Chapter 8.2, Problem 8E

Interpretation Introduction

Interpretation:

The nullclines are to be sketched for a system $\dot{\text{x}}\text{ = x}\left(\text{x}\left(\text{1 - x}\right)\text{-y}\right)$, $\dot{\text{y}}\text{ = y}\left(\text{x-a}\right)$ if $\text{ x, y }\ge \text{0}$, and the fixed points are to be classified. The phase portrait for $\text{a < 1}$ is to be sketched, and that predators go extinct is to be shown. A Hopf bifurcation occurs at ${\text{a}}_c\text{ = }\frac{1}{2}$ is to be shown, and check whether it is supercritical or subcritical as well. The frequency of limit cycle oscillations for $\text{a}$ are near the bifurcation. Also, all the topologically different phase portraits for $\text{0 < a < 1}$ are to be sketched.

Concept Introduction:

A supercritical Hopf bifurcation occurs at a fixed point when a stable spiral changes into an unstable spiral at that point.

A supercritical Hopf bifurcation occurs when the size of the limit cycle grows continuously from zero.

The fixed point is unstableif both eigenvalues are positive.

If both eigenvalues are negative, the fixed point is stable.

The fixed point is a saddle if the eigenvalues are of opposite sign.