5

Theory: Forces

There is no falsification before the emergence of a better theory.

—Imre Lakatos²²⁰

Comparative predictions arise from the fundamental forces of design. This part of the book introduces the theory toolbox. I emphasize concepts and demonstrate applications.

Later chapters present many comparative predictions. One can test those predictions without studying the theory. However, a strong theoretical foundation enhances application to challenging problems and opens new predictions.

This chapter focuses on conflict, cooperation, and life history components of fitness. The theory depends on the temporal and spatial scales of success and on variability in performance. The next chapter focuses on the nature, origin, and modification of traits. The final theory chapter introduces design principles for the regulation and control of traits.

The first section of this chapter presents the tragedy of the commons. The forces of design often favor individuals to outcompete their neighbors. Resources devoted to competition reduce the resources available for reproduction. Inefficient use of common resources for reproduction degrades the success of all individuals. This tragedy of the commons powerfully shapes the design of many microbial traits.

The second section demonstrates that similarity between neighbors reduces the tendency to compete and increases cooperative efficiency. Change in neighbors’ similarity leads to strong comparative predictions about microbial traits. Similarity sometimes arises by kinship, which correlates genotypes between neighbors. Other processes also influence similarity, modulating a common force that shapes microbial design.

The third section measures the marginal gains and losses in tradeoffs. For example, how does natural selection alter traits to balance the competitive gains against neighbors versus the cooperative efficiency of resource use? The balance typically occurs when the marginal gain for slightly better competitive success equals the marginal loss for slightly worse efficiency. If the marginal gain in competition were greater than the marginal loss in efficiency, then selection would alter traits to enhance competition. Traits change until marginal gains and losses balance.

The fourth section shows that repression of competition may enhance efficiency. If a trait prevents neighbors from competing, then individuals can increase their success only by raising the group’s shared efficiency. Repression of competition provides an alternative to similarity for reducing competitiveness and increasing efficiency.

The fifth section considers the production of public goods. A public good is something produced by an individual that benefits all neighbors. The producing individual bears the cost, whereas all neighbors share the benefit. A vigorous individual pays a smaller marginal cost of production because a unit of production takes up a smaller fraction of a vigorous individual’s total resources. The population splits into vigorous producers and weak nonproducers.

The sixth section weighs different components of fitness on the common scale of reproductive value. A fitness component’s reproductive value measures its genetic contribution to the future population. For example, we may consider the tradeoff between faster reproduction in a rare habitat versus slower reproduction in a common habitat. The benefit in the rare habitat must be weighted by the low contribution of that habitat to the future population. Similarly, the cost in the common habitat must be weighted by its high future contribution.

Demographic analysis provides the exact weighting of different fitness components by their projected future contribution. Fitness components include fecundity, survival, dispersal, and success in different habitats. Changes in demographic parameters alter the relative reproductive values of fitness components in tradeoffs. Altered reproductive values cause changes in key traits, leading to strong comparative predictions.

The seventh section evaluates traits expressed at various stages in a colony life cycle. Fitness at each stage depends on survival to that stage multiplied by the number of successful dispersers produced at that stage. Traits that increase survival at an early stage in the colony life cycle have high reproductive value because they enhance the probability of survival to all future stages. By contrast, traits that increase survival only at late stages in the colony life cycle have low reproductive value.

The exact reproductive value weighting of survival and fecundity components at each stage can be calculated by demographic analysis. Changes in reproductive value weightings across stages predict changes in stage-specific trait expression.

The eighth section reviews the three key measures of value. Similarity enhances the value of cooperative traits by increasing the shared interests of neighbors. Marginal values compare how changed trait values alter different components of success. Reproductive values weight fitness components by their relative contribution to the future population.

The final two sections raise additional forces. The spatial and temporal scaling of competition alters the fitness valuation metrics. Variability in performance alters the fitness value associated with a trait. Modulated fitness values change the design of traits.

5.1 Tragedy of the Commons

In a single battle the Peloponnesians and their allies may be able to defy all Hellas, but they are incapacitated from carrying on a war. … Slow in assembling, they devote a very small fraction of the time to the consideration of any public object, most of it to the prosecution of their own objects … and so, by the same notion being entertained by all separately, the common cause imperceptibly decays.

—Thucydides410

Cells within a clone share genes. Outcompeting clonal neighbors provides no benefit. The gained genetic transmission is offset by lost transmission of the same genes. With no chance to gain by competition, selection favors clonal traits that use common resources most efficiently.

Genetic diversity breaks common interest. Genotypes gain in the short term by outcompeting neighbors. Better competitors take more of the common resources or contribute less to the common good.

Degrading the commons reduces long-term efficiency. The better competitors initially increase but ultimately do worse over the full demographic cycle. Changed conditions that increase short-term competition between genotypes also lower long-term efficiency.

Thucydides perfectly expressed the conflict between self-interest and group efficiency in this section’s epigraph. Hardin¹⁷² named this conflict the tragedy of the commons. Frank^115,117 showed that the tragedy of the commons powerfully shapes the design of microbial traits.

5.2 Similarity Selection and Kin Selection

In the tragedy, dissimilar neighbors gain by competing. Increased similarity favors greater cooperation. More cooperation reduces wasteful competition and enhances success for all group members.

FITNESS

A simple tragedy model illustrates the fundamental force of similarity.114,115,117,122,128 The model begins by writing the expected fitness of an individual, w, in terms of the individual’s competitiveness, y, and the average competitiveness of its neighbors, z, as

$$w=\frac{y}{z}\left(1-z\right).\text{(5.1)}$$

The relative success of an individual is y∕z, the individual’s competitiveness relative to its neighbors.

Greater investment in competitiveness reduces efficiency. For example, if individuals invest more in cell surface transporters to extract resources from the commons, the cost of extra transporters takes away from resources that might otherwise have gone directly into reproduction.

Let y and z vary between 0 and 1. Values denote the fraction of maximal competitiveness. Less competitive individuals use resources for reproduction more cooperatively and efficiently. Thus, the group efficiency increases with 1 − z.

Individual fitness in eqn 5.1 is proportional to the group success, 1 − z, multiplied by the relative success of an individual within the group, y∕z. In other words, efficiency in use of the commons sets the total success of the group. Competitiveness of individuals against their neighbors sets their share of the total.

Here, neighbors means those individuals in the neighborhood, which includes the focal individual. For example, if the neighborhood has two individuals, including the focal individual, then one-half of the average value in the neighborhood comes from the focal individual.

ANALYSIS OF SIMILARITY

What level of competitiveness maximizes individual gain? Under simple assumptions,^115,122,405 we find the maximum by setting to zero the derivative of w with respect to individual competitiveness, y. Using the standard chain rule of differentiation yields

$$\frac{\mathrm{\text{d}}w}{\mathrm{\text{d}}y}=\frac{\partial w}{\partial y}+\frac{\partial w}{\partial z}\frac{\mathrm{\text{d}}z}{\mathrm{\text{d}}y}=0.\text{(5.2)}$$

Notation aids interpretation. Let

$$-Cm=\frac{\partial w}{\partial y}Bm=\frac{\partial w}{\partial z}r=\frac{\mathrm{\text{d}}z}{\mathrm{\text{d}}y},$$

so that

$$\frac{\mathrm{\text{d}}w}{\mathrm{\text{d}}y}=-Cm+rBm=0.\text{(5.3)}$$

The term  − Cm is the direct effect of the focal individual’s phenotype on its own fitness. Traditionally, in these models, one studies altruistic traits that reduce the focal individual’s fitness and provide a benefit to its neighbors. Examples include secretion of siderophores and secretion of exoenzymes. The marginal cost for the altruistic trait to the focal individual is Cm, in which a positive cost reduces fitness.

The term Bm is the direct effect of the average group phenotype on the focal individual’s fitness. In a model of altruism, Bm is the marginal benefit to the focal individual for an increase in the average altruistic trait expression of neighbors. For example, the focal individual gains as its neighbors increase their secretion of public goods, such as siderophores and exoenzymes.

The value of r is the slope of the group phenotype on the focal individual’s phenotype. That slope measures the similarity between the focal individual and its group.

If we let c = Cm and b = Bm, an increase in an individual’s trait enhances its fitness when dw / dy > 0, which occurs when rb − c > 0. That condition has the same form as Hamilton’s rule for the spread of an altruistic trait.^166–168 Technically, Hamilton’s rule differs from this expression, although the interpretation is similar (Section 15.6).^120–122

When trait values are at maximum fitness, individuals cannot do better by changing their phenotype. For simple assumptions, that means that everyone must have the same phenotype, y = z = z*, and any deviants do worse.¹²² Therefore, we can find the maximum by solving eqn 5.3 at the point y = z = z*.

