The term in front of the average goes to 1 with probability one, and the the average goes almost surely to p by the strong law of large numbers applied to the Bernoulli random variables $B_k$ (since $N_n \to +\infty$ with probability one). Finally, for the convergence in law, we exploit again the previous results, which can be gathered as :

$$ \frac{N_n - a_n^{(d)}}{b_n^{(d)}} \xrightarrow{\text{law}} W^{(d)} \qquad (B.30) $$

where

$$ W^{(1)} = \max_{0 \le s \le 1} \mathcal{B}_s - \min_{0 \le s \le 1} \mathcal{B}_s, \quad a_n^{(1)} = 0, \quad b_n^{(1)} = \sqrt{n}, $$

$$ W^{(2)} = -2\pi^2 \mathcal{L}_G, \quad a_n^{(2)} = \frac{\pi n}{\log n}, \quad b_n^{(2)} = \frac{n}{(\log n)^2}, $$

$$ W^{(3)} = \mathcal{N}(0, 1), \quad a_n^{(3)} = \kappa(3)n, \quad b_n^{(3)} = \sigma\sqrt{n \log n}, $$

see (A.9), (A.17), and (A.22).

We use Lévy's convergence theorem to get convergence in law for $N_n^{(p)}$ from the one for $N_n$. More precisely, (B.28) yields

$$ N_n^{(p)} = p N_n + \sum_{k=1}^{N_n} (V_k - p). $$

Since $E(e^{\lambda(V_k-p)}) = (1-p)e^{-\lambda p} + pe^{\lambda(1-p)}$ ($\lambda \in \mathbb{C}$), and conditioning upon $N_n$, we get

$$ \begin{aligned} E \left( e^{it(N_n^{(p)} - pa_n^{(d)})} / b_n^{(d)} \right) &= e^{-itpa_n^{(d)}/b_n^{(d)}} E \left( \left( (1-p) e^{-itp/b_n^{(d)}} + p e^{it(1-p)/b_n^{(d)}} \right)^{N_n} e^{itp N_n / b_n^{(d)}} \right) \\ &= e^{-itpa_n^{(d)}/b_n^{(d)}} E \left( e^{(itp/b_n^{(d)} + o(1)) N_n} \right) \\ &= E \left( e^{itp(N_n - a_n^{(d)})} / b_n^{(d)} \right) + o(1) \\ \xrightarrow{n \to +\infty} & E \left( e^{itpW^{(d)}} \right) \end{aligned} $$

using (B.30). By Lévy's convergence theorem, this gives

$$ \frac{N_n^{(p)} - p a_n^{(d)}}{b_n^{(d)}} \xrightarrow{\text{law}} p W^{(d)}. $$
# C Derivation of the coefficient of variation

## C.1 Reminder: Distribution of the consumption rate

The distribution of the number of prey consumed by a single predator after a time $t$ is given in App. B. Recall that the consumption rate (or functional response) after a time $t$ is defined as

$$F_t^{(d)} := \frac{R_t^{(d)}}{t}$$

which gives

$$F_t^{(d)} = \begin{cases} \frac{p}{\sqrt{\tau_e t}} W_t^{(1)} + o_p\left(\frac{1}{\sqrt{t}}\right) & \text{if } d=1 \\ \frac{\pi p}{\tau_e \ln t/\tau_e} \left(1 - \frac{1}{\ln t/\tau_e} \left(\gamma + \pi \frac{\tau_h}{\tau_e} + 2\pi W_t^{(2)}\right)\right) + o_p\left(\left(\frac{1}{\ln t}\right)^2\right) & \text{if } d=2 \\ \frac{\kappa p}{\tau_e + \kappa \tau_h} + p \sqrt{\frac{\ln t/\tau_e}{\tau_e(1+\kappa \tau_h/\tau_e)^3}} W_t^{(3)} + o_p\left(\sqrt{\frac{\ln t}{t}}\right) & \text{if } d=3 \end{cases} \quad (C.1)$$

where $p$ is the probability that a site contains a prey, $\gamma \approx 0.577$ is Euler-Mascheroni constant, $\kappa \approx 0.659$ is the probability that a simple symmetric random walk never returns to the origin in $Z^3$, and $W_t^{(d)} =_{\text{law}} W^{(d)}$ are different random variables depending on dimensionality (see App. B.2 for details): $W^{(1)}$ is the difference between the max and min position of a 1-dimensional Brownian motion: $W^{(1)} = \max_{s \le 1} B_s^{(1)} - \min_{s \le 1} B_s^{(1)}$; $W^{(2)} = L_G$ is the local time of self-intersection of a 2-dimensional Brownian motion; $W^{(3)} = N(0, \eta^2)$ is a centred Gaussian distribution with a constant variance $\eta^2$. Finally, $o_p(\eta_t)$ is defined such that for any $\varepsilon > 0$, $P(o_p(\eta_t) \ge \varepsilon\eta_t) \to 0$ as $t \to \infty$.

The mean, variance and the coefficient of variation of the consumption rate $F_t$ are summarized and illustrated in the following table and figures:

<table><caption>Table 1: Summary of statistics for the distribution of the consumption rate.</caption><thead><tr><th>Dimension</th><th>Expectation</th><th>Variance</th><th>Coefficient of variation</th></tr></thead><tbody><tr><td>1D</td><td>2p√(2/π) (1/√τ<sub>e</sub>t)</td><td>4p<sup>2</sup> (ln 2 - 2/π) / tτ<sub>e</sub></td><td>√(π/2) ln 2 - 1 ≈ 0.298</td></tr><tr><td>2D</td><td>p(π/τ<sub>e</sub>) (1 - γ + τ<sub>h</sub>π/τ<sub>e</sub>) / (ln t/τ<sub>e</sub>)</td><td>p<sup>2</sup> (2Kπ<sup>2</sup>) / τ<sub>e</sub><sup>2</sup>(ln t/τ<sub>e</sub>)<sup>4</sup></td><td>(2K/π) log t/τ<sub>e</sub> - γ - πτ<sub>h</sub>/τ<sub>e</sub> ≈ 1.3 / (ln t/τ<sub>e</sub>)</td></tr><tr><td>3D</td><td>p(κ(3)/τ<sub>e</sub> + κ(3)τ<sub>h</sub>)</td><td>p<sup>2</sup> (27κ(3)<sup>4</sup>) / (2π<sup>2</sup>) (ln t/τ<sub>e</sub>) / (τ<sub>e</sub>(1+κ(3)τ<sub>h</sub>/τ<sub>e</sub>)<sup>3</sup>)</td><td>(0.8/√1 + 0.7τ<sub>h</sub>/τ<sub>e</sub>) √(log t/τ<sub>e</sub>)</td></tr></tbody></table>
# C Derivation of the coefficient of variation

