Dynamic analysis of a fractional-order single-species model with diffusion

Li, Hong-Li; Zhang, Long; Hu, Cheng; Teng, Zhidong; Jiang, Yao‐Lin

doi:10.15388/na.2017.3.2

Cited by 17 publications

(9 citation statements)

References 28 publications

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“…Recently, some authors have investigated the importance of fractional-order differential equations in several biological systems, e.g. ecological system with delay [3,4], control based epidemiological system [5], ecological system with diffusion [6] etc. It also has applications in other fields of science and engineering [7,8,9,10,11].…”

Section: Introductionmentioning

confidence: 99%

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Mondal¹,

Biswas

Bairagi

2020

J. Appl. Math. Comput.

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This paper is focused on local and global stability of a fractional-order predator-prey model with habitat complexity constructed in the Caputo sense and corresponding discrete fractional-order system. Mathematical results like positivity and boundedness of the solutions in fractional-order model is presented. Conditions for local and global stability of different equilibrium points are proved. It is shown that there may exist fractional-order-dependent instability through Hopf bifurcation for both fractional-order and corresponding discrete systems. Dynamics of the discrete fractional-order model is more complex and depends on both step length and fractional-order. It shows Hopf bifurcation, flip bifurcation and more complex dynamics with respect to the step size. Several examples are presented to substantiate the analytical results.This model says that the prey population x grows logistically with intrinsic growth rate r to its carrying capacity K. Predator y captures the prey at a maximum rate α in absence of any habitat complexity (c = 0). In presence of complexity, predation rate decreases to α(1 − c), where the dimensionless parameter c is called the degree or strength of complexity. The value of c ranges from 0 to 1. In particular, c = 0.4 implies that predation rate decreases by 40% due to habitat complexity. If c = 0, i.e. if there is no habitat complexity then the system (1) reduces to well known Rosenzweig-MacArthur model [25]. However, if c = 1 then y → 0 as t → ∞ and the prey population grows logistically to its maximum value K. The parameter θ (0 < θ < 1) is the conversion efficiency, measuring the number of newly born predators for each captured prey and d is the death rate of predator. All parameters are assumed to be positive. For construction and more explanation of the model, readers are referred to [26].Considering the fractional derivatives in the sense of Caputo, we have the following fractional-order model corresponding to the integer order model (1): Existence and uniquenessHere we study the existence and uniqueness of the solution of our system (2). We have the following Lemma due to Li et al [30]. Lemma 2.3 Consider the system . If f (t, x) satisfies the locally Lipschitz condition with respect to x then there exists a unique solution of the above system on [t 0 , ∞) × Ω.We study the existence and uniqueness of the solution of system (2) in the region Ω × [0, T ], where Ω = {(x, y) ∈ ℜ 2 | max{|x|, |y|} ≤ M}, T < ∞ and M is large. Denote X = (x, y),X = (x,ȳ). Consider a mapping H : Ω → ℜ 2 such that H(X) = (H 1 (X), H 2 (X)), where

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Section: Introductionmentioning

confidence: 99%

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Mondal¹,

Biswas

Bairagi

2020

J. Appl. Math. Comput.

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“…13,14 In Gonzalez-Parra et al, 15 the authors proposed a SEIR epidemic fractional differential model to explain and understand the outbreaks of influenza A(H1N1). Li et al 16 investigated a fractional-order single-species model, which is composed of nðn > 2Þ patches connected by diffusion. A homogeneous-mixing population fractional model for human immunodeficiency virus (HIV) transmission was proposed in Huo et al 17 A nonlinear fractional order epidemic model was proposed in Ye et al 18 to investigate the spreading dynamical behavior of the avian influenza.…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis of a fractional-order avian-human influenza epidemic model with logistic growth for avian population

Lin

2020

Journal of Algorithms & Computational Technology

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In this paper, a fractional order avian-human influenza epidemic model with logistic growth for avian population is investigated. The dynamical behavior of this model is discussed. We first establish the existence, uniqueness, non-negativity and positive invariance of the solution. Then we analyze the existence of various equilibrium points, and some sufficient conditions are derived to ensure the global asymptotic stability of the disease free equilibrium point and endemic equilibrium point. Finally, we take some numerical simulations to validate the analytical results.

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“…Such properties are all but neglected by the classical integer-order models. Considering many intriguing results found while working on models of the dynamics of population, they were found to be exclusive to integer-order differential equations [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…Having recognized the need to use fractional-order systems, many biological phenomena and interdisciplinary fields have been precisely and successfully described through models using fractional-order systems. Due to the scarcity of theories needed to analyze the dynamics of fractional-order systems, studies of stability of fractional-order population models can only be the beginning of a vast and doubtlessly fruitful implementation [6,7]. This study is also motivated by Zika epidemic [8].…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis of a fractional SIRS model on homogenous networks

2019

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In this paper, we propose a fractional SIRS model with homogenous networks. The disease-free equilibrium point E 0 is locally and globally asymptotically stable for R 0 < 1 (the disease always disappears), and endemic equilibrium point E 1 is uniquely locally and globally asymptotically stable, but E 0 is unstable for R 0 > 1 (the disease is uniformly persistent). The main results are demonstrated by numerical simulation.

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Dynamic analysis of a fractional-order single-species model with diffusion

Cited by 17 publications

References 28 publications

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Dynamical analysis of a fractional-order avian-human influenza epidemic model with logistic growth for avian population

Dynamical analysis of a fractional SIRS model on homogenous networks

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