Specialization and evolutionary branching within migratory populations

Abstract

Understanding the mechanisms that drive specialization and speciation within initially homogeneous populations is a fundamental challenge for evolutionary theory. It is an issue of relevance for significant open questions in biology concerning the generation and maintenance of biodiversity, the origins of reciprocal cooperation, and the efficient division of labor in social or colonial organisms. Several mathematical frameworks have been developed to address this question and models based on evolutionary game theory or the adaptive dynamics of phenotypic mutation have demonstrated the emergence of polymorphic, specialized populations. Here we focus on a ubiquitous biological phenomenon, migration. Individuals in our model may evolve the capacity to detect and follow an environmental cue that indicates a preferred migration route. The strategy space is defined by the level of investment in acquiring personal information about this route or the alternative tendency to follow the direction choice of others. The result is a relation between the migratory process and a game theoretic dynamic that is generally applicable to situations where information may be considered a public good. Through the use of an approximation of social interactions, we demonstrate the emergence of a stable, polymorphic population consisting of an uninformed subpopulation that is dependent upon a specialized group of leaders. The branching process is classified using the techniques of adaptive dynamics.

As a generalized process, animal migration is observed over a broad range of temporal and spatial scales, from the diurnal, vertical migration of lake dwelling algae (1), to the annual interoceanic movements of the humpback whale (2). Animals typically navigate these routes through the use of environmental cues and various sensory modalities, responding to nutrient or thermal gradients, magnetic or electric fields, odor cues, or visual markers (3). Detecting and processing these sources of information is nontrivial and doing so requires an investment either in time and energy or in the physiological cost of developing cognitive or sensory capacity (4).

The potential cost associated with the acquisition of information raises several issues in the context of leadership dynamics and social interaction. Previous works have shown how sociality can facilitate or even enable the use of environmental cues, both theoretically (5–7) and empirically (8, 9). In ref. 10, Couzin et al. demonstrated that small groups of individuals can affect the direction choice of large numbers of uninformed followers, indeed it was shown that for a fixed proportion of leaders the accuracy in terms of the group direction increases as the size of the group grows. If considering collective migration alongside these studies of social interaction, the relation between the costly acquisition of personal information, which drives the migration, and the uninformed following of others may be mapped to a producer-scrounger dynamic (11). In this situation the commodity being generated is information (12) and it is available via two sources, directly from the environment (personal information) or through observations of the behavior of conspecifics (social information) (4).

In a recent study (13) this process was examined using an individual based model governed by localized rules of attraction, alignment etc., with differing degrees of independence and sociality. This work showed that, under certain conditions, specialized groups of leaders form. The challenge in understanding and classifying models of this type lies in identifying an appropriate methodology for the abstraction of interindividual interactions (i.e., amenable to analysis of the evolutionary process but still able to capture the higher level effects of low level behavior). This abstraction is required if complex, interacting systems are to be mapped to a framework that is compatible with the mathematical study of evolutionary dynamics. In this work we employ a unique mean-field approximation of these social interactions to analyze the frequency dependent selection dynamics created via the collective migratory process.

Model Description

In our model, individuals are able to access an imperfect source of information indicating the correct migration route. It is assumed that the effective utility of the environmental cue is an increasing, but saturating, function of the level of investment made in its detection. This represents the diminishing returns that may be associated to variable behavioral or physiological parameters, such as the frequency direction choice is evaluated, or the exploration of alternative routes. The distance travelled in the correct direction over a fixed time interval is then continuously mapped to a linear fitness benefit, which may result from long range improvements in mean abiotic conditions or nutrient availability, reduced risk from nocturnal predators by reaching safe havens earlier, first access to preferred breeding or grazing grounds etc.

A lone individual is then faced with the simple challenge of optimizing the level of return versus cost of investment, however an individual within a group may also choose to follow others and it is assumed that imitation of direction choice incurs no cost. (Note: The validity of this assumption is dependent on the cost to sociality being less than the cost of obtaining personal information but not necessarily zero.)

