Summary
Foraging animals have to locate food sources that are usually patchily distributed and subject to competition. Deciding when to leave a food patch is challenging and requires the animal to integrate information about food availability with cues signaling the presence of other individuals (e.g. pheromones). To study how social information transmitted via pheromones can aid foraging decisions, we investigated the behavioral responses of the model nematode Caenorhabditis elegans to food depletion and pheromone accumulation in food patches. We experimentally show that animals consuming a food patch leave it at different times and that the leaving time affects the animal preference for its pheromones. In particular, worms leaving early are attracted to their pheromones, while worms leaving later are repelled by them. We further demonstrate that the inversion from attraction to repulsion depends on associative learning and, by implementing a simple model, we highlight that it is an adaptive solution to optimize food intake during foraging.
Introduction
Decisions related to foraging for food are among the most critical for an animal’s survival. They can also be among the most challenging, because food is usually patchily distributed in space and time, and other individuals are attempting to find and consume the same resources (Abu Baker and Brown, 2014; Driessen and Bernstein, 1999; Stephens and Krebs, 1987).
An important decision, which has been the focus of considerable effort in models of foraging behavior, is for how long to exploit a food patch. Most models involve patch assessment by individuals and postulate that the leaving time depends on local estimates of foraging success (Eric L. Charnov, 1976; Oaten, 1977; Stephens and Krebs, 1987). As such, foragers are predicted to depart from a food patch when the instantaneous intake rate drops below the average intake rate expected from the environment, a phenomenon that has been observed in several animals, from insects (Wajnberg et al., 2008) to birds (Cowie, 1977; Krebs et al., 1974) and large mammals (Searle et al., 2005). The presence of other animals, however, affects individual foraging success so that different leaving times can be expected (Aubert-Kato et al., 2015; Giraldeau and Caraco, 2000). Once an animal leaves a food patch, another crucial decision is how to explore the environment to locate new sources of food. Since natural habitats are usually saturated with many different non-specific chemical cues, animals use pheromones and other odors to orientate their searches (Wyatt, 2014). This, however, implies determining whether pheromones point towards a resource supporting growth and reproduction or an already exploited one.
To acquire this knowledge, animals have to learn from experience. In the context of social foraging it has been shown that individuals might need to rely only on the most recent experience (Krebs and Inman, 1992). As such, the valence (positive or negative signal) of pheromones acquired during the most recent feeding activity is crucial for the success of the foraging process. While this has been shown in bumblebees feeding on transient resources (Ayasse and Jarau, 2014), we still don’t know whether it is important for other animals feeding in groups. Moreover, it is not clear if the ability to use associative learning (Ardiel and Rankin, 2010) to change the valence of pheromones could improve foraging success.
The nematode Caenorhabditis elegans is a powerful model system to investigate how information about food availability and pheromones can shape foraging in patchy habitats. C. elegans, indeed, feeds in large groups on ephemeral bacterial patches growing on decomposing plant material, a habitat that can be mimicked in a petri dish (Frézal and Félix, 2015; Schulenburg and Félix, 2017). Importantly, C. elegans can evaluate population density inside food patches using a suite of pheromones, belonging to the family of ascarosides, which are continuously excreted by worms (Greene et al., 2016; Ludewig and Schroeder, 2013). Finally, it has been shown that pheromones and food availability control the leaving times of foraging worms. In particular, the rate at which individuals abandon the patch increases when food becomes scarce and pheromones are at high concentrations (Fig. 1A) (Harvey, 2009; Milward et al., 2011).
In the present study, we experimentally investigated the behavioral responses of C. elegans to food depletion and pheromone accumulation in food patches. We confirmed that individual worms consuming a food patch leave at different times, and we found that worms leaving early have a preference for worm-secreted pheromones while those leaving late avoid the pheromones. A simple foraging model suggests that these two behaviors may optimize foraging success in the presence of competitors. Finally, using a series of behavioral assays altering worm exposure to food and pheromones, we demonstrate that associative learning underpins the change in pheromone preference.
