Virgile Brodu | Searcharxiv

arXiv2026

Individual-based stochastic model with unbounded growth, birth and death rates: a tightness result

We study population dynamics through a general growth/degrowth-fragmentation process, with resource consumption and unbounded growth/degrowth, birth and death rates. Our model is structured in a positive trait called energy (which is a proxy for any biological parameter such as size, age, mass, protein quantity...), and the jump rates of the process can be arbitrarily high depending on individual energies, which has not been considered yet in the literature. After a preliminary study to construct well-defined objects (which is necessary contrary to similar works, because of the explosion of individual rates), we consider a classical sequence of renormalizations of the underlying process and obtain a tightness result for the associated laws in large-population asymptotics. We characterize the accumulation points of this sequence as solutions of an integro-differential system of equations, which proves the existence of measure solutions to this system. Furthermore, if such a measure solution is unique, then our tightness result becomes a convergence result towards this unique process. We illustrate our work with the case of allometric rates (i.e. they are assumed to be power functions) and eventually present numerical simulations in this allometric setting.

math.PR↗

arXiv2025

An individual-based stochastic model reveals strong constraints on allometric relationships with minimal metabolic and ecological assumptions

We design a stochastic individual-based model structured in energy, for single species consuming an external resource, where populations are characterized by a typical energy at birth in $\mathbb{R}^{*}_{+}$. The resource is maintained at a fixed amount, so we benefit from a branching property at the population level. Thus, we focus on individual trajectories, constructed as Piecewise Deterministic Markov Processes, with random jumps modelling births and deaths in the population; and a continuous and deterministic evolution of energy between jumps. We are mainly interested in the case where metabolic (i.e. energy loss for maintenance), growth, birth and death rates depend on the individual energy over time, and follow allometric scalings (i.e. power laws). Our goal is to determine in a bottom-up approach what are the possible allometric coefficients (i.e. exponents of these power laws) under elementary -- and ecologically relevant -- constraints, for our model to be valid for the whole spectrum of possible body sizes. We show in particular that assuming an allometric coefficient $\alpha$ related to metabolism strongly constrains the range of possible values for the allometric coefficients $\beta$, $\delta$, $\gamma$, respectively related to birth, death and growth rates. We further identify and discuss the precise and minimal ecological mechanisms that are involved in these strong constraints on allometric scalings.

math.PR↗