## C.1 Reminder: Distribution of the consumption rate

The distribution of the number of prey consumed by a single predator after a time $t$ is given in App. B. Recall that the consumption rate (or functional response) after a time $t$ is defined as

$$F_t^{(d)} := \frac{R_t^{(d)}}{t}$$

which gives

$$F_t^{(d)} = \begin{cases} \frac{p}{\sqrt{\tau_e t}} W_t^{(1)} + o_p\left(\frac{1}{\sqrt{t}}\right) & \text{if } d=1 \\ \frac{\pi p}{\tau_e \ln t/\tau_e} \left(1 - \frac{1}{\ln t/\tau_e} \left(\gamma + \pi \frac{\tau_h}{\tau_e} + 2\pi W_t^{(2)}\right)\right) + o_p\left(\left(\frac{1}{\ln t}\right)^2\right) & \text{if } d=2 \\ \frac{\kappa p}{\tau_e + \kappa \tau_h} + p \sqrt{\frac{\ln t/\tau_e}{\tau_e(1+\kappa \tau_h/\tau_e)^3}} W_t^{(3)} + o_p\left(\sqrt{\frac{\ln t}{t}}\right) & \text{if } d=3 \end{cases} \quad (C.1)$$

where $p$ is the probability that a site contains a prey, $\gamma \approx 0.577$ is Euler-Mascheroni constant, $\kappa \approx 0.659$ is the probability that a simple symmetric random walk never returns to the origin in $Z^3$, and $W_t^{(d)} =_{\text{law}} W^{(d)}$ are different random variables depending on dimensionality (see App. B.2 for details): $W^{(1)}$ is the difference between the max and min position of a 1-dimensional Brownian motion: $W^{(1)} = \max_{s \le 1} B_s^{(1)} - \min_{s \le 1} B_s^{(1)}$; $W^{(2)} = L_G$ is the local time of self-intersection of a 2-dimensional Brownian motion; $W^{(3)} = N(0, \eta^2)$ is a centred Gaussian distribution with a constant variance $\eta^2$. Finally, $o_p(\eta_t)$ is defined such that for any $\varepsilon > 0$, $P(o_p(\eta_t) \ge \varepsilon\eta_t) \to 0$ as $t \to \infty$.

The mean, variance and the coefficient of variation of the consumption rate $F_t$ are summarized and illustrated in the following table and figures:

<table><caption>Table 1: Summary of statistics for the distribution of the consumption rate.</caption><thead><tr><th>Dimension</th><th>Expectation</th><th>Variance</th><th>Coefficient of variation</th></tr></thead><tbody><tr><td>1D</td><td>2p√(2/π) (1/√τ<sub>e</sub>t)</td><td>4p<sup>2</sup>(ln 2-2/π) / (tτ<sub>e</sub>)</td><td>√(π/2) ln 2 - 1 ≈ 0.298</td></tr><tr><td>2D</td><td>p(π/τ<sub>e</sub>) (1 - (γ+τ<sub>h</sub>π/τ<sub>e</sub>)/(ln t/τ<sub>e</sub>))</td><td>p<sup>2</sup>(2Kπ<sup>2</sup>) / (τ<sub>e</sub><sup>2</sup>(ln t/τ<sub>e</sub>)<sup>4</sup>)</td><td>(log t/τ<sub>e</sub> - γ - πτ<sub>h</sub>/τ<sub>e</sub>) / (1.3 log t/τ<sub>e</sub>)</td></tr><tr><td>3D</td><td>p(κ(3)/(τ<sub>e</sub>+κ(3)τ<sub>h</sub>))</td><td>p<sup>2</sup>(27κ(3)<sup>4</sup>) / (2π<sup>2</sup>) (ln t/τ<sub>e</sub>) / (tτ<sub>e</sub>(1+κ(3)τ<sub>h</sub>/τ<sub>e</sub>)<sup>3</sup>)</td><td>(0.8) / (√(1+0.7τ<sub>h</sub>/τ<sub>e</sub>)) √(log t/τ<sub>e</sub>) / (t/τ<sub>e</sub>)</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Figure C.1: Illustration of the mean and standard deviation of the consumption rate $F_t$ (parameters: $t = 10^3$, $\tau_e = 1$, $\tau_h = 0.045$).</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure C.1: Illustration of the mean and standard deviation of the consumption rate $F_t$ (parameters: $t = 10^3$, $\tau_e = 1$, $\tau_h = 0.045$).</figcaption></figure>
## C.2 Coefficient of variation of the consumption rate with between individuals variability

Suppose now that the consumption rate is a random variable which depends on two independent sources of random variations denoted $\omega_1$ and $\omega_2$, such that $F_t(\omega_1, \omega_2) := R_t(\omega_1, \omega_2)/t$. $\omega_1$ is the between-individual variability for searching and handling, i.e. parameters involved in searching $\tau_c(\omega_1)$ and handling $\tau_h(\omega_1)$ for a focal individual are randomly chosen in distributions depending on $\omega_1$. The parameters involved in the searching and handling times can have any explicit form (not necessarily Gaussian) but with finite mean and variance. $\omega_2$ is the variability due to the foraging stochastic process itself, i.e. the stochastic trajectory of the focal individual when foraging follows a random distribution $W^{(d)}(\omega_2)$. The distribution $W^{(d)}(\omega_2)$ followed by the trajectories directly emerges from the assumptions of the foraging process, here the simple random walk in dimension $d$.

