Abstract
Through its behavior, an organism intentionally or unintentionally produces information. Use of this ‘social information’ by surrounding conspecifics or heterospecifics is a ubiquitous phenomenon that can drive strong correlations in fitness-associated behaviors, such as predator avoidance, enhancing survival within and among competing species. By eliciting indirect positive interactions between competing individuals or species, social information might alter overall competitive outcomes. To test this potential, we present new theory that quantifies the effect of social information, modeled as predator avoidance signals/cues, on the outcomes from intraspecific and interspecific competition. Our analytical and numerical results reveal that social information can rescue populations from extinction and can shift the long-term outcome of competitive interactions from mutual exclusion to coexistence, or vice versa, depending on the relative strengths of intraspecific and interspecific social information and competition. Our findings highlight the importance of social information in determining ecological outcomes.
Introduction
The mere presence and even simple behaviors of an individual animal produce sensory information that becomes publicly available to surrounding individuals (Danchin et al. 2004, Dall et al. 2005, Goodale et al. 2010). Such ‘social information’ has long been a central topic of interest in select study systems in which individuals intentionally produce signals (Templeton and Giraldeau 1995, Magrath et al. 2015). However, recent empirical and theoretical evidence from various systems indicates that social information use extends far beyond intentional signaling and appears to be a general phenomenon in systems in which individuals that cohabit a landscape share needs (Seppänen et al. 2007, Goodale et al. 2010, Gil et al. 2017, Gil and Hein 2017, Kane and Kendall 2017, Gil et al. 2018). Perhaps the best studied and most common individual need that is enhanced by social information is predator avoidance: alarm calls warn of approaching predators in avian and primate systems (Zuberbühler 2001, Danchin et al. 2004, Magrath et al. 2015), postures, evasive movements or the use of predator-free space inadvertently provide information on the proximity of threats in avian, mammalian, and fish systems (Griffin 2004, Schmitt et al. 2016, Gil and Hein 2017), and even plants can use chemical cues from damaged neighboring plants to induce defenses to protect against herbivores (Karban et al. 2000, Dicke and Bruin 2001). Because social information typically enhances the fitness of receiving individuals, and because any individual in a population can repeatedly receive such benefits, social information could affect the dynamics of populations (Gil et al. 2018). Thus, understanding the degree to which social information can affect population dynamics is a pressing question in ecology.
Social information creates the potential for indirect positive interactions within and across species and might drive positive density dependence. Positive density dependence (i.e., an ‘Allee effect’) occurs when a greater density of individuals in a population enhances the growth rate of that population (Courchamp et al. 1999, Stephens et al. 1999). This simple process can drive profound changes to the dynamics of a population, affecting not only a population’s carrying capacity but also its likelihood of sudden change or collapse (Stephens and Sutherland 1999, Schreiber 2003). For example, under positive density dependence, loss of individuals (e.g., due to harvesting) can become increasingly detrimental to a population, even leading to negative population growth when a population falls below a critical threshold (i.e., a ‘strong Allee effect’; Stephens et al. 1999). Positive density dependence and the critical population thresholds they can cause are putatively common though difficult to rigorously identify in natural systems and are, therefore, of particular interest to natural resource conservation and management (Stephens and Sutherland 1999, Berec et al. 2007, Gregory et al. 2010). Positive density dependence has been classically attributed to non-information-mediated mechanisms, such as mate limitation or habitat amelioration, and to information-mediated mechanisms in species that form cohesive groups, such as flocks or schools (Courchamp et al. 1999, Stephens and Sutherland 1999, Stephens et al. 1999, Gascoigne and Lipcius 2004b). Yet, positive density dependence can arise due to social information regardless of whether or not individuals form cohesive groups or are conspecifics. Social information typically enhances individual survival or reproduction and increases with the density of information-producing individuals (Kenward 1978, Jackson et al. 2008, Kazahari and Agetsuma 2010, Lister 2014, Berdahl et al. 2016, Gil et al. 2017, Gil et al. 2018).
Social information use is most likely between individuals in similar guilds (e.g., those on the same tropic level with shared predators) and, thus, typically occurs in the context of intraspecific and interspecific competition for resources. As a negative interaction, competition counters the effects of social information. Effects of both competition and social information are density dependent, but in opposing ways. Social information typically is most beneficial at low to intermediate population densities, where information is less redundant or its benefits less ephemeral. In contrast, competition typically is most detrimental at higher densities, where resources are more limited (Gil et al. 2018). Thus, we expect social information to have stronger net per capita effects when population densities are low, as they often are in human-altered landscapes (Courchamp et al. 1999). Nonetheless, to measure the net impact of social information requires knowing the strength of competition. Furthermore, competition and the exchange of social information can occur to varying degrees both within species and across species (Monkkonen et al. 1999, Seppänen et al. 2007, Goodale et al. 2010). Therefore, to understand the ecological consequences of social information requires that we examine the joint effects of intraspecific and interspecific social information and intraspecific and interspecific competition.
Population models offer a framework through which to explore the population- and community-level consequences of social information use in wild animals. Classic models that measure the demographic effects of predator functional response, positive density dependence, and facilitation provide conceptual precursors to the study of social information. Noy-Meir (1975) showed that the deceleration of a generalist predator’s attack rate across low prey densities (a Type II functional response; (Holling 1966) can generate a strong Allee effect (Noy-Meir 1975). In this model, and in most population models, this deceleration of the predator’s attack rate with prey density is attributed to properties of the predator (e.g., satiation, handling time; (Oaten and Murdoch 1975). However, this deceleration could be driven by properties of the prey themselves (i.e., if more prey better help one another avoid predation). More recently, models exploring the demographic effects of mutualism have shown that even when positive interspecific interactions are constrained to low and narrow population density ranges, they can quantitatively and qualitatively affect the fate of one or both interacting populations (Hernandez 1998, Hernandez and Barradas 2003, Zhang 2003, Zhang et al. 2007, Hernandez 2008, Holland and DeAngelis 2009, Holland and DeAngelis 2010). Social information provides a possible mechanism for density dependent mutualism, but with the added complexity of being shared not only between species but also within species (Gil et al. 2018). The two existing population models that explicitly account for social information (Schmidt et al. 2015, Schmidt 2017) focus on the case of enhanced breeding habitat selection among conspecifics and show that social information can drive strong Allee effects. Evaluating whether such critical thresholds occur in multi-species systems with social information requires building on this theory to explicitly model social information in a multispecies context.
