Global Dynamics in a Beddington–DeAngelis Prey–Predator Model with Density Dependent Death Rate of Predator

2019 ◽  
Author(s):  
Koushik Garain ◽  
Udai Kumar ◽  
Partha Sarathi Mandal
Keyword(s):  
Death Rate ◽  
Global Dynamics ◽  
Density Dependent ◽  
Predator Model

Rich Global Dynamics in a Prey–Predator Model with Allee Effect and Density Dependent Death Rate of Predator

2015 ◽  
Vol 25 (03) ◽  
pp. 1530007 ◽  
Author(s):  
Moitri Sen ◽  
Malay Banerjee
Keyword(s):  
Death Rate ◽  
Allee Effect ◽  
Growth Function ◽  
Global Dynamics ◽  
Density Dependent ◽  
Strong Allee Effect ◽  
Local And Global Bifurcations ◽  
Predator Model ◽  
Type Ii Functional Response ◽  
Comprehensive Study

In this work we have considered a prey–predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation respectively.

Bifurcation Analysis of a Prey–Predator Model with Beddington–DeAngelis Type Functional Response and Allee Effect in Prey

2020 ◽  
Vol 30 (16) ◽  
pp. 2050238
Author(s):  
Koushik Garain ◽  
Partha Sarathi Mandal
Keyword(s):  
Functional Response ◽  
Death Rate ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Growth Function ◽  
Global Dynamics ◽  
Density Dependent ◽  
Existence And Stability ◽  
Predator Model ◽  
The Impact

The article aims to study a prey–predator model which includes the Allee effect phenomena in prey growth function, density dependent death rate for predators and Beddington–DeAngelis type functional response. We notice the changes in the existence and stability of the equilibrium points due to the Allee effect. To investigate the complete global dynamics of the Allee model, we present here a two-parametric bifurcation diagram which describes the effect of density dependent death rate parameter of predator on dynamical changes of the system. We have also analyzed all possible local and global bifurcations that the system could go through, namely transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Finally, the impact of the Allee effect in the considered system is investigated by comparing the dynamics of both the systems with and without Allee effect.

scholarly journals Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Dahlia Khaled Bahlool ◽  
Huda Abdul Satar ◽  
Hiba Abdullah Ibrahim
Keyword(s):  
Equilibrium Points ◽  
Global Dynamics ◽  
Persistence Conditions ◽  
Local Bifurcation Analysis ◽  
Uniqueness Of The Solution ◽  
Harvesting Process ◽  
Predator Model ◽  
The Stability ◽  
Predator System ◽  
Type Ii Functional Response

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

scholarly journals Méthode d'agrégation des variables appliquée à la dynamique des populations

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Pierre Auger ◽  
Abderrahim El Abdllaoui ◽  
Rachid Mchich
Keyword(s):  
Stable Limit Cycle ◽  
Stable Limit ◽  
Dynamique Des Populations ◽  
Globally Asymptotically Stable ◽  
Density Dependent ◽  
Migration Rates ◽  
First Case ◽  
International Audience ◽  
Predator Model ◽  
Predator System

International audience We present the method of aggregation of variables in the case of ordinary differential equations. We apply the method to a prey - predator model in a multi - patchy environment. In this model, preys can go to a refuge and therefore escape to predation. The predator must return regularly to his terrier to feed his progeny. We study the effect of density-dependent migration on the global stability of the prey-predator system. We consider constant migration rates, but also density-dependent migration rates. We prove that the positif equilibrium is globally asymptotically stable in the first case, and that its stability changes in the second case. The fact that we consider density-dependent migration rates leads to the existence of a stable limit cycle via a Hopf bifurcation. Nous présentons les grandes lignes de laméthode d'agrégation des variables dans les systèmes d'équations différentielles ordinaires. Nous appliquons laméthode à un modèle proie-prédateur spatialisé. Dans ce modèle, les proies peuvent échapper à la prédation en se réfugiant sur un site. Le prédateur doit aussi retourner régulièrement dans son terrier pour nourrir sa progéniture. Nous étudions les effets de migration dépendant de la densité des populations sur la stabilité globale du système proie-prédateur. Nous considérons des taux de migration constants, puis densité-dépendants. Dans le cas de taux constants il existe un équilibre positif toujours stable alors que dans le cas de taux de migration densité-dépendants, il existe un cycle limite stable via une bifurcation de Hopf.

scholarly journals The Effects of Media Coverage on the Dynamics of Disease in Prey-Predator Model

