Abstract
The Marginal Value Theorem (MVT) is a cornerstone of biological theory. It connects the quality and distribution of patches in a fragmented habitat to the optimal time an individual should spend exploiting them, and thus its optimal rate of movement. However, predictions regarding how habitat alterations should impact optimal strategies have remained elusive, with heavy reliance on graphical arguments. Here we derive the sensitivity of realized fitness and optimal residence times to general habitat attributes, for homogeneous and heterogeneous habitats, retaining the level of generality of the MVT. We provide new predictions on how altering travel times, patch qualities and/or relative abundances should affect optimal strategies, and study the consequences of habitat heterogeneity. We show that knowledge of average characteristics is in general not sufficient to predict the change in the average rate of movement. We apply our results to examine the conditions under which the optimal strategies are invariant to scaling. We prove a previously conjectured form of invariance in homogeneous habitats, but show that invariances to scaling are not generic in heterogeneous habitats. We also consider the relative exploitation of patches that differ in quality, clarifying the conditions under which it is adaptive to stay longer on poorer patches.
Highlights
The Marginal Value Theorem (MVT) is an important and popular tenet of biological theory (Stephens and Krebs 1986), combining high generality and a relatively simple mathematical formulation
We consider two forms of scaling invariance of the optimal strategies that have been attributed to the MVT based on particular functions, the first corresponding to scaling the gain function vertically, the second corresponding to scaling time
One prediction often attributed to the MVT is that, in a given heterogeneous habitat, optimal residence times should rank in the same order as patch qualities, where quality is intended as having a ’better’ gain function (Parker and Stuart 1976; Kelly 1990; Wajnberg et al 2000)
Summary
The Marginal Value Theorem (MVT) is an important and popular tenet of biological theory (Stephens and Krebs 1986), combining high generality and a relatively simple mathematical formulation. When resources are distributed as discrete patches throughout the habitat, the MVT predicts how long an individual should spend exploiting each patch before moving to another, depending on the kinetics of fitness accumulation within patches, and on the time it takes to move between patches (the travel time; Charnov 1976) This question has many applications in evolutionary biology, and beyond (Hayden et al 2011; Rijnsdorp et al 2011). Computing the optimal residence times requires specifying a specific functional form for the accumulation of gains in patches, and, even so, it is usually impossible to solve the equations analytically This is at best feasible for some simple functions in homogeneous habitats (i.e. if all patches are identical; Stephens and Krebs 1986) or using tractable approximations (Parker and Stuart 1976; McNair 1982; Stephens and Dunbar 1993; Charnov and Parker 1995; Ranta et al 1995). We will use our results to reanalyze the main predictions attributed to the MVT, in particular the effect of varying travel time, the consequences of improving quality, the invariance of the optimal strategies upon vertical and horizontal scalings, and the relative time individuals should spend on patches of different qualities
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