Behavioral Ecology Vol. 14 No. 6: 876-886
© 2003 International Society for Behavioral Ecology
When should males lek? Insights from a dynamic state variable model
Department of Zoology, 223 Bartram Hall, P.O. Box 118525, University of Florida, Gainesville, FL 32611–8525, USA
Address correspondence to K. Isvaran. E-mail: kavita{at}zoo.ufl.edu.
Received 13 July 2001; revised 11 November 2002; accepted 24 January 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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A central focus in the study of lek evolution is to understand the clustering of male mating territories. Lekking males typically defend small clumped territories and experience intense competition associated with dense aggregations. We used dynamic state variable modeling to evaluate three alternative selective pressures proposed to explain the evolution of lekking. These are female mating bias for large clusters, reduction in predation risk in large clusters, and male harassment of estrous females. We modeled male mating decisions during a single breeding season using a lekking ungulate as a model system. Males could choose from eight alternative tactics that included a nonreproductive tactic, territorial tactics ranging from low to high clustering, and the option to join a mixed-sex herd. The model predicted a state- and time-dependent strategy that maximizes mating success over the course of the season. We then simulated a population of 100 males that used the optimal strategy and calculated the proportion of the population that adopted each tactic. Our model generated unique predictions for the three selective pressures we considered. Female mating bias, when nonlinearly related to cluster size, had the greatest potential to generate large clusters of territorial males, whereas predation risk and harassment of females typically did not promote male clustering. More generally, our model highlights the conditions that will favor lekking. Lek-like clustering was consistently produced when the benefits in clustering increased in specific nonlinear ways. Our model thus emphasizes clarifying the shapes of relationships between potential selective factors and the size of territory clusters.
Key words: dynamic optimization, female preference, lek evolution, predation risk, sexual harassment.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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The lek mating system is perhaps the least understood of all vertebrate mating systems. Lekking males typically defend small aggregated territories. Because these territories do not contain resources attractive to females and there is no paternal care, females are thought to visit leks for the sole purpose of mating (Bradbury, 1981
). One of the most puzzling aspects of the lek mating system is the distribution of male mating territories. Males establish territories in aggregations (leks), and lekking is often distinguished from other territorial systems by the extreme clustering of territories. Thus, a principal question concerning the evolution of lekking is why competing individuals choose to defend clumped territories and thus incur the great costs of aggression associated with such dense aggregations.
The most prominent hypotheses proposed to explain the evolution and maintenance of such extreme clustering of territories can be broadly classified into those that invoke some form of female mating bias and those that do not (Höglund et al., 1993). There are two classes of female bias models. First, leks may form because females prefer to mate with clustered males (e.g., Gibson et al., 1990). Several reasons for a female bias for clustering have been proposed: average male quality may be correlated with cluster size, costs of mate-searching and mate-sampling may decline with cluster size, predation risk may be reduced in clusters, etc. (reviewed in Höglund and Alatalo, 1995). Second, the hotshot models propose that females seek out and try to mate with high quality males; leks are formed when less attractive males establish territories around popular males to try to intercept females traveling to popular males (e.g., Beehler and Foster, 1988).
Of the models that do not invoke any female mating bias, one group suggests that leks develop because males establish territories in areas of maximal overlap of either female ranges or resources used by females (hot-spot models; see Bradbury et al., 1986). Another model that assumes a lack of female preference is the black hole model (Stillman et al., 1993). This model was motivated by observations from several lekking ungulates that estrous females on leks are harassed by nonterritorial, usually young males who intrude onto lek-territories (Clutton-Brock et al., 1992; Nefdt, 1995). This harassment may drive females out of territories or even out of the lek. Thus, females may be forced to move among multiple territories before mating. Based on these observations, the black hole model assumes that females move among territories at random (to reflect movement in response to harassment) until they eventually mate. Under such conditions, close neighbors are more likely to receive females from each other's territories; thus, males in clusters have a higher mating success than do those on solitary territories. Territory clustering is selected for because clusters are better able to retain estrous females than are solitary territories, and larger clusters receive and retain more females than do smaller ones (Clutton-Brock et al., 1992). The black hole model is not strictly related to harassment; it is applicable under any conditions that create this pattern of female movement among territories. Several other hypotheses also propose benefits to males (rather than to females) from clustering their territories (e.g., a reduction in predation risk; reviewed in Höglund and Alatalo, 1995).
