We consider a Markov chain of point processes such that each state is a super position of an independent cluster process with the previous state as its centre process together with some independent noise process. The model extends earlier work by Felsenstein and Shimatani describing a reproducing population. We discuss when closed term expressions of the first and second order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.
[(vii)] In particular, @cite_8 considered the case @math and when @math or equivalently when the intensities @math are invariant over generations, and then he showed that as @math , the PCF for @math diverges and all offspring will eventually have the same ancestor''. It follows from item (iii) that the model is equivalent to a replicated Neyman-Scott process; this becomes a replicated Thomas process when each cluster size is Poisson distributed, and hence the result in @cite_8 agrees with the results in @cite_13 and @cite_15 . Note that @cite_8 implicitly assumed that a cluster can have more than one point. Otherwise the PCF of @math becomes equal to 1; we discuss this rather trivial case again in and ; see also Section 3 in @cite_15 .
@cite_15 @cite_13 @cite_8
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Malecot's models of isolation by distance seem to assume random distribution in space, Poisson distribution of offspring number, and independent migration of individuals. These assumptions are inconsistent. By calculating the probability of joint occupancy of two locations at a distance x, it is shown that a haploid model embodying the latter two assumptions violates the first, forming larger and larger clumps of individuals separated by greater and greater distances. Thus the model is biologically irrelevant, even though it is possible to use it to calculate probabilities of genetic identity at different distances. The formation of these clumps is verified by computer simulation. The same phenomenon occurs in two- and three-dimensional models. In models of finite regions (a circle or a torus), the formation of clumps takes the form of the certainty of ultimate local extinction of the organism in each region. This does not occur if we hold the number of individuals on the circle or torus constant, but con... Abstract Spatially explicit models relating to plant populations have developed little since Felsenstein (1975) pointed out that if limited seed dispersal causes clustering of individuals, such models cannot reach an equilibrium. This paper aims to resolve this issue by modifying the Neyman–Scott cluster point process. The new point processes are dynamic models with random immigration, and the continuous increase in the clustering of individuals stops at some level. Hence, an equilibrium state is achieved, and new individual-based spatially explicit neutral coalescent models are established. By fitting the spatial structure at equilibrium to individual spatial distribution data, we can indirectly estimate seed dispersal and effective population density. These estimates are improved when genetic data are available, and become even more sophisticated if spatial distribution and genetic data pertaining to the offspring are also available.