Killing the atto-fox: A new algorithm for deterministic simulation in discrete state space

Background/Question/Methods

Commonly used ecological models, in both discrete and continuous time, assume state space to be continuous. However, organisms are discrete. This concern is often answered by saying that large populations can be approximated by continuous quantities, while small populations require stochastic models. However, when a stochastic simulation method that assumes a discrete state space, such as Gillespie's algorithm, is used, attributing an effect seen in such a simulation to either stochasticity or discreteness is problematic.

Even when initial quantities are “large”, a population can approach extinction and then rebound. This gives rise to a deep problem. It is possible to say that “36.24 white-tailed deer” simply means that, when random fluctuations are taken into account, the average value is 36.24. However, this move becomes dangerous when the model predicts “0.003624 deer”, because populations can rebound from these fractional numbers to large ones, a problem sometimes termed the atto-fox effect. Thus, the predictions of the continuous model are artifactual and qualitatively wrong. Rounding is not a satisfactory way to address this problem because the very fact that a simulation method generates non-integer quantities despite integer initial conditions and biologically reasonable parameters indicates that something is fundamentally wrong with the method.

Results/Conclusions

A simple dynamical simulation algorithm that depicts flows as discrete pulses was developed. Using this algorithm to simulate Lotka-Volterra predator-prey models yields results very similar to those obtained by numerically integrating the differential equations, even when populations are small. Other dynamically interesting phenomena, such as oscillations arising exclusively from the discreteness of flow pulses, have also been observed. The dynamical consequences of small, frequent flows vs. large, infrequent ones will also be discussed. The new algorithm raises theoretical and empirical research questions, such as how to distinguish oscillations resulting from discreteness from those resulting from stochasticity and what stability concepts are meaningful in a discrete world.