# A multi-class extension of the mean field Bolker–Pacala population model

Abstract

Abstract

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References7

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A Markov evolution of a system of point particles in ℝ d is described at micro- and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and under the effect of each other (competition). The microscopic description is based on an infinite chain of equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a Vlasov-type kinetic equat...

A microscopic probabilistic description of a locally regulated population and macroscopic approximations

We consider a discrete model describing a locally regulated spatial population with mortality selection. This model was studied in parallel by Bolker and Pacala, [2] and Dieckmann, Law and Murrell [9], [4], [10]. We first generalize this model by adding spatial dependence. Then we give a path-wise description in terms of Poisson point measures. We show that different re-normalizations may lead to different macroscopic approximations of this model. The first approximation is deterministic and giv...

Abstract: A variety of models have shown that spatial dynamics and small‐scale endogenous heterogeneity (e.g., forest gaps or local resource depletion zones) can change the rate and outcome of competition in communities of plants or other sessile organisms. However, the theory appears complicated and hard to connect to real systems. We synthesize results from three different kinds of models: interacting particle systems, moment equations for spatial point processes, and metapopulation or patch m...

Spatial Moment Equations for Plant Competition: Understanding Spatial Strategies and the Advantages of Short Dispersal

abstract: A plant lineage can compete for resources in a spatially variable environment by colonizing new areas, exploiting resources in those areas quickly before other plants arrive to compete with it, or tolerating competition once other plants do arrive. These specializations are ubiquitous in plant communities, but all three have never been derived from a spatial model of community dynamics—instead, the possibility of rapid exploitation has been either overlooked or confounded with coloniza...

In [3] this author gave conditions under which a sequence of jump Markov processes X n ( t ) will converge to the solution X ( t ) of a system of first order ordinary differential equations, in the sense that for every δ > 0.

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.