As shown in (C.1), when the foraging time $t$ is large, the distribution of the consumption rate in a $d$-dimension space after a time $t$ for a focal individual takes the general form

$$F_t^{(d)}(\omega_1, \omega_2) = m_t^{(d)}(\omega_1) + s_t^{(d)}(\omega_1)W_t^{(d)}(\omega_2) + o_P(s_t^{(d)}(\omega_1)), \quad (C.2)$$

where $m_t^{(d)}(\omega_1)$ and $s_t^{(d)}(\omega_1)$ are functions which depend on time $t$, the dimension of space $d$, and the properties of individuals given by parameters $\tau_h(\omega_1)$ and $\tau_c(\omega_1)$. When these parameters are compactly supported in $(0, \infty)$, the proof of uniform integrability, namely Proposition B.4, can be directly extended, since all bounds are uniform with respect to parameters. We obtain that the mean and variance of the consumption rate satisfy, for $t \to \infty$ (thereafter notations $\omega_1$ and $\omega_2$ are dropped for the sake of simplicity):

$$ \begin{aligned} \mathbb{E}[F_t^{(d)}] & \sim \mathbb{E}[m_t^{(d)}] + \mathbb{E}[s_t^{(d)}W^{(d)}], \\ \operatorname{Var}[F_t^{(d)}] & \sim \mathbb{E}\left[(m_t^{(d)} + s_t^{(d)}W^{(d)})^2\right] - \mathbb{E}\left[(m_t^{(d)} + s_t^{(d)}W^{(d)})\right]^2 \\ & \sim \mathbb{E}\left[(m_t^{(d)})^2\right] - \mathbb{E}\left[(m_t^{(d)})^2\right] \\ & \quad + \mathbb{E}\left[(s_t^{(d)}W^{(d)})^2\right] - \mathbb{E}\left[(s_t^{(d)}W^{(d)})\right]^2 \\ & \quad + 2\left(\mathbb{E}\left[(m_t^{(d)}s_t^{(d)}W^{(d)})\right] - \mathbb{E}[m_t^{(d)}]\mathbb{E}[s_t^{(d)}W^{(d)}]\right). \end{aligned} $$

As the sources of variability $\omega_1$ and $\omega_2$ are independent, we have

$$ \begin{aligned} \mathbb{E}[s_t^{(d)}W^{(d)}] &= \mathbb{E}[s_t^{(d)}]\mathbb{E}[W^{(d)}], \\ \operatorname{Var}[s_t^{(d)}W^{(d)}] &= \mathbb{E}\left[(s_t^{(d)})^2\right]\operatorname{Var}[W^{(d)}] + \operatorname{Var}[s_t^{(d)}]\mathbb{E}[W^{(d)}]^2. \end{aligned} $$

In addition, as we have $\mathbb{E}[m_t^{(1)}] = 0$ and $\mathbb{E}[W^{(2)}] = \mathbb{E}[W^{(3)}] = 0$ (Eqs. C.1), the expression of the variance of the consumption rate $F_t^{(d)}$ further simplifies to
## C.2 Coefficient of variation of the consumption rate with between individuals variability

Suppose now that the consumption rate is a random variable which depends on two independent sources of random variations denoted $\omega_1$ and $\omega_2$, such that $F_t(\omega_1, \omega_2) := R_t(\omega_1, \omega_2)/t$. $\omega_1$ is the between-individual variability for searching and handling, i.e. parameters involved in searching $\tau_c(\omega_1)$ and handling $\tau_h(\omega_1)$ for a focal individual are randomly chosen in distributions depending on $\omega_1$. The parameters involved in the searching and handling times can have any explicit form (not necessarily Gaussian) but with finite mean and variance. $\omega_2$ is the variability due to the foraging stochastic process itself, i.e. the stochastic trajectory of the focal individual when foraging follows a random distribution $W^{(d)}(\omega_2)$. The distribution $W^{(d)}(\omega_2)$ followed by the trajectories directly emerges from the assumptions of the foraging process, here the simple random walk in dimension $d$.

As shown in (C.1), when the foraging time $t$ is large, the distribution of the consumption rate in a $d$-dimension space after a time $t$ for a focal individual takes the general form

$$F_t^{(d)}(\omega_1, \omega_2) = m_t^{(d)}(\omega_1) + s_t^{(d)}(\omega_1)W_t^{(d)}(\omega_2) + o_P(s_t^{(d)}(\omega_1)), \quad (C.2)$$

where $m_t^{(d)}(\omega_1)$ and $s_t^{(d)}(\omega_1)$ are functions which depend on time $t$, the dimension of space $d$, and the properties of individuals given by parameters $\tau_h(\omega_1)$ and $\tau_c(\omega_1)$. When these parameters are compactly supported in $(0, \infty)$, the proof of uniform integrability, namely Proposition B.4, can be directly extended, since all bounds are uniform with respect to parameters. We obtain that the mean and variance of the consumption rate satisfy, for $t \to \infty$ (thereafter notations $\omega_1$ and $\omega_2$ are dropped for the sake of simplicity):

$$\begin{aligned}
\mathbb{E}[F_t^{(d)}] & \sim \mathbb{E}[m_t^{(d)}] + \mathbb{E}[s_t^{(d)}W^{(d)}], \\
\mathrm{Var}[F_t^{(d)}] & \sim \mathbb{E}\left[(m_t^{(d)} + s_t^{(d)}W^{(d)})^2\right] - \mathbb{E}\left[(m_t^{(d)} + s_t^{(d)}W^{(d)})\right]^2 \\
& \sim \mathbb{E}\left[(m_t^{(d)})^2\right] - \mathbb{E}\left[(m_t^{(d)})\right]^2 \\
& \quad + \mathbb{E}\left[(s_t^{(d)}W^{(d)})^2\right] - \mathbb{E}\left[(s_t^{(d)}W^{(d)})\right]^2 \\
& \quad + 2\left(\mathbb{E}\left[(m_t^{(d)}s_t^{(d)}W^{(d)})\right] - \mathbb{E}[m_t^{(d)}]\mathbb{E}[s_t^{(d)}W^{(d)}]\right).
\end{aligned}$$

As the sources of variability $\omega_1$ and $\omega_2$ are independent, we have

$$\begin{aligned}
\mathbb{E}[s_t^{(d)}W^{(d)}] &= \mathbb{E}[s_t^{(d)}]\mathbb{E}[W^{(d)}], \\
\mathrm{Var}[s_t^{(d)}W^{(d)}] &= \mathbb{E}\left[(s_t^{(d)})^2\right]\mathrm{Var}[W^{(d)}] + \mathrm{Var}[s_t^{(d)}]\mathbb{E}[W^{(d)}]^2.
\end{aligned}$$

In addition, as we have $\mathbb{E}[m_t^{(1)}] = 0$ and $\mathbb{E}[W^{(2)}] = \mathbb{E}[W^{(3)}] = 0$ (Eqs. C.1), the expression of the variance of the consumption rate $F_t^{(d)}$ further simplifies to
$$ \mathrm{Var}[F_t^{(d)}] \sim \mathbb{E}\left[(m_t^{(d)})^2\right] - \mathbb{E}\left[(m_t^{(d)})\right]^2 + \mathbb{E}\left[(s_t^{(d)}\mathcal{W}^{(d)})^2\right] - \mathbb{E}\left[(s_t^{(d)}\mathcal{W}^{(d)})\right]^2 \quad (C.3) $$

$$ \sim \mathrm{Var}[m_t^{(d)}] + \mathrm{Var}[s_t^{(d)}\mathcal{W}^{(d)}]. \quad (C.4) $$

