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

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.

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Impact of Additive Allee Effect on the Dynamics of an Intraguild Predation Model with Specialist Predator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420502399 ◽

2020 ◽

Vol 30 (16) ◽

pp. 2050239

Author(s):

Udai Kumar ◽

Partha Sarathi Mandal

Keyword(s):

System Dynamics ◽

Equilibrium Point ◽

Intraguild Predation ◽

Allee Effect ◽

Equilibrium Points ◽

Dimensional System ◽

Global Dynamics ◽

Interior Equilibrium ◽

Trivial Equilibrium ◽

The Impact

Many important factors in ecological communities are related to the interplay between predation and competition. Intraguild predation or IGP is a mixture of predation and competition which is a very basic three-dimensional system in food webs where two species are related to predator–prey relationship and are also competing for a shared prey. On the other hand, Allee effect is also a very important ecological factor which causes significant changes to the system dynamics. In this work, we consider a intraguild predation model in which predator is specialist, the growth of shared prey population is subjected to additive Allee effect and there is Holling-Type III functional response between IG prey and IG predator. We analyze the impact of Allee effect on the global dynamics of the system with the prior knowledge of the dynamics of the model without Allee effect. Our theoretical and numerical analyses suggest that: (1) Trivial equilibrium point is always locally asymptotically stable and it may be globally stable also. Hence, all the populations may go to extinction depending upon initial conditions; (2) Bistability is observed between unique interior equilibrium point and trivial equilibrium point or between boundary equilibrium point and trivial equilibrium point; (3) Multiple interior equilibrium points exist under certain parameters range. We also provide here a comprehensive study of bifurcation analysis by considering Allee effect as one of the bifurcation parameters. We observed that Allee effect can generate all possible bifurcations such as transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Taken bifurcation and Bautin bifurcation. Finally, we compared our model with the IGP model without Allee effect for better understanding the impact of Allee effect on the system dynamics.

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

Differential Equations and Dynamical Systems ◽

10.1007/s12591-019-00469-9 ◽

2019 ◽

Author(s):

Koushik Garain ◽

Udai Kumar ◽

Partha Sarathi Mandal

Keyword(s):

Death Rate ◽

Global Dynamics ◽

Density Dependent ◽

Predator Model

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

Journal of Applied Mathematics ◽

10.1155/2020/5373817 ◽

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.

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Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421500504 ◽

2021 ◽

Vol 31 (03) ◽

pp. 2150050

Author(s):

Demou Luo ◽

Qiru Wang

Keyword(s):

Growth Rate ◽

Allee Effect ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Nonlinear Growth ◽

Trivial Equilibrium ◽

Existence And Stability ◽

Stable Equilibria ◽

Theoretical Results

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

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

Complexity ◽

10.1155/2020/2376374 ◽

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.

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Modeling the impact of early interventions on the transmission dynamics of coronavirus infection

F1000Research ◽

10.12688/f1000research.54268.1 ◽

2021 ◽

Vol 10 ◽

pp. 518

Author(s):

Christopher Saaha Bornaa ◽

Baba Seidu ◽

Yakubu Ibrahim Seini

Keyword(s):

Death Rate ◽

Deterministic Model ◽

Reproduction Number ◽

Equilibrium Points ◽

Transmission Probability ◽

Transmission Dynamics ◽

Early Interventions ◽

Coronavirus Infection ◽

The Impact ◽

Disease Free

A deterministic model is proposed to describe the transmission dynamics of coronavirus infection with early interventions. Epidemiological studies have employed modeling to unravel knowledge that transformed the lives of families, communities, nations and the entire globe. The study established the stability of both disease free and endemic equilibria. Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. The global stability of the disease-free equilibrium point of the model is established whenever the basic reproduction number R0 is less than or equal to unity. The reproduction number is also shown to be directly related to the transmission probability (β), rate at which latently infected individuals join the infected class (δ) and rate of recruitment (Λ). It is inversely related to natural death rate (μ), rate of early treatment (τ1), rate of hospitalization of infected individuals (θ) and Covid-induced death rate (σ). The analytical results established are confirmed by numerical simulation of the model.

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

ISRN Biomathematics ◽

10.5402/2012/796783 ◽

2012 ◽

Vol 2012 ◽

pp. 1-8 ◽

Cited By ~ 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.

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Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2260-y ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Xinxin Liu ◽

Qingdao Huang

Keyword(s):

Prey Species ◽

Equilibrium Points ◽

Maximum Sustainable Yield ◽

Predator Prey ◽

Prey System ◽

Existence And Stability ◽

Predator Model ◽

Model Equations ◽

The Given

AbstractA new way to study the harvested predator–prey system is presented by analyzing the dynamics of two-prey and one-predator model, in which two teams of prey are interacting with one team of predators and the harvesting functions for two prey species takes different forms. Firstly, we make a brief analysis of the dynamics of the two subsystems which include one predator and one prey, respectively. The positivity and boundedness of the solutions are verified. The existence and stability of seven equilibrium points of the three-species model are further studied. Specifically, the global stability analysis of the coexistence equilibrium point is investigated. The problem of maximum sustainable yield and dynamic optimal yield in finite time is studied. Numerical simulations are performed using MATLAB from four aspects: the role of the carrying capacity of prey, the simulation about the model equations around three biologically significant steady states, simulation for the yield problem of model system, and the comparison between the two forms of harvesting functions. We obtain that the new form of harvesting function is more realistic than the traditional form in the given model, which may be a better reflection of the role of human-made disturbance in the development of the biological system.

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Modeling the impact of early interventions on the transmission dynamics of coronavirus infection

F1000Research ◽

10.12688/f1000research.54268.2 ◽

2021 ◽

Vol 10 ◽

pp. 518

Author(s):

Christopher Saaha Bornaa ◽

Baba Seidu ◽

Yakubu Ibrahim Seini

Keyword(s):

Death Rate ◽

Deterministic Model ◽

Reproduction Number ◽

Equilibrium Points ◽

Transmission Probability ◽

Transmission Dynamics ◽

Early Interventions ◽

Coronavirus Infection ◽

The Impact ◽

Disease Free

A deterministic model is proposed to describe the transmission dynamics of coronavirus infection with early interventions. Epidemiological studies have employed modeling to unravel knowledge that transformed the lives of families, communities, nations and the entire globe. The study established the stability of both disease free and endemic equilibria. Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. The global stability of the disease-free equilibrium point of the model is established whenever the basic reproduction number R0 is less than or equal to unity. The reproduction number is also shown to be directly related to the transmission probability (β), rate at which latently infected individuals join the infected class (δ) and rate of recruitment (Λ). It is inversely related to natural death rate (μ), rate of early treatment (τ1), rate of hospitalization of infected individuals (θ) and Covid-induced death rate (σ). The analytical results established are confirmed by numerical simulation of the model.

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