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Bifurcations of a predator-prey system of Holling and Leslie types. (English) Zbl 1156.34029

The authors study the following predator-prey model with Holling type-IV functional response and Leslie type numerical response for the predator

$\begin{array}{cc}\hfill \stackrel{˙}{x}\left(t\right)=& rx\left(t\right)\left(1-\frac{x\left(t\right)}{K}\right)-\frac{mx\left(t\right)y\left(t\right)}{b+{x}^{2}\left(t\right)},\hfill \\ \hfill \stackrel{˙}{y}\left(t\right)=& y\left(t\right)s\left(1-\frac{y\left(t\right)}{hx\left(t\right)}\right),\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $x\left(t\right)$ and $y\left(t\right)$ represent the densities of the prey and the predator population at time $t$, respectively. $r,K,b,s$ and $h$ are positive parameters. It is shown that system (1) admits two non-hyperbolic positive equilibria for some values of parameters, one is a cusp of codimension 2 and the other is a multiple focus of multiplicity one. When the parameters vary in a small neighborhood of the values of parameters, system (1) undergoes a Bogdanov-Takens bifurcation and a subcritical Hopf bifurcation in two small neighborhoods of these two equilibria, respectively. And it is further derived that by choosing different values of parameters, system (1) can have a stable limit cycle enclosing two equilibria, or an unstable limit cycle enclosing a hyperbolic equilibrium, or two limit cycles enclosing a hyperbolic equilibrium. It is also proved that system (1) never has two limit cycles enclosing a hyperbolic equilibrium each for all values of parameters.

##### MSC:
 34C23 Bifurcation (ODE) 92D25 Population dynamics (general) 37G15 Bifurcations of limit cycles and periodic orbits 34C05 Location of integral curves, singular points, limit cycles (ODE)
##### Keywords:
bifurcation; predator-prey system