# A dynamical trichotomy for structured populations experiencing positive density-dependence in stochastic environments

## TL;DR

This paper develops a mathematical framework for structured populations with positive density-dependence and environmental randomness, revealing a trichotomy of extinction, unbounded growth, or probabilistic outcomes.

## Contribution

It introduces a dynamical trichotomy for structured populations under stochastic positive density-dependence, extending understanding of long-term behaviors in such models.

## Key findings

- Populations either go extinct, grow unbounded, or have probabilistic outcomes.
- The models apply to spatially structured populations with Allee effects.
- Results are demonstrated with age-structured populations facing mate limitation.

## Abstract

Positive density-dependence occurs when individuals experience increased survivorship, growth, or reproduction with increased population densities. Mechanisms leading to these positive relationships include mate limitation, saturating predation risk, and cooperative breeding and foraging. Individuals within these populations may differ in age, size, or geographic location and thereby structure these populations. Here, I study structured population models accounting for positive density-dependence and environmental stochasticity i.e. random fluctuations in the demographic rates of the population. Under an accessibility assumption (roughly, stochastic fluctuations can lead to populations getting small and large), these models are shown to exhibit a dynamical trichotomy: (i) for all initial conditions, the population goes asymptotically extinct with probability one, (ii) for all positive initial conditions, the population persists and asymptotically exhibits unbounded growth, and (iii) for all positive initial conditions, there is a positive probability of asymptotic extinction and a complementary positive probability of unbounded growth. The main results are illustrated with applications to spatially structured populations with an Allee effect and age-structured populations experiencing mate limitation.