### Abstract

This is a mathematical study of the interactions between non-linear feedback (density dependence) and uncorrelated random noise in the dynamics of unstructured populations. The stochastic non-linear dynamics are generally complex, even when the deterministic skeleton possesses a stable equilibrium. There are three critical factors of the stochastic non-linear dynamics; whether the intrinsic population growth rate ( λ ) is smaller than, equal to, or greater than 1; the pattern of density dependence at very low and very high densities; and whether the noise distribution has exponential moments or not. If λ < 1, the population process is generally transient with escape towards extinction. When λ {greater than or slanted equal to} 1, our quantitative analysis of stochastic non-linear dynamics focuses on characterizing the time spent by the population at very low density (rarity), or at high abundance (commonness), or in extreme states (rarity or commonness). When λ > 1 and density dependence is strong at high density, the population process is recurrent: any range of density is reached (almost surely) in finite time. The law of time to escape from extremes has a heavy, polynomial tail that we compute precisely, which contrasts with the thin tail of the laws of rarity and commonness. Thus, even when λ is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When λ = 1 and density dependence is weak at low density, rarity follows a universal power law with exponent - frac(3, 2). We provide some mathematical support for the numerical conjecture [Ferriere, R., Cazelles, B., 1999. Universal power laws govern intermittent rarity in communities of interacting species. Ecology 80, 1505-1521.] that the - frac(3, 2) power law generally approximates the law of rarity of 'weakly invading' species with λ values close to one. Some preliminary results for the dynamics of multispecific systems are presented.

Original language | English (US) |
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Pages (from-to) | 351-366 |

Number of pages | 16 |

Journal | Theoretical Population Biology |

Volume | 69 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2006 |

### Keywords

- Ecological timescales
- Environmental stochasticity
- Markov chains
- Martingales
- On-off intermittency
- Population dynamics
- Power law
- Rarity
- Stochastic non-linear difference equations

### ASJC Scopus subject areas

- Ecology, Evolution, Behavior and Systematics

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## Cite this

*Theoretical Population Biology*,

*69*(4), 351-366. https://doi.org/10.1016/j.tpb.2006.01.005