Results
To investigate C. elegans behavioral responses to food depletion, we developed an assay to simultaneously assess patch-leaving behavior and pheromone preference over time (see “Choice after food” assay in the Materials and Methods section and Fig. 1B). In our assay we used a small patch of bacteria that over the course of 5 hours is gradually depleted by feeding animals (young adult hermaphrodites of the natural isolate MY1). At equal distances from the food patch there are two spots, one of which contains a pheromone blend. Shortly after worms leave the food patch, they make a decision by choosing between the two spots (Fig. S1A). This assay mimics a foraging decision in which animals must decide when to leave a food patch and after doing so whether to follow a pheromone cue, which is indicative of the presence of others. In these experiments, the pheromone blend is obtained by collecting and filtering the supernatant of well-fed worms maintained in a liquid culture (Choe et al., 2012; Harvey, 2009; White et al., 2007). In agreement with previous results (Milward et al. 2011), we found that feeding animals abandon the food patch at very different times, with some worms leaving at the beginning while others stay until the food patch is depleted (Fig 1C). In addition, the leaving time affects worms’ preference for their pheromones, with individuals leaving the food patch early (first three hours) going to the pheromone blend (positive chemotaxis index, Fig. 1D) while worms leaving the food patch later avoiding it (negative chemotaxis index, Fig. 1D). C. elegans responses to food depletion, therefore, include an inversion in pheromone preference dependent on the leaving time from the food patch.
Inside rotting fruits and stems where C. elegans forage, bacterial food is patchily distributed (Frézal and Félix, 2015; Schulenburg and Félix, 2017). We might then expect that the timing of dispersal from existing food patches and the strategies that optimize food intake are crucial for worm survival. A natural question then arises: can the behaviors we observed in our experiments provide any benefit to C. elegans foraging? We explore this question with a simple theoretical model that allowed us to explore the optimality of the inversion in pheromone preference in the context of foraging in a heterogeneous environment.
The scenario described in our model implies that there are two types of food patches, those that are occupied by worms and those that are not. We assume that the occupied patches are easier to find as a result of either being nearby or because of the accumulation of pheromones secreted by the worms. Dispersing away from the occupied food patches in search for unoccupied ones therefore gives a low average payoff (gD), not being advantageous until the occupied food patches are depleted (see Supplement). However, the occupied patches will generically be occupied in a non-equal manner, meaning that one patch will be consumed before the other. Under these conditions, we then ask which strategy maximizes individual food intake. For the sake of illustration, In Fig. 2 we depict the simplest scenario, with two patches that are colonized by worms and another that is uncolonized.
Initially, worms should stay in the occupied patches, but they can either remain in their initial one or switch to the other occupied patch (by leaving their current patch and following pheromones to the neighboring patch) (Fig. 2A). If all worms remain in their initial patch the overcrowded food patch will be depleted faster, so worms occupying it will benefit from switching to another occupied patch (Fig. 2B). Worms have no way to tell whether they are in the overcrowded or in the undercrowded patch, but since the majority of individuals are in the overcrowded one, every individual has a higher probability of being there. Therefore, all worms have an incentive to switch to the other occupied patch during the initial phase (blue area in Fig. 2B).
How many worms should then switch? If we assume that worms attempting to switch may in fact end up in any of the occupied patches (including the initial one) with equal probability, then all worms should attempt to switch. This result is independent of any other parameters, such as number of patches or initial distribution of worms (see Supplement). If we assume that switching to other occupied patches is costly, then the optimal switching probability will be lower, but the predictions of the model remain qualitatively unaltered (see Supplement). This model therefore predicts an Evolutionary Stable Strategy (that also happens to maximize food intake) in which some worms leave a patch before it is depleted and follow the pheromone cue (Fig. 2A). This initial phase helps equalize worm occupancy and feeding across food patches.
Once worm numbers are equalized in the two easy-to-find food patches, worms feed until the food becomes scarce. At this point, worms benefit from leaving the depleted patches (gray area in Fig. 2B) and avoiding pheromones, since pheromones now mark depleted food patches. The inversion of pheromone preference therefore helps worms to disperse to unoccupied food patches. This simple model thus predicts both different leaving times and the inversion of pheromone preference and highlights that, together, these phenomena might maximize food intake of worms foraging in a patchy environment.
Nevertheless, the model does not encode any specific mechanism underpinning the inversion of pheromone preference. The most parsimonious explanation is that animals leaving the patch earlier might differ from worms leaving later simply due to their feeding status. Indeed, early-leaving worms abandon the food patch when food is still abundant and therefore, they are more likely to be well-fed. By contrast, worms leaving later—when the food is scarce—are more likely to be famished.