Assuming that all individuals have the same phenotype is not realistic. But the goal of this model is not to match reality. Instead, we clarify how various forces act to shape the design of phenotypes. If we can identify the broad characteristics of those forces, then we can make testable comparative predictions.

Eqn 5.3 applies when we can write individual fitness, w, in terms of individual phenotype and group average phenotype, y and z. The equation suggests how various forces shape the design of traits. In particular, the force of similarity expressed by r interacts with the forces of marginal costs and benefits, Cm and Bm.

APPLICATION TO THE TRAGEDY

The tragedy of the commons illustrates the role of similarity in the balance of forces. Applying the methods in the previous subsection to eqn 5.1 yields

$$-Cm=\frac{1-z^{*}}{z^{*}}rBm=-\frac{r}{z^{*}}.$$

From eqn 5.3 we obtain^114,115,117

$$z^{*}=1-r.\text{(5.4)}$$

The competitiveness, z^*, rises toward its maximum value as r becomes small and the similarity of an individual to its group declines.

When all individuals express the maximal fitness trait value, z^*, their fitness is 1 − z* = r. That result follows from the fitness expression in eqn 5.1 evaluated at the maximum, y = z = z*.

Thus we have the tragedy. Lower similarity, r, favors greater competitiveness, z^*. Everyone’s fitness decreases because fitness is 1 − z*.

Greater similarity rescues the tragedy: as r increases, z^* declines. Everyone’s competitiveness declines and their fitness rises.

We could have a more complicated functional relation between competitiveness and fitness. The tragedy remains whenever greater competitiveness enhances individual success relative to neighbors and greater competitiveness degrades the efficiency of the group.

In summary, similarity modulates the design of competitive traits. Greater similarity between neighbors alleviates the tragedy, reducing competitiveness and increasing individual and group success.

APPLICATION TO PUBLIC GOODS

A public good is something produced by an individual that benefits all neighbors. The producer pays the cost of production. Everyone gains the benefit. An individual that reduces its production lowers its own cost but still shares in the benefits of public goods produced by others. Cheating nonproducers raise their competitiveness against their neighbors.

We can match the public goods problem to the fitness expression in eqn 5.1. Lower individual production raises competitiveness against neighbors. Thus, we can think of z as reduced production and 1 − z as the average level of public goods production. Then the favored value of public goods production from eqn 5.4 is

1 − z* = r.

Once again, greater similarity, r, favors more cooperation. In this case, cooperation means the level of public goods production, 1 − z*.

Other mechanistic assumptions lead to different fitness expressions. For example, suppose 1 − y is the public goods productivity of a randomly chosen individual, and group success is proportional to the average public goods productivity, 1 − z. In this case, assume that an individual’s reproductive vigor is equal to one minus its public goods production, 1 − (1 − y) = y.

Here, vigor directly affects an individual’s reproduction rather than affecting its ability to compete with neighbors. Fitness is an individual’s intrinsic vigor, y, multiplied by the neighborhood’s quality determined by its public goods productivity, 1 − z, such that

w = y(1 − z).

Following the standard procedure, the favored level of public goods production is

$$1-z^{*}=\frac{r}{1+r}.\text{(5.5)}$$

We have the same qualitative comparative result. Rising similarity, r, predicts increased public goods production, 1 − z*.

PHENOTYPIC SIMILARITY

The similarity in the prior subsections depends only on phenotype. The individuals could be members of the same species or members of different species.

For example, the habitat may be divided into many small resource patches. In each patch, two individuals may compete for a common resource. Similarity describes, on average, how closely matched the level of competitiveness is between patchmates. Similarity is high when strong competitors tend to match with strong competitors and weak competitors with weak competitors. The weak competitors can be thought of as strong cooperators.

The cause of matching does not matter. It may be that, for members of different species, strong cooperators use similar cues to find resources. Or, for members of the same species, similar phenotypes may be associated spatially because they share common genotypes.

Whatever the cause of phenotypic similarity, the association within patches influences success in reproduction. In the tragedy model, the more similar the trait values, the more individuals are favored to reduce their competitiveness and share in enhanced efficiency.

GENOTYPIC SIMILARITY TO NEIGHBORS’ PHENOTYPE

Success in reproduction only influences evolutionary pattern when the associated traits transmit to future generations. If the descendants do not inherit the successful traits, then no evolutionary response occurs.

In the prior phenotypic analysis, we focused on an individual with trait y in a group with average trait value, z. Suppose our focal individual has a combination of heritable factors that, on average, causes their bearers to express trait values, g, such that^121,122

y = g + 𝜖.   (5.6)

The trait value depends on the genetic value, g, plus an environmental or error term, 𝜖. The average of 𝜖 is zero. The average trait value is the average genetic value, ȳ = ḡ. Here, genetic means transmissible factors that influence trait expression.

Changes in g determine the evolution of average trait values. We can rewrite the general expression for trait evolution in eqn 5.2 as

$$\frac{\mathrm{\text{d}}w}{\mathrm{\text{d}}g}=\frac{\partial w}{\partial y}\frac{\mathrm{\text{d}}y}{\mathrm{\text{d}}g}+\frac{\partial w}{\partial z}\frac{\mathrm{\text{d}}z}{\mathrm{\text{d}}g}=0.\text{(5.7)}$$

From eqn 5.6, we have dy/dg = 1. Let r = dz/dg, the slope of average group phenotype on the transmitted genetic value of the focal individual. Then we recover all of the results above but with r now describing the transmitted component of phenotype.¹²²

CAUSES OF SIMILARITY

In many cases, similarity between the focal individual and its neighbors arises because of genetic similarity. For example, the group may share a recent common ancestor, forming a kinship group. Or the genetic similarity may arise because similar genotypes tend to aggregate spatially, independently of common descent.

The method described here does not depend on the cause of similarity, r, between the neighbors’ average phenotype and the focal individual’s genetic value. Restrictive assumptions about similarity arise in other methods, such as Hamilton’s inclusive fitness, strictly defined notions of kinship, or group selection.^122,136

Sometimes the more restrictive assumptions and methods provide special insight. However, in most cases, one gains little practical value by the special assumptions of the other approaches. Here, I will use the simpler and typically more general analysis, leading to expressions such as eqn 5.7.

MUTATION DEGRADES SIMILARITY

A single cell may colonize a resource patch. As the clone expands, the genetically identical cells have perfect similarity, r = 1. Inevitably, mutations arise within the clone, degrading similarity. As mutations increase in frequency and r declines, the cohesive force of similarity breaks down. Selection favors greater competition within the group, reducing cooperation and efficiency.

Common descent and kinship set a tendency for similarity and cooperation. But the true causal force depends only on current similarity. Past history does not matter for selection. Only current phenotype and future genetic transmission matter. “The Pedigree of Honey/Does not concern the Bee.”⁸⁵

If mutation and selection enhance phenotypic similarity, then r rises even though nucleotide divergence increases. Once again, the causal force depends on the phenotypic similarity of neighbors relative to the actor’s genetically transmissible trait value, measured by r, rather than on history, kinship, or nucleotide similarity.¹²²

CHOICE OF PHRASE: SIMILARITY SELECTION OR KIN SELECTION

Similarity selection provides the most accurate phrase. However, similarity often arises by common descent and kinship. Kin selection is widely used. Few people recognize similarity selection.

Anyone who understands the basic principles should immediately recognize the historical broadening of concepts that derived from the original usage of kin selection.¹³⁶ However, the literature strongly suggests that kin selection ignites wasteful controversy and misunderstanding.

Social selection is sometimes used^280,352 for the same concepts as similarity selection, emphasizing that similar social partners do not have to be kin.^113,120,122 However, social selection is also used in other contexts,^255,442 and the phrase does not emphasize the essential factor of similarity.

In this book, I will use both similarity selection and kin selection, which I regard as interchangeable phrases.

COOPERATION BETWEEN SPECIES

For the r coefficient in eqn 5.7, only the average phenotypic value of neighbors matters. Those neighbors, comprising z, may be from different species.¹¹³ The “relatedness” coefficient is the slope of the average phenotypic value of neighbors on the focal individual’s genetic value.

The fact that correlated trait values between different species can drive cooperation between those species raises interesting questions in theory and application.^{73,87,110,113,116,119,170,171,460}

Are correlated traits between species necessary for cooperative evolution between species? That depends on how one defines cooperation between species. Suppose, for example, that species A gains from something produced by species B. Then, if an individual of species A, at a cost to itself, provides something that enhances the growth of species B, the additional beneficial product made by species B returns a benefit to members of species A.

In this case, species B acts passively as a part of the environment, and we do not consider this as an evolutionary problem of cooperation between species. But, in terms of ecological process, we may sometimes wish to view this process as an aspect of mutual benefit between species because A is enhancing B’s growth, and B is enhancing A’s growth.

