Chapter 2.3, Problem 5E

Interpretation Introduction

Interpretation:

$\text{x(t)}$ increases monotonically and approaches 1 as $\text{t}\to \infty$ is to be shown by solving for $\text{X(t) and Y(t)}$ and also by deriving a differential equation for $\text{x(t)}$.

Concept Introduction:

The model which describes population growth of organisms is $\begin{array}{l}\dot{\text{N}}\text{ = rN ,}\\ \text{where N(t) is population at time t and r is growth rate}\text{.}\end{array}$

The more accurate model is given by Verhulst, known as the logistic model, which postulates that the relative growth rate $\frac{\dot{\text{N}}}{\text{N}}$ decreases linearly with N. The corresponding logistic equation is

$\begin{array}{l}\dot{\text{N}}\text{ = rN }\left(1-\frac{N}{K}\right)\\ \end{array}$