Resource ephemerality influences effectiveness of altruistic behavior in collective foraging

Abstract

In collective foraging, interactions between conspecifics can be exploited to increase foraging efficiencies. Many collective systems exhibit short interaction ranges, making information about patches rich in resources only locally available. In environments wherein these patches are difficult to locate, collective systems might exhibit altruistic traits that increase average resource intake compared to non-interacting systems. In this work, we show that resource ephemerality and availability highly influence the benefits of altruistic behavior. We study an agent-based model wherein foragers can recruit others to feed on patches, instead of exploiting these individually. We show that the net gain by recruiting conspecifics can be estimated, effectively reducing the decision on patch detection to one based on a threshold. Patches with qualities above this threshold are expected to increase foraging efficiencies and should therefore induce recruiting of others. By letting foragers assume Lévy searches, we show that recruitment strategies with contrasting diffusion characteristics optimize conspecific encounter rates. Our results further indicate that active recruitment is only beneficial when patches are scarce and persistent. Most interestingly, the effect of choosing suboptimal threshold values is small over a wide range of resource ephemeralities. This suggests that the decision of whether to recruit others is more impactful than fine-tuning the recruitment decision. Finally, we show that the advantages of active recruitment depend greatly on both forager density and their interaction radius, as we observe passive strategies to be more efficient, but only when forager densities or interaction ranges are large.

Appendices

Appendix A: The interaction graph

Here, we aim to get an understanding of the effect of the interaction radius r on the macroscopic behavior of the collective system. Possible interactions between individuals, and thereby their collective behavior, are fully characterized by the resulting proximity graph given this interaction radius. In this interaction graph interpretation, each individual represents a vertex where edges between vertices denote interaction and only exist when the distance between two vertices is smaller than r. The foragers’ positions define a random geometric graph albeit that the distribution over time is not necessarily uniform due to the ephemeral aggregations on patches. Below, we illustrate that an initial uniform distribution makes certain values of the interaction radius uninteresting to (artificial) collective system studies.

Let us consider a uniform distribution of vertices (forager positions). Formally, one can define the connectivity (or degree) of a random geometric graph as the average number of connections per vertex:

$$\begin{aligned} \kappa = 2E / N, \end{aligned}$$

(A.1)

where E is the number of edges within the graph. Let us furthermore define the size of the giant component to be $NG(\kappa )$, where G indicates the fraction of vertices present in the giant component. It is known that there exists a critical connectivity $\kappa _c$ for which, in the limit of $N\rightarrow \infty$, we have that $G\rightarrow 1$ for any $\kappa > \kappa _c$ (Dall and Christensen 2002). In two-dimensional systems, the value of $\kappa _c$ can be numerically computed to be $\kappa _c \approx 4.5$. Even though this behavior formally only holds in the limit of $N\rightarrow \infty$, the phase transition is apparent even at relatively small N (Fig. 7).

In this work, individual foragers can be thought of as circles within the environment, each occupying an area of $V=\pi r^2$, which is related to the connectivity through

$$\begin{aligned} \kappa = NV. \end{aligned}$$

(A.2)

From these equations, we can express the interaction radius in terms of the connectivity

$$\begin{aligned} \frac{r}{L} = \sqrt{ \frac{\kappa }{\pi N} }, \end{aligned}$$

(A.3)

where we have substituted $r \leftarrow r/L$ to express the interaction radius in terms of the environment size L. From this equation, we can immediately compute the critical radius by simply substituting $\kappa =\kappa _c$, and can therefore extract a critical interaction radius $r_c$ above which the network has a giant component containing all individuals.

When the communication network is fully connected, information (e.g. on patch locations) is not locally bound. Therefore, a fully connected network can be assumed to possess global information properties. This regime is out of our current interest, since both natural systems and artificial systems do not possess global information, but instead rely on locally available information to base their decisions on (see e.g., Brambilla et al. 2013; Hamann 2018). Therefore, we focus solely on systems with interaction radii $r<r_c$. In particular, we choose $r = \frac{1}{2}r_c \approx 0.0375L$ (see Sect. 3).