However, worms leaving earlier are also exposed to pheromones in the presence of abundant food, while worms leaving later experience high levels of pheromones in association with scarce food. These conditions are analogous to those that have been shown to support associative learning in C. elegans (Ardiel and Rankin, 2010). Similarly to the well-known case of associative learning with salt (Hukema et al., 2008; Saeki et al., 2001), in our experiment worms could be initially attracted to pheromones because of the positive association with the presence of food. Attraction could later turn into repulsion if worms start associating pheromones with food scarcity.
To distinguish between the change in pheromone preference being caused by feeding status alone or by associative learning, we performed experiments in which young adult hermaphrodites were conditioned for five hours in the four scenarios corresponding to the combinations of +/− food and +/− pheromone blend. After conditioning, animals were assayed for chemotaxis to the pheromone blend (see Materials and methods section, Fig. 3A and S1B). We found that worms go to the pheromone blend when they are conditioned with + food + pheromone blend whereas they avoid it when they are conditioned with − food + pheromone blend (Fig. 3B, blue and yellow bars, mean CI++ = 0.38 ± 0.07 vs mean CI−+ = – 0.15 ± 0.04). Interestingly, worms conditioned without the pheromone blend do not exhibit a particular preference for their pheromones (Fig. 3B, red and turquoise bars, mean chemotaxis index for the + food − pheromone blend and the − food − pheromone blend scenarios are 0 ± 0.07 and – 0.02 ± 0.06, respectively). Worms therefore exhibit attraction when pheromones are paired with abundant food and aversion when pheromones are associated with absence of food. Otherwise, C. elegans does not show a specific preference for pheromones. These findings are consistent with the hypothesis that the C. elegans preference for the pheromone blend changes due to associative learning.
Conditioning in the two scenarios without pheromone blend added had to be performed at low worm density due to uncontrolled pheromone accumulation. Indeed, when conditioned at high worm density, animals in the + food − pheromone blend scenario were still exposed to the pheromones that they kept excreting during the 5-hour conditioning period (Sakai et al., 2013) and therefore exhibited attraction to the pheromone blend, albeit variable (mean CI+− = 0.25 ± 0.07, Fig. 3B inset, red bar). Worms conditioned in− food − pheromone blend displayed no significant attraction to the pheromone (mean CI−− = 0.10 ± 0.15, Fig. 3B inset, turquois bar). The variation here is even bigger, likely due to the fact that the pheromone cocktail produced by starved worms can be different from the pheromone blend we used, which was obtained from well-fed worms (Kaplan et al., 2011).
The outcome of this series of experiments with the pheromone blend can be recapitulated with pure synthetic ascarosides (Fig. S2). This allowed us to address the fact that the pheromone blend contains, in addition to a cocktail of ascarosides, other products of worm metabolism, compounds deriving from the decomposition of dead worms and bacteria, and perhaps other unknown substances. Overall, these results provide additional evidence that it is associative learning with ascaroside pheromones that underlies the inversion of pheromone preference observed in our original foraging experiment (Fig. 1).
To provide further support that the C. elegans preference for pheromones can change through associative learning, we asked whether the change in preference occurs also via the association with a repellent compound, namely glycerol (Hukema et al., 2008). To answer this question, we performed a learning experiment in which young adult hermaphrodites were conditioned for one hour in four different scenarios deriving from all the possible combinations of +/− repellent (glycerol) and +/− pheromone blend. During conditioning, animals are free to dwell in a plate seeded with E. coli OP50, and therefore they are always exposed to a high concentration of bacterial food. Here, uncontrolled pheromone accumulation was not an issue thanks to the short conditioning period. After conditioning, worms are tested for chemotaxis to the pheromone blend (Fig. 4A).
We found that the preference for the pheromone blend, which is retained in the − repellent + pheromone blend scenario (Fig. 4B, blue bar, mean CI−+ = 0.26 ± 0.04), is lost in the + repellent + pheromone blend condition (Fig. 4B, yellow bar, mean CI ++ = 0.04 ± 0.03). Again, when animals are conditioned in the absence of pheromones, they do not show any particular preference for the pheromone blend (mean CI+− = 0.01 ± 0.05, mean CI−− = − 0.05 ± 0.08, Fig. 4B red and turquois bars respectively). In other words, animals do not exhibit a particular preference for pheromones, except when they are exposed to pheromones and food (in the absence of the repellent). Exposure to the repellent in the presence of food and pheromones is required to disrupt attraction. The outcome of this experiment provides further support that C. elegans can change its preference for the pheromone cocktail it produces through associative learning.
