Chapter 6.5, Problem 19E

Interpretation Introduction

Interpretation:

The biological meaning of the system $\dot{\text{R}}\text{ = aR - bRF}$, $\stackrel{\text{.}}{\text{F}}\text{ = -cF + dRF}$ is to be discussed and comments on any unrealistic assumptions should be done. This model can be changed in dimensionless form $\text{x' = x(1 - y)}$, $\text{y' = μy(x - 1)}$ and this model predicts cycles in the populations of both species, for almost all initial conditions is to be shown. Also, the conserved quantity in terms of dimensionless variables is to be found.

Concept Introduction:

The expression of the Rabbits versus sheep model is,

$$ $\frac{{\text{dN}}_{\text{1}}}{\text{dt}}{\text{ = r}}_{\text{1}}{\text{N}}_{\text{1}}\left(\frac{{\text{K}}_{\text{1}}{\text{ - N}}_{\text{1}}{\text{ - f( N}}_{\text{2}}\text{)}}{{\text{K}}_{\text{2}}}\right)$ and $\frac{{\text{dN}}_2}{\text{dt}}{\text{ = r}}_2{\text{N}}_2\left(\frac{{\text{K}}_2{\text{ - N}}_2{\text{ - f( N}}_1\text{)}}{{\text{K}}_{\text{2}}}\right)$

Where, ${\text{N}}_{\text{1}}$ is the population of the rabbits, ${\text{N}}_2$ is the population of sheep, r is the inherent per capita growth and $\text{K}$ is the carrying capacity.

According to the theorem $\text{6}\text{.5}\text{.1}$ mentioned in the textbook, all trajectories close to ${\text{x}}^{\text{*}}$ are closed if this fixed point satisfies the following conditions:

There exists a conserved quantity $\text{E}\left(\text{x}\right).$

${\text{x}}^{\text{*}}$ is an isolated fixed point.

${\text{x}}^{\text{*}}$ is a local minimum of $\text{E}\left(\text{x}\right).$