Long-term behaviour of a cyclic catalytic branching system
type: Item
instance of: scholarly article
title: Long-term behaviour of a cyclic catalytic branching system
zbMATH Open document ID: 1228.60065
published in: Stochastic Processes and their Applications
publication date: 2011-07-08T00:00:00Z
review text: This paper treats the long-term behaviour of a system of stochastic differential equations (SDEs) for \(d \geq 2\) types involving catalytic branching and mutation \[ dX_t^i= \sqrt{ 2 \gamma^i X_t^i X_t^{i+1}} \,dB_t^i + \sum_{j=1}^d X_t^j q_{ji} \,dt, \quad i\in \{1,\dots, d\}, \tag{1} \] where \(X_t^{d+1} \equiv X_t^1\). Here, assume that \(\gamma^i\) and \(q_{ji}\), \(i \not= j\), are positive constants and \(X_0^i \geq 0\), \(i \in \{ 1, \dots, d \}\), are initial conditions. Moreover, \(( q_{ji} )\) is a \(Q\)-matrix modelling mutations from type \(j\) to type \(i\). This system involves both cyclic branching and mutation between types so that the extension of the cyclic setup to arbitrary networks is straightforward. Existence of solutions is proven by standard methods. The results of \textit{D. A. Dawson} and \textit{E. A. Perkins} [Ill.\ J.\ Math.\ 50, No.~1--4, 323--383 (2006; Zbl 1107.60045)] take care of the weak uniqueness, once it is shown that a solution does not hit \(0 \in \mathbb R^d\) in finite time. It is interesting to note that the given system of SDEs can be understood as a stochastic analogue to a system of ordinary differential equations (ODEs) for concentrations \(y_j\), \(j=1, \dots, d\), of a type \(T_j\). Then \(y_j/ \dot{y}_j\) corresponds to the rate of growth of type \(T_j\), and the following ODEs are naturally obtained: \(\dot{y}_j\) \(=\) \(b_j y_j\) for independent replication, \(\dot{y}_j\) \(=\) \(\gamma_j y_j^2\) for autocatalytic replication, and \(\hat{y}_j\) \(=\) \(\gamma_i\) \(( \sum_{i \in C_j} y_i)\) \(y_j\) for catalytic replication. In the cycle case type, \(T_{j+1}\) catalyzes the replication of type \(j\), i.e., the growth of type \(j\) is proportional to the mass of type \(j+1\) present at time \(t\). The cyclic catalytic case represents the simplest form of mutual help between different types. The author's main interest consists in the long-time behaviour of the above system, and in particular, survival and coexiste…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 5918964
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: degenerate operator
zbMATH Keywords: catalytic branching network
zbMATH Keywords: diffusion
zbMATH Keywords: mutation
author: Sandra Kliem
MaRDI profile type: Publication
full work available at URL: j.spa.2010.10.004
OpenAlex ID: W2126718561
cites work: Degenerate stochastic differential equations arising from catalytic branching networks
cites work: Strong uniqueness for cyclically symbiotic branching diffusions
cites work: The renormalization transformation for two-type branching models
cites work: On the uniqueness problem for catalytic branching networks and other singular diffusions
cites work: Q5711199
cites work: A cyclically catalytic super-Brownian motion
cites work: Strong uniqueness for a class of singular SDEs for catalytic branching diffusions
cites work: Q3992992
cites work: Q4002114
cites work: Degenerate stochastic differential equations for catalytic branching networks
cites work: Q4508926
DOI: 10.1016/J.SPA.2010.10.004
Recommended article: Long-time behavior and coexistence in a mutually catalytic branching model
Recommended article: The longtime behavior of branching random walk in a catalytic medium
Recommended article: Q3347088
Recommended article: Longtime Behavior for Mutually Catalytic Branching with Negative Correlations
Recommended article: Longtime behavior of a branching process controlled by branching catalysts
Recommended article: Permanence of sparse catalytic networks
Recommended article: Catalytic and mutually catalytic branching
Recommended article: Domain instability in heterogeneous catalytic reactions controlled by branched-chain kinetics
description: scientific article; zbMATH DE number 5918964