Discussion
We have assessed the response of C. elegans to food depletion and how this influences worms’ response to their pheromones. In agreement with previous studies, our findings indicate that worms exhibit different leaving times when feeding in groups on transient bacterial food patches. Interestingly, the leaving time affects C. elegans preference for its pheromones, with animals leaving early being attracted to their pheromones and worms leaving later being repelled by them. We showed that this inversion from attraction to repulsion depends on associative learning and appears to be an adaptive solution to optimize food intake during foraging.
Our model shows that a change from pheromone attraction to repulsion is required to optimize food intake when three different factors are combined. First, food patches that give diminishing returns, which should be abandoned when the environment provides a better average intake rate. This first factor has been thoroughly studied both theoretically and experimentally in the context of optimal foraging and the marginal value theorem (Eric L. Charnov, 1976; Krebs et al., 1978; Oaten, 1977; Stephens and Krebs, 1987; Watanabe et al., 2014), and in this respect our model simply reproduces previous results.
The second factor is competition for limited resources, which in our model creates the need to switch between pheromone-marked food patches in order to distribute the individuals more evenly. This redistribution closely resembles the Ideal Free Distribution, which postulates that animals should distribute across patches proportionally to the resources available at each source (Bautista et al., 1995; Fretwell and Lucas, 1969; Houston and McNamara, 1987; Kennedy and Gray, 1993). However, here, our model does depart from previous studies. The Ideal Free Distribution applies to cases in which the benefit per unit time decreases with the number of individuals exploiting the same resource. This is the case for habitat choice (where the distribution was first proposed, Fretwell and Lucas, 1969), or if the instantaneous feeding rate decreases with the number of feeders (Houston and McNamara, 1987). It is not however the case in many foraging scenarios, including the one represented by our model (and implicitly by most optimal foraging models), in which animals can feed unimpeded by each other. In these cases, a higher number of animals simply means that the resource is depleted faster (this can be seen in Fig. 2B: At time t=0, animals feed at the same rate in both food patches). Simply adding competition to standard optimal foraging models will not change their results qualitatively. Animals will stay in each food source until the food is so scarce that the instantaneous feeding rate falls below the environment’s average. This will happen earlier for more crowded food sources, but animals will never need to switch across food sources before they are depleted.
The third key factor in our model is non-stationarity: We assume that all pheromone-marked food patches were colonized and will be depleted at roughly the same time. This fact creates the need to switch before the current patch is depleted, because by then most of the benefit from undercrowded (but pheromone-marked) food patches will be gone. This non-stationary environment has received less attention than the previous factors. It is typical of species with boom-and-bust life cycles such as C. elegans (Frézal and Félix, 2015), but may also be applicable to other cases, such as migratory species (which arrive synchronously to a relatively virgin landscape), fast-dispersing invader species or, in general, species that occupy a non-stationary ecological niche.
C. elegans individuals use stimuli coming from the environment (smells, tastes, temperature, oxygen and carbon dioxide levels) and from other individuals (pheromones) to efficiently navigate their habitat. An important evolutionary adaptation in this regard is that the C. elegans preference for each of these stimuli can change through experience, including acclimation (Fenk and de Bono, 2017) and associative learning phenomena (Ardiel and Rankin, 2010; Colbert and Bargmann, 1995; Rankin, 2004). We have identified associative learning as the most plausible phenomenon underpinning the change in pheromone preference. During feeding, worms learn to give a positive or negative preference to pheromones depending on the context in which they experience them, in particular the presence or absence of food (Wyatt, 2014). A similar learning process occurs in bumblebees that, in their natural habitat, do not land or probe flowers that have been recently visited and marked by chemical footprints left by themselves or other bees. It has been shown that only experienced foragers, i.e. those that learnt to associate the chemical footprints with the absence of nectar in marked flowers, can successfully avoid them and increase their overall nectar intake (Ayasse and Jarau, 2014). This suggests that associative learning based on pairing pheromones or similar chemical signals with food availability might be frequently observed in animals feeding in groups, not only eusocial insects, as a strategy to increase food intake.
