Bifurcations in the dynamics of Rosenzweig-MacArthur predator-prey model considering saturated refuge for the preys

Almanza-Vasquez, Edilber; Ortiz-Ortiz, Ruben-Dario; Marin-Ramirez, Ana-Magnolia

doi:10.12988/ams.2015.510640

Cited by 8 publications

(5 citation statements)

References 6 publications

(7 reference statements)

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“…More than 50 years after the model has been proposed, the Rosenzweig-MacArthur predator-prey model [1] has been consistently developed by many scholars to approach the real world phenomena with more realistic mathematical models. The commonsensical modified Rosenzweig-MacArthur models are accomplishable by considering the biological perspectives of ecosystem conditions, for instance the stage structure [2,3], the refuge effect [4][5][6][7][8], the fear effect [9], the Allee effect [10,11], the intraspecific competition [12,13], the cannibalism [14], the infectious disease [15][16][17], and so forth.…”

Section: Introductionmentioning

confidence: 99%

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Panigoro

Suryanto

Kusumawinahyu

et al. 2020

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The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator.

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Section: Introductionmentioning

confidence: 99%

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Panigoro

Suryanto

Kusumawinahyu

et al. 2020

Axioms

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“…Almanza et al (2015) [6] proposes the monotonous growing and bounded function to express the dams that take refuge, where is the maximum capacity of the refuge and is the amount of dams needed to occupy half of the maximum refuge capacity, this function presents a solution alternative to the previous objections and also satisfies that: )…”

Section: The One That Protects a Fixed Amount Of Prey Whose Equation mentioning

confidence: 99%

Rosenzweig-MacArthur model considering the function that protects a fixed amount of prey for population dynamics

Canate-Gonzalez¹,

Fong-Silva²,

Severiche-Sierra³

et al. 2018

ces

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Populations can grow, disappear or remain in time and this depends on their birth rates, mortality, immigration and emigration. For a long time, species have disappeared and others are at the same risk if protective policies are not implemented to guarantee their survival. It is necessary to consider mathematical models that help conserve the environment. The analysis of the Rosenzweig-MacArthur model was performed considering the function that protects a fixed amount of prey. For the analytical study, the qualitative theory of the systems of differential equations is used. Considering the use of shelters by dams with the functions proposed by Maynard-Smith. It is shown that the refuge influences the stability of the only equilibrium point inside the first quadrant and produces important effects in the dynamics of the prey-prey model.

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“…Furthermore, the Rosenzweig-MacArthur model has modified with various factors, e.g. the stage-structure [10][11], the refuge effect [12] [13], the harvesting to one or more population [14] [15]. From several studies described above, no one (1) considering anti-predator behavior factors.…”

Section: Introductionmentioning

confidence: 99%

Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

Djakaria

Gaib

Resmawan

2021

CAUCHY

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This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

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Bifurcations in the dynamics of Rosenzweig-MacArthur predator-prey model considering saturated refuge for the preys

Cited by 8 publications

References 6 publications

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Rosenzweig-MacArthur model considering the function that protects a fixed amount of prey for population dynamics

Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

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