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Zbl 1156.34029
Li, Yilong; Xiao, Dongmei
Bifurcations of a predator-prey system of Holling and Leslie types. (English)
[J] Chaos Solitons Fractals 34, No. 2, 606-620 (2007). ISSN 0960-0779

The authors study the following predator-prey model with Holling type-IV functional response and Leslie type numerical response for the predator $$\aligned \dot x(t)=& rx(t)\left(1-\frac{x(t)}{K}\right)-\frac{mx(t)y(t)}{b+x^2(t)},\\ \dot y(t)=& y(t) s\left(1-\frac{y(t)}{hx(t)}\right), \endaligned\tag1$$ 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.
[Rui Xu (Shijiazhuang)]

MSC 2000:: *34C23 Bifurcation (periodic solutions)
92D25 Population dynamics
37G15 Bifurcations of limit cycles and periodic orbits
34C05 Qualitative theory of some special solutions of ODE

Keywords: bifurcation; predator-prey system

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Zentralblatt MATH Berlin [Germany]