Fig. 7

Numerically computed connectivity $\kappa$ and giant component parameter G for different vertex numbers N. a The normalized connectivity of the resulting random geometric graphs. Note that all vertex numbers collapse onto the same value where $\kappa /N = \pi (r/L)^2$ as per Eq. (A.2). b The resulting giant component parameter G as a function of the normalized interaction radius. Dotted vertical lines correspond to the critical radius defined by Eq. (A.1). c The same giant component parameter expressed in terms of the connectivity. Note the collapse of the distinct vertex numbers onto the same curve for which any $\kappa > \kappa _c \approx 4.5$ the resulting giant component contains all vertices. In all plots error bars represent 1 s.d. computed over 1000 separate random geometric graphs

Appendix B: Optimal recruitment for Lévy searchers

Below, we perform a scaling analysis and show that the decision to recruit upon patch detection is a threshold decision where patches with qualities above the threshold result in an expected positive gain and should therefore encourage recruiting. We shall show that the threshold depends on both the forager density, the movement of others, and the range at which foragers can perceive one another.

Let us consider a system of N foragers where one of the foragers detects a patch at time $t_0$. Without loss of generality, we set $t_0=0$, and the forager has to decide whether to recruit others or exploit the patch individually. Since we consider collective foraging in this study, we assume that successful foragers only start recruiting if the expected net gain by recruiting is positive. Recall (see Sect. 2.1) that the quality of the patch is defined by its (remaining) duration $\tau$. Then, assuming a fixed consumption rate $\epsilon$, we define the net gain g as the difference between individual exploitation and the expected intake rate by recruiting conspecifics:

$$\begin{aligned} g&= -g_\mathrm{exploit} + g_\mathrm{recruit} \nonumber \\&= -\epsilon \tau + \epsilon \int _0^\tau n(t,\varvec{\alpha }) {\mathrm {d}}t, \end{aligned}$$

(B.4)

where $n(t,\varvec{\alpha })$ is the expected (average) number of conspecifics feeding on the detected patch at time $t>0$. Note the dependence on the vector $\varvec{\alpha } = (\alpha , \alpha ')$, where $\alpha$ and $\alpha '$ the Lévy parameters of the searchers respectively the recruiter(s) (see Sect. 3.1). The first term in Eq. (B.4) is simply the resource intake for a single forager feeding on the patch. The second term describes the expected number of resources consumed (by others) over the remaining time before the patch disappears. We can rewrite this term by considering the fact that only conspecific encounters up to some time $s(\tau )$ are ‘successful’ encounters, wherein the recruited forager has enough time to still feed on the patch. Thus we find that

$$\begin{aligned} g_\mathrm{recruit} = \epsilon \int _0^{s(\tau ,\alpha ')} n(t,\varvec{\alpha }){\mathrm {d}}t, \end{aligned}$$

(B.5)

where $s(\tau ,\alpha ')$ depends on the distance the focal forager displaces itself from the detected patch. We would like to emphasize that, for estimating $n(t,\varvec{\alpha })$, one not only needs to consider the expected encounter rate with conspecifics, but also the expected displacement from the patch for the recruiter (Sect. 3.1).

1.1 B. 1 Scaling analysis

The expected time over which the message on the patch location should be disseminated depends on both the remaining time $\tau$ and the Lévy parameter of the recruitment search $\alpha '$. If we consider the focal forager having a displacement $\delta (t,\alpha ')$ after some time $t < \tau$, we find that

$$\begin{aligned} s(\tau ) = \tau - \delta (t,\alpha ') / \ell _0, \end{aligned}$$

(B.6)

where $\ell _0$ (the step size) the constant velocity of the forager. When assuming time scales are relatively short, i.e. $1 \ll t \ll L/\ell _0$, we know that the spatial moments of the Lévy walk scale as¹ (Nakao 2000; Vahabi et al. 2013)

$$\begin{aligned} \left\langle {\left| {x} \right| ^k(t)} \right\rangle \simeq {\left\{ \begin{array}{ll} t^{k/(\alpha -1)}, \quad & 1<\alpha<3 \;\text {and}\; 0<k<\alpha -1, \\ t, \quad & 1< \alpha < 2 \;\text {and}\; k\ge \alpha -1, \\ t^{k/2}, \quad & \alpha =3 \;\text {and}\; k>0 \end{array}\right. } \end{aligned}$$

(B.7)

Note that the appropriate timescale wherein the above results hold are applicable to ephemeral landscapes, assuming patch duration is finite and truncated (see Sect. 2.1). One recovers the expected displacement with $\alpha '$ for $k=1$,

$$\begin{aligned} \delta (t,\alpha ') = \left\langle {\left| {x} \right| (t)} \right\rangle \simeq {\left\{ \begin{array}{ll} t^{1/(\alpha '-1)}, \quad & 2< \alpha '< 3, \\ t, \quad & 1 < \alpha ' \le 2, \\ t^{1/2}, \quad & \alpha '\ge 3 \end{array}\right. } \end{aligned}$$

(B.8)

We find our results to match this type of scaling (Fig. 8a).

