Complexity and behavioral ecology

Abstract

Reductionism, the practice of predicting large system properties by adding up the measured behaviors of components, has had a long and successful run in recent science. However, in the last 2 decades, fields as diverse as physics, ecology, neurobiology, and economics have recognized that many complicated systems have emergent and self-organized properties that cannot be explained as the linear sum of components, but instead must be viewed as the potentially diverse outcomes of system nonlinearity. Despite some pioneering efforts to apply complexity theory to group movements and decision making, most behavioral ecologists have avoided invoking complexity perspectives in their research. In this essay, we argue that the reductionist focus on dyads in our field has largely run its course and that the next frontier is to examine whether and how social and communication networks function as complex nonlinear systems. To this end, we provide a sampling of topics within behavioral ecology where we think complexity theory may be illuminating.

INTRODUCTION

In the early days of ethology and animal behavior research, dominance hierarchies were “the rage.” Although chickens and primates received the most attention, dominance hierarchies were soon found in a wide variety of taxa. And in a surprising number of species, the hierarchies were structurally linear: The α animal always dominated everyone, the β dominated everyone but α, etc. Although many researchers took this linearity of hierarchies for granted, some even suggesting it had been favored by selection to minimize conflicts, there was a serious problem underlying such linear orderings. Landau (1951a, 1951b) was an early critic who showed that it was mathematically impossible for animals that settled contests based on normally distributed traits such as body size to sort themselves into a linear order. The large numbers of animals at and near the average body size would have no accurate way to do so. Although Landau’s concern was dutifully noted in later reviews and textbook treatments of the topic, its general significance was rarely discussed. The issue surfaced again when Chase (1974, 1985, 1986) made a similar point decades later. But again, the broader implications were often ignored by subsequent researchers.

An early clue to resolving this puzzle was provided by Alexander (1961). He let 4 cricket males establish a linear hierarchy and then isolated the 2 top-ranked animals together and the 2 bottom animals together. The second-ranked cricket now lost all contests, as there was no third-ranking animal for it to dominate. The third-ranking cricket now won all contests with the lowest ranking one. When the 4 were reunited, the second and third crickets switched ranks. These results suggested that crickets used recent contest success as a rule of thumb to estimate their relative fighting ability. In the absence of better information, this was not a bad option. But it could lead to mistakes: Should 1 of 2 equally competent fighters slip during a confrontation and then lose the contest, the loser would reset its self-rank lower and the winner would adjust its self-rank higher. Such adjustments would “spread out” what were otherwise overlapping rankings for this pair of animals. The adjusted self-ranks would then bias each animal’s willingness to escalate in subsequent interactions with other group members. Although this might stabilize dyadic relationships, it is not immediately obvious whether the cumulative interactions among many group members relying on such a process would stabilize and generate the observed linear hierarchies.

In fact, recent models have shown that this process can converge to a stable hierarchy (Beacham 2003; Chase and Seitz 2011). In these models, the final hierarchies are more likely to be linear when relationships are first established sequentially within subgroups, and the functions relating combatant self-ranking to the probability of winning a contest are nonlinear. Put more generally, the linearity of hierarchies is here seen as an emergent property, an artifact if you will, of complex nonlinear interactions between group members.

Given the early concerns and clues, why has it taken so long for this interpretation of hierarchy linearity to appear? We think the major impediment was the obsession with reductionist science that arose in the 1960s and is only now beginning to yield to alternative approaches. The basic assumption of reductionism is that complicated systems can be understood by characterizing the properties and behaviors of their component parts and then adding these up to generate the whole. Look at any issue of Behavioral Ecology or Animal Behaviour and you will find many successful applications of this approach: Mating systems, parent–offspring conflicts, contest behavior, cooperation, and communication are commonly broken down into dyadic interactions and the properties of any larger ensemble predicted by adding up the dyadic behaviors.

