Bifurcation analysis of a modified Leslie-Gower predator-prey model with hunting cooperation and favourable additional food for predator (original) (raw)
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Ain Shams Engineering Journal, 2018
In the present article, a modified Leslie-Gower predator-prey model with double Allee effect, affecting the prey population, is proposed and analyzed. We have considered both strong and weak Allee effects separately. The equilibrium points of the system and their local stability have been studied. It is shown that the dynamics of the system are highly dependent upon the initial conditions. The local bifurcations (Hopf, saddle-node, Bogdanov-Takens) have been investigated by considering sufficient parameter(s) as the bifurcation parameter(s). The local existence of the limit cycle emerging through Hopf bifurcation and its stability is studied by means of the first Lyapunov coefficient. The numerical simulations have been done in support of the analytical findings. The result shows the emergence of homoclinic loop. The possible phase portraits and parametric diagrams have been depicted.
Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II
2019
The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (without) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings.
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International journal of applied and computational mathematics, 2017
The present paper analyzes a predator-prey model in which the predator is provided with additional food and subjected to Allee effect. The conditions for the existence of equilibrium points and their local stability have been investigated. Under certain conditions, it is found that the solutions depend highly on the initial values. The existence of the bifurcations such as Bogdanov-Takens, Hopf-Andronov, Transcritical and Saddle-node, for the system have been shown. Further, the appearance of homoclinic loop, emerging through Hopf-bifurcation has been shown through numerical simulation. Numerical simulations have been proposed to confirm the analytical results.
Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey
Frontiers in Applied Mathematics and Statistics, 2022
In this manuscript, we study a Leslie–Gower predator-prey model with a hyperbolic functional response and weak Allee effect. The results reveal that the model supports coexistence and oscillation of both predator and prey populations. We also identify regions in the parameter space in which different kinds of bifurcations, such as saddle-node bifurcations, Hopf bifurcations and Bogdanov–Takens bifurcations.
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Qualitative Analysis of a Leslie-Gower Predator-Prey System with Nonlinear Harvesting in Predator
International Journal of Engineering Mathematics, 2016
This paper deals with the study of the stability and the bifurcation analysis of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting. It is shown that the proposed model exhibits the bistability for certain parametric conditions. Dulac’s criterion has been adopted to obtain the sufficient conditions for the global stability of the model. Moreover, the model exhibits different kinds of bifurcations (e.g., the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov-Takens bifurcation, and the homoclinic bifurcation) whenever the values of parameters of the model vary. The analytical findings and numerical simulations reveal far richer and complex dynamics in comparison to the models with no harvesting and with constant-yield predator harvesting.
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Applied Mathematics and Computation, 2002
In this paper, Hopf bifurcation is demonstrated in an interacting one-predator-twoprey model with harvesting of the predator at a constant rate. Here harvest rate is used as a control parameter. It is found that periodic solutions arise from stable stationary states when the harvest rate exceeds a certain limit. The stability of these periodic solutions is investigated with the variation of this control parameter. The approach is analytic in nature and the normal form analysis of the model is performed. Ó
Bifurcations of a predator-prey system with weak Allee effects
We formulate and study a predator-prey model with nonmonotonic functional response type and weak Allee effects on the prey, which extends the system studied by Ruan and Xiao in [Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001Math. 61 ( ), no. 4, 1445Math. 61 ( -1472 but containing an extra term describing weak Allee effects on the prey. We obtain the global dynamics of the model by combining the global qualitative and bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and the subcritical Hopf bifurcations, and the homoclinic bifurcation, as the values of parameters vary. In the generic case, the model has the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation).