Stochastic evolution models are widely used to study the dynamic behavior of biological and chemical systems where uncertainty, random fluctuations, or inherent variability are essential characteristics. These models often take the form of stochastic partial differential equations (SPDEs) or stochastic differential equations (SDEs) and are crucial for capturing the complex interplay between deterministic laws and random influences. Below, we explore some of the key aspects and examples of stochastic evolution models in these fields.

1. General Framework of Stochastic Evolution Models

• Definition: Stochastic evolution models describe systems where quantities evolve over time according to both deterministic rules (e.g., reaction kinetics, diffusion) and stochastic influences (e.g., random environmental factors or noise).
• Mathematical Form: Such models often take the form:

$\frac{\partial u(x, t)}{\partial t} = \mathcal{L}[u(x, t)] + f(u, x, t) + \eta(x, t),$

where $u(x, t)$ represents the evolving quantity (e.g., concentration of a chemical, population of a species), $\mathcal{L}$ is an operator defining deterministic dynamics (e.g., diffusion or reaction terms), $f(u, x, t)$ may be a non-linear interaction term, and $\eta(x, t)$ is a noise term modeling stochastic effects.

2. Biological Systems

1. Population Dynamics

• Stochastic Lotka–Volterra Models: These are adaptations of classical predator-prey models that include stochastic terms to account for environmental variability or random interactions between species:

$\frac{dN_1}{dt} = N_1(a - bN_2) + \sigma_1 W_1(t), \quad \frac{dN_2}{dt} = -N_2(c - dN_1) + \sigma_2 W_2(t),$

where $N_1$ and $N_2$ are the population sizes of the prey and predator, $a, b, c, d$ are interaction coefficients, $\sigma_1$ and $\sigma_2$ are noise intensities, and $W_1(t)$ and $W_2(t)$ are Wiener processes modeling stochastic influences.

2. Gene Expression

• Stochastic Gene Regulatory Networks: Gene expression often exhibits random fluctuations due to the low number of molecules involved, modeled using stochastic differential equations:

$\frac{dX(t)}{dt} = f(X(t)) + \sigma(X(t)) \eta(t),$

where $X(t)$ is the concentration of a gene product, $f(X(t))$ represents deterministic regulation (e.g., transcription and translation rates), $\sigma(X(t))$ modulates the noise intensity, and $\eta(t)$ is a stochastic process representing intrinsic noise from biochemical reactions.

3. Epidemic Modeling

• Stochastic SIR Models: Variants of the classic susceptible-infected-recovered (SIR) model that include noise to capture random contact patterns or environmental factors:

$\frac{dS}{dt} = -\beta S I + \sigma_S W_S(t), \quad \frac{dI}{dt} = \beta S I - \gamma I + \sigma_I W_I(t),$

where $S$ , $I$ , and $R$ represent the numbers of susceptible, infected, and recovered individuals, respectively, and $\beta$ and $\gamma$ are the transmission and recovery rates. The $\sigma$ terms and $W(t)$ processes account for random fluctuations.

3. Chemical Systems

1. Reaction-Diffusion Systems

• Stochastic Reaction-Diffusion Equations: These describe chemical reactions coupled with the spatial diffusion of reactants. The general form is:

$\frac{\partial u(x, t)}{\partial t} = D \nabla^2 u(x, t) + R(u) + \eta(x, t),$

where $D$ is the diffusion coefficient, $R(u)$ represents the reaction kinetics, and $\eta(x, t)$ is the noise term modeling stochastic influences like fluctuations in reactant concentrations.

2. Stochastic Chemical Kinetics

• Master Equation Approach: Chemical reactions at the molecular level are often modeled using the chemical master equation, which governs the probability distribution of molecule numbers over time. In complex systems, this can be approximated by SDEs such as the stochastic chemical Langevin equation:

$\frac{dX_i}{dt} = \sum_{j} \nu_{ij} r_j(X) + \sum_{j} \sigma_{ij} \sqrt{r_j(X)} \eta_j(t),$

where $X_i$ is the number of molecules of species $i$ , $\nu_{ij}$ are stoichiometric coefficients, $r_j(X)$ are reaction rates, and $\eta_j(t)$ are noise terms reflecting the stochastic nature of reactions.

4. Types of Noise in Models

• Intrinsic Noise: Results from the stochastic nature of biochemical interactions or low molecule numbers. Often modeled using discrete stochastic simulations such as Gillespie’s algorithm or incorporated into SPDEs.
• Extrinsic Noise: Arises from fluctuations in external conditions like temperature or nutrient availability, typically modeled as additive noise or correlated (colored) noise.
• White Noise: Uncorrelated in time, often represented by Wiener processes.