MICROHABITAT VARIATION AND SEXUAL SELECTION CAN MAINTAIN MALE COLOR POLYMORPHISMS

Authors


Abstract

Male color polymorphism may be an important precursor to sympatric speciation by sexual selection, but the processes maintaining such polymorphisms are not well understood. Here, we develop a formal model of the hypothesis that male color polymorphisms may be maintained by variation in the sensory environment resulting in microhabitat-specific selection pressures. We analyze the evolution of two male color morphs when color perception (by females and predators) is dependent on the microhabitat in which natural and sexual selection occur. We find that an environment of heterogeneous microhabitats can lead to the maintenance of color polymorphism despite asymmetries in the strengths of natural and sexual selection and in microhabitat proportions. We show that sexual selection alone is sufficient for polymorphism maintenance over a wide range of parameter space, even when female preferences are weak. Polymorphisms can also be maintained by natural selection acting alone, but the conditions for polymorphism maintenance by natural selection will usually be unrealistic for the case of microhabitat variation. Microhabitat variation and sexual selection for conspicuous males may thus provide a situation particularly favorable to the maintenance of male color polymorphisms. These results are important both because of the general insight they provide into a little appreciated mechanism for the maintenance of variation in natural populations and because such variation is an important prerequisite for sympatric speciation.

Polychromatism is widespread in nature (Barlow 1973; Anderson 1994; Hoffman and Blouin 2000; Gray and McKinnon 2006, 2007) and has long been central to studying the maintenance of variation within natural populations of animals (Ford 1965; Roulin 2004). Male color polymorphisms (MCPs), a type of polychromatism in which males within the same population exhibit different, discrete color morphs, is of particular interest both because of the role sexual selection may play in the evolution of such polymorphisms (Eakley and Houde 2004; Seehausen and Schluter 2004) and because MCPs may play an important role in sympatric speciation (Seehausen et al. 1999; Allender et al. 2003).

Several hypotheses have been proposed to explain the maintenance of MCPs. These include negative frequency-dependent male–male competition (in cichlids, Seehausen and Schluter 2004), negative frequency-dependent predation on male color morphs (in guppies, Olendorf et al. 2006) a balance between female preference and male aggression (in swordtails, Xiphophorus pygmaeus,Kingston et al. 2003), female preferences for unfamiliar or novel males (Hughes et al. 1999; Eakley and Houde 2004; Kokko et al. 2007), and spatial and temporal habitat heterogeneity (Fuller et al. 2005). As male color is potentially selected on by both predators and females, both types of receivers must be considered to explain the maintenance of MCPs within a population; predation often favors inconspicuous traits, whereas sexual selection often favors conspicuousness (Endler 1980; Anderson 1994; Zuk and Kolluru 1998).

Given that signal perception is highly habitat dependent (Endler 1980; Bradbury and Vehrencamp 1998; Chiao et al. 2000), it is impossible to interpret the signal outside the context of the environment. Color perception depends on the properties of the signal, the light under which the signal is perceived, the background against which it is viewed, the medium through which the signal is sent (i.e., air or water), and the sensory capabilities of the receiver (Endler 1991; Chiao et al. 2000; Endler and Mielke 2005). Therefore, if an environment is heterogeneous in substrate type, light intensity, or other factors that influence signal perception, the potential exists for several different color morphs to persist, as each morph will represent the optimal balance between natural and sexual selection within a given visual microhabitat (Gamble et al. 2003). Indeed, empirical evidence that habitat variation affects signal conspicuousness and the evolution of male signals has now been obtained for several sensory systems from diverse taxa including birds, lizards, frogs, insects, and fish (for a review, see Boughman 2002).

Although empirical evidence suggests a role for fine-scale habitat variation in the maintenance of MCPs, a formal theoretical analysis of how differential selection between microhabitats may lead to the maintenance of MCPs has not previously been developed. Here we present a population genetic model that looks at the conditions under which habitat heterogeneity in qualities affecting color perception will lead to the existence of a stable MCP. In doing so, we answer three specific questions about the likelihood of this mechanism maintaining a polymorphism in natural populations: (1) Can MCPs be maintained by either natural selection or sexual selection alone? (2) How robust are the conditions for MCP maintenance to asymmetries in habitat frequencies and selection strengths? and (3) How does changing the biological assumptions of the conditions under which selection occurs, for both natural and sexual selection, influence the outcome of the model? Although we focus throughout on MCPs as our “case study,” our goal is to develop a relatively general model of how environmental variation might contribute to the maintenance of variation in signals. For this reason, we keep our model of signal conspicuousness relatively simple and general.

The Model

In this haploid model, we consider a population of sexually dimorphic animals. Males are polymorphic for color with two distinct morphs. For convenience, we refer to them here as blue (occurring with frequency pb) and yellow (occurring with frequency py). These color patterns are common in fish MCPs (e.g., Seehausen and Schluter 2004; Gray and McKinnon 2006) and are also seen in lizards (e.g., Sinervo and Lively 1996). A great deal of work has been done on the genetic basis of blue and yellow color polymorphisms, and in several of these systems, color expression is controlled in large part by a single locus with multiple alleles (in swordtails, X. pygmaeus, Baer et al. 1995; in killifish, Lucania goodei, Fuller and Travis 2004; in side-blotched lizards, Uta stansburiana, Sinervo and Zamudio 2001). Therefore, we modeled color as being controlled by a single locus with two alleles. We assume that females are monomorphic for color and that color is entirely genetically controlled in both sexes. Although color does not indicate genetic or phenotypic quality, it does affect fitness in that males that are more conspicuous within their environment are both more vulnerable to predation and more likely to be chosen by females.

