Advection-diffusion equations for generalized tactic searching behaviors

Many organisms search for limiting resources by using repeated responses to local cues, which cumulatively cause movement towards more favorable parts of their environment. This paper presents a general asymptotic expression, derived under the assumption of shallow environmental gradients, for the population-level flux of organisms moving at a constant speed and reorienting at rates determined by the environmental conditions experienced since the last reorientation. The expression takes the form of an advection-diffusion equation, in which the diffusivity and advection velocity are determined by statistics of the turning algorithm that are directly comparable to empirical observations. This work provides a mechanism with which to systematically evaluate a wide variety of tactic and kinetic strategies for determining turning behaviors. The model is applied to searchers on spatially-variable, random distributions of discrete resource patches. Such algorithms are functions of the integrated resource density encountered between turns. It is shown that behaviors in which the turning time distribution is a function of integrated density cannot result in taxis. In contrast, behaviors in which the turning rate is a function of integrated density can result in taxis. These two classes of search algorithm differ in that the latter requires the searcher to "learn" about its local environment, whereas the former requires no such assessment. This suggests neural or physiological mechanisms for remembering previous encounters may be a biological requirement for searchers on discrete resource distributions.