A simple model for resource dependent diffusion

Background/Question/Methods

Numerous contributions to ecological research have steamed from the incorporation of classical models from physics (e.g., diffusion, random-walk and ideal gas models) as null hypotheses in the interpretation of individual and population movement patterns. Ideal gas-based models (e.g., Holling’s disk equation) focus on prey/host/mate encounter dynamics whereas diffusion based models (e.g., Fisher’s diffusion model) focus on rates and dynamics of population spread.  Here, we demonstrate a synthetic approach that integrates these fundamentally-different processes, predicting how diffusion rates for a mobile consumer depend on the density and distribution of resources. Our model assumes a population of consumers moving at a constant speed and in independent directions while searching for resources. After a resource is encountered and handled, each successful consumer resumes movement in a new, uniformly random direction. We use computer simulation to show that the Gamma distribution adequately represents the distance between spatial locations of resources under a variety of realistic conditions. Our simulation shows that the spatial heterogeneity of the resource is denoted by the reciprocal of the shape parameter. We then employ an extension of the ideal gas model to express the expectancies for the step lengths and squared step lengths as functions of the resources’ Gamma shape and scale and the predator’s detection radius. Finally, we use Einstein’s formulation for Brownian motion to derive the diffusion coefficient of the population as function of the density and spatial heterogeneity of resources.

Results/Conclusions

The diffusion coefficient is an important predictor of the spatiotemporal dynamics of population distribution. Our model explicitly predicts that the diffusion coefficient of randomly moving consumers exponentially declines with an increase in the abundance and spatial aggregation of their resource. The displacement rate of moving particles decreases linearly with an increase in the abundance of the resource due to increased handling time. However, the displacement rate decreases exponentially with resource abundance because of shorter step lengths associated with more frequent resource encounters. As a result, animals are expected to spend more time in resource-rich areas and move quickly through resource-poor ones. The utility of this simple model lies in its potential to illuminate the processes and interactions governing the dynamics of individual and population movement. Conspecific interactions, local resource depletions, predator cognitive capacities and interactions with other environmental factors are just a few of many possible processes that may be revealed by examining deviations from the model’s predictions in natural systems.