Emergent properties, by definition, do not fit into this paradigm. But then, how often does the reductionist strategy fail? Surely, we have done pretty well with the reductionist approach for decades. Are emergent properties common enough to worry about? In this essay, we argue that it is time for behavioral ecologists to move beyond our exclusive focus on dyads and examine how much we might be missing by not treating complexity head on. Many other fields such as physics, neurobiology, computer science, and economics have already made this transition (for a lucid and nonmathematical survey, see Strogatz 2004). Thanks to May (1976) and his students, theoretical ecology made this leap long ago. For some reason, it has been slow to appear in behavioral ecology. When attending recent ISBE and ABS meetings, we have been surprised at how few colleagues invoke or even seem aware of complex systems theory. There are indeed exceptions, which we note below, but even here the researchers seem so focused on their own specific topic that they often do not make any effort to tie their results into the general predictions of complexity theory. We do not claim to be experts in this approach. But we have learned enough recently to realize that, were we starting our careers now, this is a key approach in which we would want to invest. We think complexity is one of the major remaining frontiers in our field. This essay seeks to provide some basic background on the topic and suggest just a few of the behavioral ecological topics where the application of complexity approaches may prove enlightening.

SOME BACKGROUND

One problem with any evolving field of research is that workers adopting different entry points to that field coin their own names for equivalent or at least overlapping processes. This is certainly true for complexity theory. Below, we provide an initial definition of complexity and then summarize several different entry points to complexity that are likely to be relevant to behavioral ecology.

A definition of complexity

A number of authors have tried to provide broad definitions of complexity. For our purposes, we shall adopt a version of one proposed by Mitchell (2011): Given that a system is an ensemble of interacting entities, then a complex system is one that exhibits at least some properties that cannot be explained as the linear sum (superposition) of properties of the component elements. The exceptional properties are said to be emergent. This definition sets us up to introduce our first entry point, nonlinear dynamic systems.

Nonlinear system dynamics

Although behavioral ecologists often focus on stable equilibria (e.g., ESS analysis), most natural systems in the world can change with time. The temporal changes are called the dynamics of the system. Typically, one identifies key variables that describe the current state of the system and then derives equations that predict how these variables will change in the next time interval. The dimension of the system depends on how many key variables are invoked. The equations can be deterministic (no chance involved) or stochastic (in which new values for the key variables are drawn at random from some distribution). The right sides of these equations include extrinsic parameters that are currently fixed in value. They usually also include the values of the key variables in the current state. Given some starting values for the key variables, one can successively apply the dynamic equation again and again to plot the trajectory of the system over time. The trajectory may be quite different depending on the initial starting point and the values of the included parameters. See Strogatz (1994) for a detailed introduction to dynamic systems analysis.

In a linear system, none of the variables on the right side of the dynamic equation have any exponents other than 1, there are no products or ratios of key variables, and none of the key variables is present as the argument of a trigonometric, exponential, or similar function. It is easy to predict the trajectory of a linear system as each key variable changes independently; the next state of the system is just the linear sum of the next states of each key variable. If a parameter is varied, the system responds proportionally.

Reductionism assumes linear systems: Here, you break a system down into its components, see how each component changes over time, and add these changes up to predict the overall state of the system at each successive time point. There are no emergent outcomes in a linear system. This does not mean that linear systems are boring: Linear system trajectories can progress to an equilibrium where further change stops, spiral off into infinity, or exhibit oscillations at some fixed frequency set by the parameters and initial conditions. However, each of these trajectories is entirely predictable given the equations and the values of the extrinsic parameters and initial variable values.

A nonlinear system is one in which one or more of the conditions required for linear systems is violated. Like linear systems, nonlinear dynamics can move a system to a stable equilibrium point or spiral off into infinity. However, variation in parameter values may not result in proportional variation in the system but, instead, trigger major qualitative changes into totally different states. Such shifts in state are called bifurcations. Nonlinear systems can exhibit oscillations, but unlike the harmonic oscillations of linear systems, where the frequency is set by the external parameters and initial conditions, the oscillations of nonlinear systems are limit cycles whose frequencies depend on the system itself. If the dimension of the system is sufficiently large, changing the parameters can cause the system to go into deterministic or stochastic chaos.

