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Spinal study of a population model for colonial species with interactions and environmental noise

Sylvain Billiard, Charles Medous, Charline Smadi

TL;DR

This work introduces and study a stochastic model for the dynamics of colonial species, which reproduce through fission or fragmentation, and derives a $\psi$-spine construction and a Many-to-One formula, extending previous works on interacting branching processes.

Abstract

We introduce and study a stochastic model for the dynamics of colonial species, which reproduce through fission or fragmentation. The fission rate depends on the relative sizes of colonies in the population, and the growth rate of colonies is influenced by intrinsic and environmental stochasticities. Our setting thus captures the effect of an external noise, correlating the trait dynamics of all colonies. In particular, we study the effect of the strength of this correlation on the distribution of resources between colonies. We then extend this model to a large class of structured branching processes with interactions in which the particle type evolves according to a diffusion. The branching and death rates are general functions of the whole population. In this framework, we derive a $ψ$-spine construction and a Many-to-One formula, extending previous works on interacting branching processes. Using this spinal construction, we also propose an alternative simulation method and illustrate its efficiency on the colonial population model. The extended framework we propose can model various ecological systems with interactions, and individual and environmental noises.

Spinal study of a population model for colonial species with interactions and environmental noise

TL;DR

This work introduces and study a stochastic model for the dynamics of colonial species, which reproduce through fission or fragmentation, and derives a -spine construction and a Many-to-One formula, extending previous works on interacting branching processes.

Abstract

We introduce and study a stochastic model for the dynamics of colonial species, which reproduce through fission or fragmentation. The fission rate depends on the relative sizes of colonies in the population, and the growth rate of colonies is influenced by intrinsic and environmental stochasticities. Our setting thus captures the effect of an external noise, correlating the trait dynamics of all colonies. In particular, we study the effect of the strength of this correlation on the distribution of resources between colonies. We then extend this model to a large class of structured branching processes with interactions in which the particle type evolves according to a diffusion. The branching and death rates are general functions of the whole population. In this framework, we derive a -spine construction and a Many-to-One formula, extending previous works on interacting branching processes. Using this spinal construction, we also propose an alternative simulation method and illustrate its efficiency on the colonial population model. The extended framework we propose can model various ecological systems with interactions, and individual and environmental noises.

Paper Structure

This paper contains 20 sections, 15 theorems, 144 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. A colonial population correlated through a common environment.
  3. Description of the model
  4. Spinal Construction
  5. Long-time behavior of the population
  6. Spinal algorithm
  7. General population model and spinal construction
  8. Definition of the population
  9. $\psi$-spine construction
  10. Proofs
  11. Proofs of Section \ref{['section: general spine']}
  12. Proof of Proposition \ref{['prop: generator G']}
  13. Proof of Proposition \ref{['prop: generator']}
  14. Proof of Theorem \ref{['thm: sampling']}
  15. Proofs of Section \ref{['section: example']}
  16. ...and 5 more sections

Key Result

Theorem 2.1

For $U_t$ a colony picked uniformly at random in $\mathbb{G}(t)$ and for every measurable function $F$ on the set of càdlàg trajectories on $\mathbb{R}_+^*\times\mathbb{U}$, we have

Figures (5)

  • Figure 1: Monte-Carlo estimation of $\mathbb{E}[N_t]$ using $M = 10^4$ trajectories, for different values of the division rate $\lambda$. Other parameters are $\mu=1$ and $N_0= 1$.
  • Figure 2: Monte-Carlo estimation of $\mathbb{V}ar[R_t]$ using $M = 10^6$ trajectories, for different values of $\lambda, \mu$ and $\sigma$. Other parameters are $R_0= 15, a= 5$ and $\delta = 1$.
  • Figure 3: Comparison of the mean sharing of resources at different times $T$ and different values of $\sigma^2/(1+\delta^2)$. The x axis gives the label of the individuals, sorted by trait in a decreasing order. The other parameters are $\lambda = 3, \mu = 1, a = 1, N_0 = 10$ and $R_0 = 10$.
  • Figure 4: Monte-Carlo estimations of $\mathbb{V}ar[R_t]$ with $10^6$ trajectories from the direct and spinal methods. Parameter values are $a = 5$, $\lambda = 6$, $\delta = 1$, $R_0 = 30$.
  • Figure 5: Comparison of simulation times and relative error for Monte Carlo estimates from direct and spinal methods. (A) $\mathbb{V}ar[R_t]$ estimation. Parameters are $a = 1$, $\lambda = 1$, $\mu = 0.3$, $\sigma = 1$, $\delta = 0.5$, $T = 2$. (B) Estimation of $\mathbb{E}[\mathbbm{1}_{\mathbb{G}(t)\neq 0}X^{U_t}]$. Parameters are $a = 1$, $\lambda = 2$, $\mu = 0.5$, $\sigma = 0.5$, $\delta = 0.5$, $T = 10$.

Theorems & Definitions (21)

  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 3.3
  • Theorem 3.4
  • ...and 11 more