Finally, a general expression for the coefficient of variation of the consumption rate is

$$ \begin{aligned} \mathrm{CV}[F_t^{(d)}] & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(d)}] + \mathrm{Var}[s_t^{(d)}\mathcal{W}^{(d)}]}}{\mathbb{E}[m_t^{(d)}] + \mathbb{E}[s_t^{(d)}\mathcal{W}^{(d)}]} \\ & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(d)}] + \mathbb{E}\left[(s_t^{(d)})^2\right]\mathrm{Var}[\mathcal{W}^{(d)}] + \mathrm{Var}[s_t^{(d)}]\mathbb{E}[\mathcal{W}^{(d)}]^2}}{\mathbb{E}[m_t^{(d)}] + \mathbb{E}[s_t^{(d)}]\mathbb{E}[\mathcal{W}^{(d)}]} \end{aligned} $$

which finally gives (Eqs. C.1)

$$ \left\{ \begin{aligned} \mathrm{CV}[F_t^{(1)}] & \sim \frac{\sqrt{\mathbb{E}\left[(s_t^{(1)})^2\right]\mathrm{Var}[\mathcal{W}^{(1)}] + \mathrm{Var}[s_t^{(1)}]\mathbb{E}[\mathcal{W}^{(1)}]^2}}{\mathbb{E}[s_t^{(1)}]\mathbb{E}[\mathcal{W}^{(1)}]} = \frac{1}{\mathbb{E}[s_t^{(1)}]} \sqrt{\mathrm{Var}[s_t^{(1)}] + \mathbb{E}\left[(s_t^{(1)})^2\right]\frac{\mathrm{Var}[\mathcal{W}^{(1)}]}{\mathbb{E}[\mathcal{W}^{(1)}]^2}}, \\ \mathrm{CV}[F_t^{(2)}] & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(2)}] + \mathbb{E}\left[(s_t^{(2)})^2\right]\mathrm{Var}[\mathcal{W}^{(2)}]}}{\mathbb{E}[m_t^{(2)}]} = \frac{1}{\mathbb{E}[m_t^{(2)}]} \sqrt{\mathrm{Var}[m_t^{(2)}] + \mathbb{E}\left[(s_t^{(2)})^2\right]\mathrm{Var}[\mathcal{W}^{(2)}]}, \\ \mathrm{CV}[F_t^{(3)}] & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(3)}] + \mathbb{E}\left[(s_t^{(3)})^2\right]\mathrm{Var}[\mathcal{W}^{(3)}]}}{\mathbb{E}[m_t^{(3)}]} = \frac{1}{\mathbb{E}[m_t^{(3)}]} \sqrt{\mathrm{Var}[m_t^{(3)}] + \mathbb{E}\left[(s_t^{(3)})^2\right]\mathrm{Var}[\mathcal{W}^{(3)}]}. \end{aligned} \right. $$

Roughly speaking, the coefficient of variation of the consumption rate $F_t^{(d)}$ is a function of two variance components such as: $c_1 \mathrm{Var}[\omega_1] + c_2 \mathrm{Var}[\omega_2] + c_3 \mathrm{Var}[\omega_1] \mathrm{Var}[\omega_2]$, where $c_1, c_2, c_3$ are constants. This shows that if the between-individuals variability has a lower order than the variability of the foraging process itself ($\mathrm{Var}[\omega_1] \ll \mathrm{Var}[\omega_2]$), then the coefficient of variation of the consumption rate $\mathrm{CV}[F_t]$ should be of the same order than the coefficient of variation of the foraging process, i.e. of order 1 in 1D, $1/\ln t$ in 2D, or $1/\sqrt{t}$ in 3D. In other words, if the main driver of the variation of the consumption rate is the foraging process itself, then the coefficients of variation estimated from data should be at most of order 1 (or more precisely it should fall within the range given in Tab. 1). If its order of magnitude is larger than 1 (or more precisely if it falls outside the range given in Tab. 1), then we could conclude that the main driver of the variation of the consumption rate is the between-individual variability (i.e. $\mathrm{Var}[\omega_1] \gg \mathrm{Var}[\omega_2]$).

Two other alternatives are possible, yet less plausible. First, the coefficient of variation estimated from data could be of order lower than 1, which would mean that both the between-individual variability and the variability due to the foraging process itself are of order lower than 1. Second, the coefficient of variation estimated from data could be of order 1 and at the same time the between-individual variability be of order larger than the variability
$$ \mathrm{Var}[F_t^{(d)}] \sim \mathbb{E}\left[(m_t^{(d)})^2\right] - \mathbb{E}\left[(m_t^{(d)})\right]^2 + \mathbb{E}\left[(s_t^{(d)}\mathcal{W}^{(d)})^2\right] - \mathbb{E}\left[(s_t^{(d)}\mathcal{W}^{(d)})\right]^2 \quad (C.3) $$

$$ \sim \mathrm{Var}[m_t^{(d)}] + \mathrm{Var}[s_t^{(d)}\mathcal{W}^{(d)}]. \quad (C.4) $$

Finally, a general expression for the coefficient of variation of the consumption rate is

$$ \begin{aligned} \mathrm{CV}[F_t^{(d)}] & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(d)}] + \mathrm{Var}[s_t^{(d)}\mathcal{W}^{(d)}]}}{\mathbb{E}[m_t^{(d)}] + \mathbb{E}[s_t^{(d)}\mathcal{W}^{(d)}]} \\ & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(d)}] + \mathbb{E}\left[(s_t^{(d)})^2\right]\mathrm{Var}[\mathcal{W}^{(d)}] + \mathrm{Var}[s_t^{(d)}]\mathbb{E}[\mathcal{W}^{(d)}]^2}}{\mathbb{E}[m_t^{(d)}] + \mathbb{E}[s_t^{(d)}]\mathbb{E}[\mathcal{W}^{(d)}]} \end{aligned} $$

which finally gives (Eqs. C.1)

