February  2011, 4(1): 67-82. doi: 10.3934/dcdss.2011.4.67

Some remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains

1. 

UMR CNRS 5251, I.M.B. and INRIA Bordeaux Sud-ouest Anubis, case 36, UFR Sciences et Modélisation, Université Victor Segalen Bordeaux 2, 3 ter, place de la Victoire - 33076 Bordeaux cedex, France, France

2. 

INRIA Bordeaux Sud-ouest Anubis, case 36, Université Victor Segalen Bordeaux 2, 3 ter, place de la Victoire - 33076 Bordeaux cedex, France

Received  February 2009 Revised  March 2010 Published  October 2010

We are interested in the dynamical behaviour of the solution set to a two component reaction--diffusion system posed on non coincident spatial domains. The underlying biological problem is a predator--prey system featuring a non local numerical response to predation involving an integral kernel. Quite interesting while complex dynamics emerge from preliminary numerical simulations, driven both by diffusivities and by the parametric form or shape of the integral kernel. We consider a simplified version of this problem, with constant coefficients, and give some hints on the large time dynamics of solutions.
Citation: Arnaud Ducrot, Vincent Guyonne, Michel Langlais. Some remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 67-82. doi: 10.3934/dcdss.2011.4.67
References:
[1]

S. Aniţa, W. E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains,, DCDS B, 11 (2009), 805. Google Scholar

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H. Berestycki and P. L. Lions, Une méthode locale pour l’existence de solutions positives de problèmes semi-linéaires elliptiques dans $R^n$,, J. Analyse Math., 38 (1980), 144. Google Scholar

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R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction Diffusion Equations,", John Wiley and Sons, (2003). Google Scholar

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K. S. Cheng, Uniqueness of a limit cycle for a predator-prey system,, SIAM J. Math. Anal., 12 (1981), 541. Google Scholar

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W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains,, in, 1936 (2008), 115. Google Scholar

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S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources,, Mathematical Modelling of Natural Phenomena, 1 (2006), 65. Google Scholar

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S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation,, J. Math. Biol., 41 (2000), 272. Google Scholar

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C. S. Holling, Some characteristics of simple types of predation and parasitism,, Canadian Entomol., 91 (1959), 385. Google Scholar

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S. B. Hsu, On global stability for a predator-prey system,, Math. Biosci., 39 (1978), 1. Google Scholar

[11]

S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators,, SIAM J. Appl. Math., 135 (1978), 617. Google Scholar

[12]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in Biology,, Taiwanese Journal of Mathematics, 9 (2005), 151. Google Scholar

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W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge,, J. Diff. Eq., 5 (2006), 534. Google Scholar

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J. D. Murray, "Mathematical Biology, I & II," 3rd edition,, Springer-Verlag, (2003). Google Scholar

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A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," 2nd edition,, Springer-Verlag, (2001). Google Scholar

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H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003). Google Scholar

show all references

References:
[1]

S. Aniţa, W. E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains,, DCDS B, 11 (2009), 805. Google Scholar

[2]

H. Berestycki and P. L. Lions, Une méthode locale pour l’existence de solutions positives de problèmes semi-linéaires elliptiques dans $R^n$,, J. Analyse Math., 38 (1980), 144. Google Scholar

[3]

R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction Diffusion Equations,", John Wiley and Sons, (2003). Google Scholar

[4]

K. S. Cheng, Uniqueness of a limit cycle for a predator-prey system,, SIAM J. Math. Anal., 12 (1981), 541. Google Scholar

[5]

W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains,, in, 1936 (2008), 115. Google Scholar

[6]

S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources,, Mathematical Modelling of Natural Phenomena, 1 (2006), 65. Google Scholar

[7]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation,, J. Math. Biol., 41 (2000), 272. Google Scholar

[8]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European sawfly,, Canadian Entomol., 91 (1959), 293. Google Scholar

[9]

C. S. Holling, Some characteristics of simple types of predation and parasitism,, Canadian Entomol., 91 (1959), 385. Google Scholar

[10]

S. B. Hsu, On global stability for a predator-prey system,, Math. Biosci., 39 (1978), 1. Google Scholar

[11]

S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators,, SIAM J. Appl. Math., 135 (1978), 617. Google Scholar

[12]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in Biology,, Taiwanese Journal of Mathematics, 9 (2005), 151. Google Scholar

[13]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge,, J. Diff. Eq., 5 (2006), 534. Google Scholar

[14]

J. D. Murray, "Mathematical Biology, I & II," 3rd edition,, Springer-Verlag, (2003). Google Scholar

[15]

A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," 2nd edition,, Springer-Verlag, (2001). Google Scholar

[16]

H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003). Google Scholar

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