scholarly journals Complexity of a Discrete-Time Predator-Prey Model Involving Prey Refuge Proportional to Predator

2022 ◽  
Vol 4 (1) ◽  
pp. 50-63
Author(s):  
P. K. Santra ◽  
Hasan S. Panigoro ◽  
G. S. Mahapatra
Keyword(s):  
Time Series ◽  
Discrete Time ◽  
Local Stability ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Predator Density ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

In this paper, a discrete-time predator-prey model involving prey refuge proportional to predator density is studied. It is assumed that the rate at which prey moves to the refuge is proportional to the predator density. The fixed points, their local stability, and the existence of Neimark-Sacker bifurcation are investigated. At last, the numerical simulations consisting of bifurcation diagrams, phase portraits, and time-series are given to support analytical findings. The occurrence of chaotic solutions are also presented by showing the Lyapunov exponent while some parameters are varied.

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.

scholarly journals The dynamics of a Leslie type predator–prey model with fear and Allee effect

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. Vinoth ◽  
R. Sivasamy ◽  
K. Sathiyanathan ◽  
Bundit Unyong ◽  
Grienggrai Rajchakit ◽  
...  
Keyword(s):  
Local Stability ◽  
Allee Effect ◽  
Jacobian Matrix ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Lyapunov Coefficient ◽  
Points Of View ◽  
Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

Global analysis of a Holling type II predator–prey model with a constant prey refuge

2013 ◽  
Vol 76 (1) ◽  
pp. 635-647 ◽  
Author(s):  
Guangyao Tang ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Global Analysis ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

Dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang
Keyword(s):  
Lyapunov Function ◽  
Iterative Method ◽  
Functional Response ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.

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