Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

Tuladhar, Rohisha; Santamarı́a, Fidel; Stamova, Ivanka

doi:10.3390/e22090970

Cited by 9 publications

(7 citation statements)

References 63 publications

(154 reference statements)

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“…With the use of the Lotka-Volterra prey-predator model and logistic growth of prey species, a discrete fractional-order model was introduced in [30]. Several stability characteristics of the states, including Mittag-Lefer stability, practical stability, and stability with regard to sets, were addressed via impulsive control techniques in [31]. For a good survey on the LV model, see [32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect

El-Mesady

Bazighifan

Aracı

2023

Journal of Mathematics

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For over a hundred years, the model of prey–predator has been handled extensively. It has been proved that Lotka–Volterra (LV) models are not stable from all previous studies. In these models, there is divergent extinction of one species or cyclic oscillation. There is no solution for asymptotic stability for the prey–predator models. In this paper, a Caputo fractional-order one prey-two predators’ model with an immigration effect is investigated. Then, the effect of adding a small immigration term to the prey or predators’ populations is studied. The extension is both the inclusion of a second predator and the use of Caputo derivatives. Although LV systems have been extensively investigated, there are no previous studies on our proposed nonlinear modifications. The Grünwald–Letnikov method was applied to find numerical solutions for the proposed model to validate our findings. Several different cases were studied to reach confirmed conclusions. The effect of the variation of fractional-order derivative is handled in this study. From all the results, the effect of adding the small immigration terms is positive for stability. Hence, it can be concluded that the proposed small immigration terms can stabilize the natural populations of the one prey-two predators’ model. An optimal control strategy for the proposed model is shown at the end of the paper.

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Section: Introductionmentioning

confidence: 99%

Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect

El-Mesady

Bazighifan

Aracı

2023

Journal of Mathematics

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“…Many authors answered the question of how short-term (impulsive) perturbations can be used to create impulsive control strategies for the qualitative properties of such systems [27][28][29][30][31][32][33]. There are also several existing results on impulsive fractional-order Lotka-Volterra models, although their number is very small [25,34] The most popular definitions for fractional derivatives are the Caputo, Riemann-Liouville and Grunwald-Letnikov types [8,10,11]. However, the application of these derivatives to the qualitative analysis of fractional-order models is related to some limitations due to the absence of a simple chain rule formula, locality and singularity properties.…”

Section: Introductionmentioning

confidence: 99%

Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis

et al. 2023

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In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population dynamics’ models is applied and fixed moments impulsive perturbations are considered. The combined concept of practical stability with respect to manifolds is adapted to the introduced model. Sufficient conditions for boundedness and generalized practical stability of the solutions are obtained by using an analogue of the Lyapunov function method. The uncertain case is also studied. Examples are given to demonstrate the effectiveness of the established results.

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“…In addition, [9], in which delay-dependent stability switches in fractional differential equations were studied and obtained results were illustrated via a fractional Lotka-Volterra population model. Moreover, [10] as a biological fractional n-species delayed cooperation model of Lotka-Volterra type was presented. Examples to recent studies on numerical solutions of model problems in fractional structure with both stiff and nonstiff components and the leading-edge model problem can be given to [11,12], respectively.…”

Section: Introductionmentioning

confidence: 99%

Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation

2021

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The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with Oh2+τ2 of accuracy on the grids. Here, h and 32h are the step sizes in space variables x1 and x2, respectively and τ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for r=ωτh2≤37, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order Oh2+τ2 is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures.

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Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

Cited by 9 publications

References 63 publications

Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect

Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect

Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis

Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation

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