$$ \left\{ \begin{aligned} \mathrm{CV}[F_t^{(1)}] & \sim \frac{\sqrt{\mathbb{E}\left[(s_t^{(1)})^2\right]\mathrm{Var}[\mathcal{W}^{(1)}] + \mathrm{Var}[s_t^{(1)}]\mathbb{E}[\mathcal{W}^{(1)}]^2}}{\mathbb{E}[s_t^{(1)}]\mathbb{E}[\mathcal{W}^{(1)}]} = \frac{1}{\mathbb{E}[s_t^{(1)}]}\sqrt{\mathrm{Var}[s_t^{(1)}] + \mathbb{E}\left[(s_t^{(1)})^2\right]\frac{\mathrm{Var}[\mathcal{W}^{(1)}]}{\mathbb{E}[\mathcal{W}^{(1)}]^2}}, \\ \mathrm{CV}[F_t^{(2)}] & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(2)}] + \mathbb{E}\left[(s_t^{(2)})^2\right]\mathrm{Var}[\mathcal{W}^{(2)}]}}{\mathbb{E}[m_t^{(2)}]} = \frac{1}{\mathbb{E}[m_t^{(2)}]}\sqrt{\mathrm{Var}[m_t^{(2)}] + \mathbb{E}\left[(s_t^{(2)})^2\right]\mathrm{Var}[\mathcal{W}^{(2)}]}, \\ \mathrm{CV}[F_t^{(3)}] & \sim \frac{\sqrt{\mathrm{Var}[m_t^{(3)}] + \mathbb{E}\left[(s_t^{(3)})^2\right]\mathrm{Var}[\mathcal{W}^{(3)}]}}{\mathbb{E}[m_t^{(3)}]} = \frac{1}{\mathbb{E}[m_t^{(3)}]}\sqrt{\mathrm{Var}[m_t^{(3)}] + \mathbb{E}\left[(s_t^{(3)})^2\right]\mathrm{Var}[\mathcal{W}^{(3)}]}. \end{aligned} \right. $$

Roughly speaking, the coefficient of variation of the consumption rate $F_t^{(d)}$ is a function of two variance components such as: $c_1 \mathrm{Var}[\omega_1] + c_2 \mathrm{Var}[\omega_2] + c_3 \mathrm{Var}[\omega_1] \mathrm{Var}[\omega_2]$, where $c_1, c_2, c_3$ are constants. This shows that if the between-individuals variability has a lower order than the variability of the foraging process itself ($\mathrm{Var}[\omega_1] \ll \mathrm{Var}[\omega_2]$), then the coefficient of variation of the consumption rate $\mathrm{CV}[F_t]$ should be of the same order than the coefficient of variation of the foraging process, i.e. of order 1 in 1D, $1/\ln t$ in 2D, or $1/\sqrt{t}$ in 3D. In other words, if the main driver of the variation of the consumption rate is the foraging process itself, then the coefficients of variation estimated from data should be at most of order 1 (or more precisely it should fall within the range given in Tab. 1). If its order of magnitude is larger than 1 (or more precisely if it falls outside the range given in Tab. 1), then we could conclude that the main driver of the variation of the consumption rate is the between-individual variability (i.e. $\mathrm{Var}[\omega_1] \gg \mathrm{Var}[\omega_2]$).

Two other alternatives are possible, yet less plausible. First, the coefficient of variation estimated from data could be of order lower than 1, which would mean that both the between-individual variability and the variability due to the foraging process itself are of order lower than 1. Second, the coefficient of variation estimated from data could be of order 1 and at the same time the between-individual variability be of order larger than the variability
from the foraging process. If so, it would remain to explain how it is possible that the between-individual variability
is such that the coefficient of variation falls exactly in this order of magnitude. Those two alternatives would seem
even less plausible when the estimated coefficients of variation cover a large number of species and experimental
and ecological contexts, as in our case and the datasets we compiled.

Therefore, we finally make two global predictions: either the coefficients of variation estimated from data are
at most of order 1, or they are of order larger than 1. We would then conclude that the main driver of the variation
of the consumption rate is the foraging process in the former case, or that it is external sources of variations, such
as the between-individual variability, in the latter.
from the foraging process. If so, it would remain to explain how it is possible that the between-individual variability
is such that the coefficient of variation falls exactly in this order of magnitude. Those two alternatives would seem
even less plausible when the estimated coefficients of variation cover a large number of species and experimental
and ecological contexts, as in our case and the datasets we compiled.

Therefore, we finally make two global predictions: either the coefficients of variation estimated from data are
at most of order 1, or they are of order larger than 1. We would then conclude that the main driver of the variation
of the consumption rate is the foraging process in the former case, or that it is external sources of variations, such
as the between-individual variability, in the latter.
# D Stochastic simulations

## D.1 Standard setup

We first evaluated the accuracy and the convergence rate of the approximation of $CV_t^{(d)}$, the coefficient of variation of the consumption rate, defined as the number of prey per unit of time consumed by a predator foraging in a $d$-dimensional space during a total foraging duration $t$.

To achieve this, we conducted numerical stochastic simulations of a consumer randomly and symmetrically walking on a $d$-dimensional homogeneous square lattice where each node initially contains a prey (Fig. D.1). The consumer moves from one node to any of the four other adjacent nodes at constant speed $v$. When a node with a prey is encountered, the consumer feeds on the prey, which takes a constant handling time $\tau_h$, and the node is depleted (we assume there is no regeneration of the prey). If the node was already visited, the consumer randomly moves to any of the adjacent node without spending handling time on the currently visited node.

Denoting $x$ the initial density of prey in the lattice, and $L$ an arbitrary length unit, the distance between two prey in the lattice is

$$y^{(d)}(x) = \frac{L}{x^{1/d} - 1} \qquad (D.1)$$

The time to cover this distance is then given by $\tau_e^{(d)}(x) = y^{(d)}(x)/v$. A simulation run (i.e. one trajectory or realization of the stochastic process) is stopped when the total foraging duration $t$ is reached, time at which the total number of prey consumed $R_t^{(d)}(x)$ is counted, and the consumption rate $F_t^{(d)}(x) = R_t^{(d)}(x)/t$ is recorded. $10^4$ realizations per initial density $x$ were run with the following parameters values: $t = 10^7$, $v = 1$, $\tau_h = 0.1$, $L = 1000$.

<figure><img src="image_6.png" /><figcaption>Figure D.1: Illustration of the lattice for $d = \{1, 2, 3\}$. The red dot denotes the starting position of the walker. $L$ is an arbitrary length unit; $y$ is the distance between two nodes; $d$ is the dimension of space.</figcaption></figure>
# D Stochastic simulations

## D.1 Standard setup

We first evaluated the accuracy and the convergence rate of the approximation of $CV_t^{(d)}$, the coefficient of variation of the consumption rate, defined as the number of prey per unit of time consumed by a predator foraging in a $d$-dimensional space during a total foraging duration $t$.