2021 ◽  
pp. 981-996
Author(s):  
Walaa Madhat Alwan ◽  
Huda Abdul Satar
Keyword(s):  
Media Coverage ◽  
Global Dynamics ◽  
Local Bifurcation ◽  
Persistence Conditions ◽  
Proposed Model ◽  
Disease In Predator ◽  
Predator Model ◽  
The Stability ◽  
Effects Of Media ◽  
Disease In Prey

In this paper, an eco-epidemiological model with media coverage effects is established and studied. An -type of disease in predator is considered.  All the properties of the solution of the proposed model are discussed. An application to the stability theory was carried out to investigate the local as well as global stability of the system. The persistence conditions of the model are determined. The occurrence of local bifurcation in the model is studied. Further investigation of the global dynamics of the model is achieved through using a numerical simulation.

scholarly journals Poincaré Map Approach to Global Dynamics of the Integrated Pest Management Prey-Predator Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zhenzhen Shi ◽  
Qingjian Li ◽  
Weiming Li ◽  
Huidong Cheng
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Poincaré Map ◽  
Global Dynamics ◽  
Poincare Map ◽  
Existence And Stability ◽  
Sufficient And Necessary Conditions ◽  
Predator Model ◽  
Order One Periodic Solution

An integrated pest management prey-predator model with ratio-dependent and impulsive feedback control is investigated in this paper. Firstly, we determine the Poincaré map which is defined on the phase set and discuss its main properties including monotonicity, continuity, and discontinuity. Secondly, the existence and stability of the boundary order-one periodic solution are proved by the method of Poincaré map. According to the Poincaré map and related differential equation theory, the conditions of the existence and global stability of the order-one periodic solution are obtained when ΦyA<yA, and we prove the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution when ΦyA>yA. Furthermore, we prove the existence of the order-kk≥2 periodic solution under certain conditions. Finally, we verify the main results by numerical simulation.

Dynamics of a Discrete Nonlinear Prey–Predator Model

2020 ◽  
Vol 30 (04) ◽  
pp. 2050055
Author(s):  
JinRong Wang ◽  
Michal Fečkan
Keyword(s):  
Fixed Points ◽  
Asymptotic Properties ◽  
Invariant Sets ◽  
Global Dynamics ◽  
Type Result ◽  
Numerical Computations ◽  
Predator Model

The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results are obtained on asymptotic properties of fixed points and Neimark–Sacker bifurcations. Then global dynamics is studied by finding invariant sets. Also some achievements on attraction are shown for certain trivial invariant sets including a shadowing type result, and estimates on boundedness of orbits. Some ergodic type results are also derived. Certain issues are extended to systems with impulses by showing the influence of impulses on dynamics. Moreover, backward dynamics is investigated as well. All these results are derived analytically and numerical computations are presented to support them.

scholarly journals Effects of Additional Foods to Predators on Nutrient-Consumer-Predator Food Chain Model

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Banshidhar Sahoo
Keyword(s):  
Death Rate ◽  
Local Stability ◽  
Equilibrium Points ◽  
Chain Model ◽  
Additional Food ◽  
Boundedness Property ◽  
Interior Equilibrium ◽  
Predator Model ◽  
Food Chain Model ◽  
Unstable Dynamics

We have proposed a nutrient-consumer-predator model with additional food to predator, at variable nutrient enrichment levels. The boundedness property and the conditions for local stability of boundary and interior equilibrium points of the system are derived. Bifurcation analysis is done with respect to quality and quantity of additional food and consumer’s death rate for the model. The system has stable as well as unstable dynamics depending on supply of additional food to predator. This model shows that supply of additional food plays an important role in the biological controllability of the system.

scholarly journals Global Dynamics of a Prey-Predator Model with Antipredator Behavior and Two Predators

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Chentong Li ◽  
Yingying Zhang ◽  
Jinhu Xu ◽  
Yicang Zhou
Keyword(s):  
Antipredator Behavior ◽  
Periodic Oscillation ◽  
Global Dynamics ◽  
Boundedness Of Solutions ◽  
New Model ◽  
Basic Properties ◽  
Predator Model ◽  
Existence Of Equilibria ◽  
Insight Into ◽  
Positivity And Boundedness

In this work, we establish a new model of one prey and two predators with antipredator behavior. The basic properties on the positivity and boundedness of solutions and the existence of equilibria are established. Through analyzing the global dynamics, we find that there exist some values of the parameters such that one of the predators can be driven into being extinct by another. Furthermore, the coexistence of the three species is investigated which shows that the antipredator behavior makes the species coexist by periodic oscillation. The results give a new insight into the influence of antipredator behavior in nature selection.

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