To describe and quantify the social information source we note an equivalence between the interacting, migratory population defined by an infinite-dimensional system of orientation headings, and a collection of coupled chemical or biological oscillators. This leads us to an approach proposed by Kuramoto (14, 15) in his study of the onset of synchronization in large-scale coupled oscillator systems. By using Kuramoto’s method, the population is described by two key properties; the average heading of the group and the degree of coherence; i.e. the level of consensus around the average heading.

The result is two well-defined sources of information, an external fixed source that represents an environmental cue, and a social source dependent on the behavior of the population. Individuals can then freely evolve sociality, meaning aggregation and the imitation of others, or leadership, the tendency to acquire information and respond autonomously.

For simplicity we assume that if θ defines an individual’s orientation relative to a fixed axis and it moves at a constant speed in this direction, the orientation θ = 0 corresponds to the heading associated with maximum benefit; i.e. along the defined migration route. The quality of the environmental cue is controlled by a noise term in the dynamics of an individual’s orientation heading, representing fluctuating external fields or internal errors in their detection. Increased investment in detection reduces this noise and brings the orientation toward the optimal direction θ = 0. This leads to an Ornstein–Uhlenbeck process of the form

[1]

where x_g is the evolutionarily tunable parameter that defines the degree of investment into gradient detection, dW_t is a standard Wiener process and σ encompasses the noise inherent in detecting the gradient. Hence, increasing the amount of investment made means greater average migration speed.

Considering first the situation for a single individual, we use a Fokker–Planck (16) representation of Eq. 1,

[2]

from which it is possible to obtain the probability density function ρ(θ) that defines the time averaged distribution in θ for a given value of investment. A stationary solution to Eq. 2 can be found if it is assumed ρ( ± π) → 0; i.e., an individual spends a vanishingly small amount of time moving opposite to the desired direction. In this case the periodic boundary conditions due to the angular rotation may be neglected and an analytical approximation of the probability density function can be obtained as

[3]

This equation represents an approximation of the distribution only. It is exact in the limit of x_g/σ² → ∞, however it remains a good approximation even for small values of x_g (see Fig. 1 for a comparison of the average migration speed calculated using this approximation and the numeric values, and SI Text for comparisons of Eq. 3 to the actual distribution). Effectively, by using Eq. 3, we are neglecting the periodic boundaries, assuming ρ(θ) is defined on an infinite domain, and then extracting the only region that is meaningful to our system θ∈[-π,π].

Fig. 1.

(A) Phase coherence r_x as a function of x, lines are analytical result, points are from the numerics, black dash-dot (asterisk) is for a lone individual (the asocial case of Eq. 4), red solid (square) η = 2, green dash (circle) η = 2.5, blue dot (triangle) η = 3. Outline shapes, average migration speed; filled shapes, population order parameter modulus r, which may not be centered around the correct direction. (B) Local fitness gradient as a function of x. Lines are different values of cost, negative values indicate the population will move toward the left, positive to the right.

Using this density function, the distance traveled within a given period can be calculated by projecting the density at each point on to the optimum directional vector. This is then normalized to give a mean velocity along the migration route that lies in the interval (0,1)

[4]

If ρ(θ) were defined by a Dirac delta function then the individual would spend all its time perfectly aligned along the correct heading and the average migration speed would be maximized. In the absence of any cost for personal information x_g = ∞ is an evolutionary attractor and all individuals increase their investment so that ρ(θ) → δ(θ). In our model the average migration speed is scaled by a reducing factor dependent on the level of investment x_g. The fitness, F associated to a given strategy is

[5]

where a reducing factor has been introduced with a strength defined by c and a quadratic dependence on x_g so that higher values of investment become increasingly costly. As an additive fitness differential (i.e., in units of fitness), this cost is defined as . Our choice of this cost function is motivated by analytical tractability; the results do not appear to depend on its exact form. For example, numerical simulations with linear additive cost demonstrate equivalent behavior (see also ref. 13).

