Long-term behaviour of a cyclic catalytic branching system
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 coexistence of types. It is shown that, in the SDE setup, the overall sum \(S_t\) \(=\) \(\sum_{i=1}^d X_t^i\) converges to zero but does not hit zero in finite time almost surely. To further analyze the relative behaviour of types while they approach zero, the author turns attention to the normalized process \(Y_t^i\) \(\equiv\) \(X_t^i / \sum_j X_t^j\) and proves weak convergence to a unique stationary distribution that does not put mass on the set where at least one of the coordinates is zero, by adapting the proof of Theorem 2.3 of \textit{D. A. Dawson} et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 44, No.~6, 1038--1077 (2008; Zbl 1181.60122)]. Finally, the complete analysis of the case \(d=2\) is given as well by using methods of speed and scale. The cyclic catalytic case was firstly introduced by \textit{M. Eigen} and \textit{P. Schuster} [The hypercycle: a principle of natural self-organization. Berlin: Springer (1979)]. Similar questions naturally arise in biological competition models, and, for instance, \textit{K. Fleischmann} and \textit{J. Xiong} [Ann. Probab. 29, No.~2, 820--861 (2001; Zbl 1017.60099)] investigated a cyclically catalytic super-Brownian motion, for which they proved global segregation (noncoexistence) of neighbouring types in the limit and other results on the finite time survival or extinction. The author's previous work [Ann. Inst. Henri Poincaré, Probab. Stat. 45, No.~4, 943--980 (2009; Zbl 1201.60058)] is employed to establish the weak uniqueness of the system on the state space \(Z\) \(=\) \(\{ x \in\mathbb R_+^d\) : \(\prod_{i=1}^d (x_i + x_{i+1}) > 0 \}\).
scientific article; zbMATH DE number 5918964
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