Here, we use models of a single species and of competing species to build a theory of the demographic consequences of social information use in wild animals. We focus on the widespread use of social information about predators. We modify a framework developed in Gil et al (2018), where we demonstrated that social information can alter qualitative expectations for population and community dynamics in specific cases, to thoroughly and comprehensively quantify, and develop metrics for, when such qualitative changes are expected to occur. We first quantify the intraspecific effects of this social information using a reparameterization of the classic Noy-Meir model to address the question: under what conditions does this common form of social information affect the existence of critical thresholds, equilibrium densities or persistence of a population? We then expand to a two-species population model, in which competition and the exchange of social information can occur within and between species, to address the question: how does social information affect the nature and outcome of species interactions? Our study reveals that social information can alter competitive outcomes and generate multiple alternative stable states in a predictable manner depending on the relative strengths of intra- and interspecific competition, intra- and interspecific social information, and predation. Our modeling framework is general, meaning it is not system-specific, and our findings lay the groundwork for further theoretical and empirical investigation of how social information scales up to affect the ecology and conservation of natural systems.
Methods
Effects of social information on single species dynamics
To lay the groundwork for our two species model, we begin with the dynamics of a single species with population size N that exhibits logistic growth, determined by intrinsic per-capita growth rate r and intraspecific competition coefficient α (the carrying capacity is 1/α). The population also experiences per-capita mortality due to a generalist predator at a maximal rate p, which decays with prey density through the sharing of social information (e.g., alarm calls, evasive movements; Danchin et al. 2004, Goodale et al. 2010, Magrath et al. 2015) and the per-capita strength of social information b. Thus, the single species dynamics are
The model assumes a generalist predator whose population size remains constant independent of the focal prey species’ density N. Implicit in our model development is that the generalist predator has a Type I functional response and social information reduces the predator’s attack rate. However, if we assume the predator exhibits a Type II functional response, then we still get the same functional form of the predation term in Eq. 1 (Type II) but with new parameters (see Appendix S1 for details). Furthermore, while Eq. 1 allows social information to drive the predation rate to zero, an unlikely outcome in most natural systems, this equation is mathematically equivalent to a functional form in which social information causes predation rate to level off at a nonzero value, determined by an additional parameter (see Appendix S1 for details). Thus, all of our results about Eq. 1 also apply to models with a Type II predator functional response and a minimal predation level even when social information is high.
Effects of social information on competing species with shared predators
We expand upon the model presented in Eq. 1 to measure how social information can affect the long-term dynamics of two competing species. We follow population sizes Ni of each species i, where within-species population growth, ri, density dependence, αii, and a maximal per-capita predation rate, pi, follow the same dynamics and notation as Eq. 1, but these species compete with one another at a rate αij, which represents the per-capita negative effect of the j-th species on the i-th species, where i ≠ j. Both species experience per-capita mortality due to predation at a rate that decays with increasing densities of both species (the two-species analog of the mortality term in Eq. 1); i.e., both conspecifics and heterospecifics share and use social information (e.g., from alarm calls or evasive movements) to enhance predator avoidance (Danchin et al. 2004, Goodale et al. 2010, Magrath et al. 2015). Note that this functional form of mortality could also be used to model non-information-mediated interactions, such as species-specific prey handling times by the predator, or group defenses that increase with density. Here, bii represents the magnitude of the effect of intraspecific social information and bij (i ≠ j) that of interspecific social information, such that the dynamics of the i-th species are
Analysis of models
For the single species model in Eq. 1, we conduct a global bifurcation analysis for different values of b, to determine under what conditions social information about predators can alter the persistence of a population, generate a strong Allee effect, and alter the equilibrium density of a persisting population. For the mathematical and numerical analysis of the competing species model in Eq. 2, we focus on the case that the species are symmetric, i.e., r1 = r2, α11 = α22, α12 = α21, b11 = b22, b12 = b21, p1 = p2. For this two-species model, we analytically derive conditions for different community outcomes and develop an analytically-based numerical method to identify all equilibria and their stability. We use these methods in conjunction with numerically computed isoclines to determine how social information affects the nature and outcome of species interactions. Specifically, we compare individual and combined effects of intraspecific and interspecific social information under different relative strengths of intraspecific and interspecific competition.
Results
Single-species model of social information use
In the single-species model, social information can enhance persistence likelihood, with threshold dynamics, and equilibrium population size (Fig. 1). When the intrinsic per-capita growth rate r is greater than the maximal per-capita predation rate p, the population persists at a stable equilibrium for all positive initial densities. Our mathematical analysis (see Appendix S2 for details) implies that this stable equilibrium density always increases with social information (Fig. 1b; Fig. S2-1a in Appen).