Evolutionarily, the problem concerns which individuals in species A receive the return benefit. If, initially, a particular individual of species A performs a cooperative act that benefits species B, and the return benefit from B does not come directly to the initially cooperative A individual but instead to other A individuals, can that cooperative behavior increase?

This setup is equivalent to a public goods problem. The initial species A actor effectively releases a factor that is beneficial to all of its species A neighbors that can receive the return benefit from B.

If the A recipients are related to the initial A actor, then the cooperative behavior can increase. In this case, species B is a passive reflector of the behavior, and the neighborhood comprising z is the group of A individuals that may act cooperatively and receive return benefits. We could refine that a bit. But typically what matters is that the A actor and the A recipients have sufficiently positive relatedness, r.

What processes create trait correlations between species? If species B varies phenotypically in the return benefit provided to species A, then trait correlations between species can matter.^113,116,119 A correlation may, for example, be between the greater than average tendency of local A individuals to provide a benefit to nearby B individuals and the greater than average tendency of those nearby B recipients to provide a return benefit to the original A cooperators.

What causes such correlations between species? Familial identity by descent is out because kinship does not occur between species. Instead, spatial associations likely arise by physical linkage or codispersal between individuals of different species.

Spatial linkage between species creates genetic associations in the same way that, in classical Mendelian genetics, physical linkage of genes on chromosomes creates linkage disequilibria. Mixing the paired members of different species by uncorrelated dispersal is similar to chromosomal recombination.

Selection of favored combinations between species also creates associations.^113,116,119 When genetic variants in each of two species work well together and are spatially near each other, the paired individuals reproduce more successfully. Bad gene combinations between species reproduce less.

The expansion of good pairs and loss of bad pairs creates genetic associations between species in the same way that positive epistasis between genetic loci creates linkage disequilibria in Mendelian genetics. Overall, the associations depend on the balance between the enhancement by physical linkage or positively selected pairings and the degradation by uncorrelated dispersal.

How do novel cooperative codependencies between species arise? It could happen in a stepwise process.¹⁷¹ First, species B acts as a passive reflector of A’s behavior. Some genetic variants of A act relatively more cooperatively toward B. The enhanced growth of B returns benefits to individuals of A who are genetically correlated with the initial cooperators.

After this first step, in which the cooperative behavior of A toward B rises to a high level, the reciprocal cooperative behavior of B toward A evolves by the same process. Variants of B enhance A’s growth. Those A individuals with enhanced growth return additional benefits to B individuals who are genetically correlated with the cooperative behavioral tendency toward A.

As the mutually beneficial traits become common in each species, the pairs may evolve to depend on each other. Such dependency arises because each species becomes part of the environment of the other species.

Instead of sequential steps, the two processes may overlap because both A and B vary genetically in the tendency to produce traits that enhance the growth of the partner species. In this case, there will be a transient period during which particular pairings between species work well together because the pairs carry mutually enhancing genetic variants.¹¹³

That positive synergism will create genetic associations between species in cooperative behaviors. Such synergism may allow mutual cooperation and codependency to arise in cases for which such mutualism would not evolve without the extra impetus provided by the genetic correlations between species.

Once strong synergism evolves between species, the genetic variation in cooperative tendency on each side may decline. The genetic correlations act as a transient impetus to push the species over the required threshold for the evolution of mutually beneficial traits and possible codependency. With strong codependency, the mutualism becomes irreversible.¹¹⁶

5.3 Tradeoffs and Marginal Values

In the tragedy model, increased competitive success against neighbors reduces the efficiency of resource use. Such tradeoffs between fitness components often occur.

Typically, the maximum success arises when the marginal gains between the alternative fitness components become equal. Suppose, for example, that two fitness components trade off against each other. If investing a little more in one component provides a gain that is greater than the loss for investing a little less in the other component, then it pays to shift investment toward the first component.

A maximum occurs only when marginally shifting investment between the two components does not alter overall success. In other words, the marginal changes for each component must be the same.

For example, consider a tragedy model in which individual competitiveness against neighbors depends on the resource uptake rate, y. The average resource uptake rate in the local group is z. Let the focal individual’s share of local group success be I(y, z), and the group efficiency in using resources be G(z). Individual fitness is122

w = I(y, z)G(z),

the product of the individual’s competitive share of group success, I, and the overall group’s success, G.

In the prior section, I = y/z and G = 1 − z. Here, we consider the more general functional forms, which may include nonlinear relations.

Normalizing fitness to be one at the evolutionarily favored trait value often helps to obtain a consistent interpretation of forces. Writing

$$w=\frac{I\left(y,z\right)}{I\left(z^{*},z^{*}\right)}\frac{G\left(z\right)}{G\left(z^{*}\right)}$$

yields a normalized fitness of one when evaluated at the fixed point, y = z = z*.

We obtain the trait value that maximizes fitness by following the steps in eqn 5.7, yielding

$$\frac{\mathrm{\text{d}}w}{\mathrm{\text{d}}g}=\frac{I_y}{I}+r\left(\frac{I_z}{I}+\frac{G_z}{G}\right)=0,$$

in which a subscript means a partial derivative with respect to that variable. All functions are evaluated at the fixed point. Matching marginal costs and benefits to eqn 5.3 yields

$$-Cm=\frac{I_y}{I}\mathrm{\hspace{1em}B}m=\frac{I_z}{I}+\frac{G_z}{G}.$$

We weight the marginal benefits by r to measure the marginal valuation with respect to the focal individual’s fitness. The marginal costs and benefits equalize at the maximum, yielding

C_m = rB_m.

A simple example of nonlinearity arises when I = y/z and G = 1 − z^s for 0 < s ≤ 1. This example describes the earlier tragedy model but with more rapid degradation of group success, G, as average group competitiveness, z, rises from zero. With these assumptions

$$z^{*}={\left[\frac{1-r}{1-r\left(1-s\right)}\right]}^{1/s}.$$

When s = 1, we recover z* = 1 − r in eqn 5.4. As s declines, the degradation in group success rises at an increasing rate as group competitiveness, z, increases from zero. That greater loss in group efficiency for small increases in competitiveness reduces the favored level of competitiveness and, equivalently, increases the favored level of cooperation.

5.4 Repression of Competition

Competition degrades efficiency. In a competitive group, everyone’s success may decline. All would do better if some mechanism repressed competition.7,123 Repression of competition is sometimes referred to as policing^115,336 or cheater control.⁴¹¹

Consider a simple extension of the tragedy model in eqn 5.1,

$$w=\left(1-c\alpha \right)\left(a+\left(1-a\right)\frac{y}{z}\right)\left(1-\left(1-a\right)z\right).$$

Individual and average group competitiveness, y and z, remain the same. The trait α is an individual’s investment in repressing competition between group members. The group average for repression of competition is a. The level of repression in the group varies between a = 0 for free competition and a = 1 for complete repression of competition.

In the first term, c is the cost to an individual for investment in policing competition.

In the second term, an individual’s competitive success in the group depends on the fraction of resources divided fairly in the absence of competition a, plus the fraction of resources divided under competition, 1 − a, multiplied by the relative success of the focal individual in competition, y∕z.

In the third term, group success degrades in proportion to the fraction of resources allocated by open competition, 1 − a, multiplied by the average competitive level of group members, z.

Following the prior section, we assume that all variation vanishes at the maximum of fitness, so that y = z = z* and α = a = a*. We find those maximum values by evaluating how fitness changes with individual competitiveness, y, and investment in policing to repress competition within the group, α. The derivatives dw/dg_y = 0 and dw/dg_α = 0 express the changes in fitness with the genetic values for the traits.^115,123 Let the similarity coefficients be the same for the two traits, r = dz/dg_y = da/dg_α.

When r > 1 − c, investing in policing to repress competition does not provide sufficient benefit to individuals, and a* = 0. With no policing, competitiveness rises to the tragedy of the commons value z* = 1 − r, as in eqn 5.4.

We can write the condition equivalently as c > 1 − r. We then see that when similarity, r, is sufficiently high, the amount of competition, z* = 1 − r, that could be repressed and provide gains for policing falls below the cost of policing, c. Thus, strong similarity and an intrinsic tendency to cooperate disfavor repressing competition because there is relatively little intrinsic competitive tendency to repress.

As similarity declines, the tendency to compete rises. With more competition, the potential gains for repressing competition increase. Figure 2 of Frank¹²³ shows the quantitative analysis of this model. The joint evolution of policing mechanisms that repress competition, a^*, and competitiveness, z^*, respond in interesting ways to changes in the costs of policing and competitiveness.

