0.2425 0.4472


D =

2 0
0 -1

>> C= V\xhistory(:,1)

C =

4.1231
-2.2361

>> xpprox = C(2)*V(:,2)*D(2,2).^(time)

xpprox =

Columns 1 through 3

2.0000 -2.0000 2.0000
-1.0000 1.0000 -1.0000

Columns 4 through 6

-2.0000 2.0000 -2.0000
1.0000 -1.0000 1.0000

Columns 7 through 9

2.0000 -2.0000 2.0000
-1.0000 1.0000 -1.0000

>> subplot(2,1,1)
>> plot(time,xpprox,'r')
>> subplot(2,1,2)
>> plot(xpprox(1,:),xpprox(2,:),'r')

Answer 2

>> x=[6; 0]; xhistory=x; initial conditions
>> A=[1 4; 0.5 0]; projection matrix.

Answer 3

just wondering if it's correct

@dan815

I already computed this problem by hand and I got lambda -1 and 2 and when I found my c1 and c2 values it was c1=-1 and c2=1

but the main focus is the graph for the Juveniles and Adults population with timesteps and phase planes.

Answer 4

is X=adults?

Answer 5

is x adults? is the labeling off?