When the maximal per-capita predation rate exceeds the intrinsic rate of growth, the extinction equilibrium is stable and the population tends to extinction whenever the initial population density is low. Social information, however, can generate a strong Allee effect and allow the population to persist whenever the maximal per-capita predation rate lies below threshold value (see Appendix S2 for details)
Equation 4 implies that social information that is strong relative to competition (b > α) can prevent extinction for a population at sufficiently high density (Fig. 1b, Fig. S2-1b; Supplement). When the maximal per-capita predation rate, p, exceeds the critical threshold p*, the population goes extinct for all initial population densities (Fig. S2-1c); population persistence is not possible. When there is a strong Allee effect (i.e. r < p < p*), social information has opposing effects on the unstable equilibrium (below which the population tends to extinction) and the positive stable equilibrium. The population density at the unstable equilibrium decreases with increasing social information, while the density at the stable equilibrium increases (see Appendix S2 for a proof; compare dashed and solid curves in Fig.1b). Thus, with more social information, a population can recover from larger disturbances that reduce their densities and can ultimately approach higher densities. This pattern of social information causing positive density dependence and rescuing populations under high predation is robust to the functional form of the reduction of predation due to social information (Appendices S1 & S2; Fig. S2-1 & S2-2, including the functional form used in Gil et al 2018, where the results here indicate the level of social information necessary to produce the type of qualitatively distinct behavior in the example case study of Gil et al. 2018 Box 2).
Two-species model of social information use
Whether and how social information changes the qualitative outcome from competition within and between species depends on its strength and type. Our mathematical analyses of the symmetric two-species model (Eq. 2; Fig. 2a) provide information about the invasability of the single-species equilibria and the multiplicity of equilibria on the single-species axes and on the two-species symmetric (N1 = N2) axis (see Appendix S3 for details). These analyses identify under what conditions increasing the maximal per-capita predation rate changes the ecological dynamics in two ways. First, we identify when increasing the predation rate shifts the system from coexistence via mutual invasibilty (i.e. each species can invade the equilibrium determined by the other species) to mutual exclusion (i.e. each of the single species equilibria are stable), or vice versa. Second, we identify when increasing this predation rate leads to alternative stable states supporting both species or alternative states only supporting a single species. This analysis reveals that the dynamics of the system depend qualitatively on the joint effects of social information and competition via two simple net interaction indices whose form depends on the strength of social information. When both intraspecific and interspecific social information are weak (i.e., ; see Appendix S3 for details), the interaction index equals
This form of the interaction index is positive when interspecific information exceeds intraspecific information by a greater amount than the difference between inter- and intraspecific competition. When at least one form of social information (intraspecific and/or interspecific) is strong (i.e., ), the interaction index equals
This form of the interaction index is positive when the ratio of inter- to intra-specific information is greater than the ratio of inter- to intra-specific competition (see Appendix S3 for details). Generally, as detailed below, positive values of either of these interaction indices promote coexistence and persistence while negative values favor mutual exclusion or extinction. Note that these interaction indices, therefore, indicate the direction in which social information can change a competitive outcome (towards coexistence or competitive exclusion), not the threshold value for that change itself (i.e., when a transition between coexistence and competitive exclusion occurs). We first explore the effect of social information under neutral competition and then evaluate the full array of outcomes under non-neutral competition, closing with a description of the contexts in which we would expect social information to alter competitive outcomes.
Effects of social information in competitively neutral communities
For the neutral dynamics (Fig. 2b), because αij - αii = 0 and , the signs of Iw and Is always agree, and this index is positive only if interspecific social information is stronger than intraspecific social information (i.e., bij − bii > 0, or, equivalently, ). In this case, interspecific social information, which effectively decreases competitiveness between species by countering this negative effect, causes each species to have a positive per-capita growth rate when it is rare and its competitor is common (Fig. 2c). At high densities, diminishing returns of social information (e.g., due to redundant information and ephemeral benefits; Kenward 1978, Seppänen et al. 2007, Jackson et al. 2008, Lister 2014, Berdahl et al. 2016) will saturate the positive effects of heterospecific density (Fig. 2c) and competition will ultimately constrain population growth. Therefore, for Iw > 0 or equivalently Is > 0, social information promotes coexistence: e.g., even weak interspecific social information (; see Appendix S3 for details) shifts competitively neutral Lotka-Voltera dynamics (Fig. 2b) to coexistence (Fig. 2c). Conversely, if intraspecific social information is greater than interspecific social information (i.e., ), such that Iw < 0 or equivalently Is < 0, then intraspecific social information, which effectively increases competitiveness between species by countering negative interactions within species, causes each species to have a negative per-capita growth rate when rare and its competitor is common. Therefore, even weak intraspecific social information (; Appendix S3) shifts neutral coexistence to exclusion (Fig. 2d).
Effects of strong social information in competitively non-neutral communities
Under non-neutral Lotka-Volterra competitive dynamics (i.e., that lead to coexistence or mutual exclusion, depending on competitive strength, given our assumption of symmetric competitors), social information interacts with the relative strengths of intraspecific and interspecific competition for symmetric species to determine the sign of Is, and the outcomes further depend on the maximal per-capita predation rate (pi). Increasing pi strengthens the effects of both intraspecific and interspecific social information relative to the effects of competition and, thus, leads to different qualitative outcomes for the effect of social information on competitive dynamics. Below, we evaluate the effects of social information first when neither form (intraspecific or interspecific) is strong, then when only one form is strong, and, finally, when both forms are strong.
When neither form of social information is strong, prey populations persist only when ri >pi and, in this case, the sign of Iw (Eq. 4) determines whether one prey species will drive the other to extinction or the two competing species will coexist (Fig. 3a). When either form of social information is strong or both forms are strong, the sign of Is (Eq. 5) determines whether competing prey species will coexist or go extinct (Fig. 3b) and the suite of possible dynamics the system can exhibit, as is detailed below.
When only one form of social information is strong, that form determines the sign of Is. If intraspecific social information is strong and interspecific social information is weak (i.e., and ), then Is is negative and social information promotes mutual exclusion as pi increases (Fig. 4a). Alternatively, if interspecific social information is strong and intraspecific social information is weak (i.e., and ), then Is is positive and social information promotes coexistence aspi increases (Fig. 4b, where Fig. 4biii is the specific case study in Gil et al 2018 Box 3 using a different functional form for the social information feedback).
