Chapter 6.4, Problem 4E

Interpretation Introduction

Interpretation:

To show how the simplest model ${\dot{\text{N}}}_{\text{1}}{\text{= r}}_{\text{1}}{\text{N}}_{\text{1}}{\text{- bN}}_{\text{1}}{\text{N}}_{\text{2}}{\text{ , }\dot{\text{N}}}_{\text{2}}{\text{= r}}_{\text{2}}{\text{N}}_{\text{2}}{\text{- b}}_{\text{2}}{\text{N}}_{\text{1}}{\text{N}}_{\text{2}}$ is less realistic. With suitable rescaling of ${\text{N}}_{\text{1}}{\text{, N}}_{\text{2}}$ and $\text{t}$, show that the model can be non-dimensionalized to $\text{x' = x(1- y), y' = y(ρ - x)}$. Determine the formula for $\text{ρ}$ dimensionless group. Sketch the nullclines and vector field for the system. Draw the phase portrait and comment on the biological implications. Show that all trajectories are curves of the form $\text{ρlnx - x = lny - y + C}$

Concept Introduction:

Express the equation in dimensionless form and determine dimensionless parameter.

Analyze fixed points for the system equation.

Draw vector field for the system.

Sketch phase portrait for system equation, and show the nature of curve trajectories using $\frac{\text{dx}}{\text{dy}}$.