Acta mathematica scientia,Series A ›› 2019, Vol. 39 ›› Issue (3): 545-559.

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Positive Solutions of a Predator-Prey Model with Cross Diffusion

Hailong Yuan1,2,*(),Yuping Wang1,Yanling Li3   

  1. 1 School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021
    2 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
    3 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119
  • Received:2018-04-20 Online:2019-06-26 Published:2019-06-27
  • Contact: Hailong Yuan
  • Supported by:
    the NSFC(11271236);the NSFC(61672021);the NSFC(61872227);the Natural Science Foundation of Shaanxi University of Science and Technology(2017BJ-44)


A predator-prey model with cross diffusion under homogeneous Dirichlet boundary conditions is investigated. Firstly, the existence of positive solutions can be established by the Leray-Schauder degree theory. Secondly, we consider that the existence of positive solutions of the regular perturbation system and the singular perturbation system when m=β is sufficiently large, respectively, and moreover, we show that the positive solutions of the singular perturbation system will blow up along the continuum at a* by the bifurcation theory. Finally, the multiplicity results of positive solutions of system is also considered.

Key words: Cross diffusion, Bifurcation, Positive solutions

CLC Number: 

  • O175.26