Stochastic two-species mutualism model with jumps

Integrability of the multi-species TASEP with species-dependent rates

Assume that each species $l$ has its own jump rate $b_l$ in the multi-species totally asymmetric simple exclusion process. We show that this model is \textit{integrable} in the sense that the Bethe Ansatz method is applicable to obtain the transition probabilities for all possible $N$-particle systems with up to $N$ different species.

Concentration inequalities for the hydrodynamic limit of a two-species stochastic particle system

We study a stochastic particle system which is motivated from grain boundary coarsening in two-dimensional networks. Each particles lives on the positive real line and is labeled as belonging to either Species 1 or Species 2. Species 1 particles drift at unit speed toward the origin, while Species 2 particles do not move. When a particle in Species 1 hits the origin, it is removed, and a randomly selected particle mutates from Species 2 to Species 1. The process described is an example of a high-dimensional piecewise deterministic Markov process (PDMP), in which deterministic flow is punctuate…

Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms

Biochemical reactions can happen on different time scales and also the abundance of species in these reactions can be very different from each other. Classical approaches, such as deterministic or stochastic approach, fail to account for or to exploit this multi-scale nature, respectively. In this paper, we propose a jump-diffusion approximation for multi-scale Markov jump processes that couples the two modeling approaches. An error bound of the proposed approximation is derived and used to partition the reactions into fast and slow sets, where the fast set is simulated by a stochastic differe…

State and parameter estimation from exact partial state observation in stochastic reaction networks

We consider chemical reaction networks modeled by a discrete state and continuous in time Markov process for the vector copy number of the species and provide a novel particle filter method for state and parameter estimation based on exact observation of some of the species in continuous time. The conditional probability distribution of the unobserved states is shown to satisfy a system of differential equations with jumps. We provide a method of simulating a process that is a proxy for the vector copy number of the unobserved species along with a weight. The resulting weighted Monte Carlo sim…

Reaction-subdiffusion equations with species-dependent movement

Reaction-diffusion equations are one of the most common mathematical models in the natural sciences and are used to model systems that combine reactions with diffusive motion. However, rather than normal diffusion, anomalous subdiffusion is observed in many systems and is especially prevalent in cell biology. What are the reaction-subdiffusion equations describing a system that involves first-order reactions and subdiffusive motion? In this paper, we answer this question. We derive fractional reaction-subdiffusion equations describing an arbitrary number of molecular species which react at fir…

具有均匀速率的可积多物种全非对称随机相互作用粒子系统

Integrable multispecies totally asymmetric stochastic interacting particle systems with homogeneous rates

本文研究具有均匀速率的排斥相互作用一维随机粒子系统，将 28 种两物种 Yang-Baxter 可积规则中的 7 种推广到任意物种数，并发现含参数的新可积模型，同时为另外 8 种情形提出替代推广方案。