Spatial patterns of the Holling–Tanner predator–prey model with nonlinear diffusion effects

Abstract

In this article, we study the Holling–Tanner predator–prey model with nonlinear diffusion terms under homogeneous Neumann boundary condition. The nonlinear diffusion terms here mean that the prey runs away from the predator, and the predator chases the prey. Nonexistence and existence of nonconstant positive steady states are obtained, which reveal that cross-diffusion can create spatial patterns even when the random diffusion fails to do so. Moreover, asymptotic behaviour of positive solutions as the cross-diffusion tends to ∞ is shown.

Keywords:

• cross-diffusion
• nonconstant positive solution
• predator–prey model
• spatial pattern

AMS Subject Classifications::

• 35J55
• 35K57
• 92D25

Acknowledgements

The first author was supported by FRFCU (lzujbky-2011-148) and the second author was supported by NSF of China (No. 11031003) and FRFCU (lzujbky-2011-k27).