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

Abstract

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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Further Study on Dynamics for a Fractional‐Order Competitor‐Competitor‐Mutualist Lotka–Volterra System

“…In particular, a great deal of interesting fruits on various dynamical natures of prey-predator systems has sprung up. For instance, Mondal et al [32] derived the condition to ensure the stability behavior for a class of fractional-order prey-predator system; El-Saka et al [33] analyzed the local stability and bifurcation of fractionalorder predator-prey models; Li et al [34] dealt with the dynamical property of the solutions and global asymptotic stability for a class of fractional-order prey-predator system. For more relational publications, one can see [35][36][37].…”