The habitat in which natural and sexual selection occurs is divided into microhabitats that differ in physical properties that influence color perception, such as light intensity, light spectrum, and/or substrate color and pattern. Because of these differences in visual properties, each microhabitat provides a functionally different setting within which male color is perceived by females and by predators. Environmental heterogeneity encountered by an individual is not the result of migration; rather, variable visual backgrounds within the general environment result from fine-scaled spatial or temporal differences (e.g., Gamble et al. 2003).

We specifically consider an environment with two microhabitats. The blue male color morph is more conspicuous than the yellow male morph in habitat Hb (occurring with frequency hb), whereas in habitat Hy (occurring with frequency hy= 1 −hb), the reverse is true. Both sexes move freely and randomly between habitats (neither males nor females actively choose a habitat) so that each sex has a probability of being in each microhabitat according to relative habitat area.

The life cycle begins with the zygote stage. Natural selection follows birth and acts exclusively on males. The more-conspicuous morph in each microhabitat suffers a greater loss from predation. Specifically, sb represents the selection coefficient against blue males in habitat Hb, whereas sy is the selection coefficient against yellow males in habitat Hy. Because habitat varies on a small (microhabitat) scale, we model natural selection as occurring across habitats that organisms can freely move between (see Dempster 1955). Therefore, in males, the absolute fitness of the blue morph from natural selection (superscript “ns”) is wnsb=hb (1 −sb) +hy, which simplifies to wnsb= 1 −hbsb, and the absolute fitness of the yellow morph is likewise wnsy= 1 −hysy. The frequency of the blue morph (pb) after males undergo natural selection (p*b), is given by equation (1)

image(1)

Because females do not express the color alleles, the frequency of the blue allele carried by females at this stage of the life cycle remains pb.

After natural selection, mating (sexual selection) occurs. We assume a polygynous system in which females have equal mating success (e.g., Kirkpatrick 1982) and males provide no resources to females other than their genetic contribution. The sole advantage males of each color possess over differently colored competitors is their microhabitat-dependent conspicuousness. This advantage is represented by a factor ai (where i=b or y depending on the microhabitat that the female is in), where females are ai times more likely to mate with a more-conspicuous morph than a less-conspicuous morph if they encounter one of each. For now, we assume that females choose between blue and yellow males within their current microhabitat. That is, females are either in habitat Hb or Hy when mating decisions are made, and they can only view one background at a time. Mating success of each male morph is thus determined separately within each habitat (see Levene 1953).

The proportion of each type of cross is shown in the mating table (Table 1). Here, the mating table is a matrix Mij, where i represents female type in each habitat (i.e., row number in Table 1) and j represents male type in each habitat (i.e., column number in Table 1). The frequency of the blue color allele in the following generation (pb (t+ 1)) is shown by equation (2)

image(2)
Table 1.  Mating table—Within habitat mate choice. Frequencies of matings involving blue and yellow male morphs in two different habitats when females choose males from within one habitat. Females either carry a blue allele or a yellow allele, and they can be found either in a habitat in which blue is conspicuous (Hb, occurring with frequency hb) or a habitat in which yellow is conspicuous (Hy, occurring with frequency hy). Males are either blue or yellow in body color, and again they can be found in either habitat Hb or habitat Hy. Females carrying the blue allele occur with frequency pb, whereas blue males occur with frequency of p*b. Because males and females move between habitats at random and mating is influenced by habitat, the frequency of each type of mated pair is weighted by the proportion of each type of habitat. Blue males are favored by females in habitat Hb with a preference strength of ab, whereas yellow males are favored in habitat Hy by a preference strength of ay. Matings are normalized so that both types of females have equal mating success, and mating success is not affected by the habitat in which a female mates.
 HabitatMales
Blue morphYellow morph
HbHyHbHy
  1. z=p*bab+p*y.

  2. v=p*b+p*yay.

Females 
  Carrying blue alleleHbinline image0inline image0
 Hy0inline image0inline image
  Carrying yellow alleleHbinline image0inline image0
 Hy0inline image0inline image

The final recursion can be obtained by substituting the appropriate values of the cells of Table 1 into equation (2). Equilibrium frequencies were found by setting the offspring frequencies at time t equal to the frequencies at time t+ 1 and solving the resulting recursion equation.