Fig. 8

Scaling analysis of the conspecific encounter efficiency. The appropriate scaling is valid in the asymptotic limit $1 \ll t \ll L/\ell _0$ (see text). a The displacement $\delta (t, \alpha )$ from a patch detected at $t=0$ for different stable recruitment parameters $\alpha$. b The number of conspecifics encountered $n_e$ within a time t for $\varvec{\alpha }=(\alpha ,\alpha ')$. Here, searchers execute ballistic motion with $\alpha =1.1$ and recruiters walk with Lévy parameter $\alpha '=3$ (see text and Sect. 2.2). We find $n_e\simeq t$ for all values of $\alpha , \alpha '$ that we studied. c The conspecific search efficiency $\zeta = n_e/\ell _0t \simeq t^{-1}$ (see text). d The normalized conspecific search efficiency $\zeta _\delta = n_e/\delta$. In all figures, dashed lines are fits obtained with non-linear least squares analysis. Scaling of the quantities with time is indicated. Results are obtained by averaging over 250 realizations of an appropriately sized system with $L=1000$, $N=256$, $\ell _0=1$ and $r=0.0375L$ (see Sect. 3)

The expected number of conspecifics feeding at the patch due to having been recruited can be estimated as

$$\begin{aligned} n(t,\varvec{\alpha }) \simeq n_e(t-\delta (t,\alpha ')/\ell _0,\varvec{\alpha }), \quad 0< t < s(\tau ,\alpha ') \end{aligned}$$

(B.9)

where $n_e(t,\varvec{\alpha })$ is the expected number of conspecifics encountered within some time t. In other words, the number of foragers feeding on the patch at time t, is approximately equal to the number of encountered foragers at time $t-t'$, with $t'=\delta (t,\alpha ')/\ell _0$ the time needed to travel to the patch from distance $\delta (t,\alpha ')$. Estimating the number of conspecific encounters requires one to estimate search efficiencies for other Lévy searchers, which to the best of our knowledge has not been done analytically. Numerical simulations reveal linear scaling $n_e \simeq t$ for all values of $\varvec{\alpha }$ that we have studied (Fig. 8b), and subsequently $n \simeq t$ (Fig. 9a). A more thorough analytical scaling analysis is considered to be out of scope of this work.

Fig. 9

a The number of recruited conspecifics feeding on the patch at time t for different Lévy parameters $\alpha '$ and passive recruitment (static recruiter, see Sect. 3.4). Dashed lines are linear fits obtained with non-linear least squares analysis. Averages are obtained over 250 realizations. b The expected number of conspecifics feeding on the patch due to having been recruited, $n(t,\varvec{\alpha })$, as a function of $\varvec{\alpha }=(\alpha ,\alpha ')$ with fixed $\alpha =1.1$. Different colors indicate different forager densities, as the number of available foragers depends greatly on the patch distribution (Fig. 4). Dotted lines indicate passive recruiters, effectively approximating group-like foraging (see Sect. 3.4). Note that for $\alpha '\gtrsim 3$ the number of foragers feeding on the patch displays a plateau due to the Lévy walk asymptotically converging to Brownian motion as per the central limit theorem. Averages are obtained over 1000 realizations for $t=500$. Lines are a guide to the eye

Having established how conspecific encounter rates scale, let us consider the conspecific search efficiency (see Sect. 3.2)

$$\begin{aligned} \zeta (t,\varvec{\alpha }) = \frac{n_e(t,\varvec{\alpha })}{\ell _0 t} \end{aligned}$$

(B.10)

as the number of conspecifics found per distance traveled. It acts as a primer for the choice of $\alpha '$, i.e. what kind of diffusion should a forager that aims to maximize the number of conspecifics encountered within the remaining patch duration $\tau$. As is known (see e.g.,Viswanathan et al. 1999; Bartumeus et al. 2005), ballistic motion for $\alpha ^* \rightarrow 1$ maximizes the search efficiency in ephemeral landscapes where patch locations are uniform. When assuming the remainder of the collective is executing a Lévy search with $\alpha ^*$, we find that a parameter leading to contrasting diffusion, i.e. $\alpha ' \ge 3$, optimizes search efficiencies for conspecifics in the short timescale (Fig. 1). Using this and the above scaling analysis for the displacement and expected number of encountered conspecifics, we find $\zeta (t,\varvec{\alpha }) \simeq t^{-1}$ (Fig. 8c).