Comparatively, mechanisms that repress competition tend to be more strongly favored as the similarity between neighbors declines.¹¹⁵ Similarity by itself favors self-restraint and reduced competitiveness. Thus,

r → self-restraint ⊣ repression of competition.

Many articles discuss repression of competition in microbes.^397,411,439 Despite the potentially powerful force favoring repression of competition, it remains unclear how often such mechanisms occur in nature.

5.5 Heterogeneity in Vigor and Public Goods

Public goods arise when an individual bears the cost for a trait and all group members share equally in the gains.

Repression of competition provides a public good. Individuals pay the cost to repress competition. Group members share the gains for reduced competitiveness and increased efficiency. Similarly, secreted molecules also provide public goods. Secreting individuals bear the cost of production. All neighbors share the benefits.

The costs for producing a public good may vary between individuals. Some individuals may be more vigorous or have access to greater resources. The relative cost to an individual for expressing a public good declines as vigor increases. More vigorous individuals may be more likely to express public goods because of their lower relative costs.118 This section summarizes the models in Frank.¹²⁷

NO HETEROGENEITY

We first establish the basic setup without heterogeneity between individuals. Let fitness be

$$w=\left[\frac{1-c\left(y\right)}{1-c\left(z^{*}\right)}\right]\frac{b\left(z\right)}{b\left(z^{*}\right)}.\text{(5.8)}$$

Individual production of the public good, y, reduces the direct individual component of fitness by the cost, c(y). We normalize the individual fitness component by 1 − c(z*) to get a meaningful scale for costs, where z^* is the average of y across all groups in the population.

The average of individual contributions to public goods within the focal group is z. The group’s public goods provide a benefit to individual fitness by the group efficiency term, b(z). We normalize the benefit by the population average value, b(z^*).

We evaluate dw/dg = 0 at y = z = z* to find the trait favored by selection, as in prior sections. The marginal costs and benefits equilibrate at C_m = rB_m, yielding

$$\frac{c^{\prime }}{1-c}=r\frac{b^{\prime }}{b},$$

in which primes denote the slopes of each function obtained by differentiation. For linear costs and benefits, c(y) = y and b(z) = z. When evaluated at y = z = z*, we obtain

$$z^{*}=\frac{r}{1+r}.$$

This result differs from eqn 5.5 because I switched from considering z as competitiveness in the prior model to considering z as cooperative public goods production in this model. I switched notation here to match the analysis in Frank,¹²⁷ which the following subsections summarize.

BASELINE SUCCESS AND STARTUP COSTS

There may be some productivity in the absence of the public good. For the linear case, we may write benefits as b(z) = s + z, so that there is a fixed productivity of s in the absence of the public good.

Producing a public good may require turning on a complex pathway. Making a low level of a public good may be significantly costly because of the startup costs of production. Increasing production from low levels may not add much additional expense. For the linear case, assume that c(y) = k + y for y > 0 and c(0) = 0, in which k is the startup cost for producing the public good.

Using these benefit and cost assumptions in eqn 5.8, we obtain

$$z^{*}=\frac{r\left(1-k\right)-s}{1+r}.\text{(5.9)}$$

Higher baseline success, s, and startup costs, k, reduce production of public goods.

INDIVIDUAL HETEROGENEITY

Suppose individuals divide into classes, j, with resource or vigor level, 1 + δ_j, such that δ_j describes the class deviation in vigor from the central value of one. Then we can write individual fitnesses as

$$w_j=\left[\frac{1+{\delta }_j-c\left(y_j\right)}{1+{\delta }_j-c\left(z_j^{*}\right)}\right]\frac{b\left(z\right)}{b\left(z^{*}\right)},$$

in which y_j is the contribution to public goods for a focal individual in class j, and $z_j^{*}$ is the optimal value for class j individuals at equilibrium. The values of z and z^* are the group average and population average values of the trait.

If we assume linear costs and benefits with baseline success, s, and startup costs, k, as in the prior subsection, then following our usual methods and the details in Frank,¹²⁷ we obtain

$$z_j^{*}=1+{\delta }_j-k-\frac{s+z^{*}}{r},$$

in which z^*is given by eqn 5.9. If the parameters satisfy $z_j^{*}\ge 0$ for all of the classes, j, and we assume a symmetric distribution centered at zero for deviations in resources or vigor, δ_j, then

$$z_j^{*}-z^{*}={\delta }_j.$$

Class j individuals deviate in their public goods expression from the central value of z^* by δ_j, their deviation in vigor from the average.

Comparatively, we obtain the two predictions given in Section 4.4 for heterogeneity. First,

vigor ⊣ marginal costs ⊣ secretion.

Increasing vigor reduces the marginal costs of production, which favors greater production of public goods. Marginal costs decline with vigor because, as δ_j rises, a small change in costs, c, has proportionately less effect on baseline individual fitness, $1+{\delta }_j-c$. Second,

het vigor → het costs → het secretion.

Increasing heterogeneity (het) in vigor increases heterogeneity in the marginal costs of production, which favors greater heterogeneity in the production of public goods.

5.6 Demography and Reproductive Value

A trait often influences different components of fitness. For example, faster growth of a pathogen within a host increases the pathogen’s number of progeny and the dispersal to other hosts. Faster pathogen growth may also decrease the host’s lifespan, reducing the survival of the pathogen.

To study microbial traits that trade off dispersal versus survival, we must consider the relative valuation of those two distinct fitness components. Life history theory analyzes the reproductive values of different fitness components.61,403

Similarity selection often affects the various reproductive value components of fitness in different ways. Thus, we need to combine the life history analysis of reproductive value with the analysis of similarity.^122,405

This section briefly illustrates the main concepts. Chapter 8 of Frank¹²² provides details, extending Taylor & Frank’s⁴⁰⁵ original analysis.

PRINCIPLES

Different fitness components associate with different classes of reproduction. For example, we may label dispersers as class 1 individuals and nondispersers as class 2 individuals. The transmission of trait values to the future flows separately through the two classes.

We wish to study the total fitness effect caused by a change in trait value. To obtain the total effect, we analyze the consequences for each class and then combine the results into an overall effect.

The fitness consequence for each class depends on that class’s contribution to the future population. The contribution has three aspects.

First, the number of individuals in class j influences the contribution of that class. We write u_j for the frequency of class j.

Second, when class j individuals contribute to class i, the value of that contribution must be weighted by the reproductive value of class i, written as v_i.

For example, if class i represents dispersing individuals, then we must weight the contribution to class i by the expected relative contribution of a disperser to the future population.

Third, the relative contribution of class j to class i is w_ij. For example, we may be interested in w_ij(y, z), expressing the effect of an individual’s trait, y, and the group average trait, z, on the contribution from j to i.

The overall fitness valuation for the contribution of class j to class i is v_iw_iju_j. Summing all transitions yields

$$W=\sum _{ij}{v}_i{w}_{ij}{u}_j=\mathrm{vAu}.\text{(5.10)}$$

Here, v is the row vector of reproductive values per individual for each class, u is the column vector of class frequencies, and A is the matrix of w_ij fitness values.

We can study the direction of change in traits and find trait values that maximize fitness by analyzing dW/dg, as in earlier sections. The extended method here accounts for the different numbers of individuals in various classes and the different reproductive valuations for various components of fitness.

DISPERSAL VERSUS SURVIVAL

Consider a tradeoff between the production of dispersing progeny and the future survival in the current habitat. This brief summary follows the model in section 8.3 of Frank.¹²²

This example has two classes. Dispersers that successfully colonize a new patch form class 1. The new colonizers and their nondispersing descendants form class 2. Let the fitness components be

$$\mathrm{A}=\left[\begin{array}{cc}\hfill 0\hfill & \hfill \beta \left(y\right)/D\hfill \\ \hfill 1-t\hfill & \hfill 1-\delta -z\hfill \end{array}\right].\text{(5.11)}$$

Entries in row i and column j denote w_ij, the contribution of class j individuals to class i. Thus, w₁₁ is zero because newly arrived colonizers of class 1 do not make dispersers but instead survive locally at rate w₂₁ = 1 − t to form the surviving lineage of colonizers as class 2.

The component w₁₂ = β(y)/D describes the contribution of the local lineage to dispersers that successfully colonize a new patch. The local lineage’s investment in making dispersers is y, and β(y) is the functional relation between dispersal investment and dispersal success. Dispersal success is normalized by the density-dependent factor, D, in which greater density-dependent limitation reduces dispersal success.

The component w₂₂ = 1 − δ − z describes the survival of the colonizing lineage within its patch. The intrinsic loss rate is δ, which combines destruction of the patch, loss of the colonizers from a continuing resource patch, or death of a host when the colonizers are parasites.

