Dynamical Analysis of Two‐Microorganism and Single Nutrient Stochastic Chemostat Model with Monod‐Haldane Response Function

A stochastic chemostat model with an inhibitor and noise independent of population sizes

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.

Stochastic characteristics of a chemostat model with variable yield

To investigate the effect of information transmission, Lévy jumps and contact heterogeneity of individuals on the asymptotic behavior of epidemic, a stochastic SIQR epidemic model with non-monotone incidence rate and Lévy jumps on scale-free networks is constructed. At first, the global dynamics of the deterministic model is studied by constructing appropriate Lyapunov functions. Then the stochastic model is made in accordance with the ecological significance, the existence and uniqueness of the global positive solution of the stochastic SIQR model is manifested. Next, by constructing suitable stochastic Lyapunov functions and applying Itô formula with jump, the asymptotic behavior of solutions of stochastic model around equilibrium of the corresponding deterministic model is checked. At last, the correctness of the analytical results is verified by numerical simulations.

Ergodic stationary distribution and extinction of a hybrid stochastic SEQIHR epidemic model with media coverage, quarantine strategies and pre-existing immunity under discrete Markov switching

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

By using the Ornstein-Uhlenbeck (OU) process to simulate random disturbances in the environment, and considering the influence of jump noise, a stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps was established, and the asymptotic behaviors of the stochastic Gilpin-Ayala mutualism model were studied. First, the existence of the global solution of the stochastic Gilpin-Ayala mutualism model is proved by the appropriate Lyapunov function. Second, the moment boundedness of the solution of the stochastic Gilpin-Ayala mutualism model is discussed. Third, the existence of the stationary distribution of the solution of the stochastic Gilpin-Ayala mutualism model is obtained. Finally, the extinction of the stochastic Gilpin-Ayala mutualism model is proved. The theoretical results were verified by numerical simulations.

The asymptotic behavior of a stochastic vaccination model with backward bifurcation

Noise-induced bifurcations in the stochastic chemostat model with general nutrient uptake functions

Dynamics of the Stochastic Chemostat Model with Monod-Haldane Response Function

Malaria remains one of the most life-threatening infectious diseases worldwide, causing hundreds of thousands of deaths annually, particularly among children and pregnant women in tropical and subtropical regions. To better understand its complex transmission dynamics, this study develops and analyzes both deterministic and stochastic mathematical models of malaria based on systems of differential equations. The main objective of this research is to study the stability of both the deterministic and stochastic malaria models analytically and numerically using the Lyapunov function, the Euler method, and the Euler–Maruyama method. For the deterministic model, the disease-free and endemic equilibrium points are derived, and their local and global stability are investigated. Numerical simulations are conducted using the mentioned numerical methods to demonstrate the dynamic behavior of the system. The stochastic extension of the model incorporates random perturbations to represent environmental and demographic fluctuations that influence disease spread. The analytical and numerical results reveal that stochastic effects significantly influence malaria dynamics, potentially reducing disease persistence and stabilizing the system under certain conditions. These findings provide deeper insights into malaria transmission mechanisms and contribute to the development of more effective and evidence-based control strategies.

We look at the behavior of biological oscillators, specifically analyzing a genetic circuit that has oscillatory behavior. Implementing the system in deterministic and stochastic models, we compare the two models using various tests and analyze the effect of stochastic noise on these oscillations. We first investigate the effect of volume and find that at small system volumes, stochastic events cause the system to exhibit more sporadic oscillations and a longer period of oscillation. Next, we develop classification systems for discerning the boundary in the parameter space where the model begins to show oscillations. This is more challenging in the stochastic model, due to persistent fluctuations. We compare different methods of analysis of our deterministic and stochastic models and visualize the conditions which lead to oscillation in each model. We find that the parameter range of oscillation is larger in the stochastic model than in the deterministic model. Finally, we find an increased rate of mRNA production can create greater noise in a system and amplify the difference between a stochastic and deterministic system.

The presence of a pathogen among multiple competing species has important ecological implications. For example, a pathogen may change the competitive outcome, resulting in replacement of a native species by a non-native species. Alternately, if a pathogen becomes established, there may be a drastic reduction in species numbers. Stochastic variability in the birth, death and pathogen transmission processes plays an important role in determining the success of species or pathogen invasion. We investigate these phenomena while studying the dynamics of deterministic and stochastic models for n competing species with a shared pathogen. The deterministic model is a system of ordinary differential equations for n competing species in which a single shared pathogen is transmitted among the n species. There is no immunity from infection, individuals either die or recover and become immediately susceptible, an SIS disease model. Analytical results about pathogen persistence or extinction are summarized for the deterministic model for two and three species and new results about stability of the infection-free state and invasion by one species of a system of n-1 species are obtained. New stochastic models are derived in the form of continuous-time Markov chains and stochastic differential equations. Branching process theory is applied to the continuous-time Markov chain model to estimate probabilities for pathogen extinction or species invasion. Finally, numerical simulations are conducted to explore the effect of disease on two-species competition, to illustrate some of the analytical results and to highlight some of the differences in the stochastic and deterministic models.

This article is an introduction to capacity planning and basic models used to support capacity planning and expansion decisions. The importance of making good capacity decisions by firms is described followed by definitions of various measures of capacity. The main elements of capacity decisions such as size, timing, and location of capacity additions are discussed. The impact of flexibility of capacity and the level of automation on a firm's productivity, the nature of capacity decisions as a function of the production processes, and implications of interactions between capacity levels at firms in a supply chain are discussed. Some standard deterministic and stochastic models in the literature for making capacity decisions are presented. A key insight from continuous time deterministic models is that it is optimal to add capacity at equal time intervals under several demand and capacity cost scenarios. A basic stochastic capacity expansion model is provided that can be transformed to a deterministic model with a different discount rate under fairly general conditions on the demand process. Finally, a stochastic programming model and heuristic solution approaches are described—this is a flexible approach to model capacity expansion decisions when there is uncertainty in both demand and other parameters.

In this study, a glucose-insulin model with the Michaelis-Menten function as the rate of insulin degradation is analyzed using stochastic differential equations. Further, we solve the stochastic glucose-insulin model using the Milstein method, which is based on truncated Ito-Taylor expansions. Comparison of the approximation solution of a stochastic and deterministic model is illustrated by comparing the approximation solution with the deterministic model. A stochastic model allows random fluctuations in glucose-insulin diseases. Furthermore, the stochastic glucose-insulin model's numerical solution provides insight into its variability. The model predicts glucose-insulin dynamics accurately, which is a powerful tool for managing diabetes. Analytical and simulation results are consistent. Improved treatment strategies and personalized medical interventions could result. Treatments and insulin injections are sensitive to these parameters. Numerical simulation corroborates theoretical results.