scholarly journals Bifurcation structure of nonconstant positive steady states for a diffusive predator-prey model

2019 ◽  
Vol 16 (5) ◽  
pp. 3988-4006
Author(s):  
Dongfu Tong ◽  
◽  
Yongli Cai ◽  
Bingxian Wang ◽  
Weiming Wang
Keyword(s):  
Steady States ◽  
Bifurcation Structure ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Steady States

Asymptotic behaviour of positive steady states to a predator—prey model

2006 ◽  
Vol 136 (4) ◽  
pp. 759-778 ◽  
Author(s):  
Yihong Du ◽  
Mingxin Wang
Keyword(s):  
Asymptotic Behaviour ◽  
Spatial Effect ◽  
Steady States ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Steady States ◽  
Predator Prey Models ◽  
Limiting Case

To understand the heterogeneous spatial effect on predator–prey models, we study the behaviour of the positive steady states of a predator–prey model as certain parameters are small or large. We compare the case when the model has a spatial degeneracy with the case when it does not have such a degeneracy. Our results show that the effect of the degeneracy can be clearly observed in one limiting case, but not in the others.

scholarly journals Existence of positive steady states for a predator-prey model with diffusion

2013 ◽  
Vol 12 (5) ◽  
pp. 2189-2201 ◽  
Author(s):  
Wenshu Zhou ◽  
Hongxing Zhao ◽  
Xiaodan Wei ◽  
Guokai Xu
Keyword(s):  
Steady States ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Steady States

scholarly journals Positive steady states of a density-dependent predator-prey model with diffusion

2017 ◽  
Vol 22 (5) ◽  
pp. 16-16
Author(s):  
Kaigang Huang ◽  
Yongli Cai ◽  
Feng Rao ◽  
Shengmao Fu ◽  
Weiming Wang
Keyword(s):  
Steady States ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Steady States

BIFURCATION AND CHAOS IN A DISCRETE PREDATOR–PREY MODEL WITH HOLLING TYPE-III FUNCTIONAL RESPONSE AND HARVESTING EFFECT

2021 ◽  
pp. 1-28
Author(s):  
ANURAJ SINGH ◽  
PREETI DEOLIA
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Flip Bifurcation ◽  
Bifurcation Structure ◽  
Bifurcation And Chaos ◽  
Predator Prey ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey Model ◽  
Wide Range

In this paper, we study a discrete-time predator–prey model with Holling type-III functional response and harvesting in both species. A detailed bifurcation analysis, depending on some parameter, reveals a rich bifurcation structure, including transcritical bifurcation, flip bifurcation and Neimark–Sacker bifurcation. However, some sufficient conditions to guarantee the global asymptotic stability of the trivial fixed point and unique positive fixed points are also given. The existence of chaos in the sense of Li–Yorke has been established for the discrete system. The extensive numerical simulations are given to support the analytical findings. The system exhibits flip bifurcation and Neimark–Sacker bifurcation followed by wide range of dense chaos. Further, the chaos occurred in the system can be controlled by choosing suitable value of prey harvesting.

scholarly journals Drivers of pattern formation in a predator–prey model with defense in fearful prey

2021 ◽  
Author(s):  
Purnedu Mishra ◽  
Barkha Tiwari
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Direct Method ◽  
Turing Instability ◽  
Steady States ◽  
Predator Prey ◽  
Random Movement ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fear And Anxiety

AbstractExistence of predator is routinely used to induce fear and anxiety in prey which is well known for shaping entire ecosystem. Fear of predation restricts the development of prey and promotes inducible defense in prey communities for the survival. Motivated by this fact, we investigate the dynamics of a Leslie–Gower predator prey model with group defense in a fearful prey. We obtain conditions under which system possess unique global-in-time solutions and determine all the biological feasible states of the system. Local stability is analyzed by linearization technique and Lyapunov direct method has been applied for global stability analysis of steady states. We show the occurrence of Hopf bifurcation and its direction at the vicinity of coexisting equilibrium point for temporal model. We consider random movement in species and establish conditions for the stability of the system in the presence of diffusion. We derive conditions for existence of non-constant steady states and Turing instability at coexisting population state of diffusive system. Incorporating indirect prey taxis with the assumption that the predator moves toward the smell of prey rather than random movement gives rise to taxis-driven inhomogeneous Hopf bifurcation in predator–prey model. Numerical simulations are intended to demonstrate the role of biological as well as physical drivers on pattern formation that go beyond analytical conclusions.

Stationary patterns for a Leslie–Gower type three species model with diffusion

2014 ◽  
Vol 07 (06) ◽  
pp. 1450069 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Shiping Lu
Keyword(s):  
Global Stability ◽  
Positive Constant ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Steady States ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Positive Steady States ◽  
The Stability

In this paper, a diffusive three species predator–prey model with two Leslie–Gower terms is considered. The stability of the unique positive constant equilibrium for the reaction–diffusion system is obtained. Sufficient conditions are derived for the global stability of the positive constant equilibrium. In particular, we establish the existence and non-existence of non-constant positive steady states of this system. The results indicate that the large diffusivity is helpful for the appearance of the non-constant positive steady states (stationary patterns).

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