Equilibrium behavior of the sexual reproduction process with rapid diffusion
Consider the sexual reproduction process characterized by a birth rate \(\lambda\) and a scale parameter \(\varepsilon\). The state of the process at time \(t\) is denoted by \(_ \varepsilon\xi_ t\subset\varepsilon Z\), where points in \(_ \varepsilon\xi_ t\) are considered as being occupied by a single particle and others are vacant. The case of \(\varepsilon\downarrow 0\) of this model is usually referred to as hydrodynamic limit [cf. \textit{A. DeMasi, P. A. Ferrari} and \textit{J. L. Lebowitz}, J. Stat. Phys. 44, 589-644 (1986; Zbl 0629.60107); \textit{J. L. Lebowitz, E. Presutti} and \textit{H. Spohn}, ibid. 51, 841-862 (1988)]. The main results are that the existence of nonzero equilibrium \((\lim_{t\to\infty} P\{x\in{_ \varepsilon\xi_ t}\}\neq 0)\) depends on the value of \(\lambda\), an upper bound for the critical value of \(\lambda\) is given by \(\limsup_{\varepsilon\downarrow 0}\lambda_ c(\varepsilon)\leq 4.5\) and it is also conjectured that \(\liminf_{\varepsilon\downarrow 0}\lambda_ c(\varepsilon)\geq 4.5\). For all noncritical values of \(\lambda>4.5\) or \(\lambda<4.5\) the equilibrium density of the process is also well studied and estimated.
scientific article; zbMATH DE number 89786