Relaxing the mean-field assumption and scaling-up models of predator-prey interactions

Predator-prey interactions are often modelled without including a spatial dimension: the most common “well-mixed” models assume, for mathematical convenience, that any individual experiences densities equal to the landscape-scale average densities (also called the “mean-field approximation“). This can happen for instance when predators have access to all prey items with equal probability. Although this is more likely if the area considered is small, predator-prey models are typically used to model interaction over large areas (i.e. a hundred to a million times the animal's perceptual radius). That is why we may need to `scale-up' properly predator-prey models.

Note that even when spatial structure is prese

nt, the mean-field approximation might be a good caricature at times (for instance, if there is a lot more temporal than spatial variation). That depends on the particulars of the spatial structure, however, as we know that numerous predators have a home range or territory and can concentrate their foraging effort around their nest or burrow (distance effect). We actually illustrated what kind of home range / territorial structure can give rise to non-mixing in Fig. 1 of Barraquand F. & Murrell D. 2013. Scaling up predator-prey dynamics with spatial moment equations. Methods in Ecology and Evolution, 4: 276–289. And when predator-prey systems are not well mixed, model results can substantially change (e.g. development of negative spatial correlations between predators and prey, decrease or increase of the amplitude of predator-prey cycles). Two routes to introduce such spatial effects into predator-prey theory are possible. The simplest option is to subdivide the landscape into a number of units, and use the mean-field approximation on those smaller spatial units. Another consists in explicitly introducing probabilities of attack that decay with the distance between predators and prey (Murrell Am Nat 2005), thanks to stochastic and spatial models.