We have shown that dispersal of feeding stages of C. elegans from occupied patches is regulated by the recent experience of food availability and pheromones, which indicates, at any time, whether it is better to follow the scent of pheromones or to avoid it. A mechanism based on the synergistic interaction between food and pheromones also regulates C. elegans dispersal over longer time scales and, in general, its boom-and-bust life cycle (Edison, 2009; Frézal and Félix, 2015). Indeed, scarce food and high concentration of pheromones promote the entry into a resting stage (the dauer larva), allowing worms to survive unfavorable seasons and disperse to uncolonized rotten material, where abundant food, in turn, resumes development to adulthood (Frézal and Félix, 2015). Our findings establish an interesting parallel between mechanisms promoting dispersal over short and long temporal scales and highlight the important role that non-dauer stages play in exploiting transient bacterial patches.
As a final remark, our results suggest that C. elegans preference for pheromones might not be innate, as it was previously stated (Greene et al., 2016; Macosko et al., 2009; Pungaliya et al., 2009; Simon and Sternberg, 2002; Srinivasan et al., 2012, 2008) and question what it means to be a naïve worm. Worms that we call “naïve” are directly assayed for chemotaxis after being simultaneously exposed to both bacterial food and ascaroside pheromones, which are continuously excreted by the animals during their growth (Kaplan 2011). Hence, it is possible that the attraction that “naïve” worms exhibit is due to the positive association with food that they learn to make during growth on the plate.
Author contributions
MDB, APE and JG conceived the study. MDB performed the experiments and analyzed the data. APE built the model. FCS contributed the synthetic pheromones. All authors wrote the manuscript.
Competing interests
The authors declare no competing interests.
Material and methods
Strains and culture conditions
We used a Caenorhabditis elegans strain recently isolated from the wild, MY1 (Lingen, Germany). The strain has been obtained from the Caenorhabditis Genetic Centre (CGC). Animals were grown at 21-23 °C (room temperature) on nematode growth media (NGM) plates (100 mm) seeded with 200 μl of a saturated culture of E. coli OP50 bacteria (Stiernagle, 2006). As for OP50 culture, a single colony was inoculated into 5ml of LB medium and grown for 24 h at 37 °C.
Pheromones
We obtained the crude pheromone blend by growing worms in liquid culture for 9 days (at room temperature and shaking at 250 rpm) (Von Reuss et al., 2012). Individuals from one plate were washed and added to a 1-liter flask with 150ml of S-medium inoculated with concentrated E. coli OP50 pellet made from 1 liter of an overnight culture. Concentrated E. coli OP50 pellet was added any time the food supply was low, i.e. when the solution was no longer visibly cloudy (Stiernagle, 2006). The pheromone blend was then obtained by collecting the supernatant and filter-sterilizing it twice. A new pheromone blend was produced every 3 months. Pure synthetic ascarosides (ascr#5 and icas#9) were obtained from the Schroeder lab and kept at −20°C in ethanol. Each time an experiment was performed, an aqueous solution at the desired molar concentration was prepared (10 μM for ascr#5 and 10 pM for icas#9). The control solvent for the pheromone blend is S-medium, while the control solvent for the pure ascarosides is an aqueous solution with the same amount of ethanol present in the ascaroside aqueous solution (Srinivasan et al., 2012).
Choice after food assay
It is a chemotaxis assay modified from Bargmann et al. (1991) and Saeki et al. (2001), performed on naïve worms that encounter a food patch before making the choice between the pheromone blend and the control solvent. We used 100 mm NGM plates in which we deployed 20 μl of the pheromone blend, 20 μl of control solvent and 15 μl of a diluted OP50 E. coli culture at equal distance from each other (Fig. S1A). In the pheromone and control spots, 2 μl of 0.5 M sodium azide was added in order to anaesthetize the animals once they reached the spots. Since the anesthetic action of sodium azide lasts for about 2 hours in this set-up, another 1μl was added two hours after the beginning of the assay in both spots. ~200 naïve animals were placed close to the patch of bacteria, so that they stop and feed in the patch before they chemotaxis towards the two cues. Worms are left to wander freely on the assay plate for 5 hours. The number of worms around the two spots was counted every hour and the chemotaxis index was calculated based on the number of new worms that reached the two spots during each hour.