Because individuals in the asocial case are noninteracting, the optimum value of x_g can be easily found as

[6]

In the absence of any social interaction, there can be no frequency dependent effects and represents an evolutionary stable strategy that all initial strategy values will evolve toward. However social interactions dramatically alter this situation as the direction choice of others is another source of information that can be exploited through copying, without expending energy in detecting environmental cues directly.

Exactly how this second source of information will be accessed is dependent on the system in question, but for many migratory systems the degree of ordering and the average orientation within the population will be the key parameters that determine how useful and accessible the social information is. When the population is disordered, little information about the average heading can be obtained. As global ordering increases the quality of information increases, and even short-range, local observations of near neighbors give an accurate picture of the global mean orientation.

The degree of ordering and the average orientation can be encapsulated by a single complex order parameter (14). In a monomorphic population the system is essentially ergodic, therefore, for sufficiently large population sizes, the time averaged orientation distribution defined by ρ(θ) is equivalent to the instantaneous probability density function averaged over the population, and Eq. 4 can be related to the complex order parameter of the Kuramoto model

[7]

where r is a measure of the degree of alignment or phase coherence ranging from zero to one, and ψ is the average orientation.

It is therefore possible to define a process equivalent to Eq. 1 where an individual may decide to pay attention to social information with the amount of noise being dependent on r. Precisely how the quality of information relates to r is unknown but it can be assumed to be a decreasing function, with the more disordered the population the more difficult it is to estimate the correct direction. Here we select the simplest function, a linear relation between the variance of the noise term and the population phase coherence, r,

[8]

where x_s represents the degree to which an individual observes and imitates its neighbors and η measures the difficulty of extracting information from social interactions. The magnitude of η determines how the noise term scales with the modulus of the order parameter and so may represent several factors such as the population density or the interaction range of the organisms.

The variables x_g and x_s therefore define a trait space through which a population may evolve. A given strategy is based on a combination of using both social and environmental information, so that Eqs. 1 and 8 are combined through a weighted average in proportion to the investment allocation made

[9]

where dθ_g and dθ_s are defined by Eqs. 1 and 8 respectively.

Finally, the two dimensional trait space defined by x_s and x_g is reduced to one dimension by constraining the two traits to be negatively correlated so that x_s = 1 - x_g. This space is restricted to [0,1] and an individual is defined by a single value. Dropping subscripts we refer to the evolvable trait as x where x = x_g; i.e., x = 0 corresponds to a behavior where an individual pays no attention to any environmental gradients and instead solely copies the direction choice of others. Similarly x = 1 corresponds to an asocial individual, with maximum investment in detecting environmental cues.

Evolutionary Dynamics

Before studying the dynamics of the model, it is necessary to define an appropriate region in parameter space in which results will be of interest. Specifically we note that a condition for evolving higher environmental information processing capacity is that there should be an increase in mean velocity as more investment is made.

The system defined by Eqs. 7 and 8 exhibits a transition to a highly aligned state analogous to that observed in the self-propelled particle model of Vicsek et al. (17). In our model, as the social noise term η is reduced a transition occurs from a disordered, unaligned population, to a perfectly aligned state. (See SI Text for figures showing this transition.) There exists a bifurcation point where the stable, disordered state disappears and the only attractor is at r = 1. By matching derivatives this point can be found as η = 2.

If η < 2, the population has a high level of ordering even in the absence of directional information. For nonzero x the result is a separation of time scales. Firstly the population becomes highly ordered, then due to the asymmetry created by x > 0, it will converge on the optimum direction (see SI Text). In this situation there is little incentive for greater investment and migration ability is weak due to long transient effects. A necessary condition for evolving toward interior solutions in the region x∈[0,1] is that η > 2.

Incorporating this constraint, simulations are performed with an initially homogeneous, monomorphic population in which mutations around the mean phenotype occur. Individuals perform the migratory process until an accurate picture of fitness is obtained. The fitness, average migration speed less cost paid, is calculated and a standard roulette wheel selection algorithm (18) is used to determine which strategies will proceed to the next generation.

