How can I change my code so that time series plots show oscillations, rather than steady states?

I am trying to model the Lotka-Volterra Predator-Prey system, using the coupled DE's:

dy(1)/dt = rx(1-x/k) - ay(1)y(2) % prey population

dy(2)/dt = aby(1)y(2) - dy(2) % predator population Here is the code I have :

      %Solves equations using numerical ODE solver 45 (nonstiff runge kutta)

      runtime = 1000; % Duration time of simulation in seconds.

      % Parameter values used in simulation %

      r = 0.5; % exponentional growth rate of prey in absence of predator

      a = 0.01; % conversion efficiency of predator

      b = 0.02; % attack rate

      d = 0.10; % death rate

      k = 750; % carrying capacity

      y0 = [10, 10] % initial conditions y(1)= 10, y(2) = 10

      deq1=@(t,y) [r.*y(1)*(1-(y(1)./k))- a.*y(1)*y(2); a.*b.*y(1).*y(2)-    d.*y(2)];

      [t,sol] = ode45(deq1,[0 runtime],y0);

How can I change my code so that time series plots (y(1) vs. time and y(2) vs. time) show oscillations, rather than steady states? It seems like something is going wrong in the integration step, as the plots are structured the way I want them but the behavior of the functions is not what I expected.

Steady state time series plots

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matlabcentral/answers/uploaded_files/61108/time%20series%20equilibrium.jpg>>

Phase plot reaching equilibrium

[David Goodmanson]

Take a look at what happens if you increase the carrying capacity to, say, 1e6.

[Charlotte Coates]

Yes this helps, now my graphs are showing oscillations but they still reach equilibrium before my program is finished running.

Answer 1

[Mischa Kim]

Charlotte, the graph above shows a plot y1 versus y2. Sort of "position" versus "velocity". In other words

 plot(sol(:,1),sol(:,2))

Both y1 and y2 are showing oscillations if you plot versus time

 plot(t,sol(:,1),t,sol(:,2))

In general, the oscillatory behavior of the system depends on the system parameters. So, for example, a = 0.1 results in more pronounced oscillations.

[Charlotte Coates]

Thanks Mischa, when I plot the time series plotting y1 and y2 against time, no oscillations occur, the y1 population increases rapidly and the y2 population reaches equilibrium.

The image I posted in my question is supposed to show the y1 and y2 populations returning to their initial conditions, appearing like concentric rings in the phase plot.

I will try changing the parameter values to see if this changes, but I am not sure what else I can do to change the behavior of the functions.