With either form of strong social information, once the predation rate (pi) exceeds the intrinsic rate of growth (ri) of the two prey species, alternative stable states occur (as in the single-species model; Fig. 1), while both species would go extinct without social information. When Is is negative, there are three alternative stable states: each species persisting in isolation or mutual extinction, and coexistence can no longer occur (Fig. 4aiv). In other words, when predation is sufficiently high, strong intraspecific social information alone can cause competitors to become ‘obligate excluders’ (i.e., the only equilibria require competitive exclusion). Conversely, when Is is positive, single-species equilibria are eliminated, and there are two alternative stable states: species coexistence or extinction of all species (Fig. 4biv). Thus, strong interspecific social information alone can cause competitors to become obligate mutualists (i.e., the only equilibria require coexistence) at this critical level of predation.
Further increasing the predation rate results in the extinction of all species as the only outcome (see single-species analog in Fig. S2-1c). Where exactly the extinction threshold for the two-species system occurs depends, again, on the sign of Is. When Is is negative, system-wide extinction occurs when the predation level pi exceeds the critical predation level p* that we found for the single-species model (Eq. 3). When Is is positive, the critical predation level is (see Appendix S3 for additional details).
When both forms of social information are strong ( and ), their opposing effects generate greater dynamical complexity with the introduction of additional alternative stable states. As before, when Is is negative (i.e., intraspecific social information is stronger than interspecific social information; Fig. 5a, Fig S3-2a) and as pi increases, even a system with low interspecific competition will shift from outcomes at equilibrium that can include stable coexistence (Fig 5ai,ii & Fig. S3-2ai-iii) to those that include only mutual exclusion or extinction (Fig. 5aiii,iv & Fig. S3-2aiv-viii). Conversely, when Is is positive (i.e., intraspecific social information is weaker than interspecific social information; Fig. 5b, Fig. S3-2b) and as pi increases, even a system with high interspecific competition will shift from outcomes at equilibrium that can include mutual exclusion (Fig. 5bi,ii & Fig. S3-2bi,ii) to those that include only coexistence or extinction (Fig. 5biii,iv & Fig. S3-2biii.-viii). As before (Fig. 4), the system goes extinct when Is is negative and predation exceedsp* (Eq. 3; Fig. S3-2aviii), or when Is is positive and predation exceeds p** (Eq. 6; Fig. S3-2bviii).
When predation rate exceeds the intrinsic rate of growth, ri, strong social information of both forms further increases the range of parameters where persistence and coexistence can occur (Fig. 5) in comparison to when only one form of social information is strong (Fig. 4). When Is is negative, coexistence equilibria do not vanish until predation rate exceeds p** (Eq. 6, the system-wide extinction threshold when only interspecific social information is strong), while single-species equilibria remain. When Is is positive, single-species equilibria do not vanish until predation rate exceeds p* (Eq. 3, the system-wide extinction threshold when only intraspecific social information is strong), while coexistence equilibria remain. Thus, when both forms of social information are strong, a greater diversity of prey community states are possible at high predation rates (Fig. 5), relative to cases when only one form of social information is strong (Fig. 4), or neither form is strong (in which case, system-wide extinction occurs when pi>ri, which is represented by the vertical dashed line in Fig. 4, 5, S3-1, S3-2). Furthermore, while conditions that give rise to obligate excluders or obligate mutualists also emerge when both forms of social information are strong (Fig. S3-2avi,vii, and Fig. S3-2bvi,vii), they generally do so over a narrower range of predation rates than when only one form of social information is strong.
Context dependent effects of social information on competitive outcomes
Overall, qualitative shifts in competitive outcomes can occur under each of two conditions (Fig. 6, S3-3): (1) predation exceeds population growth such that the positive effects of social information can rescue the system from extinction (Fig. 6; S3-3a,b), or (2) strengths of intraspecific and interspecific social information are asymmetric in favor of an outcome that opposes that of competition (i.e., low interspecific competition with high intraspecific social information promoting mutual exclusion [below the dashed line in Fig. 6c] or high interspecific competition with high interspecific social information promoting coexistence [above the dashed line in Fig. 6d]; Fig. S3-3), with increasing asymmetry necessary as the overall strength of social information increases.
Discussion
Our theoretical models reveal that the simple and ubiquitous use of social information by individual animals (e.g., using the alarm calls or flight responses of others to avoid danger) can scale up to qualitatively affect population and community outcomes. Specifically, our results indicate that by having positive effects on per capita population growth, even when net positive effects are restricted to low population densities, social information typically raises equilibrium population sizes and allows persistence, with Allee effects, when extinction would otherwise occur (due to either predation or interspecific competition in our models; Fig. 1, 6). These effects of social information on population and community stability arise because social information can decrease mortality and can give rise to critical population thresholds, and if a population or community falls below such a threshold it will have insufficient information from conspecifics or heterospecifics to grow and, thus, will be susceptible to sudden and rapid collapse (Fig. 1, 4, 5). Furthermore, we show that the community-level consequences of social information are strongly context dependent where new metrics, the net interaction indices (Eq. 4 & 5), which measure relative strengths of intraspecific and interspecific social information and competition, determine the direction in which social information influences competition (towards coexistence or mutual exclusion) and, therefore, the suite of qualitative outcomes that are possible in a multi-species system (Fig. 3-6). Therefore, social information can qualitatively change the long-term outcome of species interactions from mutual exclusion to coexistence or from coexistence to mutual exclusion, by allowing systems to overcome net effects of competition (i.e., intraspecific social information counters effects of intraspecific competition, and interspecific social information counters effects of interspecific competition; Fig. 6).