To achieve this, we conducted numerical stochastic simulations of a consumer randomly and symmetrically walking on a $d$-dimensional homogeneous square lattice where each node initially contains a prey (Fig. D.1). The consumer moves from one node to any of the four other adjacent nodes at constant speed $v$. When a node with a prey is encountered, the consumer feeds on the prey, which takes a constant handling time $\tau_h$, and the node is depleted (we assume there is no regeneration of the prey). If the node was already visited, the consumer randomly moves to any of the adjacent node without spending handling time on the currently visited node.

Denoting $x$ the initial density of prey in the lattice, and $L$ an arbitrary length unit, the distance between two prey in the lattice is

$$y^{(d)}(x) = \frac{L}{x^{1/d} - 1} \qquad (D.1)$$

The time to cover this distance is then given by $\tau_e^{(d)}(x) = y^{(d)}(x)/v$. A simulation run (i.e. one trajectory or realization of the stochastic process) is stopped when the total foraging duration $t$ is reached, time at which the total number of prey consumed $R_t^{(d)}(x)$ is counted, and the consumption rate $F_t^{(d)}(x) = R_t^{(d)}(x)/t$ is recorded. $10^4$ realizations per initial density $x$ were run with the following parameters values: $t = 10^7$, $v = 1$, $\tau_h = 0.1$, $L = 1000$.

<figure><img src="image_5.png" /><figcaption>Figure D.1: Illustration of the lattice for $d = \{1, 2, 3\}$. The red dot denotes the starting position of the walker. $L$ is an arbitrary length unit; $y$ is the distance between two nodes; $d$ is the dimension of space.</figcaption></figure>
## D.2 Robustness

We evaluated, under various ecological contexts, the robustness of the analytical approximations of $CV_t^{(d)}$, the coefficient of variations of the consumption rate. We ran numerical stochastic simulations of modified versions of the standard algorithm described in the previous section. Specifically, we ran simulations with one the four following additional mechanisms (detailed below): Walk with stochastic length jumps, walk with drift, self-avoiding walk, and walk in an initially heterogeneous and random repartition of prey on the lattice.

**Jumps with random length.** The consumer moves in the lattice following a symmetrical random walk with jumps of length following a truncated power-law distribution $P(k) \sim k^{-\theta}$ with $k \in [1, k_{max}]$, where $k_{max}$ is the maximum step-size. Simulations were run for the following parameter values: $\theta \in \{2.0, 2.5, 3.0, 3.5\}$, $k_{max} = 10$. Other parameters were the same than in the standard setup (see previous section).

**Walk with drift.** The consumer moves on the lattice in a preferred direction, denoted $\to$. The drift parameter $\mu \in [0, 1]$ is defined such that, each step, the consumer has the probability

$$ \mathbb{P}(\text{jump} = \to) = \frac{1 + (2d - 1)\mu}{2d} $$

to move in this preferred direction, and a probability

$$ \mathbb{P}(\text{jump} \neq \to) = \frac{1 - \mu}{2d} $$

to move in any of the $2d - 1$ remaining directions: $\leftarrow$ in 1D, $\leftarrow$, $\downarrow$ or $\uparrow$ in 2D, $\leftarrow$, $\downarrow$ or $\uparrow$, $+1$ level or $-1$ level in 3D. If $\mu = 0$ the movement is symmetrical and resumes to the standard random walk. If $\mu = 1$, the movement is unidirectional. Simulations were run with following parameter values, corresponding to an increasing biased random walk $\mu \in \{0.05, 0.1, 0.25, 0.5\}$.

**Self-avoiding walk with a short term memory.** The consumer had a short-term memory avoiding the node it just left.

**Walk on a lattice with a random initial repartition of prey.** Each node of the lattice initially contains a prey with probability $p$. In practice, each time the consumer visited an unvisited node, a Bernoulli trial with probability $p$ were drawn. If $p = 1$, the walk resumes to a typical random walk on a uniform graph (as detailed in the previous section), if $p = 0$ the lattice is initially empty. By default simulations were run for $p = 1/2$, unless indicated.
## D.2 Robustness

We evaluated, under various ecological contexts, the robustness of the analytical approximations of $CV_t^{(d)}$, the coefficient of variations of the consumption rate. We ran numerical stochastic simulations of modified versions of the standard algorithm described in the previous section. Specifically, we ran simulations with one the four following additional mechanisms (detailed below): Walk with stochastic length jumps, walk with drift, self-avoiding walk, and walk in an initially heterogeneous and random repartition of prey on the lattice.

**Jumps with random length.** The consumer moves in the lattice following a symmetrical random walk with jumps of length following a truncated power-law distribution $P(k) \sim k^{-\theta}$ with $k \in [1, k_{max}]$, where $k_{max}$ is the maximum step-size. Simulations were run for the following parameter values: $\theta \in \{2.0, 2.5, 3.0, 3.5\}$, $k_{max} = 10$. Other parameters were the same than in the standard setup (see previous section).

**Walk with drift.** The consumer moves on the lattice in a preferred direction, denoted $\to$. The drift parameter $\mu \in [0, 1]$ is defined such that, each step, the consumer has the probability

$$ \mathbb{P}(\text{jump} = \to) = \frac{1 + (2d - 1)\mu}{2d} $$

to move in this preferred direction, and a probability

$$ \mathbb{P}(\text{jump} \neq \to) = \frac{1 - \mu}{2d} $$

to move in any of the $2d - 1$ remaining directions: $\leftarrow$ in 1D, $\leftarrow$, $\downarrow$ or $\uparrow$ in 2D, $\leftarrow$, $\downarrow$ or $\uparrow$, $+1$ level or $-1$ level in 3D. If $\mu = 0$ the movement is symmetrical and resumes to the standard random walk. If $\mu = 1$, the movement is unidirectional. Simulations were run with following parameter values, corresponding to an increasing biased random walk $\mu \in \{0.05, 0.1, 0.25, 0.5\}$.

**Self-avoiding walk with a short term memory.** The consumer had a short-term memory avoiding the node it just left.

**Walk on a lattice with a random initial repartition of prey.** Each node of the lattice initially contains a prey with probability $p$. In practice, each time the consumer visited an unvisited node, a Bernoulli trial with probability $p$ were drawn. If $p = 1$, the walk resumes to a typical random walk on a uniform graph (as detailed in the previous section), if $p = 0$ the lattice is initially empty. By default simulations were run for $p = 1/2$, unless indicated.
# E Data

N.B.: All data, associated papers (when available) and R scripts are available on the following server: github.

