Chapter 6.4, Problem 6E

Interpretation Introduction

Interpretation:

The system ${\dot{\text{N}}}_{\text{1}}{\text{= r}}_{\text{1}}{\text{N}}_{\text{1}}\left(\text{1 - }\frac{{\text{N}}_{\text{1}}}{{\text{K}}_{\text{1}}}\right){\text{ - b}}_{\text{1}}{\text{N}}_{\text{1}}{\text{N}}_{\text{2}}$, ${\dot{\text{N}}}_{\text{2}}{\text{= r}}_{\text{2}}{\text{N}}_{\text{2}}\left(\text{1 - }\frac{{\text{N}}_{\text{2}}}{{\text{K}}_{\text{2}}}\right){\text{ - b}}_{\text{2}}{\text{N}}_{\text{1}}{\text{N}}_{\text{2}}$ is to be nondimensionalized and required dimensionless groups are to be found. For long term behavior, there are four qualitatively different phase portraits is to be shown. Also, the conditions under which the two species can stably coexist is to be found and its biological meaning is to be explained.

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.

Nullclines are the curve where either $\dot{\text{x}}\text{ = 0}$ or $\dot{\text{y}}\text{ = 0}\text{.}$

The horizontal nullclines is a set of points in the phase plane where $\dot{\text{x}}\text{ = 0}$. This represents that the flow it purely vertical. These are the points where the vectors are either straight up or straight down.

The vertical nullclines is a set of points in the phase plane where $\dot{\text{y}}\text{ = 0}\text{.}$ This represents that the flow it purely horizontal. These are the points where vector is horizontal.