scholarly journals Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey–predator model

2008 ◽  
Vol 21 (11) ◽  
pp. 1215-1220 ◽  
Author(s):  
Mingxin Wang ◽  
Peter Y.H. Pang
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Steady States ◽  
Positive Steady States ◽  
Ratio Dependent ◽  
Predator Model

Positive steady states of the Holling–Tanner prey–predator model with diffusion

2005 ◽  
Vol 135 (1) ◽  
pp. 149-164 ◽  
Author(s):  
Rui Peng ◽  
Mingxin Wang
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Steady States ◽  
Homogeneous Neumann Boundary Condition ◽  
Positive Steady States ◽  
Predator Model

This paper is concerned with the Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We obtain the existence and non-existence of positive non-constant steady states.

GLOBAL ASYMPTOTIC STABILITY FOR A TWO-SPECIES DISCRETE RATIO-DEPENDENT PREDATOR–PREY SYSTEM

2013 ◽  
Vol 06 (01) ◽  
pp. 1250064 ◽  
Author(s):  
XIANGLAI ZHUO
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
Analysis Of Stability ◽  
Appl Math

The dynamical behaviors of a two-species discrete ratio-dependent predator–prey system are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, we also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator–prey system, Indian J. Pure Appl. Math.42(1) (2011) 1–26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator–prey system with delays, Appl. Math. Comput.153 (2004) 337–351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin–Ayala competition predator–prey discrete system, Appl. Math. Comput.190 (2007) 500–509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.

scholarly journals Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

2019 ◽  
Vol 16 (6) ◽  
pp. 6753-6768
Author(s):  
Xiaoling Li ◽  
◽  
Guangping Hu ◽  
Xianpei Li ◽  
Zhaosheng Feng ◽  
...  
Keyword(s):  
Steady States ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System ◽  
Positive Steady States ◽  
Ratio Dependent

Qualitative Analysis of a Ratio-Dependent Chemostat Model with Holling-(n+1) Type Functional Response

2013 ◽  
Vol 641-642 ◽  
pp. 947-950
Author(s):  
Qing Lai Dong ◽  
Ming Juan Sun
Keyword(s):  
Asymptotic Stability ◽  
Qualitative Analysis ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Monod Model ◽  
Chemostat Model ◽  
Dependent Model ◽  
Ratio Dependent ◽  
Lasalle Invariant Principle ◽  
Invariant Principle

In this paper, the ratio-dependent chemostat model with Holling-(n+1) type functional response is considered. The model develops the Monod model and the ratio-dependent model. By use of the Poincar -Bendixson theory we prove the existence of limit cycle. Detailed qualitative analysis about the global asymptotic stability of its equilibria is carried out by using the Lyapunov-LaSalle invariant principle and the method of Dulac criterion.

scholarly journals STATIONARY PATTERNS OF A RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH CROSS-DIFFUSION

2011 ◽  
Vol 16 (3) ◽  
pp. 461-474 ◽  
Author(s):  
Yu-Xia Wang ◽  
Wan-Tong Li ◽  
Hong-Bo Shi
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Steady States ◽  
Flux Boundary Condition ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System ◽  
Positive Steady States ◽  
Ratio Dependent ◽  
The Cross

This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross-diffusion in a bounded domain with no flux boundary condition. We establish the existence and non-existence of non-constant positive steady states (patterns). In particular, we show that under certain hypotheses, the cross-diffusion can create stationary patterns even though the corresponding model without cross-diffusion fails.

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