Local stability analysis on Lotka‐Volterra predator‐prey models with prey refuge and harvesting

2018 ◽  
Vol 41 (17) ◽  
pp. 7711-7732 ◽  
Author(s):  
Christopher Chow ◽  
Marvin Hoti ◽  
Chongming Li ◽  
Kunquan Lan
Keyword(s):  
Stability Analysis ◽  
Local Stability ◽  
Prey Refuge ◽  
Predator Prey ◽  
Local Stability Analysis ◽  
Predator Prey Models

Stability and Bifurcation Analysis of a Delayed Discrete Predator–Prey Model

2018 ◽  
Vol 28 (09) ◽  
pp. 1850116 ◽  
Author(s):  
A. M. Yousef ◽  
S. M. Salman ◽  
A. A. Elsadany
Keyword(s):  
Stability Analysis ◽  
Local Stability ◽  
Chaotic Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Stability Analysis ◽  
Prey Model ◽  
Stability And Bifurcation Analysis ◽  
Different Types ◽  
Two Parameters

A discrete predator–prey model with delayed density dependence in the rate of growth of the prey is considered. In particular, we analyze the model presented by Kot [2005] which consists of three coupled difference equations and contains two parameters. Existence and local stability analysis of fixed points of the model are addressed. The normal form technique and perturbation method are applied to the different types of bifurcations that exist in the model being investigated. It is proved that the existence of transcritical and Neimark–Sacker bifurcations can occur in the model. In addition, the chaotic behavior of the model in the sense of Marotto is proved. To verify the results obtained analytically, we perform numerical simulations which also explore further the richer dynamics of the model.

scholarly journals Local Stability Analysis On Lotka-Volterra Predator-Prey Models With Prey Refuge And Harvesting

2021 ◽  
Author(s):  
Christopher Chow
Keyword(s):  
Local Stability ◽  
Positive Equilibrium ◽  
Stable Node ◽  
Prey Refuge ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models ◽  
Positive Boundary

We propose a predator-prey model by incorporating a constant harvesting rate into a Lotka-Volterra predator-prey model with prey refuge which was studied recently. All the positive equilibria and the local stability of the proposed model are studied and analyzed by sorting out the intervals of the parameters involved in the model. These intervals of the parameters exhibit the effects on the dynamical behaviors of prey and predators. The emphasis is put on the ranges of the prey refuge constant and harvesting rate. We show that the model has two positive boundary equilibria and one equilibrium. By using the qualitative theory for planar systems, we show that the two positive boundary equilibria can be saddles, saddle-nodes, topological saddles or stable or unstable nodes, and the interior positive equilibrium is locally asymptotically stable. Under suitable restrictions on the parameters, we prove that the positive interior equilibrium is a stable node.

scholarly journals Local Stability Analysis On Lotka-Volterra Predator-Prey Models With Prey Refuge And Harvesting

2021 ◽  
Author(s):  
Christopher Chow
Keyword(s):  
Local Stability ◽  
Positive Equilibrium ◽  
Stable Node ◽  
Prey Refuge ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models ◽  
Positive Boundary

We propose a predator-prey model by incorporating a constant harvesting rate into a Lotka-Volterra predator-prey model with prey refuge which was studied recently. All the positive equilibria and the local stability of the proposed model are studied and analyzed by sorting out the intervals of the parameters involved in the model. These intervals of the parameters exhibit the effects on the dynamical behaviors of prey and predators. The emphasis is put on the ranges of the prey refuge constant and harvesting rate. We show that the model has two positive boundary equilibria and one equilibrium. By using the qualitative theory for planar systems, we show that the two positive boundary equilibria can be saddles, saddle-nodes, topological saddles or stable or unstable nodes, and the interior positive equilibrium is locally asymptotically stable. Under suitable restrictions on the parameters, we prove that the positive interior equilibrium is a stable node.

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