Most of the hypotheses described above have been modeled mathematically (Bradbury et al., 1986; Gibson et al., 1990; Stillman et al., 1996), and several predictions of the alternative models are supported by available data. However, many of the predictions (typically involving the spatial distribution of territorial males) are shared by more than one model. For example, both the hot-spot and black hole models predict that territory clustering should be positively correlated with female range size (Bradbury et al., 1986; Stillman et al., 1996). Similarly, a female bias model (Bradbury, 1981)and the black hole model (Stillman et al., 1996) predict that leks will be evenly spaced one female-range diameter apart. Further, most modeling efforts have focused on one or a few costs and/or benefits while generating hypotheses about lek evolution. Field studies of leks, however, report that many variables that potentially affect the payoffs to lekking correlate with territory clustering (Höglund et al., 1993; Nefdt and Thirgood, 1997). Clearly, to understand lek evolution, we need a modeling approach that can incorporate multiple costs and benefits and evaluate their relative importance (as well as interactions among them).
One such approach is dynamic state variable modeling (Clark and Mangel, 2000; Houston and McNamara, 1999; Mangel and Clark, 1988). This modeling approach allows one to predict sequences of optimal decisions. Multiple fitness components can be incorporated while calculating the payoffs to alternative tactics; these components can even be measured in different currencies. Dynamic modeling provides a common framework within which the effects of several potential selective forces and their interactions can be compared. More generally, the value of modeling approaches goes beyond their ability to produce testable quantitative predictions. Models allow one to examine the consequences of assumptions made about how hypothesized selective factors affect the behavior of interest. One can also identify the general conditions under which these factors are likely to lead to the evolution of the behavior of interest.
We used dynamic state variable modeling to examine several hypothesized selective pressures leading to lekking. Like previous theoretical efforts (Gibson et al., 1990; Stillman et al., 1996), we treat lekking as a matter of the clustering of territories and evaluate the ability of multiple selective factors to produce territory clustering. Our model is based on lekking ungulates. Ungulate leks are very similar to classical bird leks (Höglund and Alatalo, 1995); lek-territories are aggregated, lack resources attractive to females, occur on traditional sites, and are the sites of most matings in a population. A striking difference between lekking ungulates and birds is that, in most lekking ungulates, there is considerable variation in mating strategies among and within populations (Höglund and Alatalo, 1995). Males may hold resource-based territories and may also attempt to court and mate with females in mixed-sex herds (groups containing adults of both sexes).
We assessed the ability of (1) a female mating bias for large clusters of territories, (2) male harassment of estrous females, and (3) a reduction in predation risk in large clusters to produce extreme territory clustering. The decision to focus on these three factors was driven by both the theoretical treatment and the empirical support they have received. For instance, the female bias hypothesis has received much modeling attention (Bradbury, 1981; Gibson et al., 1990; Kokko, 1997) and is supported by data from several lekking ungulates and birds (Alatalo et al., 1992; Balmford et al., 1992; Nefdt, 1995). In our model, we examined the consequences of female mating bias without regard to its cause (e.g., reduced predation risk or male harassment, increased genetic benefits).
The second factor we investigated, male harassment, is suggested to play a crucial role in the black hole model, arguably the most popular hypothesis about the evolution of ungulate leks (Clutton-Brock et al., 1993; Nefdt, 1995). We could not address the black hole hypothesis directly because ours is not a spatial model. Instead, we modeled male harassment of estrous females, the factor proposed to be most likely to promote random female movement between territories before mating (Clutton-Brock et al., 1992); this movement is the main cause of territory clustering in the black hole model. Harassment can also potentially lead to a female mating bias (e.g., females may prefer to mate in larger clusters if they experience less harassment in them). However, we were interested in harassment as an alternative to female preference and, hence, examined whether harassment can lead to lekking even in the absence of any female mating bias, as suggested in one form of the black hole model (Clutton-Brock et al., 1992). We also considered the effect of a third factor, predation risk, on optimal male mating strategy because this factor is thought to influence social grouping in ungulates (FitzGibbon and Lazarus, 1995; Jarman, 1974). Safety from predation was one of the first hypothesized benefits of lekking (e.g., Koivisto, 1965), but empirical support for this hypothesis is ambiguous. However, predation risk is reportedly low on several bird leks (Höglund and Alatalo, 1995), and some studies of lekking antelopes suggest that males and females are sensitive to predation risk while establishing territories and making mate choice decisions, respectively (Deutsch and Weeks, 1992; Gosling, 1986).