Results

Three equilibria result from this model. Two of the equilibria represent loss (inline image= 0) and fixation (inline image= 1) of the blue color morph. The third equilibrium is polymorphic. The best way to understand the polymorphic equilibrium from the full model is to first present the equilibrium under sexual selection alone (with no natural selection, sb=sy= 0). We are interested in the polymorphic equilibrium frequency of the blue morph,inline image, so we write the relative fitness of blue males, due to sexual selection alone (superscript “ss”), as wssb,Hb=ab in the habitat in which blue is conspicuous (Hb) and as wssb,Hy= 1/ay in the habitat in which yellow is conspicuous (Hy) (the relative fitness of yellow males in habitat Hy must also be rescaled throughout the results below to 1 instead of ay, changing the normalization factor v in the mating table to wssb,Hypb+py). We can then express the polymorphic equilibrium, due to sexual selection alone, as

image(3)

The term (wssb,Hb− 1) is a measure of the selection coefficient favoring the blue morph due to sexual selection in habitat Hb (this term will be positive because ab > 1), whereas (wssb,Hy− 1) represents the parallel selection coefficient against the blue morph in habitat Hy (this term will be negative because ay>1). Equation (3) therefore shows that the polymorphic equilibrium frequency of the blue allele due to sexual selection alone is a balance between the frequency of habitat Hb times the selection coefficient favoring blue in that habitat, and the frequency of habitat Hy times the selection coefficient against blue in that habitat, scaled by the product of the selection coefficients (the minus sign in front of eq. 3 can be thought of as correcting for the negative selection coefficient in the denominator).

To look at the equilibrium under the full model, we can simply replace the fitnesses of the blue morph in equation (3) with ones that represent the action of both natural and sexual selection (superscript “tot”). Therefore,

image(4)

where wtotb,Hb=Ab and wtotb,Hy= 1/Ay, and where

image

Ai is therefore equal to the strength of female preference, ai, for a color morph when it is conspicuous, multiplied by the ratio of the fitnesses due to natural selection of the conspicuous to the inconspicuous morph.

Stabilities of the equilibria were determined using a linear stability analysis. The equilibrium inline image= 1 (i.e., the fixation of the blue morph) will be unstable when

image(5)

and the equilibrium inline image= 0 (i.e., the loss of the blue morph) will be unstable when

image(6)

The eigenvalue λp1 consists of terms describing the contributions by females and males to increases in the frequency of the yellow morph, which is the potentially invading morph when blue is fixed (i.e., inline image= 1), as follows. The factor of 1/2 is present because there are separate contributions to the eigenvalue from each sex. Color in females is neutral, so the contribution from females is 1. The following two terms, hb/wtotb,Hb and hy/wtotb,Hy represent contributions from males in habitat Hb and males in habitat Hy, respectively, scaled by the relative fitnesses of the yellow morphs (the reciprocal of the fitnesses of the blue morph) in each respective habitat. Likewise, the eigenvalue λp0 consists of contributions by females and males to the spread of the blue morph, the potentially invading morph when the yellow morph is fixed (i.e., inline image= 0), where hbwtotb,Hb and hywtotb,Hy represent contributions from males in each habitat scaled by the relative fitness in that habitat of the blue morph.

Note that our solutions to this point elaborate upon the findings of Gliddon and Strobeck (1975). Using our equation (4), we can write an equation for Δpy (where py= 1 −pb),

image(7)

This is equivalent to equation (1) in Gliddon and Strobeck (1975) when wy1= 1/wtotb,Hb, wy2= 1, wy3= 1/wtotb,Hy, and wy4= 1 correspond to the fitnesses of the yellow morph in males in habitats Hb (for wy1) and Hy (for wy3) and females in habitats Hb (for wy2) and Hy (for wy4). Equation (7) contains a factor of 1/2 to account for the fact that sexual selection is treated separately in each sex. Gliddon and Strobeck's (1975) stability conditions (2) and (3), applied to equation (7) correspond exactly to our eigenvalues λp1 and λp0, respectively.

The polymorphic equilibrium will be stable when both conditions (5) and (6) hold. If we make the appropriate substitutions for the wtotb terms, we can see that condition (5) is more likely to hold (the equilibrium inline image= 1 is more likely to be unstable and yellow is more likely to invade) with lower sexual selection favoring the blue morph (ab), higher sexual selection favoring the yellow morph (ay), and higher fitness due to natural selection of the yellow versus blue morph (sy < sb). The opposite conditions will tend to promote instability of the inline image= 0 equilibrium and therefore the invasion of the blue morph into a population of yellow individuals. When the appropriate balance is reached in the strength of the natural and sexual selection parameters, given a specific ratio of the habitats to one another, the polymorphic equilibrium will be maintained.

Although equation (4) and the conditions that follow from equations (5) and (6) present analytical solutions to the model and an intuitive feel for the effects of the parameters, we examine several cases graphically and numerically to illustrate the conditions for stability in an easily interpretable manner. This is done using the analytical results, not by separate simulations.

To illustrate the results when both natural and sexual selection are acting, we began with the assumption that the strengths of natural selection and sexual selection are symmetrical for blue and yellow morphs (sb=sy and ab=ay). Under these assumptions, we find that the polymorphic equilibrium is stable under a broad range of conditions (Fig. 1). The values of preference and selection that result in a stable polymorphism are most restricted when the habitats are highly skewed, but are more permissive as habitats become more symmetrical. When habitats are exactly symmetrical (hb=hy= 0.5) in addition to the symmetry in natural and sexual selection, a stable polymorphism results regardless of the values of selection and preference. Again, however, even highly skewed habitats can lead to a stable polymorphism with strong female preferences.

Figure 1.

Regions of stability of the polymorphic equilibrium for three values of hy assuming natural and sexual selection are symmetrical for blue and yellow morphs (sb=sy and ab=ay= a). The area in between each set of patterned lines is the parameter space over which the polymorphic equilibrium is stable for each specific value of hy (0.1, 0.25, and 0.45). Dotted lines represent hy= 0.1, dashed lines represent hy= 0.25, and solid lines represent hy= 0.45.

