A Predator-Prey Model with Immigration.

Consider the following system of two equations [#!Sc:95!#]

$$\begin{array}{rcl} \dot{X} & = & RX\left(1-{\displaystyle \fr... ...- D_1Y - {\displaystyle \frac{A_2ZY^2}{B_2^2+Y^2}}. \end{array}$$ (22.1)

Figure 22.1: Parametric portrait of the predator-prey system

The values of all parameters except $(K,Z)$ are set as follows :

$$R=0.5,\ A_1=0.4,\ B_1=0.6,\ D_0=0.01,\ E_1=0.6,\ A_2=1.0,\ B_2=0.5,\ D_1=0.15.$$

The parametric portrait of the system (22.1) on the $(Z,K)$-plane is presented in Figure 22.1. It contains fold ($t_{1,2}$) and Hopf ($H$) bifurcation curves, as well as a homoclinic bifurcation curve $P$. The fold curves meet at a cusp singular point $C$, while the Hopf and the homoclinic curves originate at a Bogdanov-Takens point $BT$. Only the homoclinic curve $P$ will be considered here, the other bifurcation curves can be computed using AUTO or, for example, locbif [#!KhKuLeNi:93!#].