The Galton-Watson predator-prey process

1991 ◽  
Vol 28 (1) ◽  
pp. 9-16 ◽  
Author(s):  
John Coffey ◽  
Wolfgang J. Bühler
Keyword(s):  
Discrete Time ◽  
Sufficient Condition ◽  
Positive Probability ◽  
Necessary And Sufficient Condition ◽  
Predator Prey ◽  
Probability Of Survival ◽  
Predator Prey Model ◽  
Prey Model ◽  
Necessary And Sufficient

A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive probability of survival of both populations is given.

The Galton-Watson predator-prey process

1991 ◽  
Vol 28 (01) ◽  
pp. 9-16 ◽  
Author(s):  
John Coffey ◽  
Wolfgang J. Bühler
Keyword(s):  
Discrete Time ◽  
Sufficient Condition ◽  
Positive Probability ◽  
Necessary And Sufficient Condition ◽  
Predator Prey ◽  
Probability Of Survival ◽  
Predator Prey Model ◽  
Prey Model ◽  
Necessary And Sufficient

A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive probability of survival of both populations is given.

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 Positive Solutions for a General Gause-Type Predator-Prey Model with Monotonic Functional Response

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Guohong Zhang ◽  
Xiaoli Wang
Keyword(s):  
Functional Response ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Index Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fixed Point Index Theory ◽  
Local And Global Bifurcations ◽  
Necessary And Sufficient

We study a general Gause-type predator-prey model with monotonic functional response under Dirichlet boundary condition. Necessary and sufficient conditions for the existence and nonexistence of positive solutions for this system are obtained by means of the fixed point index theory. In addition, the local and global bifurcations from a semitrivial state are also investigated on the basis of bifurcation theory. The results indicate diffusion, and functional response does help to create stationary pattern.

scholarly journals Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Jinshu Chen
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Functional Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Convergence Of Operators

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.

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