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

Virgile Brodu

Abstract

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.

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

Abstract

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.
Paper Structure (39 sections, 31 theorems, 146 equations, 6 figures, 1 table)

This paper contains 39 sections, 31 theorems, 146 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and assumptions
  3. Individual and resource dynamics
  4. Algorithmic construction of the process with Poisson point measures
  5. Deterministic flow between random jumps
  6. Algorithmic construction of the process
  7. Classical writing of the process on $[0,J_{\infty} \wedge \tau_{\exp})$
  8. Assumption for non vanishing/exploding individual energies
  9. Assumption for a well-defined population process $(\mu_{t})_{t}$ for every $t \geq 0$
  10. Definition of the weight function $\omega$
  11. Proof of the non-accumulation of jump times
  12. Martingale properties
  13. Renormalization of the process
  14. Definition of the renormalized process
  15. Properties of the renormalized process
  16. ...and 24 more sections

Key Result

Theorem 1.1

First, we assume that Then, we assume that there exists a weight function $\omega$ such that Finally, we assume that the sequence $\left(\left(\mu^{K}_{t},R^{K}_{t}\right)_{t \geq 0}\right)_{K \in \mathbb{N}^{*}}$ of renormalized processes (formally defined in Section subsec:construcdeux) is such that Under $\mathbf{(H0)}$, $\mathbf{(H1)}$, $\mathbf{(H2)}$, $\mathbf{(H3)}$, $\mathbf{(H4)}$, for

Figures (6)

  • Figure 1: Possible shape of the weight function $\omega$ with an allometric choice of parameters.
  • Figure 2: Visual representation of the constraints on $\delta$ and $\beta$ in Lemma \ref{['lemme:allomqutre']}, with $\gamma=\alpha \in [0,1]$ (we took $\alpha=0$ on this figure). The admissible coefficients $(\delta, \beta)$ are those in the green hatched area, and verify one of the two following conditions: $(i)$ ($\delta < -1$ and $\beta \leq 2+\delta$); $(ii)$ ($-1 \leq \delta \leq 0$ and $\beta \leq 1$). Remark that the green area always contains the particular case $\beta=\delta=\alpha-1=\gamma-1$ highlighted by the Metabolic Theory of Ecology, represented by a green dot on the figure.
  • Figure 3: Visual representation of the constraints on $\kappa_{1}$ in Lemma \ref{['lemme:allomqutre']}, depending on the value of $\delta$ (we took $\alpha=1$ on this figure). This represents the fact that if $\delta < -1$, then $\kappa_{1}=-\delta$, and if $-1 \leq \delta \leq 0$, then $-\delta \leq \kappa_{1} \leq (1-\delta)/2$. Note that $\kappa_{2}$ verifies the same conditions in Lemma \ref{['lemme:allomqutre']}, so that this graph is also valid to visualize the constraints on $\kappa_{2}$. To sum up, if we want to pick an admissible triplet $(\delta,\kappa_{1},\kappa_{2})$, we first choose $(\delta,\kappa_{1})$ on the green line or in the green hatched area. Then, the remaining possible values for $\kappa_{2}$ are such that the two following conditions hold true: $\kappa_{2} \geq \kappa_{1}$, and $(\delta,\kappa_{2})$ is also on the green line or in the green hatched area.
  • Figure 4: From the first row to the third row, time evolutions of $N^{K}_{t}$, $E^{K}_{t}$ and $R^{K}_{t}$ (in blue), respectively $N^{*}_{t} := \langle \mu^{*}_{t}, 1 \rangle$, $E^{*}_{t} := \langle \mu^{*}_{t}, \mathrm{Id} \rangle$ and $R^{*}_{t}$ (in red), representing the population size, the total energy of the population and the amount of resources, and associated respectively with the trajectories of 100 independent IBM simulations (in blue) and the numerical resolution of the PDE system \ref{['eq:indivedp']}, \ref{['eq:ressedp']}, \ref{['eq:boundaryedp']} with initial condition $u_{0}$ (in red). The fourth row presents the energy/resource phase portrait. The green dotted curve is the mean value of the stochastic simulations in blue. These graphs are presented for small (left), medium (middle) and large (right) initial population sizes. On the third row, the dotted black line locates the value $R_{\mathrm{eq}}$ assumed to be an equilibrium for the amount of resource.
  • Figure 5: Energy distribution on the energy window $[0,5]$ at time $t=0$ (above), 20 (middle) and 160 (bottom) for small (left), medium (middle) and large (right) initial population sizes. The red curve represents the renormalized energy distribution $\tilde{u}_{t}(.)$ for the corresponding value of $t$. The blue histogram represent the empirical energy distribution of individuals for 100 independent runs of IBM. The number of bins for the histograms is adapted to population sizes on each subfigure.
  • ...and 1 more figures

Theorems & Definitions (60)

  • Theorem 1.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.7
  • ...and 50 more