However, recall that we are not interested in the number of conspecifics encountered per distance traveled, rather as a function of the displacement from the patch, i.e.

$$\begin{aligned} \zeta _\delta = \frac{n_e(t,\varvec{\alpha })}{\delta (\alpha ')}. \end{aligned}$$

(B.11)

The reason being that recruiting conspecifics while close to the advocated patch results in faster exploitation rates, because recruited foragers arrive at the patch earlier. Since we know the displacement scales as $\delta \simeq t^{1/(\alpha '-1)}$ for $\alpha ' > 2$ (Eq. (B.8)), and encounters as $n_e\simeq t$, we find the properly normalized conspecific search efficiency scales as

$$\begin{aligned} \zeta _\delta (t,\varvec{\alpha }^*) \simeq t^{1/2}, \end{aligned}$$

(B.12)

where $\varvec{\alpha }^*=(1.1,3.0)$. In contrast, values of $\alpha ' < 2$ result in linear scaling of the displacement, $\delta \simeq t$, hence $\zeta _\delta (t,\varvec{\alpha }) \simeq \text {const.}$, i.e. the normalized conspecific search efficiency approaches a constant value as t increases. Our numerical results indeed verify this behavior, as can be seen in Fig. 8c, d.

The difference in scaling for $\alpha '\le 2$ and $\alpha '>2$ explains why the conspecific search efficiency is maximized with contrasting diffusion characteristics ($\alpha '\ge 3$ as $\alpha \rightarrow 1$). While for $\alpha '\le 2$ the rate of new conspecific encounters approaches a constant value, it grows with $t^{1/(\alpha '-1)}$ when $\alpha '>2$, hence resulting in increased $\zeta$ (Figs. 1b, 8d). It additionally raises the question if different strategies, such as simply announcing while remaining on the patch (i.e., $\delta =0$), might be more efficient. We compare active recruitment via Lévy walks with a passive strategy in Sect. 3.4 and below in Appendix C.

1.2 B. 2 Threshold decision making

Here, we wish to illustrate that the foragers can be equipped with an effective threshold for which patches with qualities above this threshold should have an expected positive net gain and thus should trigger (active) recruitment. As a result, our model effectively resembles a threshold model, where recruiting others occurs only when the forager expects the collective to benefit (see Sect. 2.2). Recall that we assume that foragers can estimate the optimal recruiting time $s(\tau ,\alpha ')$ for a given patch duration $\tau$ by estimating its displacement following the above scaling analysis. As the coefficients of both the displacement and the number of conspecifics encountered can be numerically computed, we can pre-compute $s(\tau ,\alpha ')$ from Eq. (B.6) and Eq. (B.8), and subsequently the expected net gain from Eq. (B.4), $s(\tau ,\alpha ')$ and Eq. (B.9). Since $n(t,\varvec{\alpha }) \simeq t$, we can write $n(t,\varvec{\alpha }) = d_1 t + d_2$. Then net gain g becomes

$$\begin{aligned} g&= -\epsilon \tau + \epsilon \int _0^{s(\tau ,\alpha )} n(t,\varvec{\alpha }) {\mathrm {d}}t \nonumber \\&= \epsilon \left( { \frac{1}{2} d_1s^2(\tau ,\alpha ) + d_2s(\tau ,\alpha ) - \tau } \right) , \end{aligned}$$

(B.13)

where we have simply integrated the linear approximation of $n(t,\varvec{\alpha })$. By determining the coefficients, which in artificial systems can be computed beforehand (i.e. be assumed prior knowledge to the forager), one can find critical durations for which $g(\tau _c)=0$, $\tau _c>0$. Existing numerical schemes, such as the Newton-Raphson method, can be applied to find these roots.

Then, at patch detection, foragers should recruit when $\tau > \tau _c$ and exploit individually when $\tau \le \tau _c$. Therefore, advantages of collective behavior depend heavily on the distribution over patch durations (see Sect. 3.2). The threshold $\tau _c$ ensures that foragers are not recruiting others towards patches that are not worth the effort and therefore serve as a filter on the individual level. In turn, thresholds greatly simplify decisions of recruited foragers, since instead of a (potentially complex) decision they should simply always travel towards the advocated patch.

We would like to emphasize that our results (see Sect. 3.3 and Fig. 5) appear to indicate that the specific choice of threshold does not significantly influence the resulting group search efficiency. This is possibly an artifact of the ephemeral patch distribution that we study here. Hence, different patch distributions might result in more precise estimations of $\tau _c$ to be far more beneficial for recruiting foragers than the one studied in this work.

