A class of predator-prey model with cross-diffusion and Michaelis--Menten harvest item was investigated. Firstly, the local structure of the non-constant steady-state solution of the model was studied by using the local bifurcation theory. Secondly, the global existence of steady-state solution was discussed according to the global bifurcation theory and the Leray-Schauder degree theory. Finally, the conclusions are confirmed by numerical simulation.

Topics
Computer simulation, Phase space methods, Population ecology
1.
Huang
 
J C
,
Xiao
 
D M.
 
Analyses of bifurcations and stability in a predator-prey system with Holling type-IV functional response
[J].
Acta Mathematicae Applicatae Sinica
,
2004
,
20
(
1
):
167
–
178
.
https://doi.org/10.1007/s10255-004-0159-x
Google Scholar
Crossref
Search ADS
 
2.
Sokol
 
W
,
Howell
 
J A.
 
Kinetics of phenol oxidation by washed cells
[J].
Biotechnology & Bioengineering
,
1981
,
23
(
9
):
2039
–
2049
.
https://doi.org/10.1002/bit.260230909
Google Scholar
Crossref
Search ADS
 
3.
Dan
 
Meng
,
Yanling
 
li
.
A Predator-Prey Model with Modified Leslie-Type Functional Response
[J].
Basic Sciences Journal of Textile Universities
,
2016
,
29
(
1
):
11
–
17
.
Google Scholar
 
4.
Meihua
 
Wei
,
Jinyong
 
Chang
,
Lan
 
Qi
,
Qiaowei
 
Zhang
.
Model Generation of Nonconstant Steady State Solutions for n-Dimensional Glycolysis Model
[J].
Applied Mathematics and Mechanics
,
2014
,
35
(
8
):
930
–
938
.
Google Scholar
 
5.
Conghui
 
Zhang
,
Zhiguo
 
Wang
,
Yanling
 
Li
.
Existence and stability of solutions for a predator-prey model
[J].
Journal of Shaanxi Normal University
,
2017
,
45
(
1
):
6
–
19
.
Google Scholar
 
6.
Pang
 
P Y H
,
Wang
 
M.
 
Qualitative analysis of a ratio-dependent predator-prey system with diffusion
[J].
Proc. Roy. Soc. Edinburgh A.
,
2003
,
133
:
919
–
942
.
https://doi.org/10.1017/S0308210500002742
Google Scholar
Crossref
Search ADS
 
7.
Zeng
 
X.
 
Non-constant positive steady states of a prey-predator system with cross diffusions
[J].
J. Math. Anal. Appl.
,
2007
,
332
:
989
–
1009
.
https://doi.org/10.1016/j.jmaa.2006.10.075
Google Scholar
Crossref
Search ADS
 
8.
Wonlyul
 
K
,
Kimun
 
R.
 
Coexistence states of a nonlinear Lotka-Volterra type prerdator-prey model with cross- diffusion
[J].
Nonlinear Anal.
,
2009
,
71
:
1109
–
1115
.
https://doi.org/10.1016/j.na.2009.01.097
Google Scholar
Crossref
Search ADS
 
9.
Ye
 
QiXiao
,
Li
 
Zhengyuan
,
Wang
 
mingxin
, et al. Introduction to Reaction-diffusion Equations [M].
Beijing
:
Science Press
,
1990
.
10.
Wu
 
J H
,
Guo
 
G H
,
Ma
 
C.
 
Nonlinear diffusion effect on bifurcation structures for a preydator-prey model
[J].
Differential Integral Equations
,
2011
,
24
(
1-2
):
177
–
198
.
Google Scholar
 
This content is only available via PDF.
© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.