Global stability of a Leslie-Gower-type fractional order tritrophic food chain model

Mondal, Shuvojit; Bairagi, Nandadulal; 'Guerekata, Gaston M.

doi:10.7153/fdc-2019-09-11

Cited by 5 publications

(4 citation statements)

References 49 publications

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“…Pemodelan matematika tentang masalah ekologi berkembang pesat setelah Lotka (1925) dan Volterra (1931) mengajukan model predator-prey yang dikenal sebagai model Lotka-Volterra. Model ekologi klasik ini menjadi cikal bakal diajukannya model predator-prey lain, seperti model Rosenzweig-MacArthur [1][2][3] dan model Leslie-Gower [4][5][6][7][8][9][10][11][12][13][14] yang berfokus pada sistem dengan dua spesies, yaitu satu prey dan satu predator. Lebih lanjut, agar dapat memberikan pemahaman yang lebih baik terhadap dinamika sistem ekologi yang kompleks maka kajian tentang model tiga spesies dipandang menarik untuk dilakukan.…”

Section: Pendahuluanunclassified

Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda

Savitri

Panigoro

2020

Jambura J. Biomath.

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This article discusses the one-prey, one-predator, and the super predator model with different types of functional response. The rate of prey consumption by the predator follows Holling type I functional response and the rate of predator consumption by the super predator follows Holling type II functional response. We identify the existence and stability of critical points and obtain that the extinction of all population points is always unstable, and the other two are conditionally stable i.e., the super predator extinction point and the co-existence point. Furthermore, we give the numerical simulations to describe the bifurcation diagram and phase portraits of the model. The bifurcation diagram is obtained by varying the parameter of the conversion rate of predator biomass into a new super-predator which gives forward and Hopf bifurcation. The forward bifurcation occurs around the super predator extinction point while Hopf bifurcation occurs around the interior of the model. Based on the terms of existence and numerical simulation, we confirm that the conversion rate of predator biomass into a new super-predator controls the dynamics of the system and maintains the existence of predator.

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Section: Pendahuluanunclassified

Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda

Savitri

Panigoro

2020

Jambura J. Biomath.

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“…Anderson and May [2] were the pioneers for investigating the invasion, persistence, and spread of infectious diseases by formulating an eco-epidemiological prey-predator model. At recent times, many researchers have proposed and studied epidemic and eco-epidemiological models [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] . For example, Shaikh et al [6] have investigated the dynamics of an eco-epidemiological system with disease in competitive prey species.…”

Section: Introductionmentioning

confidence: 99%

“…From the eigenvalues of the Jacobean matrix ( 4 ), which are given in(15), we can see that 3 is real, 1 , 2 are purely imaginary if and only if there is a critical value of = = 11 . Thus, at = , we have 1 =√ temporal system(3)undergoes a Hopf Bifurcation around the Disease Free Equilibrium Point at a certain critical value of the parameter = .…”

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confidence: 99%

Stability and bifurcation analysis of a diffusive modified Leslie-Gower prey-predator model with prey infection and Beddington DeAngelis functional response

Melese

Feyissa

2021

Heliyon

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In this paper, we present and analyze a spatio-temporal eco-epidemiological model of a prey predator system where prey population is infected with a disease. The prey population is divided into two categories, susceptible and infected. The susceptible prey is assumed to grow logistically in the absence of disease and predation. The predator population follows the modified Leslie-Gower dynamics and predates both the susceptible and infected prey population with Beddington-DeAngelis and Holling type II functional responses, respectively. The boundedness of solutions, existence and stability conditions of the biologically feasible equilibrium points of the system both in the absence and presence of diffusion are discussed. It is found that the disease can be eradicated if the rate of transmission of the disease is less than the death rate of the infected prey. The system undergoes a transcritical and pitchfork bifurcation at the Disease Free Equilibrium Point when the prey infection rate crosses a certain threshold value. Hopf bifurcation analysis is also carried out in the absence of diffusion, which shows the existence of periodic solution of the system around the Disease Free Equilibrium Point and the Endemic Equilibrium Point when the ratio of the rate of intrinsic growth rate of predator to prey crosses a certain threshold value. The system remains locally asymptotically stable in the presence of diffusion around the disease free equilibrium point once it is locally asymptotically stable in the absence of diffusion. The Analytical results show that the effect of diffusion can be managed by appropriately choosing conditions on the parameters of the local interaction of the system. Numerical simulations are carried out to validate our analytical findings.

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“…Secondly and more importantly, fractional order derivatives not only depend on the local conditions but also on the history of the function [14] and, therefore, fractional derivatives have become an efficient tool for those systems, where the consideration of memory or hereditary properties of the function is essential to represent the system, e.g., in the case of biological systems. In the last two decades, fractional order calculus has found its many applications in biological sciences [15,16,17,18,19,20,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

Study of a discretized fractional-order eco-epidemiological model with prey infection

Mondal¹,

Cao²,

Bairagi³

2020

Fractional Differential Calculus

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In this paper, an attempt is made to understand the dynamics of a three-dimensional discrete fractional-order eco-epidemiological model with Holling type II functional response. We first discretize a fractional-order predator-prey-parasite system with piecewise constant arguments and then explore the system dynamics. Analytical conditions for the local stability of different fixed points have been determined using the Jury criterion. Several examples are given to substantiate the analytical results. Our analysis shows that stability of the discrete fractional order system strongly depends on the step-size and the fractional order. More specifically, the critical value of the step-size, where the switching of stability occurs, decreases as the order of the fractional derivative decreases. Simulation results explore that the discrete fractional-order system may also exhibit complex dynamics, like chaos, for higher step-size.

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Global stability of a Leslie-Gower-type fractional order tritrophic food chain model

Cited by 5 publications

References 49 publications

Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda

Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda

Stability and bifurcation analysis of a diffusive modified Leslie-Gower prey-predator model with prey infection and Beddington DeAngelis functional response

Study of a discretized fractional-order eco-epidemiological model with prey infection

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