Existence theory and approximate solution to prey–predator coupled system involving nonsingular kernel type derivative

Alqudah, Manar A.; Abdeljawad, Thabet; Eiman, None; Shah, Kamal; Jarad, Fahd; Al‐Mdallal, Qasem M.

doi:10.1186/s13662-020-02970-w

Cited by 10 publications

(5 citation statements)

References 25 publications

(19 reference statements)

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“…In the fields of fractional calculus, existence and uniqueness are some of the most important qualitative studies of the fractional differential equation. Nevertheless, several researchers in the field of fractional calculus investigated the existence and uniqueness of solutions of fractional differential equation with different types of fractional operators, see among them the recent papers [9] , [41] , [3] , [35] , [5] , [4] , [1] . Hence, in this section, we demonstrate the existence and uniqueness results of the proposed model (2.6) by using the techniques of Schaefer’s and Banach fixed point theorems.…”

Section: Existence and Uniqueness Resultsmentioning

confidence: 99%

An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC-fractional operator

Ahmed

Goufo

Yusuf

et al. 2021

Alexandria Engineering Journal

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The whole world is still shaken by the new corona virus and many countries are starting opting for the lockdown again after the first wave that already killed thousands of people. New observations also show that the virus spreads quickly during the cold period closer to winter season. On the other side, the number of new infections decreases considerably during hot period closer to summer time. The geographic structure of our planet is such that when some countries (in a hemisphere) are in their winter season, others in the other hemisphere are in their summer season. However, we have observed in the world some countries undertaking national lockdown during their summer time, which result in their economy to be hugely hit. Other factors, beside the lockdown, have also impacted negatively the socio-economic situation in affected countries. These include, among others, the human immunodeficiency virus (HIV) susceptible to combine to the new corona virus. The new corona virus is indeed recent and many of its effect and impact on the society are still unknown and are still to be uncovered. Hence we use here the of Atangana-Baleanu fractional derivative to mathematically express and analyses a model of HIV disease combined with COVID-19 to assess the pandemic situation in many countries affected, such as South Africa, United Kingdom (UK), China, Spain, United States of America (USA), and Italy. A way to achieve that is to perform stability and bifurcation analysis. It is also possible to investigate in which conditions the combined model contains a forward and a backward bifurcation. Moreover, utilizing the techniques of Schaefer and Banach fixed point theorems, existence and uniqueness of solutions of the generalized fractional model were presented. Also, the Atangana-Baleanu fractional (generalized) HIV-COVID-19 con-infection model is solved numerically via well-known and effective numerical scheme and a predicted prevalence for the COVID-19 is provided. The global trend shown by the numerical simulation proves that the disease will stabilize at a later stage when adequate measures are taken.

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Section: Existence and Uniqueness Resultsmentioning

confidence: 99%

An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC-fractional operator

Ahmed

Goufo

Yusuf

et al. 2021

Alexandria Engineering Journal

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“…As a result, we may claim that fractional order dynamical systems provide temporal responses with super-fast passage and super-slow evolution towards the steady-state, which are phenomena that are difficult to achieve with classical order models. In the future, we will modify the model (4) using the definition of the Caputo-Fabrizio derivative which is based on nonsingular kernel, then study the existence and uniqueness by using fixed point theory, and finally compute the approximate solution using Laplace transform as in [23] and then compare the results.…”

Section: Discussionmentioning

confidence: 99%

Existence and Uniqueness of Caputo Fractional Predator-Prey Model of Holling-Type II with Numerical Simulations

Themairi

Alqudah

2021

Mathematical Problems in Engineering

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We suggested a new mathematical model for three prey-predator species, predator is considered to be divided into two compartments, infected and susceptible predators, as well as the prey and susceptible population based on Holling-type II with harvesting. We considered the model in Caputo fractional order derivative to have significant consequences in real life since the population of prey create memory and learn from their experience of escaping and resisting any threat. The existence, uniqueness, and boundedness of the solution and the equilibrium points for the considered model are studied. Numerical simulations using Euler’s method are discussed to interpret the applicability of the considered model.

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“…We established several fndings regarding the existence and uniqueness of solutions for system (8) using fxed point theory fndings, such as the Schauder and Banach fxed point theorems. We follow [43] in this section. Te following compensation is used for the right-hand sides of system (8):…”

Section: Existence Uniqueness and Non-negativitymentioning

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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Existence theory and approximate solution to prey–predator coupled system involving nonsingular kernel type derivative

Cited by 10 publications

References 25 publications

An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC-fractional operator

An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC-fractional operator

Existence and Uniqueness of Caputo Fractional Predator-Prey Model of Holling-Type II with Numerical Simulations

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

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