Qualitative analysis of a degenerate fixed point of a discrete predator–prey model with cooperative hunting

2021 ◽  
Author(s):  
Qi Cheng ◽  
Yanlin Zhang ◽  
Shengfu Deng
Keyword(s):  
Fixed Point ◽  
Qualitative Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Cooperative Hunting ◽  
Prey Model

scholarly journals Discrete-Time Predator-Prey Model with Bifurcations and Chaos

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
K. S. Al-Basyouni ◽  
A. Q. Khan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

scholarly journals A predator-prey model with cooperative hunting in the predator and group defense in the prey

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yanfei Du ◽  
Ben Niu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Heteroclinic Orbits ◽  
Initial Population ◽  
Continuous Transition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Cooperative Hunting ◽  
Prey Model ◽  
Existence And Stability ◽  
Group Defense

<p style='text-indent:20px;'>In this paper we propose a predator-prey model with a non-differentiable functional response in which the prey exhibits group defense and the predator exhibits cooperative hunting. There is a separatrix curve dividing the phase portrait. The species with initial population above the separatrix result in extinction of prey in finite time, and the species with initial population below it can coexist, oscillate sustainably or leave the prey surviving only. Detailed bifurcation analysis is carried out to explore the effect of cooperative hunting in the predator and aggregation in the prey on the existence and stability of the coexistence state as well as the dynamics of system. The model undergoes transcritical bifurcation, Hopf bifurcation, homoclinic (heteroclinic) bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation, and through numerical simulations it is found that it possesses rich dynamics including bubble loop of limit cycles, and open ended branch of periodic orbits disappearing through a homoclinic cycle or a loop of heteroclinic orbits. Also, a continuous transition of different types of Hopf branches are investigated which forms a global picture of Hopf bifurcation in the model.</p>

Qualitative analysis of a diffusive Crowley–Martin predator–prey model: the role of nonlinear predator harvesting

2019 ◽  
Vol 98 (2) ◽  
pp. 1169-1189 ◽  
Author(s):  
Vandana Tiwari ◽  
Jai Prakash Tripathi ◽  
Syed Abbas ◽  
Jin-Shan Wang ◽  
Gui-Quan Sun ◽  
...  
Keyword(s):  
Qualitative Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model
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