Dynamic Analysis of Stochastic Lotka–Volterra Predator‐Prey Model with Discrete Delays and Feedback Control

A stochastic process depicting the spreading dynamics of illicit drug consumption in a given population forms the crux of this work. A probabilistic cellular automaton (PCA) model is developed to examine the effects of the social interactions between nonusers and drug users. The model, called NERA, comprises four classes of individuals, namely, nonuser (N), experimental user (E), recreational user (R), and addict user (A). The stochastic process evolves in time by local transition rules. By means of dynamical simple mean field approximation, a nonlinear system of differential equations illustrating the dynamics of the PCA is obtained. The existence and uniqueness of a positive solution of the model is established, and the fixed points of the system are sought to perform the stability analysis. Furthermore, a stochastic mean field (SMF) approach to the NERA system is introduced. SMF extends the latter model to integrate the stochastic behaviour of drug consumers in a given environment. The SMF system is shown to exhibit a unique global solution which is stochastically ultimately bounded. Simulations of the cellular automaton and mean field analysis are used to study the evolution of the model. Verification and validation are carried out using data available on the consumption of cannabis in the state of Washington (Darnell and Bitney in I − 502 evaluation and benefit-cost analysis: second required report, Washington state institute for public policy. Technical Report, 2017). These numerical experiments confirm that the NERA model can help in the analysis and quantification of the spatial dynamics of illicit drug usage in a given society and eventually provide insight to policy-makers on different steps to be taken to curb this social epidemic.

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“…Following the procedures given in Guo et al (2014) and Liu and Zhao (2019), we define a non-negative C 4 function V 1 :…”

Section: Existence Of a Global Solution Of System (16)mentioning

confidence: 99%

The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

Ruhomally

Dauhoo

2020

Comp. Appl. Math.

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“…More details on the modified Lotka-Volterra models and prey-predator models can be found in [10][11][12][13][14]. In [15], a modified for Lotka-Volterra model was proposed and studied, and it represents a food chain consisting of three species.…”

Section: Introductionmentioning

confidence: 99%

On the Stability of Four Dimensional Lotka-Volterra Prey-Predator System

Farhan,

Balasim,

Al-Nassir

2023

Iraqi Journal of Science

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The aim of this work is to study a modified version of the four-dimensional Lotka-Volterra model. In this model, all of the four species grow logistically. This model has at most sixteen possible equilibrium points. Five of them always exist without any restriction on the parameters of the model, while the existence of the other points is subject to the fulfillment of some necessary and sufficient conditions. Eight of the points of equilibrium are unstable and the rest are locally asymptotically stable under certain conditions, In addition, a basin of attraction found for each point that can be asymptotically locally stable. Conditions are provided to ensure that all solutions are bounded. Finally, numerical simulations are given to verify and support the obtained theoretical results.

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“…It is very well known that this system has a stable solution in (0, 0) and that it has a nonasymptotic solution and has a limit cycle [6,7]. Also, there exist positive solutions, and there are existence and uniqueness of global positive solutions [8]. is deterministic model, however, does not take into account fluctuations in the environment, which play, in general, an important role in any real biological system, presenting disturbances caused by natural random variations in environmental conditions.…”

Section: Introductionmentioning

confidence: 99%

On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property

Romero-Meléndez¹,

Castillo-FernándezDavid²,

González-SantosLeopoldo³

2021

Complexity

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In this paper, we study some properties of the solutions of a stochastic Lotka–Volterra predator-prey model, namely, the boundedness in the mean of numerical solutions, the strong convergence for this kind of solutions, and the turnpike property of solutions of an optimal control problem in a population modelled by a Lotka–Volterra system with stochastic environmental fluctuations. Even though there are numerous results in the deterministic case, there are few results for the behavior of numerical solutions in a population dynamic with random fluctuations. First, we show, using the Euler–Maruyama scheme, that the boundedness of numerical solutions and the convergence of the scheme are preserved in the stochastic case. Second, we analyze a property of the long-term behavior of a Lotka–Volterra system with stochastic environmental fluctuations known as turnpike property. In optimal control theory, the optimal solutions dwell mostly in the neighborhood of a balanced equilibrium path, corresponding to the optimal steady-state solution. Our study shows, by means of the Stochastic Maximum Principle, that this turnpike property is preserved, when the noise in the system is small. Numerical simulations are implemented to support our results.

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Dynamic Analysis of Stochastic Lotka–Volterra Predator‐Prey Model with Discrete Delays and Feedback Control

Cited by 7 publications

References 41 publications

The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

On the Stability of Four Dimensional Lotka-Volterra Prey-Predator System

On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property

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