Appendix C: Density effects

Fig. 10

Comparison of the effect of forager density on the effectiveness and efficiency of recruitment between active and passive recruitment strategies. Results are obtained for $M=256$ and $\gamma =1.1$ in order to simulate environmental conditions under which recruitment is known to be beneficial (see text and Fig. 2). Solid lines indicate active recruitment with Lévy walks with $\alpha '=3$ for searching foragers with $\alpha =1.1$. Dashed lines in b and c indicate passive recruitment. a Total relative travel distance, computed by dividing total travel distance ${\mathcal {L}}=\sum _i{\mathcal {L}}_i$ for active recruitment by total travel distance for passive recruitment. Note that relative travel distance is always greater than 1, indicating that passive recruitment carries lower total travel distances (see text). b The recruit effectiveness as the total number of conspecifics recruited divided by the total number of recruit instances. c The recruit efficiency as the total number of resources consumed divided by the travel distance (see text and Sect. 3.4). d The relative recruit efficiency as the recruit efficiency of active recruitment divided by the recruit efficiency of passive recruitment. Dotted vertical lines indicate interaction radii above which passive recruitment has higher recruit efficiencies than active recruitment. Note that interaction radii for which this occurs are (approximately) equal to those where the forager efficiency indicates similar effects (Fig. 6)

Forager density affects patch encounter rates and subsequently the number of recruit instances. As a result, passive strategies can become more beneficial (increase foraging efficiency) as interaction radii increase (Sect. 3.4). Note that, despite the apparent increase in foraging efficiency, active recruitment result in more conspecific encounters than passive recruitment (Fig. 9). Therefore, we compare active and passive recruitment in more detail to investigate when passive recruitment might be more efficient.

To this end, we study a system of N foragers in environmental conditions wherein altruistic recruitment ($\tau =0$) outperforms selfish systems ($\tau =\infty$). This is realized for persistent patches ($\gamma =1.1$) and low patch densities ($M=256$), as indicated in Fig. 2. In such environments, we observe that total travel distances are lower for passive recruitment, as the passive recruiter is always on the patch ($\delta =0$). In contrast, active recruiters increase the distance towards the patch (as $\delta \propto t^{1/2}$, see section B and Fig. 8) and thereby increasing distances towards the patch upon recruitment. Note that total travel distances increase with the interaction radius as expected. The (small) decrease for high forager densities as r increases results from the approximately uniform forager distribution on patch detection, leading to instantaneous attraction at distances shorter than r. This induces an overall reduction in the total travel distance.

While larger distances towards the patch seem counter-productive, as the search efficiency is inversely related to the travel distance (Eq. (4)), simulations reveal that foraging efficiencies are higher when actively recruiting, but only when interaction radii are sufficiently small (Fig. 6). The reason is twofold. First, the recruit effectiveness is larger for active recruitment (Fig. 10b) as the encounter probability is higher for active recruiters than passive recruiters (Fig. 9a). Second, for sufficiently small interaction radii, the recruit efficiency is larger when actively recruiting conspecifics. The recruit efficiency is computed by the number of resources consumed after being recruited, divided by the total distance traveled, and is shown in Fig. 10c,d. When actively recruiting, conspecifics arrive on ephemeral patches earlier than when passively recruiting, thus increasing the total number of resources consumed on the patch before it disappears (see also Sect. 3.4).

The benefits of active recruitment depend strongly on the forager density and their interaction radius, because, when interaction radii are sufficiently large, advantages of active recruitment disappear. Travel distances in systems with passive recruitment decrease, due to the recruiter not moving and decreased distances towards the patch upon being recruited (Fig. 10a), hence increasing the foraging efficiency. This effect is amplified when forager densities are high, resulting in active recruitment being only beneficial when interaction radii are smaller than several body lengths (see Figs. 6b, 10d and our discussion in Sect. 3.4).

Appendix D: Distribution over individual search efficiencies

To measure distributions over individual search efficiencies presented in Fig. 3, we numerically compute histograms by computing search efficiencies for each individual forager and attributing them to 50 logarithmically spaced bins between $\eta _i=0$ and $\eta _i=\eta _\mathrm{max}$. Here, $\eta _\mathrm{max}$ is the greatest measured individual foraging efficiency encountered during our simulations for a specific parameter setting and can be empirically determined.

For generating the fits of the individual search efficiencies in Fig. 3, we use non-linear least squares to fit a log-normal distribution to the empirically obtained histograms using the SciPy Python package (Virtanen et al. 2020). To measure the statistical accuracy of the fitted curves, we compute the coefficient of determination $R^2$, and found $R^2>0.98$ for all curves shown in Fig. 3 (Table 1).

Table 1 $R^2$ values for fitted log-normal distribution shown in Fig. 3 for all shown values of $\gamma$ and M