In this way the population moves through the trait space. Mutations that are beneficial result in increased fitness and higher than average reproduction. In this case the initially rare mutant will invade the resident population and the mean phenotype will shift. The evolutionary dynamics are governed by the selection gradient, the rate of change of fitness with distance from the resident phenotype. If the selection gradient is positive (negative) the trait value will increase (decrease). A selection gradient that is zero corresponds to a singular strategy that can be classified as an evolutionary stable strategy (ESS) (19, 20) and/or a convergence stable strategy (CSS) (21, 22).

An ESS represents an evolutionary end point for a population because, by definition, once it has been established it cannot be invaded. A CSS represents an attracting point, toward which a population will evolve. These categories are neither mutually exclusive or interdependent. A population will necessarily move toward a CSS but this point may be an ESS, in which case no further evolutionary change will occur, or it may not, meaning there exists the potential for disruptive selection and branching into a polymorphic population.

Numerical simulations were performed using large populations of individuals (65,536) that process environmental information according to Eq. 1 and respond to neighbors via Eq. 8 and the global order parameter. The resulting dynamic is qualitatively equivalent to simulations that explicitly include localized interactions (13).

Fig. 2 shows how population traits change over time and illustrates the presence of an attracting convergence stable strategy at which evolutionary branching may occur. Two stable outcomes to the dynamics are possible. The first is that the population will evolve to a single evolutionary stable monomorphic strategy. The second situation occurs when the convergence stable point is evolutionarily unstable. In this case, the population will spontaneously branch and form a specialized, polymorphic population consisting of an informed group that lead and drive the migration and a social group, dependent on these leaders and the directional information they provide, that survive only by following their neighbors.

Fig. 2.

Branching process for σ = 1, η = 2.5, c = 0.9 (A) and c = 0.5 (B). Generation number is y axis and trait value x is x axis. Red line is the critical value left of which branching occurs.

Invasibility Analysis

To obtain an analytical description of the evolutionary dynamics we first make the assumption ψ ≃ 0. This is true if x > 0 (after relaxation of transients) and is entirely accurate when some level of migration ability is attained. The orientation of an individual is then described by the combination of its tendency to follow others and the environmental cue

[10]

Again introducing the approximation ρ( ± π) → 0 the average migration speed (which is equivalent to the phase coherence because ψ ≃ 0) is found as

[11]

A comparison of the values of the implicitly defined value r_x as a function of x from the individual based simulations and the analytical approximation is shown in Fig. 1. Conceptually the presence of r_x on both sides of Eq. 11 can be related to the time averaged orientation of an individual (left hand side) and the instantaneous ensemble average of the population (right hand side). It is this distinction that enables us to approximate the fitness of a single mutation in the resident population.

When considering the possible invasion of mutant phenotypes we define the resident population as having trait value x with population level order parameter, r_x. The time averaged orientation of a mutant will be different to that of the resident population, but because there are only small numbers of mutants within a large population, the ensemble average will not be affected. A mutant with trait value y will pay the cost associated with this value but will be able to use the social information contained in a population with trait value x. For the time averaged distribution this gives

[12]

The differential fitness of mutant y in a resident population x is

[13]

where

[14]

This value defines the long term exponential growth rate of the mutant in an environment containing individuals with phenotype x, therefore its gradient with respect to y defines the direction in trait space in which the population will move. If

[15]

the population will evolve to increasing values of x, and if this quantity is negative it will decrease. The evolutionary singular strategies are defined by

[16]

In Fig. 1 the selection gradient is shown as a function of the resident trait value, illustrating the presence of a convergent stable point for intermediate values of the cost parameter. For cost values that are too high, the population collapses to a nonmigratory state with zero investment, whereas for low values an informed, asocial population emerges (x = 1). In the following analysis, we neglect these scenarios and focus on the case where interior convergent stable strategies exist.

