The Theory of Individual Based Discrete-Time Processes

Authored by Alan J McKane, Joseph D Challenger, Duccio Fanelli

Date Published: 2014

DOI: 10.1007/s10955-014-0990-2

Sponsors: Italian Ministries

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

Abstract

A general theory is developed to study individual based models which are discrete in time. We begin by constructing a Markov chain model that converges to a one-dimensional map in the infinite population limit. Stochastic fluctuations are hence intrinsic to the system and can induce qualitative changes to the dynamics predicted from the deterministic map. From the Chapman-Kolmogorov equation for the discrete-time Markov process, we derive the analogues of the Fokker-Planck equation and the Langevin equation, which are routinely employed for continuous time processes. In particular, a stochastic difference equation is derived which accurately reproduces the results found from the Markov chain model. Stochastic corrections to the deterministic map can be quantified by linearizing the fluctuations around the attractor of the map. The proposed scheme is tested on stochastic models which have the logistic and Ricker maps as their deterministic limits.

Tags

Competition noise Cycles Ecological models Direct dynamical test Deterministic chaos Biological populations Stable points Series