Chapter 3.7, Problem 3E

Interpretation Introduction

Interpretation:

To show that the system can be rewritten in dimensionless form for fishery model as $\frac{\text{dx }}{\text{dτ}}\text{= x (1-x) - h}$ for dimensionless quantities $\text{x}$, $\text{τ}$, and $\text{h}$. Plot the vector field for different values of $\text{h}$. Show that a bifurcation occurs at certain value ${\text{h}}_{\text{c}}$, and classify this bifurcation. Discuss the long-term behavior of fish population for $\text{h < }\text{}{\text{h}}_{\text{c}}$ and $\text{h > }\text{}{\text{h}}_{\text{c}}$, and give biological interpretation in each case.

Concept Introduction:

Express the equation in dimensionless form using the dimensionless parameter $\text{h}$.

Plot the graph, and determine bifurcation.

Discuss the cases $\text{h < }\text{}{\text{h}}_{\text{c}}$ and $\text{h > }\text{}{\text{h}}_{\text{c}}$ with dimensionless parameters.