Uniform predation risk in nature: common, inconspicuous, and a source of error to predation risk experiments

Abstract

Previous studies indicate that when predation risk is uniform across habitats, foragers concentrate their exploitation in fewer patches. Although uniform predation risk may seem rare in nature, some scenarios might cause it. Testing all scenarios in a single experiment is unfeasible; therefore, we developed a model that points whether concentration of exploitation in specific habitats due to uniform risk requires parameter values similar to what is found in literature. This model was based on Brown’s (Behav Ecol Sociobiol 22:37–47, 1988) fitness function but rescaled to multiple habitats and predators, including uniform risk predators. Deriving function’s maximum allowed comparisons with giving-up density studies. Results showed that uniform predation risk had a u-shaped effect in habitat exploitation, causing a concentration of habitat exploitation at probabilities of survival from 0.2 to 0.8. However, the length of this interval and degree of concentration depended on the value of safety to forager fitness. Heterogeneous, nonuniform, predation risk decreases habitat exploitation where it was higher, therefore suppressing the effect of uniform risk on prey behavior. Time spent in the focal habitat and metabolic costs reduced the detectability of habitat concentration, while total time did not. We also found that uniform risk reduced accuracy of heterogeneous risk measurements. Future studies should aim to control all possible predators, as even the mild ones can induce complex behavior.

Appendix

Mathematical demonstration that uniform predation risk is proportional to variance in habitat use, considering this study model

This section is a supplementary electronic material of the following: Menezes, JFS; Kotler, BP.; and Mourão, GM. Uniform predation risk in nature: common, inconspicuous, and a source of error to predation risk experiments published at behavioral ecology and sociobiology. Corresponding author, Menezes, Jorge F.S., is affiliated to Empresa Brasileira de Pecuária e Agricultura—Rua 21 de Setembro, 1880 - Bairro Nossa Senhora de Fátima. CP 109 - Corumbá, MS- Brazil – ZIP: 79320-900. Email: jorgefernandosaraiva@gmail.com.

A relationship between variance in habitat use and uniform risk can be proven by stating that missed opportunity costs are proportional to mean resource in patches after controlling the effects of predation risk and alternate activities. This relation was reported in the beginning of foraging ecology (Charnov 1976) and has been used in several modern methods (Rieucau et al. 2009). In our model, opportunity costs are represented by $\frac{\left(i-1\right)\phi }{p_i{p}_u\left(\partial F/\partial e\right)}$, where the denominator accounts for the effect of predation risk, in the missed opportunity cost. It follows that the numerator equals the effect of mean resource patches and the effect of alternate activities. The most parsimonious way to model both effect is as follows:

$$ {\phi}_i=d+b\times \overline{f} $$

where $\overline{f}$ is the average harvest rate of all habitats j ≠ i, representing mean resource in patches; b is an unknown constant representing the influence of resources in missing opportunity costs; and d represents alternative activities that contribute to missing opportunity costs (such as grooming and sleeping).

Substituting ϕ _i in Formula 5:

$$ {f}_i=\frac{\left(F+1\right)\left( \ln {\mu}_i- \ln {\mu}_u\right)}{\partial F/\partial e}+\frac{\left(i-1\right)\times \left(d+b\times \overline{f}\right)}{p_i{p}_u\left(\partial F/\partial e\right)}+{c}_i $$

This equation can be rearranged to show the relation between f _i and $\overline{f}$:

$$ {f}_i=\frac{\left(i-1\right)\times b\times \overline{f}}{p_i{p}_u\left(\partial F/\partial e\right)}+\frac{\left(F+1\right)\left( \ln {\mu}_i- \ln {\mu}_u\right)}{\partial F/\partial e}+\frac{\left(i-1\right)\times d}{p_i{p}_u\left(\partial F/\partial e\right)}+{c}_i $$

There are no square terms in the right side of this equation. Therefore, it is proportional to f _i  − $\overline{f}$. Given that condition, uniform risk is inversely proportional to f _i  − $\overline{f}$, and consequently, directly proportional to (f _i  − $\overline{f}$)², which is the numerator of the variance formula.