The system x ˙ = rx (1 - x) - h(1 + α sin t) has no periodic solutions if h > r 4 and if r 4(1 + α) < h < r 4 is to be showed. What happens to the fish population in this case is to be explained. Using a Poincare map argument, if h < r 4(1 + α) and if r 4(1 + α) < h < r 4 there exists a 2π- periodic solution that is stable limit cycle in the strip 1 2 < x < 1 and unstable limit cycle in the strip 0 < x < 1 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 ( t, x ) are in a forward invariant region of V, f ( t, x ) < 0 . V is expanding and contains a unique periodic solution, and this cycle is a repellent if for all values of the ( t, x ) in a backward invariant region V, f ( t, x ) > 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 x ˙ = cf (t, x) for all sufficiently large c.

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Chapter 8.5, Problem 4E

Interpretation Introduction

Interpretation:

The system $x˙ = rx (1 - x) - h(1 + α sin t)$ has no periodic solutions if $h > r4$ and if $r4(1 + α) < h < r4$ is to be showed. What happens to the fish population in this case is to be explained. Using a Poincare map argument, if $h < r4(1 + α)$ and if $r4(1 + α) < h < r4$ there exists a $2π-$ periodic solution that is stable limit cycle in the strip $12 < x < 1$ and unstable limit cycle in the strip $0 < x < 12$ 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 $(t, x)$ are in a forward invariant region of V, $f(t, x) < 0$ .

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

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 $x˙ = cf (t, x)$ for all sufficiently large c.

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