The types of qualitative differences in population and community dynamics with social information illustrated for two specific case studies in Gil et al (2018) can occur under a broad range of parameters, including a range of competitive interactions, where the indices developed here provide metrics to predict how social information might influence population or community structure for a given system. As we develop this new theory, it is important to recognize the challenges of empirically measuring effects of social information in many natural systems. While notable work has been done to quantify behavioral effects of social information in the form of vocalizations in avian systems (Betts et al. 2008, Magrath et al. 2015), and this work has been used to inform demographic models (Schmidt et al. 2015, Schmidt 2017), social information is shared through more nuanced behaviors, such as movements, in many systems. Fortunately, recent advances in the collection of large, high-resolution datasets on individual behaviors in the wild, combined with probabilistic models, are able to reveal strong information-mediated behavioral effects that emerge from subtle individual movements (Strandburg-Peshkin et al. 2015, Gil and Hein 2017, Hein et al. 2018). These and other advances could aid in determining the functional form and parameter values of system-specific models that extrapolate these effects to their demographic consequences and test the theory we develop here.
Single-species model
Our findings on the effects of social information on a single species expand upon the results of (Schmidt et al. 2015, Schmidt 2017), which showed that eavesdropping on breeding habitat quality can affect the dynamics and persistence of a population. Here, we model the use of intentional signals or unintentional cues about predators and show that social information could be a driver of positive density dependence and critical thresholds in relevant natural populations (Courchamp et al. 1999, Gascoigne and Lipcius 2004a, Suding and Hobbs 2009, Kelly et al. 2015).
Our findings further suggest that social information could serve as a stabilizing mechanism for predator-prey interactions: high levels of predation that would otherwise reduce prey populations to extinction and, consequently, threaten predator populations can be sustained when there is sufficient social information available to prey (Fig. 1, 3-6). Thus, social information about predators would be most important to the coexistence of predators and prey in systems in which predators can exert high pressure on individual prey populations (e.g., Van de Koppel et al. 2005, Sandin et al. 2008). Because we assume non-dynamic predators, demographic effects of social information in the face of such factors as dynamic predators and differential social information use across trophic levels remain important, unexplored topics for further research. Nonetheless, our models are representative of instances in which predator and prey populations are demographically decoupled (e.g., due to wide-dispersing or ranging predators; Hixon et al. 2002, Van de Koppel et al. 2005) and provide an important first step toward understanding how predation pressure can interact with effects of social information and competition to shape how populations of prey species grow, persist, and interact.
Two-species model
Ecologists have long recognized that the qualitative nature of species interactions (negative, positive, neutral) are not static in space or time, but vary on a continuum in nature (Bronstein 1994, 2001). Understanding the context dependence of species interactions that shape fundamental rates at the population, community and ecosystem levels remains an open but pressing challenge in the discipline of ecology (Agrawal et al. 2007). Here, we provide theory that shows that a common driver of animal behavior, social information, can be a powerful force that shapes the strength or sign of species interactions.
We show that the fate of competing species can be determined not by the relative strengths of intraspecific and interspecific competition, per se (as we conventionally expect), but instead by the interplay between competition and social information (Fig. 4, 5), Further, we show that because positive effects of social information can strongly affect demographics at low to moderate population densities, while negative effects of competition are strongest at high densities, these opposing effects do not simply cancel one another out but instead interact to give rise to a range of stable population states. For example, we show that strong intraspecific relative to interspecific competition can fail to cause long-term coexistence, as we would otherwise expect, if the effect of intraspecific social information is stronger than the effect of interspecific social information (Fig. 4a, 5a). In nature, such a scenario could result when predators of competing species differ to enough of a degree that social information about predators is more valuable when it comes from conspecifics than when it comes from heterospecifics (Seppänen et al. 2007). Conversely, we show that weak intraspecific competition relative to interspecific competition can fail to drive a system to long-term mutual exclusion, as we would otherwise expect, if the effect of intraspecific social information is weaker than the effect of interspecific social information (Fig. 4b, 5b). In nature, such a scenario could result when individuals are distributed in space such that they are more likely to be proximate to (and, thus, privy to information from) heterospecifics than conspecifics (e.g., in mixed-species bird flocks; Graves and Gotelli 1993, Greenberg 2000, Templeton and Greene 2007, Martínez et al. 2018). Effects of interspecific (relative to intraspecific) social information can be further enhanced by phenotypic differences among species, allowing some species heightened sensory abilities and/or more effective means of transmitting information (Seppänen et al. 2007, Goodale et al. 2010). In either case, surrounding species can come to rely upon such information producers; for example, various bird species are highly responsive to even the nuances of alarm calls from keystone informant species (Templeton et al. 2005, Templeton and Greene 2007), and similarly, zebras respond strongly to simple movements of giraffes (i.e., body postures directed at predators), which possess a much higher vantage point to spot shared predators (Schmitt et al. 2016). Thus, while we may typically expect the strength of the effect of social information on prey demographics to be more pronounced in individuals that more frequently aggregate (e.g., in cohesive flocks, herds or schools), highly useful information (e.g., that prevents predation) could have strong demographic effects even when it is received infrequently.
To simplify our presentation and for analytical tractability (Appendix S3), we primarily focus on symmetrically competing populations. However, the same principles revealed above apply to cases when competing populations exhibit differences in competitive ability: social information counters effects of competition, within or between species, and, consequently, can tip the scales in favor of competitively inferior species or strengthen the dominance of competitively superior species (Appendix S4; Fig. S4-1). Furthermore, we assume that competing species share a generalist predator and that social information reduces encounter rates with this predator. However, a specialist predator would create dissimilarities in the value of social information between species, such that the prey species that is preferred by the predator would exhibit a stronger positive response to social information than the less-preferred prey species, and this can affect the long-term outcome for both prey populations (Appendix S4; Fig. S4-2). It is also true that when social information enhances resource acquisition (Dall et al. 2005, Goodale et al. 2010), instead of or in addition to enhancing predator avoidance, it could exacerbate resource or interference competition in certain contexts (Gil et al. 2017). Consequently, other factors, such as the abundance and distribution of resources, could strongly influence the overall effect of social information on competitive outcomes. In summary, our models show that when net effects of social information exceed and oppose net effects of competition, social information can affect the qualitative outcome at equilibrium, but there remains vast opportunity to expand our framework to incorporate specific features of natural history of various systems.
