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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

x ˙(t)=rx(t)1-x(t) K-mx(t)y(t) b+x 2 (t),y ˙(t)=y(t)s1-y(t) hx(t),(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.

34C23Bifurcation (ODE)
92D25Population dynamics (general)
37G15Bifurcations of limit cycles and periodic orbits
34C05Location of integral curves, singular points, limit cycles (ODE)