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Asymptotic behavior of some stochastic models in population dynamics: a Hamilton-Jacobi approach

Anouar Jeddi

Abstract

In this paper, we investigate the asymptotic behavior of individual-based models describing the evolution of a population structured by a real trait, subject to selection and mutation. We consider two different sets of assumptions: first, the case of critical or subcritical branching population processes in a regime combining a discretization of the trait space, small mutations, large time and large initial population size, where we are able to characterize using a Hamilton-Jacobi approach, the survival set of the population, and the asymptotic of the logarithmic scaling of subpopulation sizes. Second, we generalize by a direct method the convergence to the classical Hamilton-Jacobi equation obtained in the super-critical branching regime considered in \cite{CMMT} to a more general trait space and under weaker assumptions. Moreover, we establish that the stochastic and the deterministic dynamics are asymptotically equivalent in large population.

Asymptotic behavior of some stochastic models in population dynamics: a Hamilton-Jacobi approach

Abstract

In this paper, we investigate the asymptotic behavior of individual-based models describing the evolution of a population structured by a real trait, subject to selection and mutation. We consider two different sets of assumptions: first, the case of critical or subcritical branching population processes in a regime combining a discretization of the trait space, small mutations, large time and large initial population size, where we are able to characterize using a Hamilton-Jacobi approach, the survival set of the population, and the asymptotic of the logarithmic scaling of subpopulation sizes. Second, we generalize by a direct method the convergence to the classical Hamilton-Jacobi equation obtained in the super-critical branching regime considered in \cite{CMMT} to a more general trait space and under weaker assumptions. Moreover, we establish that the stochastic and the deterministic dynamics are asymptotically equivalent in large population.
Paper Structure (16 sections, 9 theorems, 82 equations)

This paper contains 16 sections, 9 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. First part:
  3. Second part:
  4. Model and motivation
  5. Critical or sub-critical case:
  6. Assumptions B
  7. Preliminary result
  8. Convergence result in the critical or subcritical dynamics
  9. Hamilton-Jacobi approach for the mean dynamics
  10. The gap between the stochastic and the deterministic dynamics
  11. Proof of Theorem 3.2
  12. Super-critical case
  13. Assumptions C
  14. Preliminary results and Hamilton-Jacobi approach for the mean dynamics
  15. Convergence result in the supercritical regime
  16. ...and 1 more sections

Key Result

Proposition 3.1

Under Assumptions A-item:1-item:2 and Assumption B-item:66. The process $(N^K_i(t),i\in \mathbb{Z},t\geq 0)$ is well-defined, and we have for all $T>0$ where $C(T,K)$ is a positive constant depending on $T,K$ but not on $i.$

Theorems & Definitions (14)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 4 more