The intrinsic loss rate is increased by z, which is the patch average of the trait value y that determines the number of successful dispersers. As successful dispersal rises, the local survival rate decreases.

When evaluating total fitness, W, from eqn 5.10, we need the individual reproductive values, v, for the classes when evaluated at demographic equilibrium, y = z = z*, derived in Frank¹²² as

v ∝ [1 − t λ],

in which “ ∝” means proportional to. The reproductive value of new colonizers is discounted by 1 − t, the probability of surviving the initial delay after colonization and before producing dispersers. The reproductive value of residents is augmented by λ, the population growth rate, because residents have average reproductive success λ during the period when new colonizers do not reproduce. The value of λ is the dominant eigenvalue of the fitness matrix A evaluated at y = z = z*.

The class frequencies at demographic equilibrium are proportional to

$$\mathrm{u}\propto \left[\begin{array}{c}\hfill \beta \left(z^{*}\right)/D\hfill \\ \hfill \lambda \hfill \end{array}\right].$$

To obtain the trait values that maximize the total fitness in eqn 5.10, we evaluate dW/dg = 0 at y = z = z*, which includes

$$\frac{\mathrm{\text{d}}\mathrm{A}}{\mathrm{\text{d}}g}=\left[\begin{array}{cc}\hfill 0\hfill & \hfill {\beta }^{\prime }\left(z^{*}\right)/D\hfill \\ \hfill 0\hfill & \hfill -r\hfill \end{array}\right]$$

and the vectors v and u at demographic equilibrium, leading to a solution that must satisfy v₁β′(z*)/D = v₂r, which yields

$${\beta }^{\prime }\left(z^{*}\right)=\frac{r\lambda D}{1-t}.$$

If we assume that dispersal success is β(z) = z^s, with s < 1, then dispersal success rises at a diminishing rate with investment in dispersal, yielding the solution

$$z^{*}={\left[\frac{s\left(1-t\right)}{r\lambda D}\right]}^{1/1-s}.\text{(5.12)}$$

The various terms interact to determine the favored dispersal rate, z^*. However, we can get a sense of partial causation by considering how z^* changes in response to partial changes in the terms. In particular, a rise in t lowers the initial survival of colonizers within a patch, decreasing investment in dispersal. Similarly, a rise in λ raises the growth of patch residents, lowering the relative value of colonizers and also decreasing investment in dispersal.

A decrease in density-dependent limitation, D, increases the opportunity for dispersers to find new patches, raising dispersal. Smaller values of s cause more rapid saturation of dispersal success, lowering dispersal investment.

This model also expresses the tragedy of the commons. Reduced similarity, r, favors more dispersal, which decreases local survival and the long-term quality of the local patch. In other words, dispersal is a competitive trait that degrades the local commons by more rapidly extracting local resources to develop dispersal-enhancing traits.

These conclusions provide a rough qualitative sense of how various forces shape dispersal. In each case, I emphasized how a change in some factor leads to a partial pathway of causation favoring either an increase or a decrease in dispersal.

ALTERNATIVE MECHANISTIC EFFECTS ON DISPERSAL AND SURVIVAL

In the fitness matrix of eqn 5.11, individual trait value, y, influences dispersal, and the group average trait, z, influences local survival. These assumptions express a tragedy type model, in which individuals compete for resources to increase the dispersal of their progeny, and competitiveness degrades the local commons.

Alternatively, successful dispersal may require joint action by neighbors, so that dispersal depends on the group average trait, z. Each individual’s cooperative contribution to joint action, y, reduces its own survival but does not affect the survival of neighbors. These assumptions create a public goods problem. Individual traits contribute to dispersal, which arises from shared public goods.

In the prior model, we change from the original tragedy assumptions to the public goods problem by switching y and z in the fitness matrix of eqn 5.11. We then obtain the same form for the favored trait value as in eqn 5.12, but with the similarity coefficient, r, now in the numerator rather than the denominator

$$z^{*}={\left[\frac{rs\left(1-t\right)}{\lambda D}\right]}^{1/1-s}.$$

In this case, increasing similarity favors greater cooperative contribution to dispersal as a public good, with a greater individual cost through lower survival.

These examples show how alternative mechanistic aspects of traits can reverse the direction of trait evolution favored by a particular force.

THE CENTRAL ROLE OF DEMOGRAPHY

Under different assumptions, a rise in population growth rate may provide better opportunities for dispersal rather than better success for residents. In that case, increasing population growth rate, λ, would associate with greater investment in dispersal.

The point is that demographic processes can strongly influence the direction of trait evolution. In studies of microbes, past work has emphasized similarity and kin selection but has paid relatively little attention to demographic aspects of populations.

5.7 Stage-Dependent Traits in Life Cycle

Suppose some microbes colonize a resource patch. They grow for many generations. They also send dispersers to colonize other patches. Those dispersers can be thought of as the reproduction or fecundity of the group. Total reproduction over the colony life cycle depends on how long the colony survives.

Over the colony life cycle, how do the fundamental forces shape competitive and cooperative traits? We must consider, at each stage in the life cycle, how traits influence an individual’s relative share of the group’s current and future genetic transmission. We must multiply that reproductive share by the total productivity of the group.

We could use the demographic methods of the previous section to analyze the various components of fitness. However, it is easier in this case to write a single expression that combines the fecundity and survival components of fitness over the full life cycle. This section briefly summarizes Frank’s128 analysis.

CYCLE FITNESS

A colony grows through j = 0,1,… temporal stages. The fitness of a focal individual in the jth stage is

$$w_j=I\left(y_j,z_j\right)\sum _{k=j}^{\infty }{\lambda }^{-k}G\left(\mathrm{z}{}_k\right).\text{(5.13)}$$

The first term, I, describes an individual’s share of the colony’s long-term success. In a tragedy model, I increases with an individual’s competitive trait expression, y_j. For example, I = y_j/z_j expresses the relative competitive success of an individual with trait y_j when competing in a group with average competitive trait value, z_j. In a public goods model, I decreases with greater individual expression of the public good, y_j.

The second term describes the reproductive value for the colony in the jth stage. That value is the sum of the colony success, G, in the current stage, j, and in all future stages. The colony success for each stage is multiplied by the discount for the amount the population size has grown, λ^− k, since colony inception at stage j = 0. We discount future reproduction by the expansion of the population size because a single progeny represents a declining share in an expanding population.

The group success in stage k can be divided into survival and fecundity components of reproductive value,

$$G\left(\mathrm{z}{}_k\right)=S\left(\mathrm{z}{}_k\right)F\left(\mathrm{z}{}_k\right).$$

The survival to stage k is $S\left(\mathrm{z}{}_k\right)$, which depends on the group average trait expression in each stage up to and including the current stage, z_k = z₀, z₁,…, z_k. Similarly, the fecundity $F\left(\mathrm{z}{}_k\right)$ also depends on the current and prior trait expression.

We find the trait vector, z*, that maximizes fitness by simultaneously evaluating dw_j/dg_j = 0 for all j when evaluated at y = z = z*.

TRAGEDY AND PUBLIC GOODS MODELS

Suppose the colony grows without producing dispersers from generations k = 0,1,…, g − 1. Then surviving colonies remain at constant size and produce migrants in proportion to their fecundity in each of the following generations.

With those assumptions, the components of individual success, group survival, and group fecundity are, respectively

$$\begin{array}{llll}\hfill I\left(y_j,z_j\right)& =\frac{y_j}{z_j}& \hfill & \\ \hfill S\left(\mathrm{z}{}_k\right)& =S\left(\mathrm{z}{}_k^{*}\right){\left[\frac{1-z_j}{1-z_j^{*}}\right]}^{\theta \left(g-1-j\right)}& \hfill & \\ \hfill F\left(\mathrm{z}{}_k\right)& =F\left(\mathrm{z}{}_k^{*}\right)\left[\frac{1-z_j}{1-z_j^{*}}\right].& \hfill & \end{array}$$

Individual success follows the standard tragedy model. An individual’s share of group success in the jth generation is the ratio of its competitive trait, y_j, relative to the group average, z_j.

Survival to generation k depends on the survival in each of the preceding generations. Thus, any cooperative enhancement of survival in a particular generation carries a benefit forward to all future generations. In this model, deviations in group trait values only influence survival during the juvenile generations, j < g − 1.

In each juvenile generation, j, the survival consequence of a deviation in group trait value, z_j, is ${\left[\left(1-z_j\right)/\left(1-z_j^{*}\right)\right]}^{\theta }$. That value multiplies for each of the g − 1 − j juvenile generations over which it acts. Any consequence to total survival over the juvenile period also affects cumulative survival to future reproductive generations. The value of $S\left(\mathrm{z}{}_k^{*}\right)$is the baseline survival rate to generation k in a group without deviant trait values.

