Skip to main content

Coexistence Solutions for a Periodic Competition Model with Singular–Degenerate Diffusion

  • Yifu Wang (a1), Jingxue Yin (a2) and Yuanyuan Ke (a3)

We investigate a system of singular–degenerate parabolic equations with non-local terms, which can be regarded as a spatially heterogeneous competition model of Lotka–Volterra type. Applying the Leray–Schauder fixed-point theorem, we establish the existence of coexistence periodic solutions to the problem, which, together with the existing literature, gives a complete picture for such a system for all parameters.

Corresponding author
*Corresponding author.
Hide All
1. Ahmad, S. and Lazer, A., Asymptotic behavior of solutions of periodic competition diffusion systems, Nonlin. Analysis TMA 13 (1989), 263284.
2. Allegretto, W., Fragnelli, G., Nistri, P. and Papini, D., Coexistence and optimal control problems for a degenerate predator–prey model, J. Math. Analysis Applic. 378 (2011), 528540.
3. Cantrell, R. S., Cosner, C. and Lou, Y., Advection-mediated coexistence of competing species, Proc. R. Soc. Edinb. A137 (2007), 497518.
4. Cirmi, G. and Porzio, M., L -solutions for some nonlinear degenerate elliptic and parabolic equations, Annali Mat. Pura Appl. 169 (1995), 6786.
5. Conti, M. and Felli, V., Coexistence and segregation for strongly competing species in special domains, Interfaces Free Bound. 10 (2008), 173195.
6. Delgado, M. and Suárez, A., On the existence of dead cores for degenerate Lotka–Volterra models, Proc. R. Soc. Edinb. A130 (2000), 743766.
7. Du, Y., Positive periodic solutions of a competitor–competitor–mutualist model, Diff. Integ. Eqns 9 (1996), 10431066.
8. Eilbeck, J. C., Furter, J. and López-Gómez, J., Coexistence in the competetion model with diffusion, J. Diff. Eqns 107 (1994), 96139.
9. Fragnelli, G., Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Analysis Applic. 367 (2010), 204228.
10. Fragnelli, G., Nistri, P. and Papini, D., Positive periodic solutions and optimal control for a distributed biological model of two interacting species, Nonlin. Analysis RWA 12 (2011), 14101428.
11. Fragnelli, G., Nistri, P. and Papini, D., Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms, Discrete Contin. Dynam. Syst. A31 (2011), 3564.
12. Fragnelli, G., Mugnai, D., Nistri, P. and Papini, D., Non-trivial non-negative periodic solutions of a system of singular–degenerate parabolic equations with nonlocal terms, Commun. Contemp. Math. 17(2) (2015), DOI: 10.1142/S0219199714500254.
13. Hess, P., Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics, Volume 247 (Longman, New York, 1991).
14. Ivanov, A. V., Hölder estimates for equations of slow and normal diffusion type, J. Math. Sci. 85 (1997), 16401644.
15. Kawohl, B. and Lindqvist, P., Positive eigenfunctions for the p-Laplace operator revisited, Analysis (Munich) 26 (2006), 545550.
16. Pao, C. V., Periodic solutions of parabolic systems with time delays, J. Math. Analysis Applic. 251 (2000), 251263.
17. Porzio, M. and Vespri, V., Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Diff. Eqns 103 (1993), 146178.
18. Suárez, A., Nonnegative solutions for a heterogeneous degenerate competition model, ANZIAM J. 46 (2004), 273297.
19. Sun, J., Yin, J. and Wang, Y., Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation, Nonlin. Analysis TMA 74 (2011), 24152424.
20. Tian, C. and Lin, Z., Asymptotic behavior of solutions of a periodic diffusion system of plankton allelopathy, Nonlin. Analysis RWA 11 (2010), 15811588.
21. Tineo, A., Asymptotic behavior of solutions of a periodic reaction–diffusion system of a competitor–competitor–mutualist model, J. Diff. Eqns 108 (1994), 326341.
22. Wang, Y., Yin, J. and Wu, Z., Periodic solutions of porous medium equations with weakly nonlinear sources, Northeastern Math. J. 16 (2000), 475483.
23. Wang, Y., Yin, J. and Ke, Y., Coexistence solutions for a periodic competition model with nonlinear diffusion, Nonlin. Analysis RWA 14 (2013), 10821091.
24. Wu, Z., Yin, J. and Wang, C., Elliptic and parabolic equations (World Scientific, 2006).
25. Yin, J. and Jin, C., Periodic solutions of the evolutionary p-Laplacian with nonlinear sources, J. Math. Analysis Applic. 368 (2010), 604622.
26. Zeidler, E., Nonlinear function analysis and its applications II/B: nonlinear monotone operators (Springer, 1989).
27. Zhou, Q., Ke, Y., Wang, Y. and Yin, J., Periodic p-Laplacian with nonlocal terms, Nonlin. Analysis TMA 66 (2007), 442453.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 59 *
Loading metrics...

Abstract views

Total abstract views: 193 *
Loading metrics...

* Views captured on Cambridge Core between 15th December 2016 - 16th August 2018. This data will be updated every 24 hours.