To illustrate the results when natural selection, sexual selection, and habitat frequency deviated from symmetry, we found the eigenvalue at each of the three equilibria (0, 1, and the polymorphic equilibrium; for the latter see (5) and (6)), solved for ay, and then substituted specific values of sb, sy,hb, hy, and ab into the resulting expression. This allowed us to see what range of asymmetry in sexual selection strengths is permissible to maintain the polymorphic equilibrium, given a set of parameter values. In doing these tests, we evaluated a wide range of values for natural selection, sexual selection, and habitat frequencies (s: 0.01–0.99; ab: 0–infinity; h: 0.01–0.99) for 60 unique combinations of parameters. Representative examples of the outcomes of different selection scenarios are presented in Table 2. We found that the polymorphic equilibrium was generally stable over a range of parameter values (Table 2), however, when the starting parameters were highly asymmetric, the range of parameter space leading to a stable polymorphism could be quite restricted.

Table 2.  Stability regions for asymmetrical parameter combinations. We present parameter values and outcomes for five different selection scenarios. The range of preference strength for yellow males (ay) required for each possible outcome (blue morph lost, polymorphism, or blue morph fixed) is shown for each combination of parameters. N/A indicates that no biologically realistic value of preference strength (i.e., ay > 0) will result in a stable equilibrium for that outcome. All other findings from other combinations of parameter values were consistent with the results presented.
ParametersStability
hysbsyab0 (blue lost)Polymorphism1 (blue fixed)Notes
0.1   0.5   0.32N/Aay > 1.17ay < 1.17 
0.1   0.1   0.91.1ay > 101.82 <ay <10ay < 1.82 
0.1   0.9   0.11.1ay > .0240 < ay < .024N/A 
0.4   0.8   0.93N/Aay > 1.53ay < 1.53 
0.4−0.8−0.93N/Aay > 2.22ay < 2.22Natural and sexual selection both favor the same morph

Finally, we evaluated a scenario in which the morph favored by females also suffered lower predation than the less-favored morph (sb and sy < 0). This could result if females favored more cryptic morphs. In this case, a stable polymorphism was again possible, although the conditions resulting in a stable polymorphism show slight numerical differences from the cases with comparable selection strengths where female preferences and natural selection favored different morphs (Table 2).

It is also illustrative to examine the effects of natural and sexual selection alone in this model. To look at sexual selection alone, we removed natural selection (sb=sy= 0). In this case, predators act indiscriminately, whereas females still preferentially choose conspicuous males. Our polymorphic equilibrium for this scenario is shown in equation (3), and the conditions for stability can be seen by setting sb=sy= 0 in the eigenvalues (5) and (6) above, which yields a stable polymorphic equilibrium when

image

Again, we can see that a stable polymorphism will be likely when a balance is struck between the strengths of sexual selection and the habitat frequencies.

Under these conditions, female preference is often sufficient to maintain a polymorphism. When preferences are symmetrical between habitats (ab=ay), even slight female preference (i.e., a= 1.01) will maintain a polymorphism, although this requires that habitat frequencies are also close to symmetrical (hbhy). As the strength of a symmetrical female preference increases, a polymorphism will be maintained under increasingly wide degrees of habitat asymmetry (Fig. 2A). The specific frequency of the blue morph under different conditions of habitat frequency and strength of preference is seen in Figure 2B. When we do not assume symmetry between female preference in each habitat (ab≠ ay), we find that a stable polymorphism is maintained under the widest range of frequencies when habitats are close to symmetrical, with increasing strength of preferences needed to maintain a polymorphism as habitat becomes increasingly skewed (Fig. 2C). Additionally, when habitat frequencies are symmetrical, a polymorphism is maintained when preferences are close to symmetrical; however, as asymmetry in habitat frequencies increases, a corresponding skew in preference strength (with stronger preferences for the morph favored in the rarer habitat) is required to maintain a polymorphism. Even under highly skewed conditions, however, a polymorphism will occur if the strength of preference for the conspicuous morph in the rarer habitat is strong enough.

Figure 2.

Figure 2.

(A) The region of stability of the polymorphic equilibrium when female preferences are symmetrical and natural selection is removed from the model (ab=ay=a, sb=sy= 0). Stronger female preferences allow a polymorphism to be maintained under a wider degree of habitat asymmetry. When a polymorphism cannot be maintained, the fixed morph is dependent on the direction of habitat skew. (B) The frequency of the blue morph at different strengths of female preference (a) and habitat frequency (hy) when natural selection is removed (sb=sy= 0) and sexual selection is symmetrical between habitats (ab=ay= a). White represents fixation of the blue morph whereas black represents loss. (C) The strength of female preference that will maintain a color polymorphism when natural selection is not acting (sb=sy= 0) for three different ratios of habitat frequency. Dotted lines represent hy= 0.1, dashed lines represent hy= 0.3, and solid lines represent hy= 0.5. A polymorphism is maintained in the region between each set of boundary lines and lost with either the blue or yellow morph becoming fixed in the regions outside the boundary lines (blue fixes in the region to the left of the vertical boundary line, and yellow is fixed in the region below the horizontal boundary line). The preference strength needed to maintain a polymorphism increases as habitat frequencies are more strongly skewed.

Figure 2.

Figure 2.

