# American Institute of Mathematical Sciences

June  2018, 15(3): 693-715. doi: 10.3934/mbe.2018031

## Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect

 Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China

* Corresponding author. Email: weijj@hit.edu.cn

Received  March 14, 2017 Accepted  September 30, 2017 Published  December 2017

Fund Project: This research is supported by National Natural Science Foundation of China (Nos.11371111 and 11771109).

A diffusive predator-prey system with a delay and surplus killing effect subject to Neumann boundary conditions is considered. When the delay is zero, the prior estimate of positive solutions and global stability of the constant positive steady state are obtained in details. When the delay is not zero, the stability of the positive equilibrium and existence of Hopf bifurcation are established by analyzing the distribution of eigenvalues. Furthermore, an algorithm for determining the direction of Hopf bifurcation and stability of bifurcating periodic solutions is derived by using the theory of normal form and center manifold. Finally, some numerical simulations are presented to illustrate the analytical results obtained.

Citation: Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
##### References:
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Grossman, Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis & Applications, 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8. [7] Y. H. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.  doi: 10.1088/0951-7715/24/1/016. [8] Y. H. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, 48 (2006), 95-135. [9] S. Ehrlinge, B. Bergsten and B. Kristiansson, The stoat and its prey: hunting behavior and escape reactions, Fauna Flora, 69 (1974a), 203-211. [10] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Transactions of the American Mathematical Society, 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7. [11] D. J. Formanowitz, Foraging tactics of an aquatic insect: Partial consumption of prey, Anim Behav, 32 (1984), 774-781.  doi: 10.1016/S0003-3472(84)80153-0. [12] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980. [13] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [14] C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7. [15] C. B. Huffaker, Exerimental studies on predation: Despersion factors and predator-prey oscilasions, Hilgardia, 27 (1958), 343-383. [16] H. Kruuk, The Spotted Hyena: A Study of Predation and Social Behavior, University of Chicago Press, Chicago, 1972. [17] H. Kruuk, Surplus killing by carnivores, Journal of Zoology, 166 (1972), 233-244.  doi: 10.1111/j.1469-7998.1972.tb04087.x. [18] X. Lin, J. So and J. H. Wu, Centre manifolds for partial differential equations with delays, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 122 (1992), 237-254.  doi: 10.1017/S0308210500021090. [19] J. L. Maupin and S. E. Riechert, Superfluous killing in spiders: A consequence of adaptation to food-limited environments?, Behavioral Ecology, 12 (2001), 569-576.  doi: 10.1093/beheco/12.5.569. [20] L. S. Mills, Conservation of Wildlife Populations: Demography, Genetics, and Management, Wiley-Blackwell, Oxford, 2013. [21] P. de Motoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems, Siam Journal on Applied Mathematics, 37 (1979), 648-663.  doi: 10.1137/0137048. [22] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869. [23] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [24] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Analysis: Theory, Methods & Applications, 48 (2002), 349-362.  doi: 10.1016/S0362-546X(00)00189-9. [25] C. V. Pao, Systems of parabolic equations with continuous and discrete delays, Journal of Mathematical Analysis and Applications, 205 (1997), 157-185.  doi: 10.1006/jmaa.1996.5177. [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [27] M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272. [28] S. G. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, Journal of Mathematical Biology, 31 (1993), 633-654.  doi: 10.1007/BF00161202. [29] S. G. Ruan and J. J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous Discrete and Impulsive Systems Series A, 10 (2003), 863-874. [30] F. Samu and Z. Bíró Z, Functional response, multiple feeding, and wasteful killing in a wolf spider (Araneae: Lycosidae), European Journal of Entomology, 90 (1993), 471-476. [31] C. T. Stuart, The incidence of surplus killing by Panthera pardus and Felis caracal in Cape Province, South Africa. Mammalia, 50 (1986), 556-558. [32] V. Volterra, Variazione e fluttuazini del numero d'individui in specie animali conviventi, Mem. Accad. Nazionale Lincei, 2 (1926), 31-113. [33] J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [34] Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Science Press, China, 1994 (in Chinese). [35] F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [36] J. T. Zhao and J. J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Analysis Real World Applications, 22 (2015), 66-83.  doi: 10.1016/j.nonrwa.2014.07.010.

