In this paper, we mainly considered the dynamical behavior of a predator-prey system with Holling type II functional response and Allee-like effect on predator, including stability analysis of equilibria and Hopf bifurcation. Firstly, we gave some sufficient conditions to guarantee the existence, the local and global stability of equilibria as well as non-existence of limit cycles. By using the cobweb model, some cases about the existence of interior equilibrium are also illustrated with numerical outcomes. These existence and stability conclusions of interior equilibrium are also suitable in corresponding homogeneous reaction-diffusion system subject to the Neumann boundary conditions. Secondly, we theoretically deduced that our system has saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation under certain conditions. Finally, for the Hopf bifurcation, we choose d as the bifurcation parameter and presented some numerical simulations to verify feasibility and effectiveness of the theoretical derivation corresponding to the existence of y k , respectively. The Hopf bifurcations are supercritical and limit cycles generated by the critical points are stable.
KeywordsPredator-Prey SystemHolling Type II Functional ResponseAllee EffectStabilityHopf Bifurcation
Holling, C.S. (1965) The Functional Response of Predator Density and Its Role in Mimicry and Population Regulations. Memoirs of the Entomological Society of Canada, 45, 3-60. https://doi.org/10.4039/entm9745fv
Pei, Y.Z., Chen, L.S., Zhang, Q.R. and Li, C.G. (2005) Extinction and Performance of One-Prey Multi-Predators of Holling Type II Function Response System with Impulsive Biological Control. Journal of Theoretical Biology, 235, 495-503. https://doi.org/10.1016/j.jtbi.2005.02.003
Misha, P. and Raw, S.N. (2019) Dynamical Complexities in a Predator-Prey System Involving Teams of Two Prey and One Predator. Journal of Applied Mathematics and Computing, 61, 1-24. https://doi.org/10.1007/s12190-018-01236-9
Huang, J.C., Ruan, S.G. and Song, J. (2014) Bifurcation in a Predator-Prey System of Leslie Type with Generalized Holling Type III Functional Response. Journal of Differential Equations, 257, 1721-1752. https://doi.org/10.1007/s12190-018-01236-9
Allee, W.C. (1927) Animal Aggregations. The Quarterly Review of Biology, 2, 367-398. https://doi.org/10.1086/394281
Allee, W.C. (1931) Animal Aggregations: A Study in General Sociology. University of Chicago Press, Chicago. https://doi.org/10.5962/bhl.title.7313
Dos Santos, L.S., Cabella, B.C.T. and Martinez, A.S. (2014) Generalized Allee Effect Model. Theory in Biosciences, 133, 117-124. https://doi.org/10.1007/s12064-014-0199-6
Stephens, P.A. and Sutherland, W.J. (1999) Consequences of the Allee Effect for Behaviour, Ecology and Conservation. Trends in Ecology & Evolution, 14, 401-405. https://doi.org/10.1016/S0169-5347(99)01684-5
Stephens, P.A., Sutherland, W.J. and Freckleton, R.P. (1999) What Is the Allee Effect? Oikos, 87, 185-190. https://doi.org/10.2307/3547011
Bascompte, J. (2003) Extinction Thresholds: Insights from Simple Models. Annales Zoologici Fennici, 40, 99-114.
