Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion

M. Guedda; R. Kersner; M. Klincsik; E. Logak

doi:10.3934/dcdsb.2014.19.1589

Article Contents

2014, Volume 19, Issue 6: 1589-1600. Doi: 10.3934/dcdsb.2014.19.1589

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Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion

1.
Université de Picardie, LAMFA, CNRS, UMR 6140, 33, rue Saint-Leu, 80039 Amiens
2.
University of Pécs, Department of Mathematics and Informatics, PMMIK, Boszorkány u. 2, Pécs, H-7624
3.
University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, F-95000 Cergy-Pontoise

Received: January 31, 2013

Revised: October 31, 2013

Published: July 2014

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Abstract

We consider a two-species competition system with nonlinear diffusion and exhibit exact solutions of the system. We first show the existence of spatially stationary solutions that are periodic patterns. In a particular case, we also provide a time-dependent solution that approximates this periodic solution. We also show that the system may sustain unbounded wavefronts above the coexistence equilibrium. In the case of equal intrinsic growth rates, we give a sharp wavefront solution with semi-finite support.

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Mathematics Subject Classification: 35B10, 35B36, 35C05, 35C07, 35K65, 35Q92.

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References

[1]	D. G. Aronson, Density-dependent interaction-diffusion systems, Dynamics and Modelling of Reactive Systems, Academic Press, New York-London, (1980), 161-176.
[2]	Zs. Biro and R. Kersner, On the compactly supported solutions of KPP or Fisher type equations, AMS/IP Studies in Adv. Math., 3 (1997), 129-137.
[3]	Zs. Biro, Stability of travelling waves for degenerate reaction-diffusion equations of KKP-type, Adv. Nonlinear Stud., 2 (2002), 357-371.
[4]	B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection-Reaction, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 2004.doi: 10.1007/978-3-0348-7964-4.
[5]	G. H. Gilding and R. Kersner, A Fisher/KPP-type equation with density-dependent diffusion and convection: travelling-wave solutions, J. Physica A, 38 (2005), 3367-3379.doi: 10.1088/0305-4470/38/15/009.
[6]	M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.doi: 10.1016/0025-5564(77)90062-1.
[7]	Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I : Singular perturbations, Discrete and Cont. Dynamical Syst., Ser. B, 3 (2003), 79-95.doi: 10.3934/dcdsb.2003.3.79.
[8]	A. S. Kalashnikov, Some problems of the qualitative theory of second-order non-linear degenerate parabolic equations, Russian Math. Surveys, 42 (1987), 135-176.
[9]	S. Kamin and P. Rosenau, Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation, Rend. Mat. Acc. Lincei., 15 (2004), 271-280.
[10]	J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlin. Anal., TMA, 27 (1996), 579-587.doi: 10.1016/0362-546X(95)00221-G.
[11]	J. D. Murray, Mathematical Biology: I. An Introduction, $3^{rd}$ edition, Springer Verlag, Berlin, 2002.
[12]	W. I. Newman, Some exact solution to a nonlinear diffusion problem in population genetics and combustion, J. Theoret. Biology, 85 (1980), 325-334.doi: 10.1016/0022-5193(80)90024-7.
[13]	A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, $2^{nd}$ edition, Springer-Verlag, New York, 2001.doi: 10.1007/978-1-4757-4978-6.
[14]	M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. of Industr. Appl. Math., 18 (2001), 657-696.doi: 10.1007/BF03167410.
[15]	P. Rosenau, Reaction and concentration dependent diffusion model, Physical Review Letters, 88 (2002), 194501, 1-4.doi: 10.1103/PhysRevLett.88.194501.
[16]	M. L. Rosenzweig, Species Diversity in Space and Time, Cambridge University Press, 2010.doi: 10.1017/CBO9780511623387.
[17]	N. Shigesada and K. Kawasaki, Biological Invasion: Theory and Practice, Oxford University Press, 1997.
[18]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biology, 79 (1979), 83-99.doi: 10.1016/0022-5193(79)90258-3.
[19]	A. M. Turing, The chemical basis of morphogenesis, Phyl. Trans. Roy. Soc. London, 237 (1952), 37-72.