scholarly journals A Predator–Prey Two-Sex Branching Process

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1408
Author(s):  
Cristina Gutiérrez ◽  
Carmen Minuesa
Keyword(s):  
Stochastic Process ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Predator Population ◽  
Predator Prey ◽  
Control Functions ◽  
Branching Model ◽  
Necessary And Sufficient ◽  
Number Of Individuals ◽  
Random Control

In this paper, we present the first stochastic process to describe the interaction of predator and prey populations with sexual reproduction. Specifically, we introduce a two-type two-sex controlled branching model. This process is a two-type branching process, where the first type corresponds to the predator population and the second one to the prey population. While each population is described via a two-sex branching model, the interaction and survival of both groups is modelled through control functions depending on the current number of individuals of each type in the ecosystem. In view of their potential for the conservation of species, we provide necessary and sufficient conditions for the ultimate extinction of both species, the fixation of one of them and the coexistence of both of them. Moreover, the description of the present predator–prey two-sex branching process on the fixation events can be performed in terms of the behaviour of a one-type two-sex branching process with a random control on the number of individuals, which is also introduced and analysed.

On the class of controlled branching processes with random control functions

2002 ◽  
Vol 39 (4) ◽  
pp. 804-815 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. Del Puerto
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Control Functions ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

The collision branching process

2004 ◽  
Vol 41 (4) ◽  
pp. 1033-1048 ◽  
Author(s):  
Anyue Chen ◽  
Phil Pollett ◽  
Hanjun Zhang ◽  
Junping Li
Keyword(s):  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hitting Times ◽  
Branching Model ◽  
Extinction Probabilities ◽  
Free Case ◽  
Necessary And Sufficient ◽  
Explicit Expressions

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

The collision branching process

2004 ◽  
Vol 41 (04) ◽  
pp. 1033-1048 ◽  
Author(s):  
Anyue Chen ◽  
Phil Pollett ◽  
Hanjun Zhang ◽  
Junping Li
Keyword(s):  
Branching Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hitting Times ◽  
Branching Model ◽  
Extinction Probabilities ◽  
Free Case ◽  
Necessary And Sufficient ◽  
Explicit Expressions

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

On the class of controlled branching processes with random control functions

2002 ◽  
Vol 39 (04) ◽  
pp. 804-815 ◽  
Author(s):  
M. González ◽  
M. Molina ◽  
I. Del Puerto
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Control Functions ◽  
Controlled Branching Processes ◽  
Necessary And Sufficient ◽  
Random Control

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

scholarly journals On a predator-prey system interaction under fluctuating water level with nonselective harvesting

2020 ◽  
Vol 18 (1) ◽  
pp. 458-475
Author(s):  
Na Zhang ◽  
Yonggui Kao ◽  
Fengde Chen ◽  
Binfeng Xie ◽  
Shiyu Li
Keyword(s):  
Water Level ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Optimal Harvesting ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Model Interaction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fluctuating Water Level

Abstract A predator-prey model interaction under fluctuating water level with non-selective harvesting is proposed and studied in this paper. Sufficient conditions for the permanence of two populations and the extinction of predator population are provided. The non-negative equilibrium points are given, and their stability is studied by using the Jacobian matrix. By constructing a suitable Lyapunov function, sufficient conditions that ensure the global stability of the positive equilibrium are obtained. The bionomic equilibrium and the optimal harvesting policy are also presented. Numerical simulations are carried out to show the feasibility of the main results.

A genealogical approach to variable-population-size models in population genetics

1986 ◽  
Vol 23 (02) ◽  
pp. 283-296 ◽  
Author(s):  
Peter Donnelly
Keyword(s):  
Population Genetics ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Process Models ◽  
Moran Model ◽  
Fisher Model ◽  
Variable Population ◽  
Exchangeable Model ◽  
Necessary And Sufficient ◽  
Variable Population Size

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.

scholarly journals Dynamical Behaviour in Two Prey-Predator System with Persistence

2016 ◽  
Vol 16 ◽  
pp. 20-37
Author(s):  
V. Madhusudanan ◽  
S. Vijaya
Keyword(s):  
Limit Cycle ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Behaviour ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Interior Equilibrium ◽  
Trivial Equilibrium ◽  
Necessary And Sufficient

In this work, the dynamical behavior of the system with two preys and one predator population is investigated. The predator exhibits a Holling type II response to one prey which is harvested and a Beddington-DeAngelis functional response to the other prey. The boundedness of the system is analyzed. We examine the occurrence of positive equilibrium points and stability of the system at those points. At trivial equilibrium E0and axial equilibrium (E1); the system is found to be unstable. Also we obtain the necessary and sufficient conditions for existence of interior equilibrium point (E6) and local and global stability of the system at the interior equilibrium (E6): Depending upon the existence of limit cycle, the persistence condition is established for the system. The numerical simulation infer that varying the parameters such as e and λ1it is possible to change the dynamical behavior of the system from limit cycle to stable spiral. It is also observed that the harvesting rate plays a crucial role in stabilizing the system.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

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