# Hopf bifurcation for delay prey-predator system

I am trying to plot the Hopf bifurcation of delay system

$$\begin{eqnarray} &&\displaystyle{\frac{dx(t)}{dt}=r\,x(t)\left(1-\frac{x(t)}{k}\right)-\frac{a\,x(t)\,y(t)}{(x(t)+A)}-E\,q\,x(t)},\\[0.1cm] &&\displaystyle{\frac{dy(t)}{dt}=s\,y(t)\,\frac{1-{y(t-r)}}{x(t-r)+A}}, \end{eqnarray}$$

where $$r=3,a=1,k=100,s=2,A=2,E=0.2,q=0.1$$

But I don't know how to do it...any idea!! Thanks

• Do you have a source for these equations? The negative population sizes in @zhk's simulations are a big red flag. I wonder about the interpretation of the predator equation, because predator growth doesn't match the type-II functional response in the prey equation. Apr 26, 2019 at 7:01
• Could it be that the r variable in the second equation is actually the delay tau? r does not seem to have the right dimensions for a time delay in the first equation. Apr 26, 2019 at 16:23

This can achieved like this,

Parameter values

a = 1; k = 100; s = 2; A = 2; E1 = 0.2; q = 0.1;


System of ODE's

sol[x0_?NumericQ] := First@NDSolve[{x'[t] == r*x[t]*(1 - x[t]/k) - a*x[t]*y[t]/(x[t] + A)
- E1*q*x[t], x[t /; t <= 0] == x0, y'[t] == s*x[t]*(1 - y[t - r])/(x[t - r] + A),
y[t /; t <= 0] == x0}, {x, y}, {t, 0, 100}];


With delay $$r=1$$

Block[{r = 1}, ParametricPlot[Evaluate[{x[t], y[t]} /. sol[#] & /@ Range[0, 5, 1]],
{t, 0, 100}, Frame -> True, PlotRange -> All]]


With delay $$r=1$$ and a different set of initial conditions

Block[{r = 1}, ParametricPlot[Evaluate[{x[t], y[t]} /. sol[#] & /@ Range[5, 10, 1]],
{t, 0, 100}, Frame -> True, PlotRange -> All]]


With delay $$r=3$$

Note: You can also use ParametericNDSolveValue but I am habitual with SetDelay.