Stochastic and spatial models and their moment approximations

Many classical approaches to spatial population dynamics (e.g. reaction-diffusion equations) focus on large spatial scales where individual discreteness, and effects due to distances between individuals can be neglected. When looking instead at small-scale spatial structure, individual-based modelling might be more appropriate. This has been done traditionally for trees or bird nests but can be important also for smaller organisms, e.g. to understand constraints in cell motion. However, using IBMs often comes at a great computational cost, and many IBMs are only specified as computer algorithms. Thus theoreticians have a great motivation to harness the complexity of IBMs, and find creative ways to connect them to earlier theory. Fortunately, some IBMs can be defined mathematically, such as dynamic spatial point processes. These dynamic point processes were initially developed for plants (e.g. Bolker and Pacala 1997, Dieckmann and Law 2000), and allow for both exact stochastic simulations and the derivation of spatial moment equations (moments = generalisations of means and covariances). Spatial moment equations are convenient because they express spatial structure in a concise way, that can be compared to field data. I work on this topic with David Murrell (UCL) and we build on the predator-prey model presented in Murrell (2005) Am Nat 166:354 - 367.

The figure on the right shows a snapshot of a dynamic spatial point process (circles = predators, dots = prey items, shading = predation intensity). The key to obtaining that pattern is that attack probability decays with the distance from the predator home range centre (circle). In the limit of large interaction and movement distances between individuals, these models are equivalent to classical mean-field ecological models (i.e., Lotka-Volterra), so we hope developing theory in that area can contribute to bridge the gap between animal movement, space-use, and population dynamics.

Note that dynamic point processes are not the only mathematically amenable IBMs, as their cousins interacting particle systems (in discrete space) are very popular both in the mathematical literature (probability theory) and the ecological field of host-parasite dynamics.