Bifurcations of a predator-prey system of Holling and Leslie types.

*(English)*Zbl 1156.34029The 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.

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

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.

Reviewer: Rui Xu (Shijiazhuang)

##### MSC:

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 |

PDF
BibTeX
XML
Cite

\textit{Y. Li} and \textit{D. Xiao}, Chaos Solitons Fractals 34, No. 2, 606--620 (2007; Zbl 1156.34029)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Wollkind, D.J.; Logan, J.A., Temperature-dependent predator – prey mite ecosystem on apple tree foliage, J math biol, 6, 265-283, (1978) |

[2] | Wollkind, D.J.; Collings, J.B.; Logan, J.A., Metastability in a temperature-dependent model system for predator – prey mite outbreak interactions on fruit trees, J math biol, 50, 379-409, (1988) · Zbl 0652.92019 |

[3] | Hoyt, S.C., Integrated chemical control of insects and biological control of mites on apple in Washington, J econ entomol, 62, 74-86, (1969) |

[4] | Hoyt SC. Population studies of five mite species on apple in Washington. In: Proc second int cong acarology. Sutton Bonington: England, 1967, p. 117-33. |

[5] | May, R.M., Stability and complexity in model ecosystems, (1973), Princeton University Press Princeton |

[6] | Leslie, P.H., Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213-245, (1948) · Zbl 0034.23303 |

[7] | Holling, C.S., The functional response of predators to prey density and its role in mimicry and population regulation, Mem ent soc can, 46, 1-60, (1965) |

[8] | Leslie, P.H.; Gower, J.C., The properties of a stochastic model for the predator – prey type of interaction between two species, Biometrika, 47, 219-234, (1960) · Zbl 0103.12502 |

[9] | Hsu, S.B.; Huang, T.W., Global stability for a class of predator – prey systems, SIAM J appl math, 55, 763-783, (1995) · Zbl 0832.34035 |

[10] | Hsu, S.B.; Hwang, T.W., Hopf bifurcation analysis for a predator – prey system of Holling and Leslie type, Taiwan J math, 3, 35-53, (1999) · Zbl 0935.34035 |

[11] | Collings, J.B., The effects of the functional response on the bifurcation behavior of a mite predator – prey interaction model, J math biol, 36, 149-168, (1997) · Zbl 0890.92021 |

[12] | Sokol, W.; Howell, J.A., Kinetics of phenol exidation by washed cells, Biotechnol bioeng, 23, 2039-2049, (1980) |

[13] | Liu, X.; Chen, L., Complex dynamics of Holling type II lotka – volterra predator – prey system with impulsive perturbations on the predator, Chaos, solitons & fractals, 16, 311-320, (2003) · Zbl 1085.34529 |

[14] | Zhang, S.; Dong, L.; Chen, L., The study of predator – prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, solitons & fractals, 23, 631-643, (2005) · Zbl 1081.34041 |

[15] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001 |

[16] | Bogdanov, R., Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta math soviet, 1, 373-388, (1981) · Zbl 0518.58029 |

[17] | Bogdanov, R., Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta math soviet, 1, 389-421, (1981) · Zbl 0518.58030 |

[18] | Takens, F., Forced oscillations and bifurcation, In applications of global analysis I, comm math inst rijksuniversitat Utrecht, 3, 1-59, (1974) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.