Chapter 7.2, Problem 13E

Interpretation Introduction

Interpretation:

It is to be shown using Dulac’s criterion with the weighing function ${\text{g = (N}}_{\text{1}}{\text{N}}_{\text{2}}{)}^{-1}$, the system

${\dot{\text{N}}}_{\text{1}}{\text{= r}}_{\text{1}}{\text{N}}_{\text{1}}\left(\text{1 - }\frac{{\text{N}}_{\text{1}}}{{\text{K}}_{\text{1}}}\right){\text{ - b}}_{\text{1}}{\text{N}}_{\text{1}}{\text{N}}_{\text{2}}$, ${\dot{\text{N}}}_{\text{2}}{\text{= r}}_{\text{2}}{\text{N}}_{\text{2}}\left(\text{1 - }\frac{{\text{N}}_{\text{2}}}{{\text{K}}_{\text{2}}}\right){\text{ - b}}_{\text{2}}{\text{N}}_{\text{1}}{\text{N}}_{\text{2}}$ has no periodic orbits in the first quadrant ${\text{N}}_{\text{1}}{\text{,N}}_{\text{2}}\text{ > 0}$.

Concept Introduction:

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

Consider a continuously differentiable vector field $\dot{\text{x}}\text{ = f(x)}$ distinct 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 $\text{R}$.