Open AccessPositive steady states of a ratio-dependent predator-prey system with cross-diffusion
Author(s) -
Xiao Ling Li,
Guang Ping Hu,
Xian Pei Li,
Zhao Feng
Publication year - 2019
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2019337
Subject(s) - steady state (chemistry) , diffusion , constant (computer programming) , neumann boundary condition , mathematics , harnack's inequality , mathematical analysis , boundary (topology) , boundary value problem , statistical physics , physics , thermodynamics , chemistry , computer science , programming language
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.