A good example of a nonlinear system is the set of vibrating membranes that create signal sounds in vertebrates (Wilden et al. 1998). Consider a terrestrial mammal in which airflow through the larynx acts as a parameter affecting the paired vocal chords on each side of the flow cavity. At very low flows, the folds remain at an immobile equilibrium. At a critical but still moderate flow, the thinner parts of each fold begin limit cycle oscillations, sweeping out a repeated 2-dimensional trajectory. Given the moderate flow and their proximity, the 2-folds act as coupled oscillators and lock into the same frequency. This is the normal vocalization mode, and because it is periodic but invariably nonsinusoidal, the resulting sound appears on a spectrogram as a harmonic series. At a somewhat higher flow, the 2-folds continue to oscillate but the coupling between them breaks down and they may adopt slightly different frequencies. The lower frequency acts as an amplitude modulator of the higher frequency “carrier,” creating sidebands around the higher frequency component in a spectrogram. This is called biphonation. At even higher flows, the entire complex of vocal folds on each side begins to oscillate, but given the larger masses and the shift to 3-dimensional trajectories, at a lower frequency. This would be seen on a spectrogram as a sudden shift from harmonics to subharmonics spaced some fraction, often half, of that seen in the prior series. Finally, at high enough flows, the system lapses into chaos and the spectrogram shows a wide band of noise. One can see several of these modes in 3 successive calls by a wild parrot in Figure 1. Like this parrot, many animals “bifurcate” their nonlinear vibratory systems from one mode to another when producing signals. Human singers do this on purpose, whereas the “voice break” of teenage boys is an unintended bifurcation.

Figure 1

Nonlinear behaviors in bird vocal organ. Three successive “peow” calls by wild male white-fronted amazon parrot (Amazona albifrons). First call on left shows typical harmonic series of stable limit cycle vibrations in syrinx. Middle call shows appearance of subharmonics in last third of call (arrow points to relevant section). Final call lapses into chaos in last two-thirds. Frequency scale (vertical axis): 0–16kHz; time scale (horizontal axis: 0–1.4 s).

Although the versatility of vertebrate sound-producing organs is itself interesting to behavioral ecologists studying communication, the broader message here is that any nonlinear system may be capable of such sudden bifurcations. And nonlinear systems must be common in behavioral ecology. As summarized by Strogatz (1994), “Whenever parts of a system interfere, cooperate, or compete, there are nonlinear interactions going on.” So, instead of the structure of sound production, we might see bifurcations such as a sudden synchronization of behaviors within a large social group (e.g., synchronous firefly flashing), a shift from solitary to aggregated dispersions (e.g., migratory locusts), or the rapid breakdown of respect for lek territories leading to chaotic dispersions of displaying and fighting males (such as we observed with eastern California sage grouse in the acute winter of 1983). Where bifurcations are possible, they will by definition lead to emergent states, and the corresponding nonlinear systems will fit our definition of complex systems. The take-home message is that knowing something about nonlinear system dynamics will help us look for possible complexity in behavioral ecology.

Networks

There has been considerable recent interest in the role of network processes in behavioral ecology (see McGregor 2005; Croft et al. 2008; Krause et al. 2009; McDonald 2009; Chapter 15 in Bradbury and Vehrencamp 2011; Pinter-Wollman et al. 2013). Nearly any interacting ensemble of animals can be modeled as a network including primate troops, males on a lek, nesting colonies of seabirds and pinnipeds, communication systems, etc. Because there is already an extensive literature on networks in general (Strogatz 2001; Albert and Barabasi 2002; Newman 2003; Barabási 2009; Pinter-Wollman 2013), measures for classifying and comparing animal social networks are immediately available.