Numerical simulations suggest that the solution converges to a singular point. Still, however, that point may not be an ESS if

[17]

In this case, evolutionary branching may occur (23, 24). It is straightforward to calculate the conditions for this branching (see SI Text) as

[18]

No closed form solution for x^∗ in terms of the parameters of the system can be obtained; however, it is fully defined by Eqs. 16 and 11. This leads to a function with transcendental terms that behaves as a quadratic in the region x^∗∈(0,1) and tends to -∞ as x → 1. Although the roots cannot be obtained directly the sign of this function at the critical value will show whether the root lies to the left or right and hence whether the singular value, x^∗ is greater or less than the value required for branching to occur (SI Text). If

[19]

then the singular strategy lies to the left of the critical value and the population will branch. After some manipulation the condition for the emergence of a polymorphic population reduces to

[20]

where and . The key to interpreting this condition lies in its influence on the location of the convergence stable strategy. If the cost c or the environmental noise value σ are low the CSS is found at higher values of x. Individuals are then locked into a high investment strategy because they are predominantly asocial and the advantage to exploitation through imitation is weak compared to the loss of environmental information. Lower values of the social noise term η moves the fixed point to the left where individuals are more social and the advantage of imitation is greater. As individuals invade the social information source is weakened, effectively increasing η and shifting the attracting strategy to the right for those that are more informed. The population therefore diverges.

Parameter values that lie either side of this condition are used in the simulations shown in Fig. 2 with the critical value x_c plotted as a red line. In Fig. 3, we include pairwise invasibility plots that show how the evolutionary dynamics shift as the parameters of the model are changed. The condition for branching is illustrated by the presence of positive differential fitness on the vertical line through x^∗.

Fig. 3.

Pairwise invasibility plots. Dark regions represent positive differential fitness (mutant will invade), light regions negative (mutant will not invade). (A, B) Numerical result, (C, D) Analytical approximation. For (A) and (C) c = 0.4 (no branching). For (B) and (D) c = 0.9 (branching occurs). The population resides on the diagonal, mutations occur along the vertical axis and the sign of the fitness differential determines if it will invade. Invasion results in this mutant becoming the resident and the population shifts along the line y = x until it reaches a convergence stable strategy, x^∗. If branching is to occur once a CSS is reached, it is required that a line drawn vertically from x^∗ will immediately encounter a region of positive differential fitness. Numerical results are obtained by evaluating the average fitness of a single individual in the resident population, analytical results stem from Eq. 13.

It should be noted the branching condition described is valid for any small but finite mutation rate. Larger mutations result in a relaxation of this condition as greater variance in the phenotype trait frequency means populations more readily evolve into a polymorphic state.

Discussion

The principle of natural selection appears to define a homogenizing force, leading to the survival of only a few well adapted species whose members compete fiercely for resources. This image, however, is inconsistent with the observed abundance of species diversity and fails to explain the cooperation, division of labor and specialization that facilitates cohesion across multiple biological scales, from human social groups (25) to the unicellular colonial origins of multicellularity (26). Much progress in reconciling these discrepancies has come through the mathematical analysis of evolutionary dynamics (23, 27–30); however, a major difficulty for the study of evolving populations arises from their inherent complexity. Often they consist of many interacting components that display emergent, collective-level phenomena that cannot be predicted or explained by an analysis of individuals in isolation (31–33).

Insight into the behavior of these systems has been gained by using techniques developed in areas such as statistical physics (17, 34, 35), but the focus has largely been on the proximate causes of behavior (i.e., from a mechanistic, functional perspective) and not the ultimate cause (i.e., how natural selection at lower levels has shaped these systems or how collective-level phenomena feeds back on the fitness of individuals). In this work we have shown that the influence of collective interactions on evolution can be dramatic, and in the framework presented results in the formation of a subpopulation of specialized leaders that drive a population level migration.

The catalyst for the evolutionary divergence in our model can be found in the creation and exploitation of information about an optimal migration route. As the production of information is often costly, and through imitation or social facilitation may be inadvertently shared with others, it can be equated to a public good. The key difference to a standard public goods game (36), is the self-dependence of the social information source (illustrated by the implicit definition of the ordering parameter of Eq. 11) that causes a steep transition to a high migration ability once a threshold level of investment is reached. It is this nonlinearity, arising from collective interactions, that creates the selection gradients required for evolutionary branching to occur, and ultimately leads to the stable coexistence of two discrete, interdependent populations.