## E.1 Dataset 1: FoRAGE database

The first dataset was obtained from the database FoRAGE [15], downloaded from https://knb.ecoinformatics.org/ in january 2021 (version 12.18.18). The database contains the predator and prey species names, the literature source, the initial prey density (either per $m^2$ or $m^3$), the consumption rate (the number of prey eaten per day per predator), the sample size per treatment (i.e. the initial density). The database FoRAGE was partly built by automatically extracting data from papers' figures, and partly populated directly from tables (see [15] for details). We discarded all data where at least one of these information was missing.

There were two categories of available data that were differently used for the calculation of the coefficient of variation estimations: i) raw data or ii) summary statistics. In the case of i) raw data: because there could have overlaps between dots in figures (i.e. when different predators fed on the same number of prey), the number of data could be lower than the sample size announced in the initial paper, in particular when the initial density of prey was low. We thus curated the database by removing the data with too large uncertainty: we only kept data where the sample size was ≥ 8 and where the number of available measured consumption rate was at least 80% of the sample size (i.e. at least 8 data points if the sample size was 10, for example). In addition, when a single dot was detected from the figure for a given experimental treatment, the FoRAGE's authors filled the database with a single value for as many independent replicates as supposed from the methods described in the original paper (for example, if the sample size was *n* = 10 but a single dot in a figure was observed, the database was populated with 10 rows, one for each replicate, with the same number of eaten prey). As a consequence, there was no variation in the consumption rate in this case but with no certainty about whether it was due to the limit of data collection itself or because there was indeed no variation. We thus decided to exclude these data when there was no variation. The coefficients of variation were estimated by calculating the mean and the standard deviation within initial density directly from raw data. This resulted in 142 estimated coefficients of variation (calculated on 1266 from 15108 initial data after curation), for 19 predator and 17 prey species.

In the case of ii) summary statistics: we only kept data where the mean, the standard error and the sample size were available, where the sample size was ≥ 8 (to ensure minimal quality for summary statistics estimations and be consistent with case i) raw data), and where the standard error was > 0 (because there was uncertainty whether it was true or false zero, and because a standard error equal to 0 was mostly encountered in low densities treatments suggesting that all prey were eaten in all replicates). We calculated the coefficients of variation directly from the mean, standard error and sample size, within an initial density, by applying the formula of the definition of the standard error: $\sigma = \sqrt{n \cdot se}$ (where *n* is the sample size, *σ* is the standard deviation, and *se* is the standard error;
# E Data

N.B.: All data, associated papers (when available) and R scripts are available on the following server: github.

## E.1 Dataset 1: FoRAGE database

The first dataset was obtained from the database FoRAGE [15], downloaded from https://knb.ecoinformatics.org/ in january 2021 (version 12.18.18). The database contains the predator and prey species names, the literature source, the initial prey density (either per $m^2$ or $m^3$), the consumption rate (the number of prey eaten per day per predator), the sample size per treatment (i.e. the initial density). The database FoRAGE was partly built by automatically extracting data from papers' figures, and partly populated directly from tables (see [15] for details). We discarded all data where at least one of these information was missing.

There were two categories of available data that were differently used for the calculation of the coefficient of variation estimations: i) raw data or ii) summary statistics. In the case of i) raw data: because there could have overlaps between dots in figures (i.e. when different predators fed on the same number of prey), the number of data could be lower than the sample size announced in the initial paper, in particular when the initial density of prey was low. We thus curated the database by removing the data with too large uncertainty: we only kept data where the sample size was ≥ 8 and where the number of available measured consumption rate was at least 80% of the sample size (i.e. at least 8 data points if the sample size was 10, for example). In addition, when a single dot was detected from the figure for a given experimental treatment, the FoRAGE's authors filled the database with a single value for as many independent replicates as supposed from the methods described in the original paper (for example, if the sample size was *n* = 10 but a single dot in a figure was observed, the database was populated with 10 rows, one for each replicate, with the same number of eaten prey). As a consequence, there was no variation in the consumption rate in this case but with no certainty about whether it was due to the limit of data collection itself or because there was indeed no variation. We thus decided to exclude these data when there was no variation. The coefficients of variation were estimated by calculating the mean and the standard deviation within initial density directly from raw data. This resulted in 142 estimated coefficients of variation (calculated on 1266 from 15108 initial data after curation), for 19 predator and 17 prey species.

In the case of ii) summary statistics: we only kept data where the mean, the standard error and the sample size were available, where the sample size was ≥ 8 (to ensure minimal quality for summary statistics estimations and be consistent with case i) raw data), and where the standard error was > 0 (because there was uncertainty whether it was true or false zero, and because a standard error equal to 0 was mostly encountered in low densities treatments suggesting that all prey were eaten in all replicates). We calculated the coefficients of variation directly from the mean, standard error and sample size, within an initial density, by applying the formula of the definition of the standard error: $\sigma = \sqrt{n \cdot se}$ (where *n* is the sample size, *σ* is the standard deviation, and *se* is the standard error;
n and *se* are given in the database). This resulted in 3039 estimated coefficients of variation after curation (from 6797 estimated mean and *se* consumption rates in total in the database), for 126 predator and 104 prey species.

Overall the coefficients of variation calculated from data compiled in the FoRAGE database may contain the following sources of variations: i) between individuals; ii) within individuals (if different or identical individuals were used for a given initial density or for different initial densities, information not available from the database); iii) between environments, as it was not known to what extent the environmental conditions of the experiment were controlled; iv) experimental and observations errors; v) errors due to the FoRAGE database build itself (in particular due to the automatic image analysis and data collection); vi) from the foraging process itself.

Fig. E.1 shows the median and quartiles of all the coefficients of variation calculated from the FoRAGE database. The median is 0.2905. 73.2% of the coefficients of variation lie between 0.1 and 1, and 95.5% lie between 0.01 and 10.

## E.2 Dataset 2: Raw data, potentially entangled between- and within- individuals variations

The datasets were obtained from

* The Dryad deposit (search in january 2021 with keywords: "foraging", "intake", "functional response", "visitation rate", 15 collected datasets).

* The database by [16] available at github.com/stoufferlab/general-functional-responses (25 datasets). From this database, we extracted and used only datasets where the number of predators was fixed to one, where there was several measurements at a given density, and a single type of prey; we excluded any dataset where inconsistencies were detected or with uncertainties; we included datasets where the mean, the sample size and the standard errors were available, from which we calculated the coefficients of variation.

* Data from direct authors sharing. We directly contacted authors for data sharing (1 dataset from [17]).