When are animal social or communication networks complex systems? Given our prior definitions, it should be clear that a network might be either a linear or a nonlinear system: It will depend on the nature of the interactive links between network members. If these relationships are essentially linear, then we would not expect to see emergent properties and the trajectories followed by these networks should be predictable by knowing the relevant equations, starting points, and ambient parameters. However, there are many reasons to believe that the complicated ways that group or network members can affect each other and then be affected in turn by the resulting feedback will generate nonlinear linkages between individuals (see Strogatz quote above). The dominance hierarchies we outlined earlier are a clear case in point. When the network links are largely nonlinear, then we should not be surprised to see the network act like a complex system showing bifurcations and emergent properties like synchronization or other qualitative changes in state. Such emergent behaviors are well known in other kinds of networks such as ecological webs, neurobiological systems, the Internet, and various physical systems (Grossberg 1988; Goldberger et al. 1990; Watts and Strogatz 1998; Strogatz 2001; Albert and Barabasi 2002; Maslov and Sneppen 2002; Barabási and Bonabeau 2003; Newman 2003; Mitchell 2006; Song et al. 2006; Gomez-Gardenes et al. 2007; Arenas et al. 2008; Ebeling et al. 2013; Suweis et al. 2013).

Fractals

Power laws, in which 1 variable is a function of some other variable (the argument) raised to some exponent, are common in nature. The inverse square laws for gravity and sound attenuation are power laws where the exponent is −2. Power laws are self-similar (also called scale free): A plot of functional results versus various values of the argument will have the same shape regardless of the scale of values used. Fractals are sets of objects (numbers, points, lines, etc.) generated by power laws with negative exponents that are not necessarily integers. The absolute value of the exponent is the dimension of the fractal. One can estimate the fractal dimension by plotting the logarithm of the function output against the log of the argument values; the absolute value of the slope should equal the fractal dimension.

The set of points defining a straight line on a plane has a dimension, both classical and fractal, of 1.0. The set of points filling a bounded area on that plane will have a classical and fractal dimension of 2.0. But a squiggling line in the plane that meets fractal criteria will have a fractal dimension between 1.0 and 2.0 because it clearly fills more of the plane than the straight line, but less than the filled area. The power function defining a fractal set can be either deterministic or stochastic; if the latter, the log–log plot will show a scattering of points, but we should still be able to discern a fixed exponent from a regression slope.

What do fractals have to do with complexity? It turns out that the dynamics of complex systems often lead to fractal sets. The more structured the fractal set, the lower the fractal dimension; more random sets have higher fractal dimensions. The efficiency of functional activities is thought to be optimal for intermediate fractal dimensions. For example, the geometric distributions of blood vessels and the pulmonary tree have an intermediate fractal dimension. Biochemical systems with fractal distributions of connections are thought to be more robust to breakdowns than other designs (Gallos et al. 2007). Foragers searching for sparse and randomly distributed prey can optimize search by drawing successive step lengths from a stochastic fractal distribution with a dimension of about 2.0 (Viswanathan et al. 2008). Even complex systems that are experiencing chaos will trace out nonrepeating trajectories that cumulatively obey a fractal rule (Strogatz 1994). For example, the relative proportions of waves of different amplitude in the waveform of a larynx oscillating chaotically will follow a fractal distribution rule.

An enormous amount of theoretical effort has focused on the appearance of fractal structures in complex networks (see summary in Chapter 15 of Bradbury and Vehrencamp (2011)). Most biological and human (e.g., Internet) networks show decreasing power functions relating the fraction of nodes in a network to the number of links to a node. However, the log–log plots are not always linear, implying that something else is going on. In many cases, for example, metabolic networks, the network is organized into hierarchical clusters and modules. Because this same modular structure recurs at various scales, these networks are structured fractals (low dimension). However, other networks include lots of links that reduce the isolation of modules (called “small worlds”; Watts and Strogatz 1998). These tend to have higher fractal dimensions. Current theory and data suggest that the more structured networks sacrifice speed and extent to which local effects are propagated throughout the network but gain robustness and resilience against functional breakdowns by isolating key hubs. The less structured systems are less functionally resilient, but communication propagates more quickly and effectively (Song et al. 2006; Ay and Krakauer 2007; Ay et al. 2007; Rozenfeld and Makse 2009). The combination of fractal and network tools thus provides some very interesting insights to the function and robustness of various network designs.

