Sexual Reproduction Process on One Dimensional Stochastic Lattice Model

Kazunori Sato

Summary. I consider the stochastic lattice model for sexual reproduction process on one-dimensional lattice investigated by Dickman and Tomé (1991), Noble (1992) and Neuhauser (1994). This model concerns the reproduction to the empty neighboring habitat by a pair of individuals on one-dimensional lattice. Noble (1992) and Neuhauser (1994) mathematically analysed the model with rapid stirring and long-range interaction, respectively. In this chapter, after reviewing the process with rapid stirring briefly, I concentrate on the case with the nearest neighboring interactions and without stirring, and the qualitative features of the dynamics for this model are studied by using pair approximation, which shows the comparative difference from the mean-field approximation.

5.1 Introduction

To date, lattice models in ecology have been successfully applied in explaining various ecological phenomena. From the points of view of mathematics, the interacting particle systems have their own long history (e. g. Liggett 1999 and references therein) based on the basic contact processes introduced by Harris (1974), which can be considered as the SIS epidemic model or logistic model with local competition for space being the limited resource. On the other hand, the procedures or methods of theoretical analyses of lattice models used in statistical physics have been successfully applied to theoretical ecology, and especially, the technique of approximation for the dynamics called pair approximation became popular and neccessary to study both qualitative and quantitative features of lattice models (Matsuda et al. 1992).

In this chapter, I consider the sexual reproduction process on the one-dimensional lattice space, originally proposed by Dickman and Tomé (1991) in the context of autocatalytic reactions, and mathematically studied by Noble (1992) and Neuhauser (1994) for the case of rapid stirring and longrange interaction, respectively. Before their studies, similar processes on two-

dimensional lattices were analyzed by several authors (see Chen 1992, Durrett 1999, 1986; Durrett and Neuhauser 1994).

As Noble pointed out, the model with rapid stirring has remarkable characteristics such as dynamical behavior independent of the initial configuration, which can be contrasted to the complete mixing model (or mean-field dynamics) with bistability. I study the dynamics of pair approximation (and triplet decoupling approximation) for this model, compare this to Monte Carlo simulation, and provide qualitative insight for the case without stirring.

5.2 Sexual reproduction process with stirring

Noble (1992) considers the sexual reproduction process with stirring as follows. It seems to be hard to find the correspondence to the biological system in the real world, but one may interpret this model as the abstract and simplest model for the mechanism of self-imcompatibility.

I consider e£t as the interacting particle system on eZ or one-dimensional (infinite size of) lattice model defined as the state of the process at time t and the unit of the spatial scale with e. If indicates the process starting from all the sites occupied by "+", then I can be confident of the existence of the equilibrium because of the attractiveness of this process defined below. Also, ££t(x) has either the value "+" or "0", which indicates the site x at time t is either an occupied site (by an individual or organism) or an empty site, respectively. Each individual dies at a constant rate. Each empty site can be given birth by an individual when the adjacent pair of sites is occupied by two individuals at a rate proportinal to the number of adjacent pair of the occupied sites. Choosing the proper time scale, I can use the death rate as 1 and the birth rate as b/2 for each pair of occupied sites.

I use the following notation as the probability measure:

Pa-m,...,a-1,ao,a1,...,an (t) = P{e£t+ (x + ke) = jk for - m < k < n} , where ji G {0, +} and the underline indicates the focal site. I also define the critical value of the birth rate for the survival as bc(e) = inf{b: survives}.

The dynamics of the model by probability measure for a single site can be written as follows:

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