PERMANENCE FOR A DELAYED DISCRETE PREDATOR–PREY MODEL WITH PREY DISPERSAL

2009 ◽  
Vol 02 (03) ◽  
pp. 311-320 ◽  
Author(s):  
CHUNQING WU ◽  
JING-AN CUI
Keyword(s):  
Discrete Time ◽  
Single Species ◽  
Time Model ◽  
Predator Prey ◽  
Discrete Time Model ◽  
Predator Prey Model ◽  
Prey Model

First, we obtain a new result for the permanence of a well known delayed discrete-time model of single species. Second, based on this new condition, we discuss the permanence of a delayed discrete-time predator–prey model in which the prey disperses in two patches with biased dispersion. The biological implications of the results are briefly discussed.

Qualitative properties and bifurcations of discrete-time Bazykin–Berezovskaya predator–prey model

2020 ◽  
Vol 13 (06) ◽  
pp. 2050040
Author(s):  
A. A. Elsadany ◽  
Qamar Din ◽  
S. M. Salman
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Initial Conditions ◽  
Qualitative Properties ◽  
Step Size ◽  
Time Model ◽  
Predator Prey ◽  
Discrete Time Model ◽  
Predator Prey Model ◽  
Prey Model

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.

scholarly journals Discrete-Time Predator-Prey Interaction with Selective Harvesting and Predator Self-Limitation

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Zhihua Chen ◽  
Qamar Din ◽  
Muhammad Rafaqat ◽  
Umer Saeed ◽  
Muhammad Bilal Ajaz
Keyword(s):  
Discrete Time ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
State Feedback Control ◽  
Time Model ◽  
Predator Prey ◽  
Selective Harvesting ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Interaction

Selective harvesting plays an important role on the dynamics of predator-prey interaction. On the other hand, the effect of predator self-limitation contributes remarkably to the stabilization of exploitative interactions. Keeping in view the selective harvesting and predator self-limitation, a discrete-time predator-prey model is discussed. Existence of fixed points and their local dynamics is explored for the proposed discrete-time model. Explicit principles of Neimark–Sacker bifurcation and period-doubling bifurcation are used for discussion related to bifurcation analysis in the discrete-time predator-prey system. The control of chaotic behavior is discussed with the help of methods related to state feedback control and parameter perturbation. At the end, some numerical examples are presented for verification and illustration of theoretical findings.

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.

A discrete-time model for cryptic oscillations in predator–prey systems

2009 ◽  
Vol 238 (22) ◽  
pp. 2238-2245 ◽  
Author(s):  
R. Willox ◽  
A. Ramani ◽  
B. Grammaticos
Keyword(s):  
Discrete Time ◽  
Time Model ◽  
Predator Prey ◽  
Discrete Time Model
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