Self-organization

A self-organizing system is one in which “pattern at the global level emerges solely from interactions among lower level components. Moreover, the rules specifying interactions among the system’s components are executed using only local information without reference to the global pattern. In short, the pattern is an emergent property of the system, rather than a property imposed on the system by an external ordering influence.” (Camazine et al. 2001). By this and our prior definitions, self-organizing systems are complex, and models of their dynamics that predict their behavior well (e.g., that for firefly flash synchronization) are invariably nonlinear. Fractals clearly fit the definition of self-organized systems: A fixed power law rule results in highly complex patterns regardless of scale. Many social networks of animals and people invoke very simple and local rules but generate complex global patterns and are thus self-organized.

Self-organizing systems with few degrees of freedom (e.g., small social networks) might show any of the usual nonlinear outcomes including chaos, whereas larger systems with stronger nonlinearities may avoid chaos but show self-organizing criticality: Here, the system spontaneously evolves to a quasi-stable state of advanced pattern and complexity but is subject to periodic breakdowns whose magnitudes and/or spatial scales are distributed as power laws and fractals. Thus, small breakdowns in a complex network’s functionality might be common, but catastrophic breakdowns would be rare (Bak et al. 1987; Creutz 1992; Bak and Creutz 1994; Turcotte 1999). The classical physics example is a trickle of dry sand grains onto a flat surface (Creutz 2004). Over time, a peaked mound of sand builds up on its own: This is a self-organized structure. When the slope of the pile walls is steep enough, the system becomes self-organized critical. Each subsequent grain of sand added to the pile has the potential of triggering an “avalanche.” Most avalanches will be small, but according to the relevant power law, there is a low but nonzero probability of a major avalanche that flattens the pile. The notion of self-organized criticality has also been applied to earthquakes, forest fires, disease epidemics, stock market crashes, and species diversity collapses in ecological communities.

Topic overlap

It should be obvious that these various phenomena all show high degrees of overlap. In practice, complexity, self-organization, and nonlinear dynamics are almost synonymous. Despite this, researchers focusing on 1 approach often fail to discuss overlapping approaches and their relevant literature. Part of the problem is that several of the key concepts such as complexity, fractals, and self-organization criticality have fuzzy or highly debated definitions (Mandelbrot 1983; Turcotte 1999; Mitchell 2011). At the same time, each of these concepts has found important applications throughout the sciences. We feel that more workers in behavioral ecology than at present should at least know what they mean and how they relate to each other.

POSSIBLE APPLICATIONS

Below, we provide just a small sample of research topics in behavioral ecology that we think are particularly likely to benefit (and in some cases already are benefitting) from complexity perspectives. Some can be, and even are currently, approximated with linear and dyadic approaches. However, this can only provide part of the picture and until researchers examine the broader view, we will not know the degree to which reductionist paradigms are sufficient.

Communication networks

The earliest work on communication networks largely focused on eavesdropping on communicating dyads by third parties (McGregor 2005). Nonlinear systems with 3 or more variables (i.e., dimensions) can exhibit a much wider range of trajectories and outcomes than systems with only 2 variables (Strogatz 1994). We should thus not be surprised were we to find that 3 party communication networks behaved distinctly differently from simple dyads (even if the latter interact nonlinearly). More recent work has examined the much larger networks typical of entire social groups (Croft et al. 2008; Krause et al. 2009). However, the main focus has been the utility of descriptive measures for comparing network structure (Croft et al. 2005; Sundaresan et al. 2007; Hamede et al. 2009; McDonald 2009; Wolf and Weissing 2010), and not on the potential for emergent properties and bifurcations (but see Lusseau 2003). The emergence of synchrony in flashing fireflies and other assemblies of displaying males is 1 case in which complexity theory has been applied to animal behavior (Strogatz 2004). Surely there are many more examples of communication networks that are sufficiently complex and nonlinear that emergent properties and states may be found once we look for them.

