A fractional model in exploring the role of fear in mass mortality of pelicans in the Salton Sea

The fear response is an important anti-predator adaptation that can significantly reduce prey's reproduction by inducing many physiological and psychological changes in the prey. Recent studies in behavioral sciences reveal this fact. Other than terrestrial vertebrates, aquatic vertebrates also exhibit fear responses. Many mathematical studies have been done on the mass mortality of pelican birds in the Salton Sea in Southern California and New Mexico in recent years. Still, no one has investigated the scenario incorporating the fear effect. This work investigates how the mass mortality of pelican birds (predator) gets influenced by the fear response in tilapia fish (prey). For novelty, we investigate a modified fractional-order eco-epidemiological model by incorporating fear response in the prey population in the Caputo-fractional derivative sense. The fundamental mathematical requisites like existence, uniqueness, non-negativity and boundedness of the system's solutions are analyzed. Local and global asymptotic stability of the system at all the possible steady states are investigated. Routh-Hurwitz criterion is used to analyze the local stability of the endemic equilibrium. Fractional Lyapunov functions are constructed to determine the global asymptotic stability of the disease-free and endemic equilibrium. Finally, numerical simulations are conducted with the help of some biologically plausible parameter values to compare the theoretical findings. The order $\alpha$ of the fractional derivative is determined using Matignon's theorem, above which the system loses its stability via a Hopf bifurcation. It is observed that an increase in the fear coefficient above a threshold value destabilizes the system. The mortality rate of the infected prey population has a stabilization effect on the system dynamics that helps in the coexistence of all the populations. Moreover, it can be concluded that the fractional-order may help to control the coexistence of all the populations.

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Dynamic Analysis of a Delayed Fractional-Order SIR Model with Saturated Incidence and Treatment Functions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501808 ◽

2018 ◽

Vol 28 (14) ◽

pp. 1850180 ◽

Cited By ~ 46

Author(s):

Xinhe Wang ◽

Zhen Wang ◽

Xia Huang ◽

Yuxia Li

Keyword(s):

Asymptotic Stability ◽

Equilibrium Point ◽

Fractional Order ◽

Global Asymptotic Stability ◽

Sufficient Conditions ◽

Endemic Equilibrium ◽

Local Asymptotic Stability ◽

Proposed Model ◽

Saturated Incidence ◽

Theoretical Results

In this paper, a delayed fractional-order SIR (susceptible, infected, and removed) epidemic model with saturated incidence and treatment functions is presented. Firstly, the non-negativity and boundedness of solutions of the proposed model are proved. Next, some sufficient conditions are established to ensure the local asymptotic stability of the disease-free equilibrium point [Formula: see text] and the endemic equilibrium point [Formula: see text] for any delay. Meanwhile, global asymptotic stability of the endemic equilibrium point [Formula: see text] is investigated by constructing a suitable Lyapunov function. Some sufficient conditions are established for the global asymptotic stability of this endemic equilibrium point. Finally, some numerical simulations are illustrated to verify the correctness of the theoretical results.

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Backward bifurcation in a fractional-order and two-patch model of tuberculosis epidemic with incomplete treatment

International Journal of Biomathematics ◽

10.1142/s1793524521500078 ◽

2020 ◽

pp. 2150007

Author(s):

Mohsen Jafari ◽

Hossein Kheiri ◽

Azizeh Jabbari

Keyword(s):

Asymptotic Stability ◽

Fractional Order ◽

Global Asymptotic Stability ◽

Multiple Equilibria ◽

Backward Bifurcation ◽

Tuberculosis Model ◽

Fractional Differential ◽

Incomplete Treatment ◽

Theoretical Results ◽

Exogenous Reinfection

In this paper, we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals, in which only susceptible individuals can travel freely between the patches. The model has multiple equilibria. We determine conditions that lead to the appearance of a backward bifurcation. The results show that the TB model can have exogenous reinfection among the treated individuals and, at the same time, does not exhibit backward bifurcation. Also, conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained. In case without reinfection, the model has four equilibria. In this case, the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations (FDEs). Numerical simulations confirm the validity of the theoretical results.

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Global asymptotic stability of fractional-order competitive neural networks with multiple time-varying-delay links

Applied Mathematics and Computation ◽

10.1016/j.amc.2020.125498 ◽

2021 ◽

Vol 389 ◽

pp. 125498 ◽

Cited By ~ 1

Author(s):

Yao Xu ◽

Jintong Yu ◽

Wenxue Li ◽

Jiqiang Feng

Keyword(s):

Neural Networks ◽

Asymptotic Stability ◽

Fractional Order ◽

Global Asymptotic Stability ◽

Multiple Time ◽

Time Varying ◽

Time Varying Delay ◽

Varying Delay

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Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives

Fractal and Fractional ◽

10.3390/fractalfract3030039 ◽

2019 ◽

Vol 3 (3) ◽

pp. 39 ◽

Cited By ~ 3

Author(s):

Ndolane Sene ◽

José Francisco Gómez Aguilar

Keyword(s):

Asymptotic Stability ◽

Fractional Derivative ◽

Analytical Solutions ◽

Global Asymptotic Stability ◽

Fractional Derivatives ◽

Mass Damper ◽

Generalized Fractional Derivatives ◽

Fractional Mass ◽

Mass Spring ◽

Generalized Fractional Derivative

This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville–Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville–Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system.

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Global asymptotic stability and S-asymptotic ω-periodicity of impulsive non-autonomous fractional-order neural networks

Applied Mathematics and Computation ◽

10.1016/j.amc.2021.126459 ◽

2021 ◽

Vol 410 ◽

pp. 126459

Author(s):

Hui Li ◽

YongGui Kao ◽

Ivanka Stamova ◽

Chuntao Shao

Keyword(s):

Neural Networks ◽

Asymptotic Stability ◽

Fractional Order ◽

Global Asymptotic Stability

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Boundedness and Stability Properties of Solutions of Mathematical Model of Measles.

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.52.2021.3367 ◽

2021 ◽

Vol 52 (1) ◽

pp. 91-112

Author(s):

Babatunde Sunday Ogundare ◽

James Akingbade

Keyword(s):

Mathematical Model ◽

Asymptotic Stability ◽

Global Asymptotic Stability ◽

Endemic Equilibrium ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Jacobian Determinant ◽

Stability Of Solutions ◽

Open Population ◽

Disease Free

In this paper, asymptotic stability and global asymptotic stability of solutions to a deterministic and compartmental mathematical model of measles infection is considered using the ideas of the Jacobian determinant as well as the second method of Lyapunov, criteria/conditions that guaranteed asymptotic stability of disease free equilibrium and endemic equilibrium were established. Also the basic reproductive number $R_0$ was obtained. The results in this work compliments existing work and provided further information in controlling the disease in an open population.

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Uniform asymptotic stability of a fractional tuberculosis model

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018015 ◽

2018 ◽

Vol 13 (1) ◽

pp. 9 ◽

Cited By ~ 15

Author(s):

Weronika Wojtak ◽

Cristiana J. Silva ◽

Delfim F.M. Torres

Keyword(s):

Mathematical Model ◽

Asymptotic Stability ◽

Numerical Simulations ◽

Fractional Order ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Uniform Asymptotic Stability ◽

Tuberculosis Model ◽

The Stability

We propose a Caputo type fractional-order mathematical model for the transmission dynamics of tuberculosis (TB). Uniform asymptotic stability of the unique endemic equilibrium of the fractional-order TB model is proved, for anyα∈ (0, 1). Numerical simulations for the stability of the endemic equilibrium are provided.

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