Plotting a Hopf bifurcation diagram

Question

Can anyone guide me on how to obtain the Hopf Bifurcation diagram? I am very new to this topic.I am trying to plot a Hopf bifurcation diagram for the system with bifurcation parameter α with the following given values:

α = 2.438571114; β = 0.2; δ = 2.3; γ = 0.7;

My system is

{x'[t] == x[t](1 - x[t]) + x[t]y[t], 
 y'[t] == y[t](δ - (β y[t])/x[t]) - (α y[t])/(γ + y[t])}

Can anyone guide me on how to obtain a Hopf bifurcation diagram. Plot should have parameter α on x-axis and xy on y-axis

http://demonstrations.wolfram.com/HopfBifurcationsInANonlinearTwoDimensionalAutonomousSystem/ — David G. Stork, Mar 23 '18 at 20:52
Closely related question: https://mathematica.stackexchange.com/questions/13277/mathematica-code-for-bifurcation-diagram — m_goldberg, Mar 23 '18 at 22:08

score 2 · Accepted Answer · answered Mar 24 '18 at 14:05

Can you give parameter values where there is a limit cycle to get started with? The parameters given result in a blowup.

α = 2.438571114; β = 0.2; δ = 2.3; γ = 0.7;
tmax = 100;

sol = NDSolve[{x'[t] == x[t] (1 - x[t]) + x[t] y[t], 
  y'[t] == y[t] (δ - (β y[t])/x[t]) - (α y[t])/(γ + y[t]),
  x[0] == 1, y[0] == 1}, {x, y}, {t, 0, tmax}][[1]];

(* NDSolve::nderr: Error test failure at t == 15.34966818880772`;
  unable to continue. *)

Plot[Evaluate[{x[t], y[t]} /. sol], {t, 0, 15.349}]

Mathematica graphics

Plotting a Hopf bifurcation diagram

1 Answers1