Chapter 8.5, Problem 3E

Interpretation Introduction

Interpretation:

To show that the system $\dot{\text{N}}\text{ = rN }\left(\text{1 - }\frac{\text{N}}{\text{K}\left(\text{t}\right)}\right)$, where the carrying capacity is positive, smooth, and $\text{T-}$ periodic in $\text{t}$, has at least one stable limit cycle of period $\text{T}$, contained in the strip ${\text{K}}_{\text{min}}\le \text{ N }\le {\text{ K}}_{\text{max}}$. To find if the cycle is necessarily unique.

Concept Introduction:

The mapping from $\text{N}$ to $\text{P}\left(\text{N}\right)$ is called the Poincaré map. It shows how the height of a trajectory changes after one laparound the cylinder.

If there is a point ${\text{N}}^{*}$ such that $\text{P}\left({\text{N}}^{*}\right){\text{ = N}}^{*}$, then the corresponding trajectory will be a closed orbit because it returns to the same location on the cylinder afterone lap.