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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A. Bjärvall and E. Nilsson, Surplus-killing of reindeer by wolves, Journal of Mammalogy , 57 (1976), p585. Google Scholar

[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.  Google Scholar

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S. S. ChenJ. 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.  Google Scholar

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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.  Google Scholar

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S. EhrlingeB. Bergsten and B. Kristiansson, The stoat and its prey: hunting behavior and escape reactions, Fauna Flora, 69 (1974a), 203-211.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980.  Google Scholar

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B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.  Google Scholar

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C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

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C. B. Huffaker, Exerimental studies on predation: Despersion factors and predator-prey oscilasions, Hilgardia, 27 (1958), 343-383.   Google Scholar

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H. Kruuk, The Spotted Hyena: A Study of Predation and Social Behavior, University of Chicago Press, Chicago, 1972. Google Scholar

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H. Kruuk, Surplus killing by carnivores, Journal of Zoology, 166 (1972), 233-244.  doi: 10.1111/j.1469-7998.1972.tb04087.x.  Google Scholar

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X. LinJ. 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.  Google Scholar

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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.  Google Scholar

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L. S. Mills, Conservation of Wildlife Populations: Demography, Genetics, and Management, Wiley-Blackwell, Oxford, 2013. Google Scholar

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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.  Google Scholar

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J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar

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C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

V. Volterra, Variazione e fluttuazini del numero d'individui in specie animali conviventi, Mem. Accad. Nazionale Lincei, 2 (1926), 31-113.   Google Scholar

[33]

J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[34]

Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Science Press, China, 1994 (in Chinese). Google Scholar

[35]

F. Q. YiJ. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Bjärvall and E. Nilsson, Surplus-killing of reindeer by wolves, Journal of Mammalogy , 57 (1976), p585. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. S. ChenJ. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

S. EhrlingeB. Bergsten and B. Kristiansson, The stoat and its prey: hunting behavior and escape reactions, Fauna Flora, 69 (1974a), 203-211.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.  Google Scholar

[14]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar

[15]

C. B. Huffaker, Exerimental studies on predation: Despersion factors and predator-prey oscilasions, Hilgardia, 27 (1958), 343-383.   Google Scholar

[16]

H. Kruuk, The Spotted Hyena: A Study of Predation and Social Behavior, University of Chicago Press, Chicago, 1972. Google Scholar

[17]

H. Kruuk, Surplus killing by carnivores, Journal of Zoology, 166 (1972), 233-244.  doi: 10.1111/j.1469-7998.1972.tb04087.x.  Google Scholar

[18]

X. LinJ. 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.  Google Scholar

[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.  Google Scholar

[20]

L. S. Mills, Conservation of Wildlife Populations: Demography, Genetics, and Management, Wiley-Blackwell, Oxford, 2013. Google Scholar

[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.  Google Scholar

[22]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869.  Google Scholar

[23]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

V. Volterra, Variazione e fluttuazini del numero d'individui in specie animali conviventi, Mem. Accad. Nazionale Lincei, 2 (1926), 31-113.   Google Scholar

[33]

J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[34]

Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Science Press, China, 1994 (in Chinese). Google Scholar

[35]

F. Q. YiJ. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  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 $m_2=\alpha m_1/K_2$.
Figure 2.  The positive equilibrium is asymptotically stable when $\tau\in[0, \tau^*)$, where $\tau=2<\tau^*\approx4.6242$.
Figure 3.  The bifurcating periodic solution is stable, where $\tau=5>\tau^*\approx4.6242$.
Figure 4.  The axial equilibrium $E_1(0,1)$ is global asymptotically stable.
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