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##### References:
 [1] A. Bjärvall and E. Nilsson, Surplus-killing of reindeer by wolves, Journal of Mammalogy , 57 (1976), p585. [2] P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Analysis: Real World Applications, 13 (2012), 1837-1843.  doi: 10.1016/j.nonrwa.2011.12.014. [3] S. S. Chen and J. P. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, Journal of Differential Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031. [4] S. S. Chen, J. P. Shi and J. J. Wei, The effect of delay on a diffusive predator-prey system with Holling Type-Ⅱ predator functional response, Communications on Pure & Applied Analysis, 12 (2013), 481-501.  doi: 10.3934/cpaa.2013.12.481. [5] R. J. Conover, Factors affecting the assimilation of organic matter by zooplankton and the question of superfluous feeding, Limnol Oceanogr, 11 (1966), 346-354. [6] K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis & Applications, 86 (1982), 592-627.  doi: 10.1016/0022-247X(82)90243-8. [7] Y. H. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.  doi: 10.1088/0951-7715/24/1/016. [8] Y. H. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, 48 (2006), 95-135. [9] S. Ehrlinge, B. Bergsten and B. Kristiansson, The stoat and its prey: hunting behavior and escape reactions, Fauna Flora, 69 (1974a), 203-211. [10] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Transactions of the American Mathematical Society, 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7. [11] D. J. Formanowitz, Foraging tactics of an aquatic insect: Partial consumption of prey, Anim Behav, 32 (1984), 774-781.  doi: 10.1016/S0003-3472(84)80153-0. [12] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980. [13] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [14] C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7. [15] C. B. Huffaker, Exerimental studies on predation: Despersion factors and predator-prey oscilasions, Hilgardia, 27 (1958), 343-383. [16] H. Kruuk, The Spotted Hyena: A Study of Predation and Social Behavior, University of Chicago Press, Chicago, 1972. [17] H. Kruuk, Surplus killing by carnivores, Journal of Zoology, 166 (1972), 233-244.  doi: 10.1111/j.1469-7998.1972.tb04087.x. [18] X. Lin, J. So and J. H. Wu, Centre manifolds for partial differential equations with delays, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 122 (1992), 237-254.  doi: 10.1017/S0308210500021090. [19] J. L. Maupin and S. E. Riechert, Superfluous killing in spiders: A consequence of adaptation to food-limited environments?, Behavioral Ecology, 12 (2001), 569-576.  doi: 10.1093/beheco/12.5.569. [20] L. S. Mills, Conservation of Wildlife Populations: Demography, Genetics, and Management, Wiley-Blackwell, Oxford, 2013. [21] P. de Motoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems, Siam Journal on Applied Mathematics, 37 (1979), 648-663.  doi: 10.1137/0137048. [22] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869. [23] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [24] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Analysis: Theory, Methods & Applications, 48 (2002), 349-362.  doi: 10.1016/S0362-546X(00)00189-9. [25] C. V. Pao, Systems of parabolic equations with continuous and discrete delays, Journal of Mathematical Analysis and Applications, 205 (1997), 157-185.  doi: 10.1006/jmaa.1996.5177. [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [27] M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272. [28] S. G. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, Journal of Mathematical Biology, 31 (1993), 633-654.  doi: 10.1007/BF00161202. [29] S. G. Ruan and J. J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous Discrete and Impulsive Systems Series A, 10 (2003), 863-874. [30] F. Samu and Z. Bíró Z, Functional response, multiple feeding, and wasteful killing in a wolf spider (Araneae: Lycosidae), European Journal of Entomology, 90 (1993), 471-476. [31] C. T. Stuart, The incidence of surplus killing by Panthera pardus and Felis caracal in Cape Province, South Africa. Mammalia, 50 (1986), 556-558. [32] V. Volterra, Variazione e fluttuazini del numero d'individui in specie animali conviventi, Mem. Accad. Nazionale Lincei, 2 (1926), 31-113. [33] J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [34] Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Science Press, China, 1994 (in Chinese). [35] F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [36] J. T. Zhao and J. J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Analysis Real World Applications, 22 (2015), 66-83.  doi: 10.1016/j.nonrwa.2014.07.010.
The critical curve on $(m_1,\gamma)$ plane. Ⅰ: $E_*(u_*,v_*)$ is global asymptotically stable; Ⅱ: $E_*(u_*,v_*)$ is local asymptotically stable; Ⅲ: $E_*(u_*,v_*)$ disappears while $E_1(0,1)$ is global asymptotically stable. The parameters are chosen as follows: $\alpha=0.3$, $K_2=0.2$, $\theta=0.5$ with $m_2=\alpha m_1/K_2$.
The positive equilibrium is asymptotically stable when $\tau\in[0, \tau^*)$, where $\tau=2＜\tau^*\approx4.6242$.
The bifurcating periodic solution is stable, where $\tau=5>\tau^*\approx4.6242$.
The axial equilibrium $E_1(0,1)$ is global asymptotically stable.
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