Wang, M.H. and Kot, M. (2001) Speeds of Invasion in a Model with Strong or Weak Allee Effects. Mathematical Biosciences, 171, 83-97. https://doi.org/10.1016/S0025-5564(01)00048-7
Aguirre, P., Flores, J.D. and Gonzalez-Olivares, E. (2014) Bifurcations and Global Dynamics in a Predator-Prey Model with a Strong Allee Effect on the Prey, and a Ratio-Dependent Functional Response. Nonlinear Analysis: Real World Applications, 16, 235-249. https://doi.org/10.1016/j.nonrwa.2013.10.002
González-Olivares, E. and Rojas-Palma, A. (2012) Limit Cycles in a Gause-type Predator-Prey Model with Sigmoid Functional Response and Weak Allee Effect on Prey. Mathematical Methods in the Applied Sciences, 35, 963-975. https://doi.org/10.1002/mma.2509
Zu, J. (2013) Global Qualitative Analysis of a Predator-Prey System with Allee Effect on the Prey Species. Mathematics and Computers in Simulation, 94, 33-54. https://doi.org/10.1016/j.matcom.2013.05.009
Zu, J. and Mimura, M. (2010) The Impact of Allee Effect on a Predator-Prey System with Holling Type II Functional Response. Applied Mathematics and Computation, 217, 3542-3556. https://doi.org/10.1016/j.amc.2010.09.029
Cai, Y.L., Zhao, C.D., Wang, W.M. and Wang, J.F. (2015) Dynamics of a Leslie-Gower Predator-prey Model with Additive Allee Effect. Applied Mathematical Modelling, 39, 2092-2106. https://doi.org/10.1016/j.apm.2014.09.038
Zhou, S.R. and Wang, G. (2004) Allee-Like Effects in Metapopulation Dynamics. Mathematical Biosciences, 189, 103-113. https://doi.org/10.1016/j.mbs.2003.06.001
Xiao, Z.W., Xie, X.D. and Xue, Y.L. (2018) Stability and Bifurcation in a Holling Type II Predator-Prey Model with Allee Effect and Time Delay. Advances in Difference Equations, Article No.: 288. https://doi.org/10.1186/s13662-018-1742-4
Sun, Y.S., Wang, K.H. and Gui, Z.J. (2017) Periodic Solution of a Nonautonomous Predator-Prey System of Holling Type II with Strong Allee Effect and Impulsive Perturbation. 6th International Conference on Energy, Environment and Sustainable Development, DEStech Transactions on Environment, Energy and Earth Sciences, Phuket, Thailand, April 2017, 159-164. https://doi.org/10.12783/dteees/eesd2017/11992
Cui, R.H., Shi, J.P. and Wu, B.Y. (2014) Strong Allee Effect in a Diffusive Predator-Prey System with a Protection Zone. Journal of Differential Equations, 256, 108-129. https://doi.org/10.1016/j.jde.2013.08.015
Wang, G., Liang, X.G. and Wang, F.Z. (1999) The Competitive Dynamics of Populations Subject to an Allee Effect. Ecological Modelling, 124, 183-192. https://doi.org/10.1016/S0304-3800(99)00160-X
Birkhoff, G. and Rota, G.C. (1962) Ordinary Differential Equations Introductions to Higher Mathematics. Ginn and Company, Boston.
Chen, F. (2005) On a Nonlinear Nonautonomous Predator-Prey Model with Diffusion and Distributed Delay. Journal of Computational and Applied Mathematics, 180, 33-49. https://doi.org/10.1016/j.cam.2004.10.001
Merkin, D.R. Afagh, F.F. and Smirnov, A.L. (1997) Introduction to the Theory of Stability. Springer, New York.
Chicone, C. (2006) Ordinary Differential Equations with Applications. World Scientific, Springer-Verlag, New York.
Andronov, A. (1973) Qualitative Theory of Second-Order Dynamic Systems. Halsted Press.
Perko, L. (2001) Differential Equations and Dynamical Systems. 3rd Edition, Springer-Verlag, New York. https://doi.org/10.1007/978-1-4613-0003-8
Pirayesh, B., Pazirandeh, A. and Akbari, M. (2016) Local Bifurcation Analysis in Nuclear Reactor Dynamics by Sotomayor’s Theorem. Annals of Nuclear Energy, 94, 716-731. https://doi.org/10.1016/j.anucene.2016.04.021
Kuznetsov, Y. (1998) Elements of Applied Bifurcation Theory. 2nd Edition, Springer-Verlag, New York.
Yi, F.Q., Wei, J.J. and Shi, J.P. (2009) Bifurcation and Spatiotemporal Patterns in a Homogeneous Diffusive Predator-Prey System. Journal of Differential Equations, 246, 1944-1977. https://doi.org/10.1016/j.jde.2008.10.024
Wan, A.Y., Song, Z.Q., Zheng, L.F. (2016) Patterned Solutions of a Homogenous Diffusive Predator-Prey System of Holling Type-III. Acta Mathematicae Applicatae Sinica (English Series), 32, 1073-1086. https://doi.org/10.1007/s10255-016-0628-z
Casten, R.G. and Holland, C.J. (1977) Stability Properties of Solutions to Systems of Reaction-Diffusion Equations. SIAM Journal on Applied Mathematics, 33, 353-364. https://doi.org/10.1137/0133023