Leks and mating swarms

Leks and mating swarms are subsets of the communication networks discussed above. However, there is an additional perspective that may be relevant in their case. As several authors have pointed out, many behavioral interactions should be modeled as “markets” where multiple parties are engaged in negotiations whose outcomes affect not only a negotiating dyad, but many others as well (Noë and Hammerstein 1994, 1995; Hammerstein and Hagen 2005). Patricelli et al. (2011) have recently applied such market analogies to leks. The links between all parties in a market are complicated and likely nonlinear. This may lead to a variety of emergent properties. For example, how much of the unanimity of female choice on classical leks may be an emergent property similar to the linearity in dominance hierarchies? Do large leks ever show self-organized criticality with power law distributed breakdowns in territoriality? Mating swarms of male insects have less structure than leks, but individuals do adjust their positions according to neighbors and dominant males often aggregate at the side where females are most likely to appear (Downes 1969; Thornhill and Alcock 1983; Bradbury 1985). Even swarms may constitute sufficiently complex networks that they show emergent properties and bifurcations. We would not know until we look.

Group coordination and decision making

This is 1 area where complexity modeling is already being applied and proving to be very instructive. The shapes of flying bird flocks and swimming fish schools are emergent properties of a complex system with only local behavior rules (Couzin and Krause 2003; Hemelrijk and Hildenbrandt 2008; Hemelrijk et al. 2010; Hildenbrandt et al. 2010). They are thus self-organized. The trails of ants and the refuges built by social insects are also examples of self-organized systems with globally emergent consequences (Camazine et al. 2001). Finally, the processes by which members of animal groups make joint decisions are often best modeled as complex systems with emergent properties (Conradt and Roper 2005; Conradt and List 2009; Seeley 2010; Sueur et al. 2011). Although these examples are breaking new ground, there are clearly many other coordination systems in other taxa in which we should be looking for similar phenomena (or perhaps different outcomes, given system nonlinearity).

Colonial breeding

Many marine vertebrates such as penguins, sea birds, and pinnipeds breed in dense colonies. Terrestrial species such as social weavers, bee-eaters, some parrots, some icterids, prairie dogs, hyraxes, etc. also breed and sometimes live permanently in colonies. Colonial breeding provides countless opportunities for extrapair mating, acquisition of helpers, competition between neighbors, and shared activities such as alarm signaling. All of these interactions can be seen as links in complex networks, and given the opportunities for feedbacks, many links will be nonlinear. Any of these contexts thus has the potential to show emergent properties, state bifurcations, and other trajectories associated with nonlinear systems. The risk is that one might observe such an emergent property and then spend a lot of effort trying to find some selective and adaptive reason for its existence when, in fact, it is a direct consequence of being a nonlinear system. The example of dominance hierarchies should always be in our minds when faced with unexpected properties and traits.

Evolution of cooperation

It is generally held that the evolution of cooperation is most easily achieved in structured populations with limited mixing (Rousset and Billiard 2000). Some recent work has put a new twist on this idea by viewing a population as a network of interacting individuals. The study argues that if the benefit/cost ratio of cooperating is greater than the number of network links experienced by the average individual, cooperation can evolve as an emergent property of the network (Ohtsuki et al. 2006; Santos et al. 2006; Nowak et al. 2010). Although this claim has stirred up a lot of debate among researchers studying cooperation and altruism (Grafen 2007; Taylor et al. 2007), the basic point remains: higher order properties such as widespread cooperation or the forming of coalitions are potentially emergent properties given appropriate network structure and nonlinearity. Clearly, there is a lot more than can be mined from these approaches.

