Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment

Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method

Stochastic SIRS models play a key role in formulating and analyzing the transmission of infectious diseases. These models reflect the environmental changes of the diseases and their biological mechanisms. Therefore, it is very important to study the uniqueness and existence of the global positive solution to investigate the asymptotic properties of the model. In this article, we investigate the dynamics of the stochastic SIRS epidemic model with a saturated incidence rate. The effects of both deterministic and stochastic distribution from infectious to susceptible are analyzed. Our findings show that the occurrence of symmetry breaking as a function of the stochastic noise has a significant advantage over the deterministic one to prevent the spread of the infectious diseases. The larger stochastic noise will guarantee the control of epidemic diseases with symmetric Brownian motion. Periodic outbreaks and re-infection may occur due to the existence of feedback memory. It is shown that the endemic equilibrium is stable under some suitable initial conditions, taking advantage of the symmetry of the large amount of contact structure. A numerical method based on Legendre polynomials that converts the given stochastic SIRS model into a nonlinear algebraic system is used for the approximate solution. Finally, some numerical experiments are performed to verify the theoretical results and clearly show the sharpness of the obtained conditions and thresholds.

A stochastic predator-prey model with Holling II increasing function in the predator

This paper is concerned with a stochastic predator-prey model with Holling II increasing function in the predator. By applying the Lyapunov analysis method, we demonstrate the existence and uniqueness of the global positive solution. Then we show there is a stationary distribution which implies the stochastic persistence of the predator and prey in the model. Moreover, we obtain respectively sufficient conditions for weak persistence in the mean and extinction of the prey and extinction of the predator. Finally, some numerical simulations are given to illustrate our main results and the discussion and conclusion are presented.

Stability and optimal harvesting of a predator-prey system combining prey refuge with fuzzy biological parameters

In this manuscript, a novel predator-prey system combining prey refuge with fuzzy parameters is formulated. Sufficient conditions for the existence and stability of biological equilibria are derived. The existence of bionomic equilibria is discussed under fuzzy biological parameters. The optimal harvesting policy, by Pontryagin's maximal principle, is also investigated under imprecise inflation and discount in fuzzy environment. Meticulous numerical simulations are performed to validate our theoretical analysis in detail.

Global dynamical behavior of a multigroup SVIR epidemic model with Markovian switching

In this paper, we are concerned with the global dynamical behavior of a multigroup SVIR epidemic model, which is formulated as a piecewise-deterministic Markov process. We first obtain sufficient criteria for extinction of the diseases. Then we establish sufficient criteria for persistence in the mean of the diseases. Moreover, in the case of persistence, we find a domain which is positive recurrence for the solution of the stochastic system by constructing an appropriate Lyapunov function with regime switching.

Dynamics of a nutrient-phytoplankton model with random phytoplankton mortality

This study formulates a stochastic nutrient-phytoplankton model which incorporates the effect of white noise on phytoplankton growth. The global existence and uniqueness of a positive solution, stochastic boundedness, and stochastically asymptotic stability are well explored. A stochastic ecological reproductive index R₀^s is formulated to characterize the global dynamics. The theoretical analysis demonstrates that, if R₀^s<1, then phytoplankton dies out with probability one; if R₀^s>1 and some other conditions hold, then there exists an invariant and asymptotically stable density of the system and the approach involves integral Markov semigroups theory. Numerical simulations are presented to illustrate the analytical findings and to investigate the long-time effect of water temperature, light, nutrients, and environmental noise on the dynamic evolution of phytoplankton.

Stability and Bionomic Analysis of Fuzzy Prey-Predator Harvesting Model in Presence of Toxicity: A Dynamic Approach

This paper deals with a prey-predator model in which both the species are infected by some toxicants which are released by some other species or source with fuzzy biological parameters. The application of fuzzy differential equation in the modeling of prey-predator populations with the effect of toxicants is presented. The dynamical behavior and harvesting of the fuzzy exploited system are studied by using the utility function method. Sufficient conditions for the local stability of the positive equilibrium are obtained by analyzing the characteristic equation. Furthermore, the possibility of the existence of bionomic equilibrium is studied under imprecise biological parameters. The study of the presence of toxic substance and harvesting in the modeling system can have significant impact on the existence of both the species, which is in line with reality. Numerical simulation results are presented to validate the theoretical analysis.

SURVIVAL ANALYSIS OF A STOCHASTIC COOPERATION SYSTEM IN A POLLUTED ENVIRONMENT

Stochastic cooperation model of two species with generalized dose-response function in a polluted environment is studied. Sufficient criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean and stochastic persistence are established. The threshold between weak persistence in the mean and extinction for each species is obtained. Certain long-run-average limits of the solutions are represented by two constants in some cases. The results also reveal that both the stochastic noises and the dose-response function play important roles in determining the persistence or extinction of the species.