(A) The region of stability of the polymorphic equilibrium when female preferences are symmetrical and natural selection is removed from the model (ab=ay=a, sb=sy= 0). Stronger female preferences allow a polymorphism to be maintained under a wider degree of habitat asymmetry. When a polymorphism cannot be maintained, the fixed morph is dependent on the direction of habitat skew. (B) The frequency of the blue morph at different strengths of female preference (a) and habitat frequency (hy) when natural selection is removed (sb=sy= 0) and sexual selection is symmetrical between habitats (ab=ay= a). White represents fixation of the blue morph whereas black represents loss. (C) The strength of female preference that will maintain a color polymorphism when natural selection is not acting (sb=sy= 0) for three different ratios of habitat frequency. Dotted lines represent hy= 0.1, dashed lines represent hy= 0.3, and solid lines represent hy= 0.5. A polymorphism is maintained in the region between each set of boundary lines and lost with either the blue or yellow morph becoming fixed in the regions outside the boundary lines (blue fixes in the region to the left of the vertical boundary line, and yellow is fixed in the region below the horizontal boundary line). The preference strength needed to maintain a polymorphism increases as habitat frequencies are more strongly skewed.

Figure 2.

Figure 2.

(A) The region of stability of the polymorphic equilibrium when female preferences are symmetrical and natural selection is removed from the model (ab=ay=a, sb=sy= 0). Stronger female preferences allow a polymorphism to be maintained under a wider degree of habitat asymmetry. When a polymorphism cannot be maintained, the fixed morph is dependent on the direction of habitat skew. (B) The frequency of the blue morph at different strengths of female preference (a) and habitat frequency (hy) when natural selection is removed (sb=sy= 0) and sexual selection is symmetrical between habitats (ab=ay= a). White represents fixation of the blue morph whereas black represents loss. (C) The strength of female preference that will maintain a color polymorphism when natural selection is not acting (sb=sy= 0) for three different ratios of habitat frequency. Dotted lines represent hy= 0.1, dashed lines represent hy= 0.3, and solid lines represent hy= 0.5. A polymorphism is maintained in the region between each set of boundary lines and lost with either the blue or yellow morph becoming fixed in the regions outside the boundary lines (blue fixes in the region to the left of the vertical boundary line, and yellow is fixed in the region below the horizontal boundary line). The preference strength needed to maintain a polymorphism increases as habitat frequencies are more strongly skewed.

We next looked at the outcome of natural selection alone (modeled here as occurring across habitats, as described above) by removing the effects of sexual selection (sb, sy > 0; ab=ay= 1). In this scenario, predators preferentially prey upon conspicuous morphs, whereas females mate randomly. In this case, a polymorphism resulted only when there were exactly symmetrical parameters between habitats (sb=sy, hb=hy) or when selection and habitat area are exactly balanced. Any deviation from these conditions leads to the fixation of either the blue or the yellow morph. These conditions of complete symmetry are highly unrealistic and unlikely to occur in nature. This result is unsurprising because several authors have demonstrated the difficulty of maintaining a polymorphism when selection occurs across habitats, as we have modeled natural selection here (e.g., Dempster 1955; Christiansen 1975; Karlin and Campbell 1981; de Meeûs et al. 1993).

Alternative Assumptions

In the model above, we consider the maintenance of a color polymorphism when natural selection is assumed to occur across habitats (males of both morphs move between habitats), whereas sexual selection occurs within habitats (females choose among the males that are present in the microhabitat that the female happens to be in when she is ready to mate). To confirm the effects of these assumptions on the outcome of the model, we also examined the outcome of selection when we considered alternative assumptions. That is, we modeled natural selection as occurring within one habitat (males remain in one microhabitat throughout the period of natural selection and potentially subsequent reproduction and predators stay within a habitat at least for each prey selection event) and we modeled sexual selection occurring across habitats (females examine males in both habitats before they choose a mate). In the discussion below, we describe when these alternative assumptions may be appropriate.

To look at natural selection occurring within habitats, we use the notation wnsi, where i is b or y, to denote fitness of the blue and yellow morphs. Here, in habitat Hb, the fitness of the blue morph is wnsb,Hb= 1 −sb and the fitness of the yellow morph is wnsy,Hb= 1. In habitat Hy, where the yellow morph is conspicuous, the fitness of the blue morph is wnsb,Hy= 1 and the fitness of the yellow morph is wnsy,Hy= 1 −sy. After natural selection, the frequency of the blue morph in microhabitat i is now

image(8)

where i is b or y depending on the microhabitat. When combining the frequencies across habitats, we must take into account the proportion of each microhabitat, so the total frequency of the blue morph is p*b=p*b,Hbhb+p*b,Hyhy. This combination of the frequencies multiplied by the proportion of each habitat is valid in two cases: (1) if there is reproduction with separate population regulation in each habitat (see Levene 1953), or (2) if the population density of our focal species between each habitat remains equivalent because of equivalent densities of a species of predator. Solving for the equilibrium condition for natural selection alone with these assumptions yields three equilibria (0, 1, and a polymorphic equilibrium). This polymorphic equilibrium is

image(9)

where inline image and inline image. Note that the structure of equation (9) is exactly parallel to that of equation (3) above, and can be explained by the same logic. The stability conditions for these equilibria are also parallel to the results of the full model above (eq. 4 and see Gliddon and Strobeck 1975). Under these conditions, polymorphism maintenance by natural selection alone is therefore quite possible. However, if microhabitat variation is truly on a small scale, the assumption that males will remain in any given microhabitat throughout the time that prey is likely to be under selection (or that the effects of predation on population density in each habitat would be exactly equivalent) is probably unreasonable except for certain organisms with very specific patterns of dispersive and nondispersive life-history stages, and predators with appropriate foraging behavior.

