Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

Jicai Huang; Yijun Gong; Shigui Ruan

doi:10.3934/dcdsb.2013.18.2101

Article Contents

2013, Volume 18, Issue 8: 2101-2121. Doi: 10.3934/dcdsb.2013.18.2101

This issue Previous Article Convergence, non-negativity and stability of a new Milstein scheme with applications to finance Next Article Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates

Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079
2.
Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250

Received: March 31, 2013

Revised: May 31, 2013

Published: September 2013

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Abstract

In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.

Keywords:

Predator-prey model,
constant-yield predator harvesting,
Bogdanov-Takens singularity of codimension 3,
Bogdanov-Takens bifurcation,
degenerate Hopf bifurcation.

Mathematics Subject Classification: Primary: 34C25, 34C23; Secondary: 37N25, 92D25.

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