Patch Choice under Predation Hazard

In this paper we study optimal animal movement in heterogeneous environments consisting of several food patches in which animals trade-off energy gain versus predation risk. We derive a myopic optimization rule describing optimal animal movements by fitness maximization assuming an animal state is described by a single quantity (such as weight, size, or energy reserves). This rule predicts a critical state at which an animal should switch from a more dangerous and more profitable patch to a less dangerous and less profitable patch. Qualitatively, there are two types of behavior: either the animal switches from one patch to another and stays in the new patch for some time before it switches again, or the animal switches between two patches instantaneously. The former case happens if animal state growth is positive in all patches, while the latter case happens if animal state growth is negative in one patch. In particular, this happens if one patch is a refuge. We consider in detail two special cases. The first one assumes a linear animal state growth while the second assumes a saturating animal state growth described by the von Bertalanffy curve. For the first model the proportion of time spent in the more profitable and more risky patch increases with profitability of this patch when state growth is positive in both patches. On contrary, if state growth is negative in the less profitable and safer patch, animals spend proportionally less time in the more profitable and more risky patch as its profitability increases. As a function of the predation risk in the more profitable patch the time spent there proportionally decreases. When animal state growth is described by the saturating curve, time spent in the more risky patch is a hump-shaped curve if state growth is positive in both patches. Our results extend the μ/f rule, which predicts that animals should behave in such a way as to minimize mortality risk to resource intake ratio.