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.
Clustering at multiple critical scales may be common for plants since many different factors and processes may cause clustering. This is especially true for tropical rain forests for which theories explaining species coexistence and community structure rest heavily on spatial patterns. We used point pattern analysis to analyze the spatial structure of Shorea congestiflora, a dominant species in a 25-ha forest dynamics plot in a rain forest at Sinharaja World Heritage Site (Sri Lanka), which apparently shows clustering at several scales. We developed cluster processes incorporating two critical scales of clustering for exploring the spatial structure of S. congestiflora and interpret it in relation to factors such as competition, dispersal limitation, recruitment limitation, and Janzen-Connell effects. All size classes showed consistent large-scale clustering with a cluster radius of ;25 m. Inside the larger clusters, small-scale clusters with a radius of 8 m were evident for recruits and saplings, weak for intermediates, and disappeared for adults. The pattern of all trees could be divided into two independent patterns: a random pattern (nearest neighbor distance . 8m ) comprising ;12 of the trees and a nested double-cluster pattern. This finding suggests two independent recruitment and or seed dispersal mechanisms. Saplings were several times as abundant as recruits and may accumulate several recruit generations. Recruits were only weakly associated with adults and occupied about half of the large-scale clusters, but saplings almost all. This is consistent with recruitment limitation. For ;70 (95 ) of all juveniles the nearest adult was less than 26 m away (53 m), suggesting a dispersal limitation that may also be related to the critical large-scale clustering. Our example illustrates the manner in which the use of a specific and complex null hypothesis of spatial structure in point pattern analysis can help us better understand the biology of a species and generate specific hypotheses to be further investigated in the field. Spatial Hawkes processes can be considered as spatial versions of classical Hawkes processes. We derive the pair correlation function of stationary spatial Hawkes processes and discuss the connection to the Bartlett spectrum and other summary statistics. Particularly, results for Gaussian fertility rates and the extension to spatial Hawkes processes with random fertility rates are discussed. Abstract Photoactivated localization microscopy (PALM) is an ingenious super-resolution imaging technique that produces 2D point patterns of proteins. Individual proteins may appear as small artificial clusters of points, due to multiple blinking of individual fluorophores. The proteins may also cluster together, and in such cases a pertinent model for a PALM point pattern describes clustering at two different scales. Despite the importance of the imaging technique, statistical methods for analysingPALM data have remained relatively under-studied. In the present paper, we develop a model-based framework for analysis of PALM data. We focus on a subclass of independent cluster processes, denoted double Cox cluster processes (DCCPs), for which both the parent process (of proteins) and the observed process are Cox cluster processes. Parametric models for DCCPs with a Neyman–Scott process as parent process are developed together with statistical inference procedures, based on moment methods. To illustrate the proposed methodology, we analyse a data set from a PALM acquisition. In contrast to earlier model-free methods, the analysis provides information, directly relating to the performance of the proteins. The paper also represents an independent contribution to point process theory.