To evaluate the three hypothesized selective factors in a lekking ungulate system, we constructed a dynamic state variable model that predicts optimal male mating decisions. The mating options available to males were to establish a solitary territory, to join territory clusters of different sizes, or to join a mixed-sex herd. We use "cluster" to mean closely spaced territories with interacting neighbors. The basic model included the major costs and benefits associated with different mating options reported from lekking ungulates. We then modeled each selective factor of interest and compared patterns of territory clustering produced to see whether our model made testable alternative predictions. We also examined interactions between factors and performed sensitivity analyses of several variables built into the basic model, such as foraging yield, the degree of synchrony of estrus, and the energetic costs of maintaining territories in clusters of different sizes.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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The basic model
Understanding the trade-offs associated with different strategies and determining the optimal strategy is not always intuitive. The costs and benefits of a strategy may vary with time and depend on the condition (state) of the animal (which may also vary with time). Further, the decision that an animal makes at one time affects its state and thus the decisions that are optimal in the future. Dynamic state variable modeling incorporates these features and allows one to model the sequence of decisions that an animal must make to maximize fitness. The basic approach entails defining a state variable, the alternative tactics that an individual can adopt, the costs and benefits associated with these tactics, the time period of interest, and the time intervals within this period in which an individual makes a single decision (e.g., a day). The optimal solution is found by iterating backward through time. In each time interval and for each state, the current reproductive increment and expected future reproduction associated with the alternative tactics are evaluated; the tactic that maximizes overall fitness (current reproduction + expected future reproductive success) is optimal. Further, in each time interval the state of the individual is modified based on the tactic chosen; that is, the individual gains or loses condition, and the modified state influences the expected future reproductive success of the individual. The basic model thus predicts a sequence of state- and time-specific tactics that maximize the overall fitness of the individual (Clark and Mangel, 2000
; Houston and McNamara, 1999).
We used energy reserves as the state variable X(t) and arbitrarily defined 24 energy levels. We considered a single breeding season (T) consisting of 15 time intervals, t, each of which corresponded to a day. In each time interval, t, the following eight tactics were available: (1) join an all-male bachelor herd (a nonreproductive option); (2) join a mixed-sex herd (groups with adults and immatures of both sexes); (3) establish a solitary territory; (4) establish a territory next to one territorial male, join 1; (5) establish a territory next to two males, join 2; (6) establish a territory next to three males, join 3; (7) establish a territory next to four males, join 4; and (8) establish a territory next to five males, join 5. The territorial tactics represent a relative degree of territory clustering from low to high, rather than absolute levels of clustering. Each tactic was associated with a set of costs and benefits. The cost structure was based on data from Uganda kob (Kobus kob thomasi), Kafue lechwe (Kobus leche kafuensis), fallow deer (Dama dama), topi (Damaliscus korrigum), and blackbuck (Antilope cervicapra) (Apollonio et al., 1992; Balmford et al., 1992; Clutton-Brock et al., 1988; Gosling and Petrie, 1990; Isvaran and Jhala, 2000; Nefdt and Thirgood, 1997). We considered two kinds of costs: an energetic cost, ci, associated with maintaining a particular reproductive tactic and a reduction in foraging yield, yi. Several studies have reported that fighting rates are lower on solitary territories than on leks (Apollonio et al., 1992; Nefdt and Thirgood, 1997), and one study of the Kafue lechwe (Nefdt and Thirgood, 1997) found that per capita male fighting rates on a lek were positively correlated with male numbers on the lek. That study also found that males on territories fought more frequently than did males in mixed-sex herds. Based on these data, we assumed that the energetic cost, ci, increased approximately linearly from tactic 2 (mixed-sex herd) to tactic 8 (join 5) (Table 1). Studies of lekking ungulates also report reduced foraging among territorial males compared with males in herds; further, males holding lek-territories forage less than do males defending solitary territories (Clutton-Brock et al., 1988; Nefdt and Thirgood, 1997). Hence, we assumed that foraging yield, yi, decreased from tactic 2 to tactic 8 (Table 1). In each time interval, the energy state (x) dynamics were specified by the equation
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For simplicity, we assumed that individuals had a low probability of surviving to breed in the next season, and thus, the expected future reproductive success at the end of the breeding season,