Chemotaxis assay
Chemotaxis assay has been performed in 60 mm NGM plates, in which worms are given the choice between pheromone (either 20 μl of the pheromone blend or 20 μl of a pure ascaroside in aqueous solution) and a control solvent (20 μl) (Bargmann and Horvitz, 1991; Saeki et al., 2001). The two spots are deployed ~3 cm apart from each other (Fig. S1B). Shortly before the start of the assay, 1 μl of 0.5 M sodium azide is added to both spots in order to anaesthetize the animals once they reach the spots. ~50 animals are placed equidistant from the two spots and left to wander on the assay plate for 1 hour at room temperature (Fig. S1B). The assay plates were then cooled at 4°C and the number of worms around each spot was counted using a lens. The chemotaxis index is then calculated as , where Np is the number of worms within 1 cm of the center of the pheromone spot, while Nc is the number of worms within 1 cm of the center of the control spot. The number of independent experiments, i.e. performed in different days, is indicated in each figure caption. For each experiment, we usually performed 10 replicated assays for each scenario.
Learning experiment
Hermaphrodite individuals of the MY1 strain are grown until they become young adults in NGM plates seeded with 200 μl of saturated E. coli OP50 bacteria. Then, animals are washed off the plates with wash buffer (M9 + 0.1% triton), transferred to an Eppendorf tube and washed twice by spinning down the worms and replacing the supernatant with fresh wash buffer each time. After that, ~1000 animals are transferred to each conditioning plate. For the experiments in which worms are conditioned at low population density, the number of individuals placed in each conditioning plate is ~100. In the first series of experiments, the four different scenarios derive from all the possible combinations of +/− food and +/− pheromone blend. Plates are prepared ~16 hours before the training starts, so that bacteria can grow and form a lawn. NGM plates are seeded with +/− 200 μl of saturated E. coli OP50 culture and +/− 200 μl of pheromone blend. Animals spend 5 hours in the conditioning plates at room temperature before being assayed for chemotaxis to the pheromone blend. In the second series of experiments (with the repellent), the four different scenarios derive from all the possible combinations of +/− repellent (glycerol) and +/− pheromone blend. Conditioning plates are prepared ~1 before the start of the experiment and are NGM plates seeded with 200 μl of saturated E. coli OP50 culture +/− 0.5 M glycerol and +/− 200 μl of pheromone blend. To keep the concentration constant, when the pheromone blend was not added, we diluted OP50 with 200 μl of S-medium. Animals stay in the conditioning plates for one hour at room temperature before being assayed for chemotaxis to the pheromone blend. In the experiments with pure ascarosides ascr#5 and icas#9, the four different scenarios derive from all the possible combinations of +/− food and +/− pure ascaroside (in aqueous solution) and are prepared as the experiment with food and the pheromone blend. However, the concentration of ascaroside that was added in the conditioning plate was higher than the concentration at which the worms were tested for chemotaxis (for ascr#5 was 10 μM, while for icas#9 was 10 pM icas#9) to compensate for the diffusion of the ascaroside throughout the agar in the conditioning plates. Ascr#5 was added at a concentration of 100 μM onto conditioning plates, while icas#9 was added at a concentration of 1 μM. Worms spend 5 hours in the conditioning plates at room temperature, after which they are assayed for pheromone chemotaxis.
Supplementary data
Foraging model
We assume that two types of food patches exist:
- Food patches marked with pheromones, which have a high average value (they are capable of sustaining worm growth and are easy to find).
- Unmarked food patches, which have a low average value as they are more difficult to find).
Initially, individuals are distributed across the pheromone-marked patches. Let K be the number of patches, ni the number of individuals in the i-th patch (for i = 1,2 … K), and N the total number of individuals (so ).
At any time, individuals can take three possible actions: Remain in the current patch, switch to another pheromone-marked patch (so they leave the patch and follow pheromones), or disperse and search for an unmarked patch (so they leave the patch and avoid pheromones). Individuals’ instantaneous feeding rate g(t) depends on their choices. Let’s start with the choice of dispersal. To model this decision we borrowed the results of classical foraging models from which the Marginal Value Theorem was derived (E L Charnov, 1976). These models describe an individual depleting a food patch, whose environment contains other food patches that remain stationary (i.e. on average the other food patches are not being depleted over time). We also assume that the unmarked food patches remain stationary. In these conditions, one can compute an average expected intake rate from dispersing and searching for unmarked patches, which we will call gD. This average intake rate takes into account the average quality of the unmarked food patches and the time needed to find and consume them. The optimal strategy is to remain in the current food patch until the instantaneous feeding rate (g(t)) falls below gD (E L Charnov, 1976). Following these models, we assume that any individual that disperses will experience a constant instantaneous feeding rate gD.
