• Previous Article
    Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors
  • DCDS-B Home
  • This Issue
  • Next Article
    Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint
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 and 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, 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.

show all references

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.

[1]

Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108

[2]

Yu Ichida. Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022114

[3]

Bingtuan Li. Some remarks on traveling wave solutions in competition models. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 389-399. doi: 10.3934/dcdsb.2009.12.389

[4]

Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227

[5]

Genni Fragnelli, Paolo Nistri, Duccio Papini. Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3831-3834. doi: 10.3934/dcds.2013.33.3831

[6]

Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35

[7]

Guo Lin, Shuxia Pan. Periodic traveling wave solutions of periodic integrodifference systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049

[8]

Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841

[9]

Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417

[10]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265

[11]

Bouthaina Abdelhedi. Existence of periodic solutions of a system of damped wave equations in thin domains. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 767-800. doi: 10.3934/dcds.2008.20.767

[12]

Rui Huang, Yifu Wang, Yuanyuan Ke. Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1005-1014. doi: 10.3934/dcdsb.2005.5.1005

[13]

Xiaoxiao Zheng, Hui Wu. Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components. Mathematical Foundations of Computing, 2020, 3 (1) : 11-24. doi: 10.3934/mfc.2020002

[14]

Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815

[15]

Aijun Zhang. Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel. Discrete and Continuous Dynamical Systems - S, 2022, 15 (10) : 3079-3095. doi: 10.3934/dcdss.2022061

[16]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[17]

Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173

[18]

Roger Lui, Hirokazu Ninomiya. Traveling wave solutions for a bacteria system with density-suppressed motility. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 931-940. doi: 10.3934/dcdsb.2018213

[19]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171

[20]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]