Deterministic versus stochastic aspects of superexponential population growth models

TL;DR

This paper compares deterministic and stochastic models of superexponential population growth, revealing a broader spectrum of behaviors in stochastic versions through selfsimilarity and time change techniques.

Contribution

It introduces two stochastic models based on selfsimilar processes and stable branching, expanding understanding of superexponential growth dynamics.

Findings

Stochastic models exhibit richer growth behaviors than deterministic ones.

Selfsimilarity and time change are key tools in analyzing these models.

Stochastic models include Lamperti constructions and stable branching processes.

Abstract

Deterministic population growth models with power-law rates can exhibit a large variety of growth behaviors, ranging from algebraic, exponential to hyperexponential (finite time explosion). In this setup, selfsimilarity considerations play a key role, together with two time substitutions. Two stochastic versions of such models are investigated, showing a much richer variety of behaviors. One is the Lamperti construction of selfsimilar positive stochastic processes based on the exponentiation of spectrally positive processes , followed by an appropriate time change. The other one is based on stable continuous-state branching processes, given by another Lamperti time substitution applied to stable spectrally positive processes.