While we can use the formalism of classical foraging models for the dispersal decision, we cannot do the same for the switching decision, because the pheromone-marked food patches are non-stationary (they all get depleted at roughly the same time, a feature characteristic of species with a boom-and-burst life cycle such as C. elegans). We will therefore model explicitly food depletion in all pheromone-marked patches.
We assume that individuals at a pheromone-marked food patch feed at a rate proportional to the amount of food left in the patch: g(t) = a(t), where a(t) is the amount of food available at the food patch at time t.1 Therefore, food patches get depleted over time as a(t) = A0e−mt, where A0 is the initial food density in the pheromone-marked patches and m is the number of individuals in the patch.2
Individuals that switch pay a cost c for switching. We assume that switching is fast compared to the depletion rate of the food patches, so switching is instantaneous in our model. Individuals that switch will then arrive to any pheromone-marked food patch with equal probability (including their initial one).
We assume that evolution has maximized the total food intake, which is the integral of the instantaneous intake rate (g(t)) over a long period of time. The exact length of this period does not actually matter, because in all relevant cases we will be comparing strategies that end with dispersal, and therefore get the same intake rate at the end. We will always work with differences between the payoffs of these strategies, so these final periods will cancel out. Therefore, for simplicity we will always integrate to t = ∞.
Finally, we assume that individuals have strong sensory constraints: They only perceive their instantaneous feeding rate (g(t)), not having information about any of the other parameters (number of patches, number of individuals per patch, etc.). However, their behavior can be adapted to be optimal with respect to the average values of these parameters over the species’ evolutionary history.
In these conditions, the following Evolutionary Stable Strategy (Maynard Smith, 1982, 1974) exists: At time t = 0, all individuals have a probability p* of switching (so a fraction p* of the individuals will switch). Then they all remain in the food patches until their instantaneous feeding rate falls below gD, at which point they disperse.
Proof
We will prove each part of the Evolutionary Stable Strategy separately:
1. Individuals will not disperse until the food patches are depleted
Dispersing gives an instantaneous average payoff of gD, so individuals should never disperse if their instantaneous intake rate is above gD (i.e. if the food density in their current patch is a(t) > gD).
2. The probability to switch (p) has a stable equilibrium (p*)
If all individuals follow the Evolutionary Stable Strategy, at time t = 0 a fraction p of them switches, changing the distribution of individuals across food patches. Let m1, m2 … mK be the number of individuals in each food patch after the switch. These numbers are related to the initial distribution as which is the number of individuals that remained in the i-th patch plus the number of individuals that arrive to the i-th patch after the switch.
After the switch, all individuals will remain in their new food patch until the instantaneous feeding rate reaches gD. This will happen at time for the i-th food patch. It’s now convenient to split the calculation of the total intake (G) in the two periods before and after dispersal. Then, for an individual that spends its time between the switch and the dispersal at the i-th patch, the total intake is
Note that the solution to the first integral is just the amount of food eaten in the patch (A0 − gD), distributed equally across the mi individuals populating it.
Now we can compute the expected payoff for each decision (〈H〉), which is the expected total food intake minus any costs incurred by the behavior. Individuals that switch have an equal probability of ending up in any of the food patches, so their expected payoff is simply the average of the payoffs across the food patches minus the cost of switching:
In contrast, individuals that remain have a probability ni/N of being in the i-th patch, so their expected payoff is
We now compute the benefit of switching,
It is now convenient to define , which is the deviation in the initial number of individuals from the average number of individuals in every food patch. We also substitute mi according to Equation 1, getting
The equilibrium value of p, or p* will be such that ΔH = 0, so
We did not find a simple analytical expression for the value of p*, but we can make several observations:
- If all Δni are zero, the first term of ΔH is always zero, so ΔH ≤ 0 and switching is never advantageous (therefore, p* = 0). This makes intuitive sense: If all Δni are zero, the individuals were initially distributed in the optimal way (equally distributed across the food patches), so switching cannot bring any benefit.