The fecundity consequence for a deviation in group trait value is $\left(1-z_j\right)/\left(1-z_j^{*}\right)$. The value of $F\left(\mathrm{z}{}_k^{*}\right)$ is the baseline fecundity in generation k in a group without deviant trait values.

This model assumes the typical tragedy of the commons form, in which increasing z corresponds to greater competitiveness and degradation of group success. We can also interpret this model as a public goods problem, in which 1 − y is an individual’s public goods production and 1 − z is the group’s average production. Then decreasing z corresponds to greater cooperative contribution to public goods and an increase in group success.

With these alternative model interpretations, we can think of $1-z_j^{*}:z_j^{*}$ as the ratio of the cooperative to competitive tendency in trait values.

Solving $\mathrm{\text{d}}{w}_j/\mathrm{\text{d}}{g}_j=0$ for all j when evaluated at y = z = z* yields $z_j^{*}$, the favored trait value in each generation j. When expressed as the cooperative to competitive tendency, $1-z_j^{*}:z_j^{*}$, we obtain

$$r\left(1+{\gamma }_j\right):1-r,$$

with the enhanced demographic component for the cooperative tendency caused by the trait’s contribution to colony survival as

$${\gamma }_j=\left\{\begin{array}{cc}\theta \left(g-j-1\right)\hspace{1em}\hfill & j<g-1\hfill \\ 0\hspace{1em}\hfill & j\ge g-1.\hfill \end{array}\right.$$

This model illustrates the increased selective force on cooperative and competitive traits during the early stages of colony growth, when j is small. More detailed mechanistic assumptions for trait action would lead to more specific predictions for particular traits. For example, a secreted public good that decays more slowly than the generation time would be strongly favored early in the colony growth cycle but would be less advantageous later in the colony life cycle.

The declining value of new secretions arises in two ways. First, public goods may already be present in the environment because of the slow decay from secretions in prior generations. Second, cooperative traits later in the colony life cycle typically have lower reproductive value.

5.8 The Three Measures of Value

This section briefly summarizes the primary measures of value. Three exchange rates transform the various effects of traits into the common currency of contribution to the future population.122

First, interacting individuals may have similar trait values or share similar genes. The coefficient, r, relates the similarity of individuals to the consequences for reproductive success and heritable transmission to the future, the primary currency.

Second, marginal values compare a trait’s effects on different components of fitness. The favored trait value, when altered by a small amount, typically causes equal marginal gains and losses between its associated fitness components. If a changed trait caused a larger marginal gain in one component than the offsetting loss in another component, then the trait value would tend to change until it settled near the favored balance of marginal gains and losses.

Third, reproductive value compares a trait’s influence on different pathways of heritable transmission to the future population. For example, a trait may influence survival, fecundity, and dispersal. A gain in one component may be offset by a loss in another component. To compare the gains and losses, each component must be expressed in terms of its ultimate contribution to the future population, the component’s reproductive value.

We may also assign reproductive values to different life stages or to different kinds of habitat. In each case, the classification can be used to analyze the class’s relative contribution to the future population, which is its reproductive value. In tradeoffs, the fundamental forces typically favor traits that positively influence classes with relatively high reproductive value.

5.9 Scaling of Time and Space

Forces acting over short timescales may oppose forces acting over longer time periods. Consider a fast-growing mutant. The mutant outcompetes its neighbors, rising in frequency immediately, over a short timescale.

Faster growth may associate with poor conversion efficiency of food into reproduction. Poor yield typically acts over a longer timescale as resources slowly become depleted (p. 138).

Inefficient resource use may, for example, lower a group’s long-term production of dispersers to colonize new habitats. Group against group competition happens more slowly than the direct competition between individuals within groups.446

The design of traits depends on the balance between within-group forces acting over short timescales and between-group forces acting over long timescales.^{133,168,269,433,447}

The spatial scale of competition influences the relative weighting of different timescales. When the spatial scale of competition is large, and individuals compete globally with each other across all spatial locations, then relative success depends only on the direct and immediate competition between individuals. The short timescale dominates.

When the spatial scale of competition is small, and individuals compete locally, then total success depends on the balance of the two forces. Competition within groups favors fast growth, acting over short timescales. Competition between groups favors high yield, acting over long timescales.

Short timescales act rapidly and ubiquitously. Long timescales act slowly and sporadically. All else equal, the short timescales dominate.⁴⁴⁶

But all else may not be equal.¹⁶⁸ If there is relatively little variation between individuals within groups, then within-group competition has relatively little consequence. The long timescale of between-group competition dominates.

By contrast, if most of the variation occurs between individuals within groups, then not much difference occurs between groups. Limited between-group variation means that there is only a small force of competition at that longer scale. The short timescale of within-group competition dominates.

The next subsection sketches the basic theory for relative variation and timescale. The following subsection considers situations in which competitive and cooperative interactions happen at different spatial scales.

TIMESCALE AND THE PROCESSES THAT GENERATE VARIATION

The force acting at each scale depends on the intensity of selection.¹³³ Within groups, we may write the opportunity for outcompeting neighbors as the intensity of selection, s_w.

Selection has consequences only when competition occurs between differing individuals. If individuals carry the same genes, then with regard to evolutionary change, it does not matter which one wins in competition. We express the differences between individuals within groups as V_w, the within-group variance.

The evolutionary force within groups scales as s_wV_w, the product of the potential for differences in success multiplied by the variance. Between groups, we write s_bV_b, the potential for differential success between groups multiplied by the variance between groups.

When the two scales oppose each other, then traits evolve toward a balance between the opposing forces,

s_b V_b = − s_w V_w.   (5.14)

Equality requires changing the sign on one side of the equation because opposing forces have opposite signs.

The variance between groups, V_b, versus the variance within groups, V_w, determines the relative weighting of selection at the global versus local spatial scales.^168,434 We can relate those variances to similarity and kin selection by expressing the values as descriptions of relative similarity. To begin, we write the total variance as

V_t = V_w + V_b.

We then define the relative similarity of individuals within groups as the fraction of the total variance that is between groups,

$$r=\frac{V_b}{V_t}.$$

The more of the total variance that occurs between groups, the lower the fraction of the total variance that occurs within groups. Less variability within groups is the same as more similarity within groups. Here, r is the correlation coefficient between individuals within groups, which expresses the coefficient of similarity within groups.

Substituting those identities for similarity and variance into eqn 5.14, the balance of opposing forces occurs when

s_br = − s_w(1 − r).   (5.15)

As similarity within groups, r, increases, the force, s_br, between groups rises and the force, s_w(1 − r), within groups declines. The balance shifts toward selection between groups. Thus, as r rises, selection increasingly favors traits that enhance competition between groups, often reducing competition or raising cooperation within groups.¹¹²

The basic tragedy of the commons model follows directly from the balance of forces. Group success in the basic tragedy model of eqn 5.1 is 1 − z, and thus s_b = − 1, the slope of group success with respect to the average trait value in groups, z. The selective intensity within groups is s_w = (1 − z)/z, which is the partial change in individual fitness, w, with respect to the change in individual character value, y, holding constant group phenotype, z. Substituting these values for selective intensity into eqn 5.15 and evaluating at the fixed point z^*yields the basic result for the tragedy, z* = 1 − r, given in eqn 5.4.

The distinction in this section arises from a focus on the relative timescales for the different forces and a clearer spatial separation of processes within and between groups. With those explicit considerations of dynamics, we get a better sense of the forces that shape traits.

For example, the previous analyses of similarity selection took r as a given value. But what, in fact, determines the value of r?

If the generation of variation happens slowly, on a long timescale compared with selection, then the distribution of variation within and between groups arises by the way in which individuals assort spatially. Common ancestry is often the most powerful cause of spatial assortment and similarity within groups. In that case, similarity and the associated value of r arise by kinship, leading to the natural interpretation of similarity selection as kin selection.

In many multicellular organisms, new variation arises slowly. Most aspects of similarity depend on kinship. Other factors may sort similar individuals into groups, but kinship typically dominates.

Microbes differ. Short generation times and large population sizes mean that mutation and selection within groups can create new variation relatively rapidly compared to spatial sorting by common descent. Thus, the processes that generate the distribution of variation may happen on the same timescale as selection. Kinship and common descent do not necessarily dominate the spatial patterns of similarity and variance.

The generation of new variation within groups degrades the local similarity and decreases r over time. The decay of within-group similarity shifts the balance of forces toward within-group competition, reducing the potential for within-group cooperation.

The dynamic changes of force that increase within-group competition may lead to microbial cancers, in which highly competitive variants arise and overgrow their neighbors, degrading the long-term success of groups (Section 4.3).