, was a small function of the state at that time: = 0.01X(T). We then sought the reproductive strategy (i.e., series of state- and time-specific tactics) that maximized lifetime mating success. Specifically, we searched for the strategy that gave the maximum expected mating success from time t to the time horizon, T, given an energetic state x at time t, by solving the following generalized dynamic state variable equation:
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The current mating success, Mi, in a time interval was zero for bachelors, and for the remaining tactics was given by the equation:
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The model predicted a matrix of state-dependent optimal decisions for each time interval. We then iterated forward such that a population of 100 males, whose initial distribution of states followed a normal distribution with mean = 12 and SD = 6 units of energy, was run through time intervals t1 to T. In each time interval, the males were assumed to adopt the optimal state- and time-specific decisions; the distribution of their energy states was correspondingly modified, and the new distribution entered the next time interval. At the end of the forward iteration, we calculated the percentage of the population that adopted each tactic, averaged across time intervals (Mangel and Clark, 1988
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Evaluating the effect of a female bias for clustered territories
We defined female mating bias as the probability that a female mates in a cluster of a given size. There are few data available on which to base our patterns of female mating bias. Hence, we modeled female mating bias for increased territory aggregation, Bi, in three ways: as linear, accelerating, and saturating functions of cluster size. In each case, we kept the basic model parameters constant (Table 1), and the mating probabilities for alternative tactics were scaled so that they summed to one. Thus, we assumed that each female mated only once per season. All tactics were assigned a survivorship (si) of 0.97, and there was no harassment of females (i.e., Hi = 1).
First, we modeled female bias as a linear increase from tactic 3 (solitary territory) to tactic 8 (join 5); that is, Bi = a + bi, where i represents tactics 3–8. We varied the slope of the linear relationship from 0.01 to 0.04. Thus, female mating bias increases proportionally with cluster size, and because there is a similar increase in male-male competition, Ci, with cluster size, there is no per capita gain in male mating success. We assumed that males in mixed-sex herds are unattractive to females (Clutton-Brock et al., 1988) and maintained female bias for this tactic at 0.005.
Second, we modeled a disproportionate increase in female bias for larger clusters by using a power function, Bi = ib, where i represents the different reproductive tactics and increases from mixed-sex herd to join 5 (i = 1 for mixed-herd males, i = 2 for solitary territory, i = 3 for join 1, etc.). The exponent b represents the strength of discrimination among the alternative tactics and was varied from 1.1 to 2.5 (Figure 2).
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Finally, we used a logistic function, Bi = (ea+bi)/(1 + ea+bi) (where i took values from one to seven, as above), to represent a saturating female mating bias for large clusters. The parameters b, which again represents the strength of discrimination between the alternative tactics, and a (the point of inflection) were varied (Figure 3).
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For a preliminary test of the model, we searched the literature for data on the probability of females mating at different cluster sizes and for cluster size distributions from the same population. We were unable to find suitable data on lekking ungulates. However, we found one study of a lekking bird (ruff, Philomachus pugnax), which reported copulation rates over a wide range of clusters (Widemo and Owens, 1995
). We assumed that copulation rates represented female mating bias for different cluster sizes because female ruffs are reported to move freely among leks and among lek-territories (Höglund et al., 1993). Widemo and Owens (1995) found that a logistic regression best described the relationship between overall copulation rates and lek size. We used the equation they presented to calculate female mating bias, Bi, in our model and used our basic model values for the other parameters. Because we were also interested in examining the importance of multiple factors in lek evolution, we then ran the model, including female mating bias from Widemo and Owens (1995) as before, and included an intermediate level of predation risk (a linear decrease of 3% in predation risk between successive clustering tactics). Including a reduction in predation risk with clustering is likely to be realistic, because several studies of lekking birds report that predation is rare at leks, and one study has shown that predation risk decreases with lek size (Höglund and Alatalo, 1995; Trail, 1987). We summarized the model outcomes as the proportion of males adopting different levels of clustering and compared them to data from Widemo and Owens (1995).
