Step By Step Calculus » 16.5 - Predator-Prey Systems

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Synopsis
It is more natural to model a pair of species together through the predator-prey equations

\displaystyle \newcommand{\index}[1]{}{\textbf{(Predator-Prey Evolution)}\index{Predator-Prey evolution}\qquad\frac{dW}{dt}=-rW+bRW\text{ and }\frac{dR}{dt}=kR-aRW}\displaystyle \newcommand{\index}[1]{}{\textbf{(Predator-Prey Evolution)}\index{Predator-Prey evolution}\qquad\frac{dW}{dt}=-rW+bRW\text{ and }\frac{dR}{dt}=kR-aRW}
subject to W(0)=W_{0},R(0)=R_{0}W(0)=W_{0},R(0)=R_{0}

Using Euler’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 Predator-Prey 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 non-equilibrium solutions, we need the ideas of phase plane and phase trajectory. The plane with WW on the yy-axis and RR on the xx-axis 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

\displaystyle \newcommand{\index}[1]{}{\textbf{(Predator-Prey Phase)}\index{Predator-Prey phase}\qquad\frac{dW}{dR}=\frac{-rW+bRW}{kR-aRW}.}\displaystyle \newcommand{\index}[1]{}{\textbf{(Predator-Prey Phase)}\index{Predator-Prey phase}\qquad\frac{dW}{dR}=\frac{-rW+bRW}{kR-aRW}.}

It might be useful to know that the solutions to the Predator-Prey Phase equation satisfy
\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.