Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2 Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA

Special Issues: Recent Advances in Mathematical Population Dynamics

In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack’s inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.
  Figure/Table
  Supplementary
  Article Metrics

References

1. Y. Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, New York: Academic Press, 191 (1993).

2. A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, Singapore: World Scientific, (1998).

3. B. Li and Y. Kuang, Heteroclinic bifurcation in the Michaelis-Menten type ratiodependent predator-prey system, SIAM J. Appl. Math., 67 (2007), 1453–1464.

4. Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389–406.

5. L. Zhang, J. Liu and M. Banerjee, Hopf and steady state bifurcation analysis in a ratio-dependent predatorprey model, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 52–73.

6. M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio- dependent functional responses and diffusion, Phys. D, 196 (2004), 172–192.

7. P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh A, 133 (2003), 919–942.

8. R. Z. Yang, M. Liu and C. R. Zhang, A delayed-diffusive predator-prey model with a ratio-dependent functional response, Commun. Nonlinear Sci. Numer. Simul., 53 (2017), 94–110.

9. X. Jiang, Z. K. She, Z. Feng, et al., Bifurcation analysis of a predator-prey system with ratio-dependent functional response, Internat. J. Bifur. Chaos., 27 (2017), 1750222–21.

10. L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differ. Equations, 224 (2006), 39–59.

11. Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differ. Equations, 203 (2004), 331–364.

12. A. Madzvamuse, H. S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domian: the effects of cross-diffusion, Discrete Cont. Dyn. Syst., 36 (2017), 2133–2170.

13. W. Ko and K. Ryu, Non-constant positive steady-state of a diffusive predator-prey system in ho- mogeneous environment, J. Math. Anal. Appl., 327 (2007), 539–549.

14. Z. Du, Z. Feng and X. Zhang, Traveling wave phenomena of n-dimensional diffusive predator-prey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.

15. Z. L. Shen and J. Wei, Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect, Math. Biosci. Eng., 15 (2018), 693–715.

16. X. Y. Chang and J. Wei, Stability and Hopf bifurcation in a diffusive predator-prey system incor-porating a prey refuge, Math. Biosci. Eng., 10 (2013), 979–996.

17. E. Avila-Vales, G. Garca-Almeida and E. Rivero-Esquivel, Bifurcation and spatiotemporal pat-terns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response, Discrete Cont. Dyn. Syst. Ser. B, 22 (2017), 717–740.

18. R. Peng and M. X. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh A, 135 (2005), 149–164.

19. P. Y. H. Pang and M. X. Wang, N-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. Lond. Math. Soc., 88 (2004), 135–157.

20. H. B. Shi, W. T. Li and G. Lin, Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response, Nonlinear Anal. Real World Appl., 11 (2010), 3711–3721.

21. L. J. Hei and Y. Yu, Non-constant positive steady state of one resource and two consumers model with diffusion, J. Math. Anal. Appl., 339 (2008), 566–581.

22. C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, 72 (1998), 1–27.

23. L. Yang and Y. M. Zhang, Positive steady states and dynamics for a diffusive predator-prey system with a degeneracy, Acta Math. Sci., 36 (2016), 537–548.

24. W. M. Ni, Cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), 9–18.

25. X. Z. Zeng, Non-constant positive steady states of a prey-predator system with cross-diffusions, J. Math. Anal. Appl., 332 (2007), 989–1009.

26. G. P. Hu and X. L. Li, Turing patterns of a predator-prey model with a modified Leslie-Gower term and cross-diffusion, Int. J. Biomath., 5 (2012), 1250060–1250082.

27. Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79–131.

28. C. R. Tian, Z. Ling and Z.G. Lin, Turing pattern formation in a predator-prey-mutualist system, Nonlinear Anal. Real World Appl., 12 (2011), 3224–3237.

29. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, New York: Springer-Verlag, (2001).

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved