
Mathematical Biosciences and Engineering, 2019, 16(6): 67536768. doi: 10.3934/mbe.2019337
Research article Special Issues
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Positive steady states of a ratiodependent predatorprey system with crossdiffusion
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
Received: , Accepted: , Published:
Special Issues: Recent Advances in Mathematical Population Dynamics
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 MichaelisMenten type ratiodependent predatorprey system, SIAM J. Appl. Math., 67 (2007), 1453–1464.
4. Y. Kuang and E. Beretta, Global qualitative analysis of a ratiodependent predatorprey system, J. Math. Biol., 36 (1998), 389–406.
5. L. Zhang, J. Liu and M. Banerjee, Hopf and steady state bifurcation analysis in a ratiodependent predatorprey model, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 52–73.
6. M. X. Wang, Stationary patterns for a preypredator model with preydependent 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 ratiodependent predatorprey system with diffusion, Proc. Roy. Soc. Edinburgh A, 133 (2003), 919–942.
8. R. Z. Yang, M. Liu and C. R. Zhang, A delayeddiffusive predatorprey model with a ratiodependent 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 predatorprey system with ratiodependent functional response, Internat. J. Bifur. Chaos., 27 (2017), 1750222–21.
10. L. Chen and A. Jüngel, Analysis of a parabolic crossdiffusion population model without selfdiffusion, J. Differ. Equations, 224 (2006), 39–59.
11. Y. H. Du and S. B. Hsu, A diffusive predatorprey model in heterogeneous environment, J. Differ. Equations, 203 (2004), 331–364.
12. A. Madzvamuse, H. S. Ndakwo and R. Barreira, Stability analysis of reactiondiffusion models on evolving domian: the effects of crossdiffusion, Discrete Cont. Dyn. Syst., 36 (2017), 2133–2170.
13. W. Ko and K. Ryu, Nonconstant positive steadystate of a diffusive predatorprey 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 ndimensional diffusive predatorprey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288312.
15. Z. L. Shen and J. Wei, Hopf bifurcation analysis in a diffusive predatorprey 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 predatorprey system incorporating a prey refuge, Math. Biosci. Eng., 10 (2013), 979–996.
17. E. AvilaVales, G. GarcaAlmeida and E. RiveroEsquivel, Bifurcation and spatiotemporal patterns in a Bazykin predatorprey model with self and cross diffusion and BeddingtonDeAngelis response, Discrete Cont. Dyn. Syst. Ser. B, 22 (2017), 717–740.
18. R. Peng and M. X. Wang, Positive steady states of the HollingTanner preypredator model with diffusion, Proc. Roy. Soc. Edinburgh A, 135 (2005), 149–164.
19. P. Y. H. Pang and M. X. Wang, Nconstant positive steady states of a predatorprey system with nonmonotonic 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 predatorprey system with modified HollingTanner functional response, Nonlinear Anal. Real World Appl., 11 (2010), 3711–3721.
21. L. J. Hei and Y. Yu, Nonconstant 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 predatorprey system with a degeneracy, Acta Math. Sci., 36 (2016), 537–548.
24. W. M. Ni, Crossdiffusion and their spikelayer steady states, Notices Amer. Math. Soc. 45 (1998), 9–18.
25. X. Z. Zeng, Nonconstant positive steady states of a preypredator system with crossdiffusions, J. Math. Anal. Appl., 332 (2007), 989–1009.
26. G. P. Hu and X. L. Li, Turing patterns of a predatorprey model with a modified LeslieGower term and crossdiffusion, Int. J. Biomath., 5 (2012), 1250060–1250082.
27. Y. Lou and W. M. Ni, Diffusion, selfdiffusion and crossdiffusion, J. Differ. Equations, 131 (1996), 79–131.
28. C. R. Tian, Z. Ling and Z.G. Lin, Turing pattern formation in a predatorpreymutualist 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: SpringerVerlag, (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)