We next considered sexual selection occurring across habitats. In this situation, females view all males across both habitats before selecting a mate. Therefore, the mating table F is a 2 × 2 matrix with only two female types (females carrying the blue allele and females carrying the yellow allele) and two male types (blue males and yellow males). The mating table is shown in Table 3. The frequency of the blue color morph in the following generation (pb (t+ 1)) is shown by equation (10)

image(10)
Table 3.  Mating table—Across habitat mate choice. The table shows frequencies of matings involving blue and yellow male morphs in two different habitats when females select males after viewing potential mates from both habitats before mating. Females either carry a blue allele or a yellow allele, whereas males are either blue or yellow in body color. Both sexes are found in either habitat Hb or habitat Hy. Blue males are favored by females in habitat Hb with a preference strength of ab, whereas yellow males are favored in habitat Hy by a preference strength of ay. Matings are normalized so that both types of females have equal mating success.
 Males
Blue MorphYellow Morph
  1. z=p*bhbab+p*bhy+p*yhb+p*yhyay.

Females 
 Carrying blue alleleinline imageinline image
 Carrying yellow alleleinline imageinline image

The final recursion can be obtained by substituting the appropriate values of the cells of Table 3 into equation (10). When sexual selection is modeled with these assumptions, the only equilibria that result are loss (pb= 0) and fixation (pb= 1) of the blue color morph. Here, we can think of the loss of polymorphism occurring because sexual selection is unidirectional. Specifically, because females sample both habitats before mating, there is only one set of conditions that all females experience. Therefore, one particular male morph will generally be more attractive to females than the other and will receive a higher proportion of matings. This contrasts with within-microhabitat selection because when females only view males from one microhabitat before mating, some females will prefer blue males and some will prefer yellow males, depending on the habitat that they are in when they make their choice. In this scenario, sexual selection will be divergent between habitats and thus more likely to lead to a polymorphism.

Discussion

It has long been known that a heterogeneous environment can be important in maintaining phenotypic and genetic variation. Previous models, however, have generally found that polymorphism maintenance in a heterogeneous environment is either quite restricted (e.g., Dempster 1955) or requires population regulation to occur separately in each habitat (e.g., Levene 1953). The model we present above differs in two important ways—(1) we include habitat-dependent sexual selection and (2) the scale at which selection occurs is quite small, so that each individual may experience several habitats, even within the course of the day. We find that including sexual selection in a model of microhabitat variation, either as the sole selective force or in conjunction with natural selection, can often lead to a stable polymorphism.

In our initial model of natural selection occurring across habitats and sexual selection occurring within habitats, we find that natural selection alone cannot maintain a polymorphism, but sexual selection can, either alone or in conjunction with natural selection. This is because in our initial model, natural selection can be thought of as being subsumed by sexual selection. For example, from the males' perspective, increasing the probability of survival is mathematically equivalent to increasing the mating success of that male in both habitats; changes in natural selection thus have effects that can be described within the context of the sexual selection model. Although natural selection alters the relative fitness of each type of male, as long as some males of each color survive, polymorphism maintenance will ultimately be determined by the fact that reproductive fitness is essentially regulated separately by female choice in each habitat (see eqs. 3 and 4).

Our findings stem from the assumptions that we make regarding whether natural selection and sexual selection are occurring within or across habitats. By selection within habitats, we refer to the case in which females or predators select males only from within the habitat that they (the agents of selection) are currently in, whereas selection across habitats refers to the scenario of females or predators sampling different habitats before finally selecting a male. With natural selection, in the former case we also assume that males stay within habitats throughout the entire period of selection, whereas in the latter case, we assume that males are moving in between habitats between individual predation events as well. Here, we can think of selection occurring within habitats as being roughly analogous to soft selection, as the processes regulating the population are occurring separately in each habitat. In contrast, selection occurring across habitats is roughly analogous to hard selection, because the processes regulating the population occur on a larger scale that spans both habitats. Under hard selection, the contribution of organisms to the next generation is absolute regardless of habitat, whereas under soft selection, organisms from each habitat contribute to the next generation in proportion to the carrying capacity of that habitat (Karlin and Campbell 1981). Previous models of hard selection find that the conditions for polymorphism maintenance are quite restrictive, whereas models of soft selection find that polymorphisms can be maintained under a much broader range of conditions (Christiansen 1975; Karlin and Campbell 1981; de Meeûs et al. 1993). We see a similar outcome in our model, where selection occurring within habitats is more conducive to polymorphism maintenance than selection occurring across habitats.

When we model sexual selection as occurring within habitats in our primary model, we find broad conditions for polymorphism maintenance. Because the mating success of females is not dependent upon the habitat in which they choose mates, females will contribute offspring to the next generation in ratios proportional to the habitat ratios themselves. Sexual selection under these assumptions thus provides a form of independent population regulation within each habitat. In contrast, we show that an across habitat sexual selection model cannot maintain a polymorphism. The conditions under which female choice occurs will determine whether modeling sexual selection as occurring within or across habitats is more appropriate. If a female chooses from among the males that are present in the habitat that the female happens to be in when she decides to mate, then modeling selection as occurring within each habitat is most appropriate. This could occur if the patch size is large or if the habitat type is temporal (perhaps with daylight changes over the course of the day). Alternatively, if females move between habitats as they evaluate potential mates, modeling sexual selection as occurring across habitats may then be the more appropriate assumption. If costs associated with searching for a mate are high, the latter situation may be less common because it requires females to view males from both habitats before making a mating decision.

