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
\[ \begin{aligned} \dot x(t)=& rx(t)\left(1-\frac{x(t)}{K}\right)-\frac{mx(t)y(t)}{b+x^2(t)},\\ \dot y(t)=& y(t) s\left(1-\frac{y(t)}{hx(t)}\right), \end{aligned}\tag{1} \]
where \(x(t)\) and \(y(t)\) 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.


34C23 Bifurcation theory for ordinary differential equations
92D25 Population dynamics (general)
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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