Step By Step Calculus » 16.5  PredatorPrey Systems
Synopsis
It is more natural to model a pair of species together through the predatorprey equations
\displaystyle \newcommand{\index}[1]{}{\textbf{(PredatorPrey Evolution)}\index{PredatorPrey evolution}\qquad\frac{dW}{dt}=rW+bRW\text{ and }\frac{dR}{dt}=kRaRW}\displaystyle \newcommand{\index}[1]{}{\textbf{(PredatorPrey Evolution)}\index{PredatorPrey evolution}\qquad\frac{dW}{dt}=rW+bRW\text{ and }\frac{dR}{dt}=kRaRW} 
subject to W(0)=W_{0},R(0)=R_{0}W(0)=W_{0},R(0)=R_{0}
Using Eulerâ€™s method, one can find approximate solutions to this evolution. \left( W,R\right) \left( W,R\right) is said to be an equilibrium solution for PredatorPrey Evolution equations if
\displaystyle
\frac{dW}{dt}=0\displaystyle
\frac{dW}{dt}=0 and \displaystyle \frac{dR}{dt}=0\displaystyle \frac{dR}{dt}=0 for all \displaystyle t\geq 0.
\displaystyle t\geq 0.

In order to investigate the equilibrium and nonequilibrium solutions, we need the ideas of phase plane and phase trajectory. The plane with WW on the yyaxis and RR on the xxaxis is the phase plane. The graph of W(t)W(t), R(t)R(t) (as tt increases) on the phase plane is called the phase trajectory or phase curve.
There is an easy way to come up with the phase trajectory. The idea is to come up with an equation in terms of just WW and RR by eliminating tt. The phase equation is the equation describing the change in WW with respect to RR. We can easily determine the phase equation by parametric differentiation \frac{dW}{dR}=\frac{dW/dt}{dR/dt}\frac{dW}{dR}=\frac{dW/dt}{dR/dt}, which yields
It might be useful to know that the solutions to the PredatorPrey Phase equation satisfy
\displaystyle \newcommand{\index}[1]{}{\textbf{(PredatorPrey Phase)}\index{PredatorPrey phase}\qquad\frac{dW}{dR}=\frac{rW+bRW}{kRaRW}.}\displaystyle \newcommand{\index}[1]{}{\textbf{(PredatorPrey Phase)}\index{PredatorPrey phase}\qquad\frac{dW}{dR}=\frac{rW+bRW}{kRaRW}.} 
\displaystyle
W^{k}R^{r}=C\exp \left( aW+bR\right)
\displaystyle
W^{k}R^{r}=C\exp \left( aW+bR\right)
for any constant C>0C>0.