Revisiting discrepancies between stochastic agent-based and deterministic models

Predicting which species will present (or absent) across geographical regions, and where, remains one of the important issues in ecology. From a methodical viewpoint, one of the concerns in examining species presence–absence across an environmental gradient is about the robustness of model-based predictions, which are given by distinct modelling frameworks used. Generally, different complexities and ecological factors are incorporated into such models, e.g. abiotic environments, spatial dispersal process, and stochasticity. Motivated by these ecological issues, we revisit a single-species logistic growth problem by employing stochastic agent-based model (ABM) and deterministic system and extend these frameworks to incorporate the effects of spatially changing environments. We observe that our ABM, which is formulated using random walk theory and birth–death process, demonstrates important qualitative behaviours that are consistent with the underlying theories of stochastic process. The results of ABM with large population sizes also agree with those of the deterministic equation. However, some discrepancies are observed when the population size is small. The ABM densities seem to underestimate the deterministic solutions, which illustrate the effects of stochasticity on small populations with some individuals may go extinct simply by chance. To quantify the underestimation of ABM as opposed to deterministic predictions, we employ certain probabilistic techniques: while the means of quasistationary probabilities distribution appear to give a counterintuitive prediction particularly near the edge of species ranges, the expected values given by state probabilities distribution are in agreement with the ABM densities observed for small population sizes across spatial locations. These salient observations depict emergent behaviours of stochastic ABM, which can contribute to additional insights on the dynamics of ecological species. It also shows how such small-scale interactions coupled with local dispersal and spatial phenomena occurring at a microscopic level can affect macroscopic-level dynamics. As such, comparing and contrasting the dynamics of different models can help in understanding the generality of ecological results and may offer important insights into the robustness of model-based predictions of species presence–absence.