A non-autonomous competitive Lotka–Volterra system

Abstract

As our first extended example we will consider a non-autonomous Lotka–Volterra model,

$$\begin{array}{rcl} \dot{u}& = u(\lambda (t) - au - bv)& \\ \dot{v}& = v(\mu - cu - dv), &\end{array}$$

(9.1)

where the parameters a, b,c,d, and μ are positive, ad>bc, and 0<λ≤λ(t)≤Λ. In line with the interpretation of this model in terms of the numbers of two competing species, we consider the dynamics only in positive quadrant $\overline{Q} :=\{ (u,v) :\ u,v \geq 0\}$; the interesting dynamics takes place in the interior of this quadrant, {(u,v) :u,v>0}, which we denote by Q. Note that the u- and v-axes are both invariant.