Chapter 8.5, Problem 4E

Interpretation Introduction

Interpretation:

The system $\dot{\text{x}}\text{ = rx (1 - x) - h(1 + α sin t)}$ has no periodic solutions if $\text{h > }\frac{\text{r}}{\text{4}}$ and if $\frac{\text{r}}{\text{4(1 + α)}}\text{ < h < }\frac{\text{r}}{\text{4}}$ is to be showed. What happens to the fish population in this case is to be explained. Using a Poincare map argument, if $\text{h < }\frac{\text{r}}{\text{4(1 + α)}}$ and if $\frac{\text{r}}{\text{4(1 + α)}}\text{ < h < }\frac{\text{r}}{\text{4}}$ there exists a $\text{2π-}$ periodic solution that is stable limit cycle in the strip $\frac{\text{1}}{\text{2}}\text{ < x < 1}$ and unstable limit cycle in the strip $\text{0 < x < }\frac{\text{1}}{\text{2}}$ is to be shown. The results should be interpreted biologically.

Concept Introduction:

V is contracting and contains a unique periodic solution, and this cycle is an attractor if all values of the $\left(\text{t, x}\right)$ are in a forward invariant region of V, $\text{f}\left(\text{t, x}\right)\text{ < 0}$.

V is expanding and contains a unique periodic solution, and this cycle is a repellent if for all values of the $\left(\text{t, x}\right)$ in a backward invariant region V, $\text{f}\left(\text{t, x}\right)\text{ > 0}$.

When the nullclines of the system equation pinch together and are no longer defined in some interval of time, then the system has no periodic solution for the system equation $\dot{\text{x}}\text{ = cf (t, x)}$ for all sufficiently large c.