Large population limit and time behaviour of a stochastic particle model describing an age-structured population

Abstract

Abstract.We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound is established via exponential tightness, the difficulty being that the marginals of our measure-valued processes are not of bounded masses. The local minoration is proved by linking the trajectories of the action functional's domain to the solutions of perturbations of the PDE obtained in the large population limit. The use of Girsanov theorem then leads us to regularize these perturbations. As an application, we study the logistic age-structured population. In the supercritical case, the deterministic approximation admits a non trivial stationary stable solution, whereas the stochastic microscopic process gets extinct almost surely. We establish estimates of the time during which the microscopic process stays in the neighborhood of the large population equilibrium by generalizing the works of Freidlin and Ventzell to our measure-valued setting.Mathematics Subject Classification. 60J80, 60K35, 92D25, 60F10.

Population-size-dependent, age-structured branching processes linger around their carrying capacity

“…Our model is closest to Oelschläger's, and the French school around Méléard (cf. [3], [18], and other papers), but formulated in terms of branching rather than birthand-death processes. As in this paper, Tran [18] added an age structure to the birth-and-death framework.…”

“…[3], [18], and other papers), but formulated in terms of branching rather than birthand-death processes. As in this paper, Tran [18] added an age structure to the birth-and-death framework. He considered large populations, scaled them, and studied the limit behaviour.…”

“…Then it levels away, and indeed is subcritical above the carrying capacity. Such behaviour underlies many phenomena in evolutionary biology and ecological theory, in particular in adaptive dynamics (see [3], [8], [15], [18], and [19], just to mention a few titles in the vast literature). Our purpose is to render this knowledge more precise.…”

“…[2], [3], [5], [6, Section 9.4], [16], [17], and [18]. We take the state space to be the finite positive Borel measures on R + with the topology of weak convergence, i.e.…”

The age structure of population-dependent general branching processes in environments with a high carrying capacity

“…Birth-and-death style process have been investigated in a series of papers by Méléard, Champagnat, Lambert, and coauthors, [2,3,4]. More general processes were treated in [12,25] and [13]. The present paper follows up the latter by studying the complete age-distribution.…”