Evaluating the risk of predation
We examined the hypothesis that predation may favor lekking if males experience a lower predation risk in larger territory clusters. In our model, predation risk was represented by the probability, si, of surviving each time interval. Bachelor and mixed-sex herd males were assigned high survivorship (si = 0.97) in all runs (assuming that survivorship was maximal in herds owing to the advantages of group living). Data on predation risk at leks of different sizes are scarce, but information on survivorship in bird and ungulate social groups suggests that survivorship might accelerate with group size (Cresswell, 1994; FitzGibbon, 1990). Because there are so few data available, we chose to model the change in survivorship across clusters owing to predation risk in several ways: (1) as a linear function of cluster size, (2) as an accelerating (power) function, and (3) as a saturating function, si = ai/(1 + bi) (as expected from dilution of predation risk); i represents tactics 3–8. In each case, we varied the magnitude of the selective factor from a weak effect in which survivorship of territorial males, s, ranged from 0.9 to 0.95 to stronger selection in which s ranged from 0.7–0.95. All other parameters were as in Table 1, and there was no differential female mating bias or harassment.
We also examined the interaction between predation risk and female mating bias by considering combinations of low and high risk and bias.
Evaluating male harassment
We could not directly address the spatial black hole model by using dynamic modeling. However, the central tenet of the black hole model is that before mating, females move among territories; the main factor proposed to cause this movement is harassment by nonterritorial males from nearby herds (Clutton-Brock et al., 1992). Hence, we used our model to evaluate whether harassment by nonterritorial males can produce territory clustering. Harassment was modeled as the probability that females arriving at a cluster of a given size were not driven away by intruders but remained to mate. We examined whether this probability might increase sufficiently with cluster size and thus favor territory clustering. We modeled harassment as a function of intruder pressure and the probability that intruders arriving at a territory cluster were successfully chased away by resident males. Both these factors varied with cluster size. For simplicity, we assumed that intruder pressure (hi, i.e., the probability that an intruder arrives at a cluster of territories) was directly proportional to the number of territorial males in a cluster and that all territorial males were equally capable of successfully driving an intruder away. Given a basic probability, d, that a territorial male chases an intruder away successfully, we calculated the probability, Pi, that, in a cluster of i territorial males, an intruder is chased away by at least one male in that cluster (following the binomial theorem):
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Sensitivity analyses
Finally, we conducted analyses of other parameters in the basic model to examine how sensitive model results were to these parameters. We increased and decreased the relative differences among tactics in foraging yield, yi (Table 2), and evaluated their effect on model outcomes. The relative energetic cost, ci, was varied in a similar manner (Table 2). In addition, we modified the distribution of estrous females (the degree of estrus synchrony, Figure 1) and evaluated its effect on model predictions. We also varied the total estrous female population and evaluated the effect of lower (half the population of the basic model) and higher abundances (double the population of the basic model) on the model predictions. Finally, we varied the terminal fitness function by varying the contribution of the state X at the final time interval T to future reproductive success (range: = 0.01X(T) to = 0.2X(T)). Each variation in a particular parameter in the basic model was run maintaining the original values of the other parameters.
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Sensitivity analyses were not performed for all sets of values of female mating bias, predation risk, and harassment. Because there were numerous potential parameter combinations, we ran the sensitivity analyses under the conditions that had the greatest effect in the model, that is, nonlinear forms of female mating bias. Specifically, we considered three forms of female mating bias, a low (Bi = i1.1) and a high intensity (Bi = i2.0) power function, and an intermediate intensity (b = 6) logistic function with the inflection point at join 3. Survival was kept uniform, and there was no harassment.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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The model predicted state- and time-specific optimal tactics. We then used those predictions and the process of forward iteration to predict the proportions of the male population that adopt the different tactics in each time interval. Here we discuss both the state-dependent decisions and the population level predictions (i.e., the percentage of males that adopt each alternative tactic, averaged over the breeding season).