Conclusions
Mounting empirical and theoretical evidence suggests that social information, a ubiquitous driver of animal behavior and fitness (Goodale et al. 2010, Magrath et al. 2015), can play a significant role in the ecology of natural systems (Gil et al. 2018). Yet, our paper is the first to our knowledge to formalize the inclusion of social information in both single and multi-species population models and to rigorously characterize the demographic effects thereof. Our study provides an important step in our understanding of the potential for social information to underlie the persistence, coexistence and diversity of species across systems.
Our study also highlights a multifaceted potential impact of social information on expectations for conservation management. Social information can raise population size and allow persistence that we would not otherwise expect (e.g., due to high predation and/or competition); however, under conditions of high predation, social information can cause putatively common critical population thresholds (Courchamp et al. 1999, Gascoigne and Lipcius 2004a, Suding and Hobbs 2009, Kelly et al. 2015) that, if not identified by resource managers, can lead to unrealized risks of sudden population collapse (Holt 2007). The probability of such information-mediated local extinctions could increase with demographic or environmental stochasticity (Gilpin and Soulé 1986, Lande 1998). Furthermore, the demographic effects of social information that we reveal suggest that environmental changes that simply inhibit social cueing/signaling among individuals (e.g., anthropogenic increases in turbidity or disruption of chemical cues in aquatic systems (Kimbell and Morrell 2015, Chivers et al. 2016), urban noise masking auditory signals in terrestrial systems (Patricelli and Blickley 2006)) could drive unexpected changes to extinction risk, the outcome of competition, and ultimately the community state (Holt 2007). Furthermore, by changing the expected community structure as it depends on competitive and predation rates, social information could affect the expected ecological outcome of an invasive predator or competitor. The directionality of this effect on expected invasiveness and invasive impact will inevitably depend on the interplay between competition and social information, as described above. Consequently, our findings point to social information as an important factor that could affect how we conserve and manage natural resources, particularly for endangered species at small population sizes where social information is more likely to influence demographic rates.
Acknowledgements
We thank Andrew Hein, Andy Sih and Orr Spiegel for helpful feedback in early discussions of this model. This research was funded by a National Science Foundation Postdoctoral Research Fellowship awarded to MAG.
Appendix S1: Re-parameterizations
In this Appendix, we show the models in the main text can account for a minimal level of predation and a type II functional response. We do these re-parameterizations separately to minimize the amount of notation.
Accounting for minimal predation
First, we show how a model which includes a minimal level of predation , for each species i = 1, 2, that occurs irregardless of social information corresponds to a re-parameterization of the model presented in the Models and Methods. As in the model presented in the main manuscript (single-species model: Eq. 1, two-species model: Eq. 2), ri is the intrinsic growth rate of species i, αij is the per-capita competition coefficient for the effect of species j on species i, pi is the additional maximal predation level that occurs when there is no social information, and bij determines the per-capita reduction of predation on species i due to social information from species j. This model is given by
Assume for i = 1, 2. Define and . Then and we get which has the same form of the model shown in Eq. 2 in the main text.
Accounting for a type II functional response
To account for a type II functional response for the generalist predator, we begin with the single species model and then discuss the two species model. For the single species model, we shall show that it is equivalent, via re-parameterization, to the model studied in the main manuscript. For the two species model, we shall show that it is equivalent to the model presented in the main text under special circumstances that allow some of the analysis to extend to the two species model with a type II functional response.
For the single species model, let h be the handling time of the predator. The attack rate of the predator is a decreasing function of intraspecific social information a/(1 + bN) where a is the maximal attack rate of the predator. Under these assumptions the functional response of the predator is given by
If P is the total density of the generalist predator, then the model becomes
Setting and , we get that is equivalent to Eq. 1 in the main text.
Now, consider the two species model where the generalist predator has a handling time hi on species i, and an attack rate ai/(1 + biiNi + bijNj) on species i. Then its functional response with respect to species i is
Unlike the functional response with a single prey species, this expression does not always simplify to an expression equivalent to the predation term in Eq. 2 in the main text. However, the bifurcation analysis of the symmetric model along the N1, N2, N1 = N2 axes in Appendix S3 still applies where h1 = h2, a1 = a2, pi is replaced with aiP, bii is replaced with hiai + bii, and bij is replaced with hiai + bij. In particular, the interaction index in this case is given by .
Appendix S2: Single species model
In this Appendix, we analyze the single-species model presented in the main text and numerically explore an alternative formulation of the model. Although the main model is mathematically equivalent to the model of Noy-Meir [1975], our analysis of the two possible types of bifurcations and the first order approximation of the positive equilibrium are novel.
The bifurcation analysis
Recall, the single species model is where N is the population density, r is the intrinsic rate of growth, α is the strength of intraspecific competition, p corresponds to the maximal predation level which can be reduced to near zero by social information, and b is the per-capita reduction in predation due to intraspecific social information. Positive equilibria of (S2-1) must satisfy or equivalently
As the left hand side is a quadratic with roots at 1/α and −1/b and a maximum of (1/α + 1/b)2/4 at (1/α − 1/b)/2, we get two possible sequences of bifurcations as p increases from zero to infinity (Figure S2-1):
Weak social information (b/α < 1): If p < r, then the population persists at a globally stable positive equilibrium n*. If p > r, then the population goes extinct as n = 0 is a globally stable equilibrium.
