Question

Q 4. With the aid of fourth order Runge-Kutta method, solve the competing species model

[20 points]

defined by

dx =x(2 − 0.4x − 0.3y), x(0) = 4 dt

dy =y(1 − 0.1y − 0.3x), y(0) = 3 dt

where the populations x(t) and y(t) are measured in thousands and t in years. Use a step size of 0.2 for 0 ≤ t ≤ 2 and plot the trajectories of the populations with Matlab or GNU Octave.

Answer 1

Answer:-

For trajectories use the MATLAB code

function model1

[T,Y] = ode45(@fun,[0:0.2:2],[4 3]);

plot (T,Y(:,1),'b','linewidth',1.5)
hold on
plot (T,Y(:,2),'r','linewidth',1.5)

legend('x','y')

xlabel('t','fontsize',12)
ylabel('x,y','fontsize',12)
function dy1 = fun(t,u)
dy1 = zeros(2,1);
dy1=[u(1)*(2-0.4*u(1)-0.3*u(2))
u(2)*(1-0.1*u(2)-0.3*u(1))]

For phase portrait use the MATLAB code

function model1

[T,Y] = ode45(@fun,[0:0.2:2],[4 3]);

plot (Y(:,1),Y(:,2),'b','linewidth',1.5)

xlabel('x','fontsize',12)
ylabel('y','fontsize',12)
function dy1 = fun(t,u)
dy1 = zeros(2,1);
dy1=[u(1)*(2-0.4*u(1)-0.3*u(2))
u(2)*(1-0.1*u(2)-0.3*u(1))]
  

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