scholarly journals The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model

2020 ◽  
Vol 2 (2) ◽  
pp. 87-96
Author(s):  
Hasan S. Panigoro ◽  
Emli Rahmi ◽  
Novianita Achmad ◽  
Sri Lestari Mahmud
Keyword(s):  
Numerical Simulations ◽  
Autonomous System ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Dimensional System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results

In this paper, the influence of additive Allee effect in prey and periodic harvesting in predator to the dynamics of the Leslie-Gower predator-prey model is proposed. We first simplify the model to the non-dimensional system by scaling the variable and transform the model into an autonomous system. If the effect Allee is weak, we have at most two equilibrium points, else if the Allee effect is strong, at most four equilibrium points may exist. Furthermore, the behavior of the system around equilibrium points is investigated. In the end, we give numerical simulations to support theoretical results.

A Markov-switching predator–prey model with Allee effect for preys

2020 ◽  
Vol 13 (03) ◽  
pp. 2050018
Author(s):  
Xiaoxia Guo ◽  
Zhiming Guo
Keyword(s):  
Numerical Simulations ◽  
Allee Effect ◽  
Global Attractivity ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results ◽  
Prey Populations

This paper concerns with a Markov-switching predator–prey model with Allee effect for preys. The conditions under which extinction of predator and prey populations occur have been established. Sufficient conditions are also given for persistence and global attractivity in mean. In addition, stability in the distribution of the system under consideration is derived under some assumptions. Finally, numerical simulations are carried out to illustrate theoretical results.

scholarly journals Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1280
Author(s):  
Liyun Lai ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

A Michaelis–Menten Predator–Prey Model with Strong Allee Effect and Disease in Prey Incorporating Prey Refuge

2018 ◽  
Vol 28 (06) ◽  
pp. 1850073 ◽  
Author(s):  
Sangeeta Saha ◽  
Alakes Maiti ◽  
G. P. Samanta
Keyword(s):  
Allee Effect ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Susceptible Population ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Lasalle Theorem ◽  
Transcritical Bifurcations

Here, we have proposed a predator–prey model with Michaelis–Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh–Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson–Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.

scholarly journals Bifurkasi Hopf pada Model Lotka-Volterra Orde-Fraksional dengan Efek Allee Aditif pada Predator

2020 ◽  
Vol 1 (1) ◽  
pp. 16-24
Author(s):  
Hasan S. Panigoro ◽  
Dian Savitri
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Biological Condition ◽  
Existence Of Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Caputo Fractional Order Derivative ◽  
Theoretical Results ◽  
Extinction Point

This article aims to study the dynamics of a Lotka-Volterra predator-prey model with Allee effect in predator. According to the biological condition, the Caputo fractional-order derivative is chosen as its operator. The analysis is started by identifying the existence, uniqueness, and non-negativity of the solution. Furthermore, the existence of equilibrium points and their stability is investigated. It has shown that the model has two equilibrium points namely both populations extinction point which is always a saddle point, and a conditionally stable co-existence point, both locally and globally. One of the interesting phenomena is the occurrence of Hopf bifurcation driven by the order of derivative. Finally, the numerical simulations are given to validate previous theoretical results.

scholarly journals Predator-Prey Model with Refuge, Fear and Z-Control

2021 ◽  
Vol 26 (1) ◽  
pp. 40-57
Author(s):  
Ibrahim M. Elmojtaba ◽  
Kawkab Al-Amri ◽  
Qamar J.A. Khan
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Predator Population ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Long Run ◽  
Constant State

In this paper, we consider a predator-prey model incorporating fear and refuge.  Our results show that the predator-free equilibrium is globally asymptotically stable if the ratio between the death rate of predators and the conversion rate of prey into predator is greater than the value of prey in refuge at equilibrium.  We also show that the co-existence equilibrium points are locally asymptotically stable if the value of the prey outside refuge is greater than half of the carrying capacity.  Numerical simulations show that when the intensity of fear increases, the fraction of the prey inside refuge increases; however, it has no effect on the fraction of the prey outside refuge, in the long run. It is shown that the intensity of fear harms predator population size. Numerical simulations show that the application of Z-control will force the system to reach any desired state within a limited time, whether the desired state is a constant state or a periodic state. Our results show that when the refuge size is taken to be a non-constant function of the prey outside refuge, the systems change their dynamics. Namely, when it is a linear function or an exponential function, the system always reaches the predator-free equilibrium.  However, when it is taken as a logistic equation, the system reaches the co-existence equilibrium after long term oscillations.

scholarly journals Spatiotemporal Complexity of a Leslie-Gower Predator-Prey Model with the Weak Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Yongli Cai ◽  
Caidi Zhao ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Allee Effect ◽  
Steady State Solution ◽  
Turing Instability ◽  
Dimensional System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonexistence And Existence ◽  
The Impact

We investigate a diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey subject to the zero-flux boundary conditions. Some results of solutions to this model and its corresponding steady-state problem are shown. More precisely, we give the stability of the positive constant steady-state solution, the refineda prioriestimates of positive solution, and the nonexistence and existence of the positive nonconstant solutions. We carry out the analytical study for two-dimensional system in detail and find out the certain conditions for Turing instability. Furthermore, we perform numerical simulations and show that the model exhibits a transition from stripe-spot mixtures growth to isolated spots and also to stripes. These results show that the impact of the Allee effect essentially increases the model spatiotemporal complexity.

scholarly journals Extinction in a generalized Lotka-Volterra predator-prey model

2000 ◽  
Vol 13 (3) ◽  
pp. 287-297 ◽  
Author(s):  
Azmy S. Ackleh ◽  
David F. Marshall ◽  
Henry E. Heatherly
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Mortality Rates ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results

In this paper we discuss the asymptotic behavior of a predator-prey model with distributed growth and mortality rates. We exhibit simple criteria on the parameters which guarantee that all subpopulations but one predator-prey pair are driven to extinction as t→∞. Finally, we present numerical simulations to illustrate the theoretical results.

scholarly journals Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 785
Author(s):  
Hasan S. Panigoro ◽  
Agus Suryanto ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Isnani Darti
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Equilibrium Point ◽  
Power Law ◽  
Equilibrium Points ◽  
Essential Difference ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

scholarly journals The dynamics of a Leslie type predator–prey model with fear and Allee effect

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. Vinoth ◽  
R. Sivasamy ◽  
K. Sathiyanathan ◽  
Bundit Unyong ◽  
Grienggrai Rajchakit ◽  
...  
Keyword(s):  
Local Stability ◽  
Allee Effect ◽  
Jacobian Matrix ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Lyapunov Coefficient ◽  
Points Of View ◽  
Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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