Strong social information: Assume b/α > 1. If p < r, then the population persists at a globally stable feasible equilibrium n*. If (the same as Eq.3 in the main text), there are two feasible equilibria n* < n*, such that initial conditions below n* go to extinction and initial conditions above n* converge to the stable equilibrium n*. If p > p*, then the population goes asymptotically extinct for all initial conditions as n = 0 is globally stable.
Note that the critical value p* is proportional to r, and increases from the value of r up to ∞ as b/α goes from 1 to ∞. Namely, the stronger the social information, the higher level of predation that the population can withstand.
Effects of social information on equilibrium densities
To understand the effect of social information b on the non-zero equilibria of the model, let R denote the per-capita growth rate of the population i.e.
Then the growth rate is G(N, b) = NR(N, b). A positive equilibrium density N*(b) for this model satisfies R(N*(b),b) = 0 and is stable if and unstable if
To understand how N*(b) varies with b, we can implicitly differentiate with respect to b
Therefore, whenever ,
As equations (S2-4)-(S2-6) imply that when N* is stable and when N* is unstable.
We can estimate the positive effect of a small amount of social information b on the globally stable equilibrium . Let N*(b) be the globally stable equilibrium for b sufficiently small. Evaluating equation (S2-6)at b = 0 yields
Thus, to first order in b, when r > p. Namely, the equilibrium increases by a factor proportional to the level b of per-capita social information and effective predation rate p, but inversely proportional to the strength of intraspecific competition and the intrinsic rate of growth r.
An alternative functional form of the model
As an alternative functional form of the effect of social information on predation, we also consider an inverse normal function to model cases in which reductions in per capita mortality due to social information manifest at low densities but are completely negated at higher densities (e.g., due to false alarms and/or occlusion of information [Rosenthal et al., 2015]). In this form, pmax sets the effect of social information on reducing mortality due to predation (analogous to p in Eq. 1 in main text), b controls the strength of the effect of social information and the (symmetric) strength of compensation, and N* is the population size at which mortality due to predation is minimized by social information, such that the effective predation rate is determined by the second term in the model:
Note that complete compensation for effects of social information is likely less common than cases of partial compensation (i.e., when benefits of social information are only partially negated at higher densities) [Kenward, 1978, Seppänen et al., 2007, Jackson et al., 2008, Lister, 2014, Berdahl et al., 2016]. Thus, the functional form of predation in Eq. S2-8 can be considered a lower bound of the demographic consequences of social information. Nonetheless, this functional form with compensation drives the same qualitative pattern as the monotonic form (without compensation; Eq. 1; Fig. S2-2): it allows for a greater carrying capacity of the population (relative to the logistic + predation model) and can prevent extinction, when predation exceeds the intrinsic rate of growth (pmax > r; Fig. S2-2).
Appendix S3: Analysis of two-species model
In this Appendix, we analyze the two-species competition model presented in the main text. Recall, this model is given by where ri is the intrinsic rate of growth of species i, αij is the strength of the competitive effect of species j on species i, pi is the maximal predation level in the absence of social information, and bij is the per-capita effect of social information from species j to species i.
Bifurcation Analysis of the Symmetric Case
Consider the symmetric case i.e. r1 = r2 =: r, p1 = p2 = p, b11 = b22 =: bii, b12 = b21 =: bij, α11 = α22 =: αii, and α12 = α21 =: αij. Under these assumptions, there is an invariant line N1 = N2 =: N for the dynamics, and the dynamics on the line are given by
Our analysis of the symmetric model is divided into two parts. First, we identify when increasing p can cause the non-zero nullclines to cross on the single species axis. When intraspecific information is weak (i.e. bii/αii < 1 cf. the analysis in Appendix S2), this bifurcation corresponds to the system switching from bistability (i.e. both single species equilibria are stable) to coexistence in the sense of mutual invasability (i.e. both single species equilibria are unstable), or vice-versa. Second, we study the equilibrium structure on the single species axes and the N1 = N2 axis. Together, these analyses provide the analytical scaffolding for the results presented in the main text. These analyses, however, do not address the structure of the asymmetric equilibria, i.e. pairs of equilibria of the form (N1, N2) = (a, b) and (N1, N2) = (b, a) with a > b > 0.
Bifurcations along the single species and symmetric axes
As equation (S3-2) is the same as the single species model but with α replaced by αii + αij and b replaced by bii + bij, we can classify the bifurcations as p increases into 4 types. For this classification, we define two critical predation levels:
As ϕ(x) in an increasing function for x ≥ 1, p** > p* when and, conversely, p** < p* when . Furthermore, notice that
Thus, the interaction index defined in Eqn. 5 in the main text is positive if and only if , and Is < 0 if and only if . Based on these observation, we get the following cases:
Weak intra and interspecific information : If r > p, then there are positive equilibria on each of these axes. If r < p, then there are no positive equilibria on these axes and extinction occurs for all initial conditions. As in this case, there is at most one positive equilibrium on each of the single species axes, our earlier analysis implies that the sign of interaction index Iw determines whether predation can shift the system from coexistence to bistability (Iw < 0) or vice-versa (Iw > 0).
Stronq intra and weak interspecific information : As one increases p, one goes from a unique positive equilibrium on each axis (when p < r), to having no equilibria on the symmetric species axes and two positive equilibria on each of the single species axes (when r < p < p*), to finally having no positive equilibria on any axis (when p > p*). Notice that Is < 0 in this case as . See Figure S3-1a.
Weak intra and strong interspecific information : As one increases p, one goes from a unique positive equilibrium on each axis (when p < r), to having no equilibria on the single species axes and two equilibria on the symmetric axis (when r < p < p**), to finally having no positive equilibria on any axis (when p > p**). Notice that Is > 0 in this case as . See Figure S3-1b.