In microbes, different species often strongly interact over short spatial and temporal scales. Similarity selection occurs between species when processes other than kinship cause similarity in trait values.^113,116,119

SPATIAL SCALE OF COMPETITION VERSUS COOPERATION

Competitive and cooperative interactions may happen over different spatial scales. For example, secreted public goods may act locally, cooperatively benefiting only close neighbors. By contrast, key resources that competitively limit growth may diffuse over relatively longer spatial scales.

Those different spatial scales influence the costs and benefits that shape cooperative traits. Consider the expression of fitness from section 7.1 in Frank,¹²²

$$w=\frac{bz-cy}{az\left(b-c\right)+\left(1-a\right)\bar{z}\left(b-c\right)},$$

in which an individual invests y in cooperative public good secretion, at cost cy to itself. The average level of altruistic public goods expression in the neighborhood is z, with beneficial effect bz on fitness. The focal individual’s reproduction is therefore proportional to bz − cy, which is the numerator.

The denominator is the intensity of competition for scarce resources. Competition increases as the average reproductive success rises. The overall level of competition combines local and global components of reproductive competition.

The average local reproduction in the neighborhood is the average of bz − cy, which is z(b − c) because the local average of y is z. The average in the population is $\bar{z}\left(b-c\right)$. The parameter a is the spatial scale of density-dependent competition. An increase in the reproductive success of neighbors by a proportion δ increases local competition by a factor aδ. An increase in the average reproductive success of the population by a proportion γ increases global competition by a factor (1 − a)γ.

Using our standard method in eqn 5.7 to find the trait value favored by natural selection, we obtain the condition for the cooperative trait to increase,¹²²

$$\frac{\mathrm{\text{d}}w}{\mathrm{\text{d}}g}=r\left[b-a\left(b-c\right)\right]-c>0,$$

in which the marginal benefit is B_m = b − a(b − c), and the marginal cost is C_m = c.

Comparatively, as a rises and competition for resources becomes increasingly local, the tendency for cooperative trait expression declines. Local competition reduces the benefit of cooperative traits because an increase in neighbors’ vigor from enhanced cooperative expression is offset by the increased competition among those more vigorous individuals for the same locally limiting resources. Put another way, limited local resources reduce the potential for enhanced success through cooperative traits.^{6,112,333,404,449}

In general, a trait may alter various components of fitness acting at different spatial scales. The changed balance of forces at the various scales modifies the trait’s design.

5.10 Variable Environments

Reproduction multiplies. If a population grows by μ in each generation, the total population growth is μ × μ = μ² after two rounds of reproduction.

Variation in reproduction lowers fitness. For the same average population growth rate of μ, if the rate goes up by δ in the first generation and down by the same amount in the second generation, then total growth is (μ + δ) (μ − δ) = μ² − δ². Variation in reproduction reduces success.99,154,176,223,304,331,341,414

The multiplicative nature of reproduction leads to the geometric mean principle. The next subsection discusses the geometric mean, which shows how variation discounts value.

The current literature emphasizes the geometric mean principle but mostly ignores other aspects of variation that influence value. We obtain a deeper sense of biological design by thinking about what traits do in organismal life history and how different kinds of variation alter value.

After introducing the geometric mean, the following subsections show other ways in which variation influences value.^{131,144,153,234,274,306,337}

GEOMETRIC MEAN

The total growth after t generations is the product of the growth, λ_i, in each generation,

$$\Lambda =\prod _{i=1}^t{\lambda }_i.$$

Because growth multiplies, there must be some value, λ, that we can multiply with itself t times to get the same total growth. In symbols, t multiplications of λ is λ^t. Thus, we can write λ^t = Λ as the total growth and then figure out what sort of average value λ is.

Taking the natural logarithm of both sides yields the same equality, now written as a sum of logarithms on the right-hand side,

$$log\left({\lambda }^t\right)=log\left[\prod _i{\lambda }_i\right]=\sum _{i}log\left({\lambda }_i\right).$$

Define m = log(λ) and note that log(λ^t) = t log(λ) = tm. Then

$$m=\frac{1}{t}\sum _{i}log\left({\lambda }_i\right)$$

is the average of the logarithmic growth rate, the Malthusian parameter. The geometric mean is defined as λ = e^m. Total growth is t multiplications of the geometric mean growth rate,

λ^t = e^mt.

Variation in the individual growth rates per generation reduces the geometric mean and the total growth. As noted above, if we let μ be the arithmetic mean growth rate per generation and suppose, over two generations, that growth fluctuates up and down by δ, then

λ² = (μ + δ) (μ − δ) = μ² − δ².

Increasing fluctuation, δ, always reduces the total growth.

No simple mathematical expression describes exactly how increasing variation causes a greater discount to the total growth. For arithmetic mean and variance in growth rate per generation, μ and σ², the geometric mean is approximately

λ ≈ μ − σ² / 2μ.

The smaller the variance relative to the mean, the better the approximation will be. Because the arithmetic mean growth rate is often near one in evolutionary models, μ ≈ 1, the geometric mean approximation is often written as

λ ≈ μ − σ² / 2.

In summary, the geometric mean measures the long-term growth rate. Increased arithmetic mean growth per generation, μ, may lead to a lower geometric mean fitness value if the enhanced growth also causes a sufficiently large increase in the variance, σ². In general, variation in performance discounts long-term value. Natural selection often favors traits that reduce variation in performance.

ABSOLUTE VERSUS RELATIVE SUCCESS

The geometric mean has often been claimed as a one-step principle for calculating the valuation discount caused by trait variability. However, other aspects also influence the relation between variability and value.^131,144

Consider the distinction between absolute and relative success. The geometric mean calculates total growth, a measure of absolute success. Relative success is what matters in biology. If one genotype increases tenfold, that increase describes significant success. However, if a competitor increases 100–fold, then the original type has greatly declined in frequency.

The traits that dominate the observable patterns of nature associate with greater relative success. To describe how relative success affects the relation between variability and the discount in value, I summarize Frank & Slatkin’s¹⁴⁴ extension of Gillespie’s¹⁵³ analysis.^131,306

Figure 5.1   Increasing variation in reproductive success reduces fitness, from eqn 5.16. The fitness loss from negative fluctuations outweighs the fitness gain from positive fluctuations. Thus, equally frequent negative and positive fluctuations return a net loss. Redrawn from Frank & Slatkin.144

Consider alternative genotypes that encode different trait values. Let q₁ and q₂ be the frequencies of the alternative genotypes. After one round of reproduction, the updated frequency for the first type is

$$q_1^{\prime }=q_1\left(R_1/\bar{R}\right)=q_1{F}_1,$$

in which R₁ is the reproductive success or absolute fitness of the first type, $\bar{R}=q_1{R}_1+q_2{R}_2$ is the average reproductive success of the two types, and F₁ is the relative fitness of the first type. This equation emphasizes that relative fitness is what controls frequency change and the evolution of traits.

Writing out the definition of relative fitness explicitly in terms of frequency and absolute reproductive success yields

$$F_1=R_1/\bar{R}=\frac{R_1}{q_1{R}_1+q_2{R}_2}.\text{(5.16)}$$

A gain in absolute success causes a smaller ultimate benefit than the loss imposed by an equal and opposite decline in absolute success (Fig. 5.1). Put another way, diminishing return causes variability in absolute success to impose a discount on relative success.

The curvature between absolute and relative success depends on frequency (Fig. 5.2). A rare type has a nearly linear relation between reproduction and relative fitness. A common type has a strongly diminishing relation between reproduction and relative fitness.

Figure 5.2   The curvature of relative fitness versus reproductive success depends on frequency. The numbers above each curve show q₁. Rising frequency increases the curvature between absolute and relative success. Greater curvature leads to a bigger fitness value discount. The curve for q₁ = 0.1 shows that there is little curvature when a type is rare, providing an advantage for rare types. Based on eqn 5.16, with R₂ = 1. Redrawn from Frank & Slatkin.144

More strongly diminishing returns cause variability in reproductive success to impose a greater penalty on relative fitness. Thus, common types, with more strongly diminishing returns between absolute and relative success, suffer a greater discount than do rare types. In general, the variability discount to relative fitness is frequency dependent.^144,274

Competition for resources makes relative success particularly important. Over time, one cannot simply multiply the reproductive successes of each type independently and then compare the long-term geometric means. Instead, each bout of density-dependent competition causes interactions between alternative types.

The fitness measure of relative success in eqn 5.16 accounts for density-dependent interactions. But that equation does not tell us the temporal and spatial scales over which density-dependent competition acts. Different scalings of competition alter the relation between trait variability and relative fitness value.

Competitive scale varies widely among microbes. How does a change in competitive scale alter the relation between trait variability and fitness? The theory for that question has not been developed in a general way.

