Existence of a polycycle in non-Lipschitz Gause-type predator-prey models
This paper studies the Gause-type predator-prey model \[ x' = F(x) - y \phi(x),\quad y' = y (\psi(x) - \mu). \] The authors show that, when the interaction between predator and prey is not locally-Lipschitz continuous in the absence of one of them, there appears a polycycle, which affects the existence of limit cycles for the system. The authors assume: {\parindent8mm \begin{itemize}\item[(a] \(F(0) = 0\), and there exists \(k>0\) such that \(F(x) (x-k)<0\) for \(0<x\not=k\), and \(F'(k) < 0 < F'(0)\). \item[(a] \(\phi(0) = \psi(0) = 0\), and \(\phi'(x), \psi'(x) > 0\) for all \(x > 0\). \item[(a] \(\mu \in \psi((0,k))\). \item[(a] For some \(x > 0\), \(\int_0^x {d z \over \phi(z)} > +\infty\). \item[(a] \(\lim_{x\to 0^+} F(x)/\phi(x) = 0\), and there exist \[ \lim_{x\to 0^+} (F/\phi)'(x)\phi(x)\in {\mathbb R},\quad \lim_{x\to 0^+} \psi'(x)\phi(x)\in {\mathbb R}. \] \end{itemize}} Under the conditions above, the authors prove the existence of a unique value of \(\mu\) such that the equation possesses a positive heteroclinic connection \(H\) joining the equilibrium \((k,0)\) to the equilibrium \((0,0)\). In particular, for this value of \(\mu\), the above system has a polycycle consisting of \(H\) plus the linear segment that joins \((0,0)\) to \((k,0)\).
scientific article; zbMATH DE number 5804526