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 and 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), 805-822.

[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), 144-187.

[3]

R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction Diffusion Equations," John Wiley and Sons, Chichester, 2003.

[4]

K. S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.

[5]

W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains, in "Structured Population Models in Biology and Epidemiology" (eds. P. Magal et S. Ruan), Lecture Notes in Mathematics (Mathematical Biosciences Subseries), vol. 1936, Springer, (2008), 115-164.

[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), 65-82.

[7]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.

[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), 293-320.

[9]

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

[10]

S. B. Hsu, On global stability for a predator-prey system, Math. Biosci., 39 (1978), 1-10.

[11]

S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators, SIAM J. Appl. Math., 135 (1978), 617-625.

[12]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in Biology, Taiwanese Journal of Mathematics, 9 (2005), 151-173.

[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), 534-550.

[14]

J. D. Murray, "Mathematical Biology, I & II," 3rd edition, Springer-Verlag, Berlin, 2003.

[15]

A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," 2nd edition, Springer-Verlag, New York, 2001.

[16]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.

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), 805-822.

[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), 144-187.

[3]

R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction Diffusion Equations," John Wiley and Sons, Chichester, 2003.

[4]

K. S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.

[5]

W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains, in "Structured Population Models in Biology and Epidemiology" (eds. P. Magal et S. Ruan), Lecture Notes in Mathematics (Mathematical Biosciences Subseries), vol. 1936, Springer, (2008), 115-164.

[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), 65-82.

[7]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.

[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), 293-320.

[9]

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

[10]

S. B. Hsu, On global stability for a predator-prey system, Math. Biosci., 39 (1978), 1-10.

[11]

S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predators, SIAM J. Appl. Math., 135 (1978), 617-625.

[12]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in Biology, Taiwanese Journal of Mathematics, 9 (2005), 151-173.

[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), 534-550.

[14]

J. D. Murray, "Mathematical Biology, I & II," 3rd edition, Springer-Verlag, Berlin, 2003.

[15]

A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," 2nd edition, Springer-Verlag, New York, 2001.

[16]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.

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