Most of the model runs resulted in the expression of not more than two alternative mating tactics. When the basic model was run while setting female mating bias and survivorship equal for all tactics and removing male harassment, joining a mixed-sex herd was the optimal tactic for all states at all times. Territorial options were optimal only when specific benefits were included, such as a female mating bias for an increased clustering of territories. Of the three selective pressures evaluated—that is, female mating bias, predation risk, and male harassment—female mating bias had the greatest effect in this model. Below, we describe the patterns in territory clustering generated by manipulating each of these factors.
Female mating bias
We considered three alternative forms of female mating bias: linear, accelerating, and saturating increases in mating probability with cluster size. When female preference was modeled as a function increasing linearly with clustering, clustered territories were never the optimal decision. The tactic with the highest fitness for all states at all times was either to join a mixed-sex herd or to establish a solitary territory.
However, when female mating bias accelerated with clustering (Bi = ib), clustering was produced at relatively low levels of nonlinearity (Figure 2). At the lowest level of nonlinearity, Bi = i1.1, holding solitary territories was the optimal tactic for all states in all time intervals (Figure 2A). As the rate of acceleration increased (i.e., b increased), clustering was predicted. Initially, only large clusters (tactic join 5) were predicted in the optimal decision matrix, and the proportion of the male population adopting clustering was low (Figure 2B). With further increases in b, a mixture of clustering tactics appeared; the majority of the population joined large clusters, and a small proportion joined small clusters (Figure 2C). As the exponent was increased further, extreme clustering (join 5) was the dominant mating tactic (males with lower energy states joined bachelor herds, which is not a mating tactic). Less than 5% of males adopted any of the other territorial tactics (Figure 2D).
When female mating bias showed a saturating relationship with male clustering (Bi followed a logistic function), patterns similar to those described for the power function were predicted. When females distinguished among tactics only weakly (low b), holding a solitary territory was the dominant tactic. As b increased, clustering also increased in frequency. However, the optimal cluster size depended on the point of inflection of the curve describing the relationship between female mating bias and cluster size. In general, the tactic succeeding the inflection point predominated (Figure 3). For example, when the inflection point occurred at join 2, the dominant tactic was join 3.
In several model runs, a mixture of clustering tactics was predicted both within a time interval and across time intervals in a breeding season. This variation in clustering was associated both with energy state and time in the breeding season. Extreme clustering was associated with relatively high male energy states, whereas joining bachelor and mixed-sex herds and joining smaller clusters was associated with lower energy states. Males with higher states adopted clustering tactics earlier than did other males. Males with lower energy states adopted high levels of clustering progressively, as the breeding season unfolded. For example, when female mating bias increased steeply with cluster size (b = 2.0), in the first four time intervals of the breeding season, clustering tactics were optimal only for males with very high energy states (greater than 20 units); for all other males, joining a bachelor herd was optimal. As the breeding season progressed, males with lower energy states adopted clustering; however, males with very low energy states (less than 3 units) rarely adopted clustering tactics. Furthermore, males who initially adopted highly clustered mating tactics typically fell in energy state and later adopted less costly tactics.
We preliminarily tested model predictions concerning female mating bias by using data from ruff leks (Widemo and Owens, 1995). When we ran the model using female mating bias from their study and using our basic model values for other parameters, model results reflected a large part of the variation in clustering seen in the field (Figure 4A). Our model predicted that males should show intermediate clustering (join 2) and that none should adopt low or high levels of clustering. In the field, more than 50% of territorial ruffs were in leks of intermediate size, and small leks were rare. However, a significant proportion of lekking males were found on large leks, a pattern not predicted by our model. When we included a second factor, predation risk, model predictions from this simulation were very similar to lek size distributions seen in the field (Figure 4C).
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Predation risk
We modeled survivorship as linear, accelerating, and saturating functions of cluster size. When survivorship increased linearly with cluster size, clustering was predicted only when the slope was relatively high (0.05; 5% increase in survivorship between successive cluster sizes). Further, the extent of clustering was low; in the above example (slope = 0.05), 70% of the population, averaged across time intervals, held solitary territories, whereas only 19% adopted the join 5 tactic. Intermediate clustering tactics were not seen.