Strong intra and interspecific information : If p < r, then there is a unique positive equilibrium on each axis. If r < p < min{p*, p**}, there are two positive equilibria on all three axes. If p > p*, then all initial conditions on the single species axis go to extinction. If p > p**, then all initial conditions on symmetric two species axis go to extinction. Depending on whether p* > p** or p* < p** one gets different orders of the bifurcations. If Is < 0, then p* > p** and one first loses the positive equilibria on the symmetric axis followed by the positive equilibria on the single species axes (Fig. S3-2a). If Is > 0, then p** > p* and one first loses the positive equilibria on the singles species axes and then the positive equilibria on the symmetric axis (Fig. S3-2b).
Bifurcations from coexistence to bistability and vice versa
The invasion growth rates change sign as one increases p if and only if there is a p value at which the N1 and N2 nullclines intersect at the same point on the N1 axis (by symmetry, this intersection also occurs on the N2 axis). When such an intersection occurs, one has that N1 = x satisfies
Equivalently,
From the first equality, we get and either x = 0 (in which case p = r) or
A nullcline crossing at N1 = x is only of interest if x > 0. Hence, we get two cases. First, if αijbij > αiibii, then x is positive if and only if Δb + Δα > 0. Notice that the quantity Δb + Δα corresponds to the interaction index Iw = bij − bii − (αij − αii) presented in Eq. 4 of the main text. Second, if αijbij < αiibii, then x is positive if and only if Iw < 0.
These observations have two implications. Recall that Δb > 0 means interspecific information is greater than intraspecific information, and Δα > 0 means that intraspecific competition is greater than interspecific competition. If Δα < 0 (i.e. bistability in the absence of predation) and Δb > 0, then predation can reverse the sign of the invasion growth rates (i.e. make them positive and thus allow for coexistence) only if Iw > 0. Second, if Δα > 0 (i.e. coexistence in the absence of predation) and Δb < 0, then the sign of the invasion growth rates are reversed (i.e. both negative resulting in bistability) only if Iw < 0.
Approximation of invasion growth rates
Using the first order approximation developed in Appendix S2, we identify under what conditions social information inhibits or facilitates these invasions. Let be the stable equilibrium density for species i. The invasion growth rate of species j at this equilibrium is
This expression is clearly increasing with interspecific information bji and indirectly decreasing with intraspecific information bii, which increases competition due to an increased density of species i. When social information is low (bii and bji are small), we get the first order approximation
Social information has a positive effect if and only if
In the symmetric case (see above), this has the nice interpretation that relative strength of interspecific information has to be greater than the relative strength of interspecific competition. Namely,
Hence, as presented in Eq. 5 of the main text, the net interaction index determines whether social information has a positive (Is > 0) or negative (Is < 0) effect on the invasion growth rates .
Appendix S4: Effects of social information on competition in the face of asymmetries
In this Appendix, we use numerical simulations to examine the effects of social information on competitive dynamics between two species when one species is competitively superior, or when one species is preferentially consumed by a shared specialist predator.
Superior competitor
To measure the demographic consequences of social information when two competing species are not competitively equivalent (i.e., symmetric), we use Eq. 2 to model species 1 as a superior competitor (α12 < α21), whose population, N1, has an advantage over the population of species 2 (N2), an inferior competitor, for most initial conditions. When interspecific competition exceeds intraspecific competition (Fig. S4-1a), species 1 outcompetes species 2 over a greater range of initial conditions (Fig. S4-1ai), and when intraspecific competition exceeds interspecific competition (Fig. S4-1b), species 1 and 2 coexist, but N1 is greater than N2 over a greater range of initial conditions (Fig. S4-1bi). These outcomes result from the absence of social information, or from the case when the effects of intraspecific and interspecific social information are equivalent and the same for both species. However, if intraspecific or interspecific social information has a greater effect on one species, this can quantitatively or qualitatively shift competitive outcomes; when social information provides greater benefits to competitively inferior species, these species can persist or reach the larger of the two population sizes over a greater range of initial conditions than competitively superior species (Fig. S4-1aii, aiv, bii, biv), and when social information provides greater benefits to competitively superior species, these species can exert greater dominance over the inferior competitor, in some cases excluding this species under all initial conditions (Fig. S4-1aiii, biii).
Specialist predator
We use Eq. 2 to model the case when the predator shared between the competing populations specializes (i.e., prefers) on one of the two species; in this case, the predator prefers species 1 over species 2 (p1 > p2). When interspecific competition exceeds intraspecific competition (Fig. S4-2a), species 2 can competitively exclude the predator-targeted species 1 over all initial conditions (ai), and when intraspecific competition exceeds interspecific competition (Fig. S4-2b), species 1 and 2 coexist, with N2 greater than N1 (Fig. S4-2bi). These outcomes result from the absence of social information, or from the case when the effects of intraspecific and interspecific social information are equivalent and the same for both species. However, if intraspecific or interspecific social information has a greater effect on one species, this can quantitatively or qualitatively shift competitive outcomes; when social information provides greater benefits to the prey species that is preferred by the predator (species 1), this species that could otherwise go extinct can persist and reach the larger of the two population sizes over a greater range of initial conditions than the species that is less preferred by the predator: species 2 (Fig. S4-2aiii, av). When social information provides greater benefits to the species less preferred by the predator (species 2), this species can exert greater dominance over the predator-targeted species (species 1), which can cause or maintain competitive exclusion of this species over all initial conditions (Fig. S4-2aii, aiv, bii).
Footnotes
↵† M.A.G.: mikegil{at}sciall.org; M.L.B.: mlbaskett{at}ucdavis.edu; S.J.S.: sschreiber{at}ucdavis.edu