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

By Nicolas G. Fournier, S. M'el'eard. 2004

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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 gives a rigorous sense to the number density. The second one is a superprocess already studied by Etheridge [6]. Finally, we study in specific cases the long time behavior of the system and of its deterministic approximation.

Subfields: Economics and Econometrics, Mathematical Physics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/667371e74227b7ffdfe8a707ed95b48c87e82dd6

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Citations in the corpus

Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior (2015, background): “([59]), and mathematically studied by Fournier and Méléard [38].”
A microscopic interpretation for adaptive dynamics trait substitution sequence models (2005, methodology): “The assumption (B) of Section 2 is the mathematical formulation of the impossibility of coexistence of two different traits, in the sense that, starting in the neighborhood of the equilibrium (n̄x, 0) of system (13), either its solution converges to this equilibrium or to the equilibrium (0, n̄y).”
Limit theorems for Markov processes indexed by continuous time Galton--Watson trees (2009, methodology): “Following the work of Fournier and Méléard [26], we can describe the evolution of Z in terms of stochastic differential equations (SDE).”
Understanding, modelling and managing longevity risk: key issues and main challenges (2012, methodology): “Nevertheless, an abundant literature on the economic aspects of long-term policy-making has been developed (see for example Gollier (2007, 2008), Hansen and Scheinkman (2009) or Breeden (1979)).”
LARGE POPULATION LIMIT AND TIME BEHAVIOUR OF A STOCHASTIC PARTICLE MODEL DESCRIBING AN AGE-STRUCTURED POPULATION (2008, methodology): “This convergence result is an adaptation of results due to Fournier and Méléard [17] and Champagnat et al.”
Polymorphic evolution sequence and evolutionary branching (2008, methodology): “For fixed K, under (A1) and (A3) and assuming that E(〈νK 0 ,1〉) < ∞, the existence and uniqueness in law of a process on D(R+,M K) with infinitesimal generator LK has been proved in [12].”
Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations (2009, background): “3 of Fournier and Méléard (2004). Letting ν̂0 = M−1δχ0 (a point mass at χ0 of size 1/M), run the process ν̂t up to time t = log(M) and define χ1 to be such that ν̂log(M)([χ1,∞)) = 1/M .”
Nonlocal anisotropic dispersal with monostable nonlinearity (2008, background): “Such kind of equations appears in various applications ranging from population dynamics to Ising models as seen in [1,6, 12,13,15,16,19,23,24] among many references.”
Large graph limit for an SIR process in random network with heterogeneous connectivity (2010, background): “The result then follows the steps of [13] and [25] (Proposition 2.”
Limit theorems for some branching measure-valued processes (2011, methodology): “Following [17, 43], we scale our empirical measure and prove that these P.”
From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution (2008, background): “Champagnat, N.; Ferrière, R.; Méléard, S. Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models.”