Nonlinearity in survivorship did not have a great effect on model predictions. When the range of survivorship probabilities was small (e.g., 0.90–0.95), clustering options were never optimal, even when survivorship accelerated steeply from solitary territories to large clusters. Clustering was seen only when the range in survivorship probabilities was large (i.e., 0.70–0.95). Within this range, increasing the rate of acceleration (i.e., the exponent of the power function) did not produce significant increases in the proportion of the male population adopting clustering tactics. This proportion was never more than 20%. Of the clustering tactics, extreme clustering (join 5) was the only optimal tactic predicted. The saturating survivorship function produced similar results.
Predation did, however, significantly modify patterns of clustering produced by other factors. Even a small linear increase in survivorship produced clustering at low levels of female bias for all functions used to model female bias. For example, when female mating bias was modeled as a weakly accelerating function (Bi = i1.1) and predation risk was equal for all tactics, holding a solitary territory was the dominant tactic and clustering was not seen. However, if this female bias function was combined with linearly increasing survivorship (slope = 0.03), 34% of the male population adopted the join 5 tactic (Figure 5).
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Male harassment
We modeled the effect of male harassment of females on male mating strategy as the probability that a nonterritorial male would enter a cluster to harass females, hi, and the probability that at least one resident male would exclude that male, d, and thus permit the females to mate. In all cases, we assumed that these probabilities, and thus the overall probability of females mating, were proportional to the number of males in the cluster. Male harassment never produced clustering, and establishing a solitary territory was always the optimal strategy.
Sensitivity analyses
We conducted sensitivity analyses of five parameters in the basic model: foraging yield, energetic cost, terminal fitness, abundance of estrous females, and the variance in the distribution of estrous females over the breeding season. These analyses were conducted for three nonlinear patterns of female mating bias, Bi (see Methods). When female mating bias was weakly nonlinear, manipulations of basic parameters did not affect model results. Therefore, the results presented below refer to sensitivity analyses conducted with relatively strong nonlinear functions of female mating bias. In general, changes in basic parameters did not result in qualitative changes in model outcomes; e.g., clustering never disappeared after a sensitivity manipulation. Further, with few exceptions, parameter manipulations had only small effects on the patterns of male clustering.
Two manipulations that yielded significant changes in clustering patterns were a reduction in the differences between tactics in foraging yield and changes in estrus synchrony. When we reduced differences in foraging yield between successive tactics, the proportion of the male population adopting clustering tactics increased, especially extreme clustering (join 5) which increased to nearly 0.6 from 0.35. In contrast, when differences in foraging yield were increased, the predicted pattern of clustering was similar to that produced under the original assumptions.
Increasing the synchrony of estrous females (Figure 1) resulted in an overall reduction in the reproductive activity of males. The proportion of bachelors increased by 20% and was matched by a decrease in the proportion of solitary males. This was largely because when synchrony was increased, there were no estrous females in the first two and the last two time intervals. Consequently, no mating tactic was chosen, and joining a bachelor herd was the optimal tactic for these time intervals for all states. When females were distributed more uniformly through the breeding season (lower synchrony), the proportion of join 5 increased by 20%, and the proportion of solitary males similarly decreased compared with the original model.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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We used a dynamic optimization approach to evaluate the relative merits of three alternative explanations for the evolution of lekking. The usefulness of this model comes not solely from its ability to generate testable predictions. The real strength of such modeling approaches is to clarify (1) the conditions under which a putative selective pressure is likely to favor the evolution of lekking and (2) the consequences of assumptions one makes about how a particular factor affects male territory clustering. We evaluated the potential of female mating bias, predation, and male harassment as selective pressures in lek evolution. Our model generated quantitatively and qualitatively different predictions for these three factors. We found that female mating bias had the greatest potential to generate high levels of clustering. Male harassment never produced clustering in our model, whereas predation risk produced very limited clustering but may be important in conjunction with other factors.
Female mating bias
One of the most popular hypotheses of lek evolution suggests that lekking has evolved owing to a female bias for mating with males in aggregations (Alexander, 1975; Bradbury, 1981; Gibson et al., 1990). In our model, we evaluated this hypothesis by manipulating female mating bias, defined as the probability that a female will mate in a cluster of a given size. This parameter had the greatest effect on male decisions, and its shape greatly influenced the outcome of the model. Linear functions did not produce any clustering largely because there was no per capita increase in male mating benefits with cluster size. However, when modeled as power and logistic functions, even a small degree of nonlinearity produced extreme clustering (large leks).