Our results highlight the potential importance of sexual selection, in this case for conspicuous male traits, in maintaining a stable polymorphism. The conditions for polymorphism maintenance under sexual selection can often be broad. We found symmetry of parameter values between habitats to be very important in determining the range of parameter space that will lead to a stable polymorphism. When sexual selection, natural selection, and habitat frequency are completely symmetrical between habitats, polymorphism is the only possible outcome. As symmetry decreases, the parameter space that leads to a polymorphism decreases as well, although not particularly rapidly. When parameters are highly skewed between habitats, a stable polymorphism is still possible, although the range of female preferences that will yield a polymorphism may be quite restricted (see Table 2). Thus polymorphism maintenance does not require symmetry, but in cases in which equilibrium frequencies in traits are expected to be highly asymmetric, stochastic forces acting in natural populations may be expected to lead to the loss of the less common allele.

The stable polymorphism maintained under sexual selection results because under certain sets of parameter values, the rare morph in the population will increase in frequency. We can think of the habitat under these conditions as essentially having “protected areas” that result from a combination of microhabitat area and female preference. In some areas (or microhabitats), one morph is favored, whereas in different areas, the other morph is favored. Under conditions in which the polymorphism is stable, the increase in frequency that a rare morph will exhibit in the areas in which it is favored will more than compensate for the decrease in frequency that it will experience in areas in which it is not favored. For example, when habitat frequency and preference strengths are symmetrical and only sexual selection is acting, half the females in a population will prefer blue males and half will prefer yellow males at any given time. Therefore, if a morph frequency falls below 50%, the rarer morph will have a mating advantage. By analogy with models of feeding polymorphisms, the rare male morph can be thought of as having more of its favored resource, the habitat in which it is more conspicuous and preferred by females, available to it. This mechanism, which is usually defined as a form of spatially varying selection (e.g., Futuyma 1997), behaves very similarly to frequency dependence in terms of the advantages gained by a rare morph. Frequency dependence is not, however, explicitly included in our equations (our selection coefficients s and a are constants and do not depend upon the frequency of the color morphs in the population). Because of the general nature of the mechanism of polymorphism maintenance, we expect the qualitative results to be robust to changes in many of our specific assumptions such as ploidy levels or the absence of sexual dimorphism.

In our primary model, we treat natural selection as occurring across habitats, and conclude that, as expected, a polymorphism cannot be maintained under these conditions when natural selection acts alone. We additionally assess the result of modeling natural selection as occurring within habitats, as in soft selection, in one of our alternative models; we find that a stable polymorphism can indeed be maintained, even when parameters are not symmetrical (e.g., Levene 1953; Christiansen 1975; de Meeûs et al. 1993; see Dempster 1955). Which of these assumptions is more appropriate depends to some degree on the biology of a particular situation. It is generally assumed that when organisms can move freely between habitats, as would be the case with the microhabitats modeled here, treating selection as occurring across habitats is more realistic (e.g., Dempster 1955). This could occur if predators move between habitats during a prey selection event and view males against different backgrounds before choosing a prey item. This may be especially likely if predators have a large body size relative to that of the prey species. More importantly, we assume that males are freely moving between selection events when selection is occurring across habitats. Thus, as predators remove selected males, the frequency of each morph changes in both habitats, regardless of where the predation took place.

If our alternative, within habitat, model of natural selection is to be appropriate, we need to assume that predators stay within a habitat during each specific predation event, at a minimum, and males remain in a specific microhabitat during the entire set of prey selection events. Moreover, to match our specific within-habitat assumptions, predators would be assumed to disperse randomly to microhabitats, rather than spending more time in one than the other. In other words, the prey capture success of predators would have to be equalized within each habitat; strongly density-dependent reproduction within each habitat may be another way to equilibrate the number of offspring produced by each habitat (e.g., Levene 1953). Because we are considering a microhabitat scale of variation without barriers to movement by males and predators, there is unlikely to be separate population regulation in each habitat and selection across habitats will be the more appropriate way of modeling natural selection for most organisms.

In developing the model, we made two simplifying assumptions that should nevertheless be biologically realistic. The first is that the visual systems of predators and females are assumed to be similar; that is, the background effects are similar whether viewed by females or by predators so that one male color is always most conspicuous in a particular habitat and vice versa. Conspicuousness to females is often related to conspicuousness to predators (Anderson 1994; Zuk and Kolluru 1998). We also evaluated the hypothesis that the environment did not affect natural selection but did influence sexual selection by having predators select prey at random whereas females choose mates in a habitat-dependent manner. In removing natural selection but keeping female preferences habitat specific, we are essentially following the assumptions of a private communication channel that allows males to transmit signals to females in a way that cannot be perceived by predators (Cummings et al. 2003). In this scenario, we found a stable polymorphism could be maintained over a range of parameter space. Finally, we briefly considered the extreme case of predators and females preferring different morphs within a habitat (i.e., females favor blue males in one habitat, whereas predators preferentially prey on yellow males in the same habitat). In this situation, we still found that a stable polymorphism could result, although the conditions are slightly numerically different than when females and predators find the same males to be conspicuous. As most empirical studies show that females prefer conspicuous males that are also most prone to predation (Anderson 1994; Zuk and Kolluru 1998), it is unlikely that the reverse case is commonly seen in nature.

