Discrete Stochastic Models in Continuous Time for Ecology

Discrete stochastic models in continuous time are the simplest way to turn survival analysis of an ecological system into a simulation of that system. They build models most faithful to a picture of individuals making choices which affect each other. This article describes how rules for individual behavior can construct a simulation of a population. It argues that, within the context of discrete stochastic simulation, hazard rates for transitions are intrinsic descriptions of individual behavior. Discrete stochastic models in continuous time are associated with the Gillespie algorithm and its variants, including the Next Reaction algorithm. The Gillespie algorithm samples to find the next state and time of a stochastic process. The process and sampling algorithm are separate. This article won’t define a new sampling algorithm but will examine the stochastic process that is sampled. Almost every discrete stochastic model in continuous time used for ecology is based upon a chemical kinetics model. These chemical kinetics models take the form of chemical species which interact according to stoichiometry at rates specified by propensities. They have been enormously successful for ecological applications such as stochastic movement models[1, 2, 3, 4] and any model that builds group behavior from individual behavior[5, 6, 7]. Clear statement of chemical kinetics as a process leads to clear specification of chemical kinetics models as matrices and vectors in Systems Biology Markup Language and computer code which, in turn, leads to their utility to solve common problems. The chemical kinetics model constrains expression of ecological problems in two ways. The first is that every state in a chemical kinetics model is a count of chemical species when an ecologist might want the state of an individual to carry properties that record its life history. The second is that propensities are usually simple rates when an ecologist might want rates to be time-dependent, so that an insect gets more hungry over time, or a cow waits to reproduce again, or a frog jumps quickly if it jumps early. These two important modifications are that individuals hold state and their hazard rates for change are timedependent.