Population Dynamics and Spatial Scale: Effects of System Size on Population Persistence

Abstract

This paper explores the limitations of ecological models based on differential equations. We test three different model implementations of the processes represented by the Lotka–Volterra predator–prey equations. We use these models to investigate the system dynamics suppressed by the assumptions contained in differential-equation-based models; specifically, we examine the effects of space and demographic stochasticity. The differential equation implementation of the Lotka–Volterra processes assumes that there are enough individuals in the system that the effects of demographic stochasticity can be ignored and that the interactions between individuals always occur as though the system were continually well mixed or spatially random. The model predicts a neutrally stable equilibrium and infinite persistence, regardless of the size of the system. We use a stochastic birth–death (SBD) model to explore the possible effects of demographic stochasticity on the basic Lotka–Volterra model. The results show that, over a wide range of system sizes, the time to extinction is finite and increases linearly with the total size of the system. We introduce a new type of spatially explicit individual-based model called the Heuristic Asynchronous Discrete Event Simulation or HADES. HADES adds an explicit component of space to the SBD version of our process. We compare the system dynamics of the SBD and HADES models over the same system sizes, using the same demographic parameters, thereby partitioning the effects of demographic stochasticity and space. As system sizes are increased, the dynamics of the HADES model with respect to the SBD model, as measured by time to extinction, move from equivalent to less stable to much more stable. We analyze the effects of space on the system dynamics using animation, statistical point process analysis, and predicted system dynamics (spatial effects on the predation rate). We then show that both the destabilizing and stabilizing effects of space with respect to the SBD dynamics can be accounted for by specific nonrandom spatial patterns. We contrast our results with three commonly invoked mechanisms that affect the stability of predator–prey systems. We comment on the strengths and limitations of differential-equation-based models.