Personality diversity

The last decade has seen a burst of interest in the existence and role of individual personalities in animals (Sih et al. 2004; Réale et al. 2007; Dingemanse and Wolf 2010; Réale et al. 2010). One obvious question is whether personality diversity is simply a reflection of residual genetic and phenotypic variation or is instead favored by selection in some way. Several recent papers suggest a third option that personality diversity is an emergent property of nonlinear interactions between individuals in a population (McNamara et al. 2009; Botero et al. 2010). These 2 papers focus on different types of social interactions (trust and communicative accuracy), but they are unlikely to have exhausted the types of interactive networks that might produce emergent personality diversity. Personality diversity has proved to be so common that there may be other trajectories that lead social networks to exhibit this property.

Foraging dispersions

Fretwell and Lucas (1969) defined the “ideal free distribution” as the result of settlement on heterogeneously distributed resources when settlers have perfect knowledge, unlimited mobility, and no hindrance to settling except local levels of competition. The final distribution is a Nash equilibrium at which it does not pay for any settled animal to move elsewhere. This concept has spawned decades of subsequent theoretical, lab, and field work (see recent reviews in Cressman et al. 2004; Flaxman and deRoos 2007; Křivan et al. 2008; Cantrell et al. 2012; van der Hammen et al. 2012; Williams et al. 2013). In the current context, it should be clear that the ideal free distribution is a self-organized emergent pattern based on local and individual behaviors. There has been great interest recently in which sampling strategies might be optimal for settlers, and which one(s) are actually used in the wild: Truly random walks, correlated walks, biased random walks, composite walks, Lévy walks, etc. (Bartumeus and Levin 2008; Raposo et al. 2009; Smouse et al. 2010; Viswanathan et al. 2011; MacIntosh et al. 2013; Reynolds 2013). Interestingly, some models that best fit the data rely on stochastic power function and/or fractal distributions to select either the distance to move next or how long to wait before changing direction. This is an area where at both the trajectory and final distribution scales, complexity approaches are likely to provide key insights.

Adaptive dynamics

Although ESS thinking now pervades much of behavioral ecology, it is important to remember that arriving at a stable equilibrium, like an ESS, is only one possible outcome of a system’s evolutionary trajectory. Nearly all evolutionary games involve nonlinear dynamics, and when more than 2 parties are involved (e.g., more than 2 dimensions), the possible trajectories and outcomes are diverse. A number of theoreticians have recognized these facts and now study adaptive dynamics in which entire trajectories are traced out based on the relevant dynamic equations and starting conditions. ESSs are not the only fixed (stopping) points in such trajectories, and examples of bifurcations, limit cycles, branching trajectories, and chaos can be found depending on the model and initial conditions (Dieckmann and Law 1996; Metz et al. 1996; Geritz et al. 1997, 1998; Leimar 2005; Nowak 2006; McGill and Brown 2007; Dercole and Rinaldi 2008; Apaloo and Butler 2009; Leimar 2009). Good examples of outcomes other than ESSs in evolutionary games are the stable limit cycles involving 3 male morphs in each of the lizards Uta stansburiana and Lacerta vivipara (Alonzo and Sinervo 2001; Sinervo et al. 2006, 2007, 2008). A few other taxa appear to show similar cycles (Frean and Abraham 2001; Kerr et al. 2002), and branching has been suggested as 1 mechanism for generating stable polymorphisms (e.g., personalities) in single populations (Geritz et al. 1998; Leimar 2005). It may be time for more of us to step back from ESSs and consider the possibilities of other nonlinear evolutionary trajectories.

CONCLUDING REMARKS

There are certainly additional phenomena within the purview of behavioral ecology that are complex systems and may show emergent self-organized patterns with all the possible side effects and complications. Our goal here was only to provide a few examples and not an exhaustive list. If this essay caused you to stop and wonder whether your favorite system is actually nonlinear and complex, then we have done our job. It may turn out that many phenomena can be adequately described by linearizing the relevant dynamics and focusing on dyads. However, our sense is that there are major discoveries awaiting those who take the broader view, look for emergent properties, and if they find them, try to understand what kind of complex system they are dealing with. As we noted earlier, the addition of this perspective has had major repercussions in other fields of science such as physics, ecology, and economics, and maybe it is time for more of us to give it a burl.

Author notes

Editor-in-Chief: Leigh Simmons