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August  2014, 19(6): 1589-1600. doi: 10.3934/dcdsb.2014.19.1589

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, France

2. 

University of Pécs, Department of Mathematics and Informatics, PMMIK, Boszorkány u. 2, Pécs, H-7624, Hungary, Hungary

3. 

University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, F-95000 Cergy-Pontoise, France

Received  February 2013 Revised  November 2013 Published  June 2014

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.
Citation: M. Guedda, R. Kersner, M. Klincsik, E. Logak. Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1589-1600. doi: 10.3934/dcdsb.2014.19.1589
References:
[1]

D. G. Aronson, Density-dependent interaction-diffusion systems,, Dynamics and Modelling of Reactive Systems, (1980), 161. Google Scholar

[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. Google Scholar

[3]

Zs. Biro, Stability of travelling waves for degenerate reaction-diffusion equations of KKP-type,, Adv. Nonlinear Stud., 2 (2002), 357. Google Scholar

[4]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection-Reaction,, Progress in Nonlinear Differential Equations and Their Applications, (2004). doi: 10.1007/978-3-0348-7964-4. Google Scholar

[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. doi: 10.1088/0305-4470/38/15/009. Google Scholar

[6]

M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations,, Math. Biosci., 33 (1977), 35. doi: 10.1016/0025-5564(77)90062-1. Google Scholar

[7]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I : Singular perturbations,, Discrete and Cont. Dynamical Syst., 3 (2003), 79. doi: 10.3934/dcdsb.2003.3.79. Google Scholar

[8]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order non-linear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 135. Google Scholar

[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. Google Scholar

[10]

J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlin. Anal., 27 (1996), 579. doi: 10.1016/0362-546X(95)00221-G. Google Scholar

[11]

J. D. Murray, Mathematical Biology: I. An Introduction,, $3^{rd}$ edition, (2002). Google Scholar

[12]

W. I. Newman, Some exact solution to a nonlinear diffusion problem in population genetics and combustion,, J. Theoret. Biology, 85 (1980), 325. doi: 10.1016/0022-5193(80)90024-7. Google Scholar

[13]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, $2^{nd}$ edition, (2001). doi: 10.1007/978-1-4757-4978-6. Google Scholar

[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. doi: 10.1007/BF03167410. Google Scholar

[15]

P. Rosenau, Reaction and concentration dependent diffusion model,, Physical Review Letters, 88 (2002), 1. doi: 10.1103/PhysRevLett.88.194501. Google Scholar

[16]

M. L. Rosenzweig, Species Diversity in Space and Time,, Cambridge University Press, (2010). doi: 10.1017/CBO9780511623387. Google Scholar

[17]

N. Shigesada and K. Kawasaki, Biological Invasion: Theory and Practice,, Oxford University Press, (1997). Google Scholar

[18]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theor. Biology, 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[19]

A. M. Turing, The chemical basis of morphogenesis,, Phyl. Trans. Roy. Soc. London, 237 (1952), 37. Google Scholar

show all references

References:
[1]

D. G. Aronson, Density-dependent interaction-diffusion systems,, Dynamics and Modelling of Reactive Systems, (1980), 161. Google Scholar

[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. Google Scholar

[3]

Zs. Biro, Stability of travelling waves for degenerate reaction-diffusion equations of KKP-type,, Adv. Nonlinear Stud., 2 (2002), 357. Google Scholar

[4]

B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection-Reaction,, Progress in Nonlinear Differential Equations and Their Applications, (2004). doi: 10.1007/978-3-0348-7964-4. Google Scholar

[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. doi: 10.1088/0305-4470/38/15/009. Google Scholar

[6]

M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations,, Math. Biosci., 33 (1977), 35. doi: 10.1016/0025-5564(77)90062-1. Google Scholar

[7]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I : Singular perturbations,, Discrete and Cont. Dynamical Syst., 3 (2003), 79. doi: 10.3934/dcdsb.2003.3.79. Google Scholar

[8]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order non-linear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 135. Google Scholar

[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. Google Scholar

[10]

J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlin. Anal., 27 (1996), 579. doi: 10.1016/0362-546X(95)00221-G. Google Scholar

[11]

J. D. Murray, Mathematical Biology: I. An Introduction,, $3^{rd}$ edition, (2002). Google Scholar

[12]

W. I. Newman, Some exact solution to a nonlinear diffusion problem in population genetics and combustion,, J. Theoret. Biology, 85 (1980), 325. doi: 10.1016/0022-5193(80)90024-7. Google Scholar

[13]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, $2^{nd}$ edition, (2001). doi: 10.1007/978-1-4757-4978-6. Google Scholar

[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. doi: 10.1007/BF03167410. Google Scholar

[15]

P. Rosenau, Reaction and concentration dependent diffusion model,, Physical Review Letters, 88 (2002), 1. doi: 10.1103/PhysRevLett.88.194501. Google Scholar

[16]

M. L. Rosenzweig, Species Diversity in Space and Time,, Cambridge University Press, (2010). doi: 10.1017/CBO9780511623387. Google Scholar

[17]

N. Shigesada and K. Kawasaki, Biological Invasion: Theory and Practice,, Oxford University Press, (1997). Google Scholar

[18]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theor. Biology, 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[19]

A. M. Turing, The chemical basis of morphogenesis,, Phyl. Trans. Roy. Soc. London, 237 (1952), 37. Google Scholar

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