Stochastic Modeling of Three-Species Prey–Predator Model Driven by Lévy Jump with Mixed Holling-II and Beddington–DeAngelis Functional Responses

Abstract

This study examines the dynamics of a stochastic prey–predator model using a functional response function driven by Lévy noise and a mixed Holling-II and Beddington–DeAngelis functional response. The proposed model presents a computational analysis between two prey and one predator population dynamics. First, we show that the suggested model admits a unique positive solution. Second, we prove the extinction of all the studied populations, the extinction of only the predator, and the persistence of all the considered populations under several sufficient conditions. Finally, a special Runge–Kutta method for the stochastic model is illustrated and implemented in order to show the behavior of the two prey and one predator subpopulations.

Threshold properties of a stochastic epidemic model with a variable vaccination rate

“…Proof Since the coefficients of system (2) satisfy the local Lipschitz property, then for any (w(0), x(0), y(0), z(0)) ∈ R 4 + , there is unique local solution (w(t), x(t), y(t), z(t)) ∈ R 4 + which may blow up at time τ e , such that t ∈ [0, τ e ) (see [21]). To show this solution is global, we only need to prove that τ e = ∞ a.s. Let ϵ 0 > 0 such that w(0), x(0), y(0), z(0) > ϵ 0 .…”

“…Mathematical treatment using differential equations is largely used to forecast the spread of infectious diseases. This approach consists of dividing the total population within which the diseases spread, into several compartments [1][2][3][4][5][6][7][8]. Most compartmental epidemic models descend from the works [1][2][3].…”

Dynamics for a Nonlinear Stochastic Cholera Epidemic Model under Lévy Noise

“…The two main methods used in epidemic modelling are stochastic and deterministic. The stochastic modelling technique is frequently preferred in biological system modelling because it can provide a more realistic model than deterministic models, as sources [23][24][25][26] emphasize. Determining a distribution of expected outcomes, such as the number of infected people over time, t, is made easier using stochastic differential equations (SDEs).…”