- If c = 0, then p* = 1 regardless of the value of the rest of the parameters (this can be checked by substitution in Equation Error! Reference source not found.). This recovers the result of our costless model, in which all individuals should switch regardless of the values of the other parameters.
- When p = 1, ΔH = −c
- When p = 0,
- ΔH decreases monotonically as p increases, for any values of the parameters.3
Therefore, ΔH decreases monotonically between and − c, and the value of p at which ΔH = 0 the equilibrium p* of our Evolutionary Stable Strategy (Figure S3). If , then p* is greater than 0 and a fraction of the population will switch. If , then p* = 0.
In the equilibrium, no mutant has an incentive to deviate from its strategy, since both switching and remaining give the same payoff. Furthermore, the equilibrium is stable: When p > p*, ΔH becomes negative, meaning that individuals that switched get lower payoff than individuals that remained, and pushing the population towards lower p. Conversely, when p < p*, individuals that switched have an advantage and the system is pushed towards higher p.
3. Individuals must not switch more than once
Individuals that switched at have the same probability of being in every food patch. A new switch will leave these probabilities unchanged4, so will not affect the expected payoff. Therefore, individuals have no incentive to switch more than once.
4. Individuals must switch at t = 0
A mutant that delays the switch to some later time tS > 0 will spend its time before the switch in an overcrowded patch (on average) and will therefore get lower final payoff than the wild-type that switches at time t = 0. Let’s see it mathematically, comparing the expected payoff for switching at t = 0 and the expected payoff for switching at t = tS:
Most of the term cancel out, leaving which is always positive5. Therefore, switching at t = 0 is advantageous.
5. Individuals must disperse once their current food patch is depleted
Once g(t) = gD, remaining in the same patch will lead to an instantaneous feeding rate below gD, because g(t) decreases over time. Therefore, worms should not remain. And neither they should switch, as we saw in sections 3 and 4. Therefore, they should disperse.
Discussion
A key assumption of the model is that the food patches give diminishing returns, meaning that for individuals remaining in a food patch h(t) decreases over time. In some cases, this may not be true, for example if food density is so high at the beginning that animals are feeding at their maximum possible rate until food depletion reaches some threshold. In this case, switching is not advantageous until this threshold is met.
Acknowledgments
The authors would like to thank the members of the Gore lab for comments on the manuscript, Ying K. Zhang for assistance with the synthesis of ascarosides and Jonathan Friedman for feedback on the model. The C. elegans strain we used was provided by the CGC, which is funded by NIH Office of Research Infrastructure Programs (P40 OD010440). This work was supported by NIH and the Schmidt Foundation. While at MIT, APE was funded by EMBO Postdoctoral Fellowship Grant ALTF 818-2014 and Human Frontier Science Foundation Postdoctoral Fellowship Grant LT000537/2015.
Footnotes
Lead contact: Jeff Gore gore{at}mit.edu
↵1 In general, we have g(t) = Cai(t), where C is a constant. But we simply amounts to re-scaling the units of time).
↵2 2 Proof: If mi worms occupy a patch, and each worm feeds at a rat a rate . Assuming that mi remains constant over time, the a(t) = A0e−mt, where A0 is the initial food density.
↵3 Proof: . This is always negative as long as A > gD, because N is always positive and all terms inside the sum are squared.
↵4 We assume that the population is large enough so that a single mutant does not alter the distributions significantly.
↵5 Proof: , where Δni is defined as in Equation (3), and we define . We will show first that Δni and αi are perfectly anticorrelated (i.e. if Δni > Δnj, then αi < αj for any i, j). Then, we will show that this implies that must be negative.
Δni and αi are perfectly anticorrelated: Both Δni and αi depend on ni. Let’s see that their derivatives with respect to it have opposite signs: From Equation (3), , so it’s positive. From Equation (1), , so has the same sign as . Given that and are always positive, this has the same sign as , which is always negative (it’s zero for mi = 0, and as long as tS ≥ 0 and mi ≥ 0, which is always true. Therefore, is always negative.
is always negative: Let’s split the sum, separating the terms according to the sign of Δni: . Now, given that all the Δni in the first sum are greater than all the Δni in the second, and that Δni and αi are perfectly anticorrelated, all the αi in the first sum must be smaller than all the αi in the second. Therefore, we can find a number α0 which is in the middle of the two groups of αi, so that ,