EXPECTED CHANGE IN FREQUENCY

The previous subsection analyzed the frequencies of two competing types, q₁ and q₂. The updated frequency of q₁ after a round of competition is $q_1^{\prime }=q_1{F}_1$, in which F₁ is the relative fitness of that type. Thus, $\Delta q_1=q_1^{\prime }-q_1=q_1\left(F_1-1\right)$. Using the definition of relative fitness in eqn 5.16, we can write the change in frequency for the first type as

$$\Delta q_1=q_1{q}_2\left(\frac{R_1-R_2}{\bar{R}}\right).$$

The reproductive successes fluctuate randomly. If the fluctuations in success are small relative to the average success and we normalize the success values to be close to one, then the approximate expected change in frequency is^131,144,153

$$E\left(\Delta q_1\right)\approx q_1{q}_2\left\{\left({\mu }_1-{\mu }_2\right)+\left[\mathrm{cov}\left(R_2,\bar{R}\right)-\mathrm{cov}\left(R_1,\bar{R}\right)\right]\right\},\text{(5.17)}$$

in which μ₁ and μ₂ are the expected reproductive successes for types 1 and 2.

HIERARCHICAL STRUCTURE OF VARIABILITY

Suppose the variance in success for an individual of genotype 1 is ${\sigma }_1^2$. Then the variance in the genotypic success is $\mathrm{var}\left(R_1\right)={\rho }_1{\sigma }_1^2$, in which ρ₁ is the correlation between randomly chosen individuals of that genotype.

When all individuals have the same success, ρ = 1, then individual and genotypic variance are the same. As individuals become less correlated, the genotypic variance declines because the variance of an average decreases with the number of uncorrelated samples. We may also write $\mathrm{var}\left(R_2\right)={\rho }_2{\sigma }_2^2$ for type 2, and $\mathrm{cov}\left(R_1,R_2\right)={\rho }_{12}{\sigma }_1{\sigma }_2$, in which ρ₁₂ is the correlation between randomly chosen individuals of types 1 and 2.

If, for simplicity, we assume ρ₁₂ = 0, then eqn 5.17 becomes^131,144

$$E\left(\Delta q_1\right)\approx q_1{q}_2\left\{\left({\mu }_1-q_1{\rho }_1{\sigma }_1^2\right)-\left({\mu }_2-q_2{\rho }_2{\sigma }_2^2\right)\right\}.\text{(5.18)}$$

On average, type 1 increases in frequency when

$${\mu }_1-q_1{\rho }_1{\sigma }_1^2>{\mu }_2-q_2{\rho }_2{\sigma }_2^2.\text{(5.19)}$$

Rare types, with smaller q, gain an advantage. That rare-type advantage occurs because the curvature between reproductive success and fitness increases with frequency (Fig. 5.2), making common types more sensitive to the fitness value discount for variability in reproductive success.

The rare-type advantage tends to push frequencies away from zero, favoring a mixture of types. However, stochastic fluctuations in frequency often cause loss of one of the types, leading to fixation of the other type.

Over time, the frequencies tend to be biased toward the type with the greater geometric mean fitness. That long-term bias can most easily be seen by starting with equal frequencies, q₁ = q₂ = 1/2. At that frequency midpoint, type 1 tends to be favored when

$${\mu }_1-{\rho }_1{\sigma }_1^2/2>{\mu }_2-{\rho }_2{\sigma }_2^2/2.$$

This expression compares the geometric means of the two types. A type can potentially lower its overall variance, ${\rho }_i{\sigma }_i^2$, and increase its success by reducing the correlation between individuals, ρ_i.

BET-HEDGING

Reducing the correlation between individuals is one type of bet-hedging. For example, if individuals stochastically express alternative traits, then the genotype increases the chance that a subset of individuals match the current state of a varying environment. In general, bet-hedging strategies tend to reduce the overall variance of a genotype’s success.^159,400

SPATIAL SCALE OF COMPETITION

In the previous subsections, competition occurs in one large population. This subsection considers a population distributed over many independent spatial locations. Competition happens within each separate location.

Temporal fluctuations within each location induce frequency dependence, favoring the rare type (eqn 5.19). When there is only a single location, one of the types typically becomes fixed after a period of time because the random fluctuations in frequency are too strong relative to the directional tendency of evolutionary change. Fixation is biased toward the type with the highest geometric mean.⁸⁰

By contrast, in a population distributed over many separate locations, the rare-type advantage typically maintains a mixture of types. The tendency for mixture arises in the following way.²³⁴

In each time period of local competition, the rare types gain on average in each patch because of their intrinsic frequency-dependent advantage. The population-wide fluctuations in each round of local competition become small because of the averaging effect over the many patches. We can therefore treat eqn 5.18 as an essentially deterministic process. The rare-type frequency dependence now dominates. The equilibrium frequency of types can be obtained from eqn 5.18 by solving E(Δq₁) = 0, which yields¹⁴⁴

$$\frac{q_1}{q_2}=\frac{{\mu }_1-{\mu }_2+{\rho }_2{\sigma }_2^2}{{\mu }_2-{\mu }_1+{\rho }_1{\sigma }_1^2}.$$

Each ρ is the correlation between individuals of a type measured within each patch. This result shows that geometric mean success does not always provide the correct fitness value.

RELATION BETWEEN TRAITS AND VARIABLE PERFORMANCE

The previous subsections assumed that an individual’s variability in reproductive success is a given parameter. This subsection briefly summarizes how an individual’s multiple traits combine to determine its overall variability in performance. See the details and examples in Frank.¹³¹

We begin with a single trait for resource acquisition, in which reproductive success is

R = 1 + f(δ).

Random fluctuations in resource acquisition, δ, with mean zero and variance, V_x, affect reproductive success by f(δ). If fluctuations are relatively small, then the approximate average reproductive success is

μ ≈ 1 + f″V_x/2,

in which f^″ is the second derivative of f evaluated at zero.³³⁷ Typically, f″ < 0 because the benefits of resource acquisition have diminishing returns. Thus, greater fluctuations, V_x, reduce expected reproductive success. All else equal, resource acquisition strategies with less variability yield higher average reproductive success than those strategies with more variability.

The variance in an individual’s reproductive success is approximately

$${\sigma }^2\approx {f{}^{\prime }}^2{V}_x,$$

in which f^′is the derivative of f evaluated at zero.

To keep the focus on trait variability within individuals, I give only the geometric mean reproductive success for an individual in this subsection. A full analysis of fitness valuation requires the additional aspects discussed in the prior subsections.

An individual’s geometric mean reproductive success is approximately

$$G\approx \mu -{\sigma }^2/2\mu \approx 1-\left(f^{″}-{f{}^{\prime }}^2\right)V_x/2.\text{(5.20)}$$

Now consider two different traits that provide additive returns. How should an individual divide its investment between those two traits? Assume that reproductive success is

R = x[1 + f(δ)] + y[(1 − γ) + g(𝜖),

in which x and y are the fractions of total resources invested in each trait, γ is the small discount in expected return for the second trait, and 𝜖 is the small random fluctuation associated with the second trait.

Assuming that the fluctuations δ and 𝜖 are uncorrelated, V_x = V_y, and f≡g, the geometric mean reproductive success for an individual is

μ − σ²/2μ ≈ G + B(x, y),

in which G is the geometric mean in eqn 5.20 for allocating all resources to the first trait, x = 1, and B(x, y) is the benefit obtained when mixing allocation of resources between the two traits, with x + y = 1 and

$$B\left(x,y\right)={f{}^{\prime }}^2\left[1-\left(x^2+y^2\right)\right]V_x/2-y\gamma .$$

Optimizing B to obtain the best mixture of allocations between the two traits yields

$$\begin{array}{llll}\hfill x^{*}& =\frac{1}{2}\left(1+\frac{\gamma }{{\sigma }^2}\right)& \hfill & \\ \hfill y^{*}& =\frac{1}{2}\left(1-\frac{\gamma }{{\sigma }^2}\right),& \hfill & \end{array}$$

in which γ is the discount in expected return for the second trait, and σ² is the variance in individual reproductive success per trait, with γ < σ².

It pays to invest some resources in y, the trait with lower expected return. The lower expected return is offset by the benefit from reduced overall variance in performance obtained from averaging the returns over the two uncorrelated traits. This mixed allocation is another type of bet-hedging, the combining of alternative traits to reduce the variation in performance.

In both biology and financial investing, returns tend to multiply over time. Thus, reduced fluctuations enhance the multiplicative (geometric) average return. In financial investing and modern portfolio theory, the geometric mean plays a key role in the allocation of resources among alternative asset classes.⁴⁰ In biology, one can think of different traits as different asset classes.