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

Zuolin Shen; Junjie Wei; Zuolin Shen; Junjie Wei

doi:10.3934/mbe.2018031

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 3: 693-715. doi: 10.3934/mbe.2018031

Previous Article Next Article

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

Zuolin Shen ,
Junjie Wei ^,

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

Received: 14 March 2017 Accepted: 30 September 2017 Published: 01 June 2018
MSC : Primary: 35K57, 35R10; Secondary: 35P15

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.
- diffusive predator-prey system,
- time delay,
- surplus killing effect,
- Hopf bifurcation
Citation: Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 693-715. doi: 10.3934/mbe.2018031

Related Papers:

Abstract

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.

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.
[3]	[ S. S. Chen,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.
[4]	[ S. S. Chen,J. P. Shi,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.
[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,Z. Grossman, Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis & Applications, 86 (1982): 592-627.
[7]	[ Y. H. Du,L. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011): 319-349.
[8]	[ Y. H. Du,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,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.
[11]	[ D. J. Formanowitz, Foraging tactics of an aquatic insect: Partial consumption of prey, Anim Behav, 32 (1984): 774-781.
[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.
[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.
[18]	[ X. Lin,J. So,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.
[19]	[ J. L. Maupin,S. E. Riechert, Superfluous killing in spiders: A consequence of adaptation to food-limited environments?, Behavioral Ecology, 12 (2001): 569-576.
[20]	[ L. S. Mills, Conservation of Wildlife Populations: Demography, Genetics, and Management, Wiley-Blackwell, Oxford, 2013.
[21]	[ P. de Motoni,F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems, Siam Journal on Applied Mathematics, 37 (1979): 648-663.
[22]	[ J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
[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.
[25]	[ C. V. Pao, Systems of parabolic equations with continuous and discrete delays, Journal of Mathematical Analysis and Applications, 205 (1997): 157-185.
[26]	[ A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
[27]	[ M. L. Rosenzweig,R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963): 209-223.
[28]	[ S. G. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, Journal of Mathematical Biology, 31 (1993): 633-654.
[29]	[ S. G. Ruan,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,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.
[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,J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009): 1944-1977.
[36]	[ J. T. Zhao,J. J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Analysis Real World Applications, 22 (2015): 66-83.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)