Differential equation model for central-place foragers with memory: implications for bumble bee crop pollination

Bumble bees provide valuable pollination services to crops around the world. However, their populations are declining in intensively farmed landscapes. Understanding the dispersal behaviour of these bees is a key step in determining how agricultural landscapes can best be enhanced for bumble bee survival. Here we develop a partial integro-differential equation model to predict the spatial distribution of foraging bumble bees in dynamic heterogeneous landscapes. In our model, the foraging population is divided into two subpopulations, one engaged in an intensive search mode (modeled by diffusion) and the other engaged in an extensive search mode (modeled by advection). Our model considers the effects of resource-dependent switching rates between movement modes, resource depletion, central-place foraging behaviour, and memory. We use our model to investigate how crop pollination services are affected by wildflower enhancements. We find that planting wildflowers such that the crop is located in between the wildflowers and the nest site can benefit crop pollination in two different scenarios. If the bees do not have a strong preference for wildflowers, a small or low density wildflower patch is beneficial. If, on the other hand, the bees strongly prefer the wildflowers, then a large or high density wildflower patch is beneficial. The increase of the crop pollination services in the later scenario is of remarkable magnitude.