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How optimal foragers should respond to habitat changes: on the consequences of habitat conversion

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Abstract

The marginal value theorem (MVT) provides a framework to predict how habitat modifications related to the distribution of resourcesover patches should impact the realized fitness of individuals and their optimal rate of movement (or patch residence times) across the habitat. The MVT theory has focused on the consequences of changing the shape of the gain functions in some patches, describing for instance, patch enrichment. However, an alternative form of habitat modification is habitat conversion, whereby patches are converted from one existing type to another (e.g., closed habitat to open habitat). In such a case, the set of gain functions existing in the habitat does not change, only their relative frequencies does. This case however has received comparatively little attention. Here we analyze mathematically the consequences of habitat conversion under the MVT. We study how realized fitness and the average rate of movement should respond to changes in the frequency distribution of patch-types and how they should covary. We further compare the response of optimal and non-plastic foragers. We find that the initial pattern of patch exploitation in a habitat, characterized by the regression slope of patch yields over residence times, can help predict the qualitative responses of fitness and movement rate following habitat conversion. We also find that for some habitat conversion patterns, optimal and non-plastic foragers exhibit qualitatively different responses, and that adaptive foragers can have opposite responses in the short- and long-term following habitat conversion. We suggest taking into account behavioral responses may help better understand the ecological consequences of habitat conversion.

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Acknowledgments

This work was supported by INRA and Université Côte d’Azur (IDEX JEDI).

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Correspondence to Vincent Calcagno.

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Appendix

Appendix

Derivation of Eq. 7

As explained in Section “Response of average movement rate to habitat conversion” in the main text, and following Calcagno et al. (2014b) to express the indirect variation of \(\left \langle {t}_{j}^{*}\right \rangle \), we can formulate the total variation of \(\left \langle T_{j}+{t}_{j}^{*}\right \rangle \) as

$$ \frac{d\left\langle T_{j}+{t}_{j}^{*}\right\rangle }{dx}=\sum\limits_{j=1}^{s}\frac{d p_{j}}{d x}\left( T_{j}+{t}_{j}^{*}\right)+\frac{{dE}_{n}^{*}}{dx}\frac{1}{H} $$

where H is the harmonic mean of the second time-derivatives, i.e.,

$$ H=1/\sum\limits_{j=1}^{s}p_{j}\left( \frac{d^{2}F_{j}({t}_{j}^{*})}{d{{t}_{j}^{2}}}\right)^{-1} $$

Replacing the variation of \({E}_{n}^{*}\) with its expression (given in Section “Response of realized fitness to habitat conversion” in the main text), we get

$$ \begin{array}{@{}rcl@{}} \frac{d\left\langle T_{j}+{t}_{j}^{*}\right\rangle }{dx}\!&=&\!\sum\limits_{j=1}^{s}\frac{dp_{j}}{dx}\left( T_{j}+{t}_{j}^{*}\right)\\&&\!+\frac{{E}_{n}^{*}}{H}\!\left( \!\frac{1}{\left\langle F_{j}({t}_{j}^{*})\right\rangle }\sum\limits_{j=1}^{s}\frac{dp_{j}}{dx}F_{j}({t}_{j}^{*}) - \frac{1}{\left\langle T_{j} + {t}_{j}^{*}\right\rangle }\sum\limits_{j=1}^{s}\frac{dp_{j}}{dx}(T_{j} + {t}_{j}^{*})\right) \end{array} $$

We can directly identify the partial log derivative of \(\left \langle F_{j}({t}_{j}^{*})\right \rangle \) and \(\left \langle T_{j}+{t}_{j}^{*}\right \rangle \) in the parenthesis, and upon factoring \({E}_{n}^{*}/\left |H\right |\) out we obtain

$$ \frac{d\left\langle {t}_{j}^{*}\right\rangle }{dx} = \frac{{E}_{n}^{*}}{\left|H\right|}\!\left( \!-\frac{\partial\ln\left\langle F_{j}({t}_{j}^{*})\right\rangle }{\partial x} + \frac{\left|H\right|}{{E}_{n}^{*}}\sum\limits_{j=1}^{s}\frac{dp_{j}}{dx}\left( T_{j}\!+{t}_{j}^{*}\right)\!+\frac{\partial\ln\left\langle T_{j}\!+{t}_{j}^{*}\right\rangle }{\partial x}\right) $$

Introducing the partial log derivative of \(\left \langle T_{j}+{t}_{j}^{*}\right \rangle \), and requiring the parenthesis to be negative (since average movement rate varies in opposite direction of \(\left \langle T_{j}+{t}_{j}^{*}\right \rangle \)) yield

$$ \frac{\partial\ln\left\langle F_{j}({t}_{j}^{*})\right\rangle }{\partial x} > \frac{\partial\ln\left\langle T_{j}+{t}_{j}^{*}\right\rangle }{\partial x}+\frac{\partial\left\langle T_{j}+{t}_{j}^{*}\right\rangle }{\partial x}\frac{\left|H\right|}{{E}_{n}^{*}} $$
(9)

This, replacing \({E}_{n}^{*}\) with its expression, yields Eq. 7.

Now, if all patches have on average the same travel time T, or if variation in travel time shows no consistent trend with habitat conversion, we have \(d\left \langle T_{j}\right \rangle /dx=0\). Note that under the second scenario (i.e., if all travel times are not equal), achieving a null derivative would in practice require specific forms of habitat conversion, owing to the constraint that the sum of the pj is constant. The constraint gradually vanishes as the number of patch types (s) gets large. From \(d\left \langle T_{j}\right \rangle /dx=0\) it follows

$$ \frac{\partial\left\langle T_{j}+{t}_{j}^{*}\right\rangle }{\partial x}= \frac{\partial\left\langle {t}_{j}^{*}\right\rangle }{\partial x} $$

Numerical simulations

In order to generate the numerical simulations presented in Fig. 4, we used a simple gradient ascent algorithm. Individuals were assumed to update each residence time (tj, omitting the asterisk as they need not be at optimal value) gradually, in the direction that (locally) increases the long-term average rate of gain (\({E}_{n}^{*}\)) and at a rate proportional to the fitness differential, i.e.,

$$ \frac{dt_{j}}{dt}=\omega \frac{\partial {E}_{n}^{*}}{\partial t_{j}} $$
(10)

where t denotes ecological time (on which habitat changes take place) and ω is a constant quantifying the speed of behavioral adjustments.

Habitat conversion is modeled by specifying how the patch relative frequencies change through time, i.e., by specifying functions pj(t). In the simulations of Fig. 4, the function was taken to be linear, so that relative frequencies changed linearly from their initial to their final values. It was assumed that individuals, prior to the onset of habitat change, had settled at the optimal residence times for the initial habitat. When ω is very large, the forager is effectively optimal and immediately adjusts its strategy to match current habitat conditions (in accordance with the MVT). When ω = 0, the individual does not adjust its strategy (non-plastic forager). Intermediate values of ω represent less-than-perfect plastic foragers that gradually adapt to habitat changes. To generate Fig. 4 in the main article, we used ω = 0 for the static forager and ω = 0.3 for the plastic forager.

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Calcagno, V., Hamelin, F., Mailleret, L. et al. How optimal foragers should respond to habitat changes: on the consequences of habitat conversion. Theor Ecol 13, 165–175 (2020). https://doi.org/10.1007/s12080-019-00437-7

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