Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior


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In the first part, we focus on monotype populations described by one-dimensional stochastic differential equations with jumps. We consider their scaling limits for large populations and study the long time behavior of the limiting processes.

Sylvie Méléard

It is achieved, thanks to martingale properties, Poisson measure representations, and stochastic calculus. These tools and results will be used and extended to measure-valued processes in the second part. The latter is dedicated to structured populations, where individuals are characterized by a trait belonging to a continuum.


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If the population is large, so many birth and death events occur that the dynamics becomes difficult to describe individual per individual. Living systems need resources in order to survive and reproduce and the biomass per capita depends on the order of magnitude of these resources.

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In this part, we consider a new class of stochastic differential equations for monotype populations, taking into account exceptional events where an individual has a large number of offspring. Combining this property with the tools developed in the first part, we describe finely the processes, their long time behavior, and the scaling limits they come from. We deal now with a new family of branching processes taking into account the effect of the environment on the population dynamics.

It may cause random fluctuations of the growth rate [12, 33] or catastrophes which kill a random fraction of the population [7].

Stochastic Models for Structured Populations

In the previous sections, the models that we considered described a homogeneous population and could be considered as toy models. A first generalization consists in considering multitype population dynamics. The polymorphic evolution sequence for populations with phenotypic plasticity Martina Baar , Anton Bovier. References Publications referenced by this paper.


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Survival and extinction in a locally regulated population Alison Etheridge. Markov Processes, Characterization and Convergence.

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Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior

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