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

TL;DR

This work develops an energy-based, individual-level stochastic model for a single species in a fixed-resource environment, formulating a piecewise deterministic Markov process (PDMP) whose energy dynamics are governed by allometric scalings. By constructing a measure-valued population process and embedding a Galton–Watson generation mechanism, the authors derive rigorous constraints on allometric exponents (β,δ,γ) relative to metabolism exponent α, identifying admissible regions I1 and I2 (for α ≤ 1) and analogous conditions for α > 1. The key results show that an allometric coefficient α for metabolism imposes strong restrictions on birth, death, and growth exponents, with explicit necessary and (in parts) sufficient conditions; simulations illustrate regimes of subcriticality, supercriticality, and possible explosion in offspring under the I2 case. The framework connects bottom-up energetic balances to interspecific allometries, offering a probabilistic foundation for observed metabolic and ecological scaling laws and suggesting directions for extending to dynamic resources and multi-species interactions. Overall, the paper provides a mathematically rigorous bridge between energy flow, demographic rates, and allometric theory, with potential implications for understanding universal scaling across body sizes.

Abstract

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 $α$ related to metabolism strongly constrains the range of possible values for the allometric coefficients $β$, $δ$, $γ$, 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.

Figures (7)

• Figure 1: Visual representation of Theorem \ref{['theo:short']}. The green dot represents the singleton $I_{1}$, and the green line represents the set $I_{2}$. The non-admissible allometric coefficients $\beta$ for our model with $\alpha \leq 1$ are highlighted in red (and also, every $(\gamma,\delta,\beta)$ with $\delta \neq \alpha -1$ or $\gamma \neq \alpha$ is non-admissible).
• Figure 2: Visual representation of Theorem \ref{['theo:sharpdeux']}. All the coefficients $(\gamma, \delta, \beta)$ verifying one of the following conditions are non-admissible: 1) $\gamma \neq \alpha$; 2) $\delta \neq \alpha -1$, except for the ones in the hatched area and its border; 3) $\gamma=\alpha$, $\delta= \alpha-1$ and $\beta$ on the red line (except for the green dot).
• Figure 3: Different shapes for individual trajectories, with parameters $x_{0}=\xi_{0}=1$, $\beta=-0.2$, $C_{\beta}=2$, $C_{\delta}=0.5$. Most of the time, we obtain the shapes of \ref{['fig:first']} and \ref{['fig:second']}, and the rarest trajectory is \ref{['fig:sixth']}. The dashed vertical orange line represents the time of death.
• Figure 4: Monte Carlo estimation of the average number of offspring $m_{x_{0},R}(x_{0})$ for $x_{0} \in [10^{-100},10^{100}]$, plotted on a log-scale, with parameters $C_{\beta}=2$, $C_{\delta}=0.5$, and different values of $\alpha - 1 = -0.25 < \beta < -0.05 = \alpha -1 + \frac{C_{\delta}}{C_{\gamma}-C_{\alpha}}$. For every value of $x_{0}$, we simulated $n =50000$ Monte-Carlo samples to obtain an estimation of $m_{x_{0},R}(x_{0})$. If at some $x_{0}$, the blue line is above the orange line, the numerical simulation suggests that the population process is supercritical for the corresponding value of $x_{0}$, and if it is below, the population process is subcritical.
• Figure 5: Monte-Carlo estimation of the mean number of offspring $m_{x_{0}}(x_{0})$ for $x_{0}=1$, $C_{\beta}=2$, $C_{\delta}=0.5$ and different values of $\beta$. The $x$-axis represents the number $n$ of independent individual trajectories simulated to estimate $m_{x_{0}}(x_{0})$.

Theorems & Definitions (93)

• Theorem 1
• Proposition 1
• proof
• Proposition 2
• Proposition 3
• Lemma 4
• proof
• Theorem 2
• Corollary 5
• Proposition 6