We further assume that neither males nor females choose their habitat but instead move between microhabitats at random. If there were habitat choice, this could change the outcome of the model, as it has been found that habitat matching will broaden the conditions leading to a stable polymorphism (e.g., Maynard Smith 1966; Ravigné et al. 2004). Some empirical evidence suggests that males can select microhabitats. Specifically, males can maximize their conspicuousness while courting females and minimize their conspicuousness at all other times by choosing the appropriate light environments within their habitat (Endler 1991; Endler and Thery 1996). The effectiveness with which males and females choose habitats, however, is unclear. Future research on habitat choice may yield important insights on conditions that will lead to polymorphism maintenance or sympatric speciation.

The basic environmental conditions assumed by our model are common in nature. Fine-scale variation in the sensory environment can be seen in both aquatic and terrestrial environments. In aquatic environments, microhabitats with different visual properties could result from different water depths (Johnsen 2002; Maan et al. 2006), substrate types (Endler 1980), amount and type of overhanging vegetation, or even time of day (Endler 1993; Johnsen 2002; Gamble et al. 2003). Water depth may be a particularly important way in which habitats can be partitioned, as the properties of visual light change rapidly with changing depth. In fact, the visual systems of many fish species are tuned to the specific light environment of their habitat (Loew and Lythgoe 1978). This match between the sensory system and the environment can even be seen at the microhabitat scale among closely related species with different foraging habitats (Cummings and Partridge 2001). In terrestrial environments, microhabitats with variation in visual properties could occur in places such as forest edges where there are differences in light profiles and substrate (Endler 1993).

Although we have framed our discussion of this model in terms of visual signals that result in selection on body color, the model we present is very general in nature and should be equally applicable to habitat heterogeneity that affects the reception of multiple kinds of signals potentially including sound, chemical, electrical, and even tactile. Future empirical studies may add to our understanding of microhabitat variation and its effects by looking carefully at the scale of environmental variation, the frequency distribution of habitat types, and the degree of symmetry in selection between microhabitats. Also, experiments that manipulate microhabitat type, scale, and frequency in replicate populations and then track MCP evolution (similar to Endler's classic work on the evolution of guppy color patterns under different selection regimes, Endler 1980) are necessary to determine the exact role of the environment in MCP maintenance.

The role of the sensory environment in maintaining polymorphism is becoming recognized as increasingly important in part because MCPs may be a precursor to sympatric speciation (Seehausen et al. 1999; Allender et al. 2003). For example, female choice for conspicuous males in a heterogeneous visual environment has been proposed as a mechanism for the rapid diversification of cichlid fish (Seehausen et al. 1997; Allender et al. 2003; Maan et al. 2006). If different sensory environments allow the maintenance of variation in male color, it is possible that divergence could occur if female preferences were also allowed to evolve. Sympatric speciation could result if genes for increasingly strong female preferences spread in the population; this will be explored in future models.

In addition to its theoretical importance, understanding the maintenance of MCPs also has practical relevance because anthropogenic change is rapidly altering the signaling environment of many organisms. If changes in the environment negatively affect discrimination of visual signals, the conditions for the maintenance of MCPs will be greatly reduced and MCPs may even collapse. For example, cichlid fish diversity may be threatened because of increasing turbidity, caused by human environmental changes, which obscures male color (Seehausen et al. 1997; Seehausen 2006). Similarly, in terrestrial environments any disturbance to forest structure can have a profound impact on the light regime in the forest, which may once again affect visual displays and thereby disturb mating behavior and reproduction (Endler and Thery 1996).

Sexual selection is increasingly being recognized as an important factor in maintaining genetic and phenotypic diversity, and our model reinforces that idea. Also, many sexually selected traits exhibit an extreme degree of continuous variation. This variation is often attributed to the condition dependence of the traits. Although we model discrete traits here, this work opens up the possibility that the variation in some continuous male traits may also result in part from trade-offs in conspicuousness in heterogeneous environments. More empirical data on preference strengths, selection from predation, and the effects of microhabitat variation on these processes will lead to a greater ability to predict the environmental and biological conditions that allow polymorphism maintenance and perhaps those that ultimately make sympatric speciation possible.

Associate Editor: T. Hansen

ACKNOWLEDGMENTS

The authors would like to thank H. Olofsson, S. Diamond, C. Ledon-Rettig, G. Ragland, J. Knies, A. Rice, M. Ferris, T. Hansen, and two anonymous reviewers for comments on the manuscript, and S. Gray, N. Hamele, N. Frey, M. Whitlock, and J. Greene for helpful discussions. M. Whitlock proposed biological situations under which hard versus soft selection is appropriate for sexual selection. This work was supported by the National Science Foundation Award DEB-0234849 and DEB-0614166 to MRS, and a National Science Foundation REU grant to UW-Whitewater. The concept for this article was proposed by JSM, model development and analyses were carried out by AJC and MRS, and all authors contributed to writing.

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