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January  2013, 12(1): 481-501. doi: 10.3934/cpaa.2013.12.481

The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response

Shanshan Chen 1, , Junping Shi 2, and  Junjie Wei 1,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China, China
2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187

Received  November 2010 Revised  April 2012 Published  September 2012

A delayed diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered here. The stability/instability of nonnegative equilibria and associated Hopf bifurcation are investigated by analyzing the characteristic equations. By the theory of normal form and center manifold, an explicit formula for determining the stability and direction of periodic solution bifurcating from Hopf bifurcation is derived.
Keywords: delay, Predator-prey, Hopf bifurcation, stability, periodic orbit..
Mathematics Subject Classification: Primary: 35K57; Secondary: 35R10, 92D2.
Citation: Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure and Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481
References:
[1]

S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Int. J. Bifurcation Chaos., 22 (2012), 1250061. doi: 10.1142/S0218127412500617.

[2]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7.

[3]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182.

[4]

T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations., 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144.

[5]

T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations., 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145.

[6]

B. C. Goodwin, "Temporal Organzization in Cells," Academic Press, London and New York, 1963.

[7]

J. K. Hale, "Theory of Functinal Differentail Equations," Second edition. Applied Mathematical Sciences, 3. Springer-Verlag, New York-Heidelberg, 1977.

[8]

B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, "Theory and Application of Hopf Bifurcation," Cambridge University Press, Cambridge-New York, 1981.

[9]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.

[10]

G. Hu and W. Li, Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonl. Anal. Real World Appl., 11 (2010), 819-826. doi: 10.1016/j.nonrwa.2009.01.027.

[11]

G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246.

[12]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type-II functional response incorporating a prey refuge, J. Differential Equations., 231 (2006), 534-550. doi: 10.1016/j.jde.2006.08.001.

[13]

X. Lin, J. W.-H. So and J. Wu, Center manifolds for partial differential equations with delay, Proc. Roy. Soc. Edinburgh., 122 A (1992), 237-254. doi: 10.1017/S0308210500021090.

[14]

R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902.

[15]

A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 900-902. doi: 10.1137/S0036144502404442.

[16]

R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations., 247 (2009), 866-886. doi: 10.1016/j.jde.2009.03.008.

[17]

M. L. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interaction, Amer. Natur., 97 (1963), 209-223.

[18]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.

[19]

H. Thieme and X. Zhao, A non-local delayed and diffusive predator-prey model, Nonl. Anal. Real World Appl., 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7.

[20]

P. K. C. Wang, Asymptotic stability of a time-delayed diffusion system, J. Appl. Mech. Ser., E 30 (1963), 500-504. doi: 10.1115/1.3636609.

[21]

Y. Wang, Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays, J. Math. Anal. Appl., 328 (2007), 137-150. doi: 10.1016/j.jmaa.2006.05.02.

[22]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119. Springer, New York, 1996.

[23]

D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations., 2001 (2009), 494-510. doi: 10.1006/jdeq.2000.3982.

[24]

X. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comput., 192 (2007), 137-150. doi: 10.1016/j.amc.2007.03.033.

[25]

X. Yan and C. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Applied Mathematical Modelling, 34 (2010), 184-199. doi: 10.1016/j.apm.2009.03.040.

[26]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024.

[27]

J. Zhang, W. Li and X. Yan, Multiple bifurcations in a delayed predator-prey diffusion system with a functional response, Nonl. Anal. Real World Appl., 11 (2010), 2708-2725. doi: 10.1016/j.nonrwa.2009.09.019.

show all references

References:
[1]

S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Int. J. Bifurcation Chaos., 22 (2012), 1250061. doi: 10.1142/S0218127412500617.

[2]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7.

[3]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182.

[4]

T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations., 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144.

[5]

T. Faria and L. T. Magalh, Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations., 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145.

[6]

B. C. Goodwin, "Temporal Organzization in Cells," Academic Press, London and New York, 1963.

[7]

J. K. Hale, "Theory of Functinal Differentail Equations," Second edition. Applied Mathematical Sciences, 3. Springer-Verlag, New York-Heidelberg, 1977.

[8]

B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, "Theory and Application of Hopf Bifurcation," Cambridge University Press, Cambridge-New York, 1981.

[9]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385-398.

[10]

G. Hu and W. Li, Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonl. Anal. Real World Appl., 11 (2010), 819-826. doi: 10.1016/j.nonrwa.2009.01.027.

[11]

G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246.

[12]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type-II functional response incorporating a prey refuge, J. Differential Equations., 231 (2006), 534-550. doi: 10.1016/j.jde.2006.08.001.

[13]

X. Lin, J. W.-H. So and J. Wu, Center manifolds for partial differential equations with delay, Proc. Roy. Soc. Edinburgh., 122 A (1992), 237-254. doi: 10.1017/S0308210500021090.

[14]

R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902.

[15]

A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 900-902. doi: 10.1137/S0036144502404442.

[16]

R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations., 247 (2009), 866-886. doi: 10.1016/j.jde.2009.03.008.

[17]

M. L. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interaction, Amer. Natur., 97 (1963), 209-223.

[18]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.

[19]

H. Thieme and X. Zhao, A non-local delayed and diffusive predator-prey model, Nonl. Anal. Real World Appl., 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7.

[20]

P. K. C. Wang, Asymptotic stability of a time-delayed diffusion system, J. Appl. Mech. Ser., E 30 (1963), 500-504. doi: 10.1115/1.3636609.

[21]

Y. Wang, Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays, J. Math. Anal. Appl., 328 (2007), 137-150. doi: 10.1016/j.jmaa.2006.05.02.

[22]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119. Springer, New York, 1996.

[23]

D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations., 2001 (2009), 494-510. doi: 10.1006/jdeq.2000.3982.

[24]

X. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comput., 192 (2007), 137-150. doi: 10.1016/j.amc.2007.03.033.

[25]

X. Yan and C. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Applied Mathematical Modelling, 34 (2010), 184-199. doi: 10.1016/j.apm.2009.03.040.

[26]

F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024.

[27]

J. Zhang, W. Li and X. Yan, Multiple bifurcations in a delayed predator-prey diffusion system with a functional response, Nonl. Anal. Real World Appl., 11 (2010), 2708-2725. doi: 10.1016/j.nonrwa.2009.09.019.

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