Overall we computed 602 coefficients of variation from 41 different datasets (from which 20 coefficients of variation were equal to 0, all in the case of low densities: suggesting that all prey were consumed by all predators, resulting in a standard deviation equal to 0; As the minimum non-zero value of the coefficient of variation was ≥ 0.029, we discarded zero values). The coefficients of variation calculated from these datasets potentially contains the following sources of variations: i) between individuals; ii) within individuals (if different or identical individuals were used for a given initial density or for different initial densities, information not available from the database by [16], or with uncertainty from datasets downloaded from Dryad, or the dataset shared by the author ); iii) between environments, as it was not known to what extent the environmental conditions of the experiment were controlled; iv) experimental and observations errors; v) the foraging process itself.
n and *se* are given in the database). This resulted in 3039 estimated coefficients of variation after curation (from 6797 estimated mean and *se* consumption rates in total in the database), for 126 predator and 104 prey species.

Overall the coefficients of variation calculated from data compiled in the FoRAGE database may contain the following sources of variations: i) between individuals; ii) within individuals (if different or identical individuals were used for a given initial density or for different initial densities, information not available from the database); iii) between environments, as it was not known to what extent the environmental conditions of the experiment were controlled; iv) experimental and observations errors; v) errors due to the FoRAGE database build itself (in particular due to the automatic image analysis and data collection); vi) from the foraging process itself.

Fig. E.1 shows the median and quartiles of all the coefficients of variation calculated from the FoRAGE database. The median is 0.2905. 73.2% of the coefficients of variation lie between 0.1 and 1, and 95.5% lie between 0.01 and 10.

## E.2 Dataset 2: Raw data, potentially entangled between- and within- individuals variations

The datasets were obtained from

* The Dryad deposit (search in january 2021 with keywords: "foraging", "intake", "functional response", "visitation rate", 15 collected datasets).

* The database by [16] available at github.com/stoufferlab/general-functional-responses (25 datasets). From this database, we extracted and used only datasets where the number of predators was fixed to one, where there was several measurements at a given density, and a single type of prey; we excluded any dataset where inconsistencies were detected or with uncertainties; we included datasets where the mean, the sample size and the standard errors were available, from which we calculated the coefficients of variation.

* Data from direct authors sharing. We directly contacted authors for data sharing (1 dataset from [17]).

Overall we computed 602 coefficients of variation from 41 different datasets (from which 20 coefficients of variation were equal to 0, all in the case of low densities: suggesting that all prey were consumed by all predators, resulting in a standard deviation equal to 0; As the minimum non-zero value of the coefficient of variation was ≥ 0.029, we discarded zero values). The coefficients of variation calculated from these datasets potentially contains the following sources of variations: i) between individuals; ii) within individuals (if different or identical individuals were used for a given initial density or for different initial densities, information not available from the database by [16], or with uncertainty from datasets downloaded from Dryad, or the dataset shared by the author ); iii) between environments, as it was not known to what extent the environmental conditions of the experiment were controlled; iv) experimental and observations errors; v) the foraging process itself.
Fig. E.1 shows the median and quartiles of all the coefficients of variation calculated from these raw datasets.
The median is 0.408, 87.8% of the coefficients of variation lie between 0.1 and 1, and 100% lie between 0.01 and 10.

List of references: Data obtained directly from the authors: [17]. Data obtained from Dryad deposit: [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Data from the database: [16].

E.3 Dataset 3: Raw data, accounting for within individuals variations

We collected three datasets directly from the authors where consumption rates were measured several times for
the same individuals. One dataset were obtained in natural conditions [33], in semi-natural conditions [34], or
in controlled conditions [35]. We directly calculated coefficients of variation within individuals from the raw
data shared by [33] and [35]. The coefficients of variation were calculated by the authors themselves and shared
with us in the case of the Imperial shag [34]. It resulted in 229 estimations of within-individuals coefficients of
variation. In natural and semi-natural conditions, the sources of possible variations were: i) within individuals; ii)
the environment (including uncontrolled prey densities); iii) experimental errors, iv) the foraging process itself.
In the controlled conditions, the sources of possible variations were: i) within individuals; ii) experimental errors;
iii) the foraging process itself. In the case of the crayfish experiments [35], we could calculate the coefficients of
variation within individuals within and across densities as the same individuals were used several times for a given
density, and for different densities.

Fig. E.1 shows the median and quartiles of all the coefficients of variation calculated from these datasets. The median is 0.273, 85.1% of the coefficients of variation lie between 0.1 and 1, and 99.1% lie between 0.01 and 10.
Fig. E.1 shows the median and quartiles of all the coefficients of variation calculated from these raw datasets.
The median is 0.408, 87.8% of the coefficients of variation lie between 0.1 and 1, and 100% lie between 0.01 and 10.

List of references: Data obtained directly from the authors: [17]. Data obtained from Dryad deposit: [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Data from the database: [16].

E.3 Dataset 3: Raw data, accounting for within individuals variations

We collected three datasets directly from the authors where consumption rates were measured several times for
the same individuals. One dataset were obtained in natural conditions [33], in semi-natural conditions [34], or
in controlled conditions [35]. We directly calculated coefficients of variation within individuals from the raw
data shared by [33] and [35]. The coefficients of variation were calculated by the authors themselves and shared
with us in the case of the Imperial shag [34]. It resulted in 229 estimations of within-individuals coefficients of
variation. In natural and semi-natural conditions, the sources of possible variations were: i) within individuals; ii)
the environment (including uncontrolled prey densities); iii) experimental errors, iv) the foraging process itself.
In the controlled conditions, the sources of possible variations were: i) within individuals; ii) experimental errors;
iii) the foraging process itself. In the case of the crayfish experiments [35], we could calculate the coefficients of
variation within individuals within and across densities as the same individuals were used several times for a given
density, and for different densities.

Fig. E.1 shows the median and quartiles of all the coefficients of variation calculated from these datasets. The median is 0.273, 85.1% of the coefficients of variation lie between 0.1 and 1, and 99.1% lie between 0.01 and 10.
<figure><img src="image_7.png" /><figcaption>Figure E.1: Boxplot of the coefficients of variation of the consumption rate of prey per single predator calculated from different datasets (see text for details).</figcaption></figure>

## References

[1] Frank Spitzer. *Principles of random walk*, volume 34. Springer Science & Business Media, 2001.

[2] Aryeh Dvoretzky and Paul Erdös. Some problems on random walk in space. In *Proceedings of the second Berkeley symposium on mathematical statistics and probability*, volume 2, pages 353–368. University of California Press Berkeley and Los Angeles, 1951.
<figure><img src="image_6.png" /><figcaption>Figure E.1: Boxplot of the coefficients of variation of the consumption rate of prey per single predator calculated from different datasets (see text for details).</figcaption></figure>

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