Chapter 7.2, Problem 18E

Interpretation Introduction

Interpretation:

For the predator-prey model, prove that the given system $\dot{\text{x}}\text{ = rx( 1- }\frac{\text{x}}{\text{2}}\text{) - }\frac{\text{2x}}{\text{1+x}}\text{y, }\dot{\text{y}}\text{ = - y +}\frac{\text{2x}}{\text{1+x}}\text{y}$ has no closed orbits by using Dulac’s criterion with the function $\text{g(x,y) = }\frac{\text{1+ x}}{\text{x}}{\text{y}}^{\alpha -1}$ for suitable choice of $\alpha$.

Concept Introduction:

The method for ruling out closed orbits that is ruling out the existence of a limit cycle based on Green’s theorem is called Dulac’s criterion.

Consider a continuously differentiable vector field $\dot{\text{x}}\text{ = f(x)}$ defined on a subset $\text{R}$ of the plane. If a continuously differentiable function, real valued function $\text{g(x)}$ and if $\nabla \text{.(g(}\dot{\text{x}}\text{))}$ has same sign throughout subset R then there will not be any closed orbit lying entirely in $\text{R}$.