Bifurcation of a delayed Gause predator-prey model with Michaelis-Menten type harvesting.

In this paper, a Gause predator-prey model with gestation delay and Michaelis-Menten type harvesting of prey is proposed and analyzed by considering Holling type III functional response. We first consider the local stability of the interior equilibrium by investigating the corresponding characteristic equation. In succession, we derive some sufficient conditions on the occurrence of the stability switches of the positive steady state by taking the gestation delay as a bifurcation parameter. It is shown that the delay can induce instability and small amplitude oscillations of population densities via Hopf bifurcations. Furthermore, the stability and direction of the Hopf bifurcations are determined by employing the center manifold argument. Finally, computer simulations are performed to illustrate our analytical findings, and the biological implications of our analytical findings are also discussed.

[1]  Jafar Fawzi M. Al-Omari.  The effect of state dependent delay and harvesting on a stage-structured predator-prey model , 2015, Appl. Math. Comput..

[2]  Moxun Tang,et al.  Coexistence Region and Global Dynamics of a Harvested Predator-Prey System , 1998, SIAM J. Appl. Math..

[3]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[4]  N. Mankiw,et al.  Principles of Economics , 1871 .

[5]  Boshan Chen,et al.  Bifurcations and stability of a discrete singular bioeconomic system , 2013 .

[6]  H. I. Freedman.  Deterministic mathematical models in population ecology , 1982 .

[7]  Teresa Faria,et al.  Normal forms and Hopf bifurcation for partial differential equations with delays , 2000 .

[8]  Yi Shen,et al.  Bifurcation analysis in a discrete differential-algebraic predator–prey system , 2014 .

[9]  Yi Shen,et al.  Hopf bifurcation of a predator–prey system with predator harvesting and two delays , 2013 .

[10]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation , 1995 .

[11]  Maja Vasilova.  Asymptotic behavior of a stochastic Gilpin-Ayala predator-prey system with time-dependent delay , 2013, Math. Comput. Model..

[12]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[13]  Jiejie Chen,et al.  Bifurcation and chaotic behavior of a discrete singular biological economic system , 2012, Appl. Math. Comput..

[14]  H. I. Freedman,et al.  A time-delay model of single-species growth with stage structure. , 1990, Mathematical biosciences.

[15]  H. I. Freedman,et al.  Nonoccurence of Stability Switching in Systems with Discrete Delays , 1988, Canadian Mathematical Bulletin.

[16]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[17]  Christopher M. Heggerud,et al.  Local stability analysis of ratio-dependent predator-prey models with predator harvesting rates , 2015, Appl. Math. Comput..

[18]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[19]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[20]  Yi Shen,et al.  Positive periodic solutions in a non-selective harvesting predator–prey model with multiple delays , 2012 .

[21]  Debadatta Adak,et al.  Complexity in a predator-prey-parasite model with nonlinear incidence rate and incubation delay , 2015 .

[22]  K. Cooke,et al.  Discrete delay, distributed delay and stability switches , 1982 .

[23]  Global asymptotic stability of periodic Lotka-Volterra systems with delays , 2001 .

[24]  Xiang-Ping Yan,et al.  Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects , 2007, Appl. Math. Comput..

[25]  Huawen Ye,et al.  A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting , 2017 .

[26]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations and Applications to Bogdanov-Takens Singularity , 1995 .

[27]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[28]  C. S. Holling,et al.  The functional response of predators to prey density and its role in mimicry and population regulation. , 1965 .

[29]  Zhigang Zeng,et al.  A New Comparison Method for Stability Theory of Differential Systems with Time-Varying Delays , 2008, Int. J. Bifurc. Chaos.

[30]  Sebastian Reich.  On the local qualitative behavior of differential-algebraic equations , 1995 .

[31]  H. I. Freedman.  Stability analysis of a predator-prey system with mutual interference and density-dependent death rates , 1979 .

[32]  T. Marchant,et al.  The diffusive Lotka-Volterra predator-prey system with delay. , 2015, Mathematical biosciences.

[33]  Dongmei Xiao,et al.  Bifurcations of a Ratio-Dependent Predator-Prey System with Constant Rate Harvesting , 2005, SIAM J. Appl. Math..

[34]  H. Gordon.  The Economic Theory of a Common-Property Resource: The Fishery , 1954, Journal of Political Economy.

[35]  H. Schättler,et al.  Local bifurcations and feasibility regions in differential-algebraic systems , 1995, IEEE Trans. Autom. Control..

[36]  A. J. Lotka.  Elements of mathematical biology , 1956 .

[37]  B Chen,et al.  NORMAL FORMS AND BIFURCATIONS FOR THE DIFFERENTIAL-ALGEBRAIC SYSTEMS , 2000 .

[38]  Boshan Chen,et al.  Stability and Hopf Bifurcation of a Predator–Prey Biological Economic System with Nonlinear Harvesting Rate , 2015 .

[39]  H. I. Freedman,et al.  Analysis of a model representing stage-structured population growth with state-dependent time delay , 1992 .

[40]  Colin W. Clark,et al.  Mathematical Bioeconomics: The Optimal Management of Renewable Resources. , 1993 .

[41]  S. Ruan,et al.  Predator-prey models with delay and prey harvesting , 2001, Journal of mathematical biology.

[42]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[43]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[44]  T. K. Kar,et al.  Non-selective harvesting in prey predator models with delay , 2006 .

[45]  Guodong Zhang,et al.  Periodic solutions for a neutral delay Hassell-Varley type predator-prey system , 2015, Appl. Math. Comput..

[46]  Xinzhu Meng,et al.  Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays. , 2006, Journal of theoretical biology.

[47]  G. F. Gause.  The struggle for existence , 1971 .

[48]  Teresa Faria,et al.  Stability and Bifurcation for a Delayed Predator–Prey Model and the Effect of Diffusion☆ , 2001 .

[49]  Zhidong Teng,et al.  Persistence in nonautonomous predator-prey systems with infinite delays , 2006 .

[50]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.