When we assumed that female mating bias did not increase exponentially but included a pattern of diminishing returns at some point, the model predicted that cluster sizes just past the point of inflection should predominate in the population (Figure 3). Thus, variation in this threshold among species or populations may explain observed variation in maximum cluster size. A simple mechanistic explanation of variation in this threshold is the influence of habitat structure on the ability of females to distinguish between clusters of increasing size. Alternatively, the shape of the relationship between female mating bias and cluster size may be governed by the way in which the payoffs (e.g., male quality, safety from predation) to females scale with cluster size.
Our model highlights the importance of explicitly specifying the shape of the relationship between a putative selection pressure (here, female mating bias) and cluster size, because the same parameter modeled in different ways led to very different patterns in territory clustering or did not produce clustering at all. However, quantifying female mating bias in the field is difficult because it may not be possible to collect data over a wide range of cluster sizes. In many species, lek size distributions are greatly skewed so that most territorial males are either on large leks or in small clusters, and intermediate cluster sizes are often not seen in the field (Deutsch, 1994; Nefdt, 1995). Interestingly, such lek size distributions are consistent with the predictions of our model for certain forms of underlying female mating bias (Figure 2B,C). Given these skewed lek distributions, quantifying the underlying bias for clusters of different sizes may only be possible by experimentally providing females intermediate clusters to assess.
For one study on birds in which these data are available (ruffs; Widemo and Owens, 1995), our model predicted a large part of the observed variation in the size of territory clusters, even though the model was run using crude estimates for all parameters other than female mating bias. Model predictions when female bias alone was incorporated predicted the large proportion of intermediate lek sizes observed but did not predict the large leks recorded in the field (Figure 4A,B). Including an additional benefit of clustering (a linear decrease in predation risk with cluster size) led to predictions very similar to lek size distributions in the field and included large leks (Figure 4C). It is currently believed that although predation risk may often decrease with increased territory clustering in lekking birds, it is unlikely to be a primary factor in lek evolution (Höglund and Alatalo, 1995). Although supporting this conclusion, our modeling exercise using data from ruffs emphasizes studying multiple selective factors, including those that appear to have little effect when acting alone, because these factors could have a dramatic effect in combination with other factors.
Nonlinear patterns of female bias, such as those reported for ruffs, have been documented for several lekking species (Alatalo et al., 1992; Lank and Smith, 1992). For example, Alatalo et al. (1992) report that per capita male mating success of black grouse increased with lek size up to around 10 males and then reached a plateau. Similar patterns have been described in the Uganda kob (Balmford et al., 1992; but see Deutsch, 1994). Such data suggest that nonlinear patterns of mating bias are not uncommon in lekking species and that female mating biases for larger clusters may, in fact, be a significant factor in the evolution and maintenance of lekking as our model indicates.
Predation risk
The hypothesis that predation risk plays an important role in the evolution of lekking has recently received renewed interest. The argument has been that the reduction in predation risk to lekking males outweighs the costs associated with competition among clustering males. We modeled predation risk in three ways: as a simple linear increase in survivorship from solitary territories to the tactic join 5, as an accelerating function, and as a saturating function. Interestingly, in contrast to female mating bias, nonlinearity in predation risk did not lead to extensive clustering. Predation risk, whether modeled as linear or as nonlinear functions, produced clustering only when differences in predation risk among behavioral options were high (e.g., a difference in survivorship probabilities of 25% or more between solitary territories [ssolitary= 0.7 in each time interval] and clusters of six males [sjoin5 = 0.95 in each time interval]). Data on predation levels in ungulate populations suggest that such high levels of predation are very unlikely. For example, in Thomson's gazelles, Gazella thomsoni, the average probability of being killed by predators ranges from 0.0004–0.0006 per day (calculated from Borner et al., 1987). Likewise, in blackbuck, this probability varies from 0.00009–0.00013 per day (calculated from Jhala, 1993).
Data on the antipredatory benefits of clustering are ambiguous. For example, in a Uganda kob population, there was no difference in predation risk on and off leks (Balmford and Turyaho, 1992), and risk may in fact be higher on leks because predators may view leks as a reliable source of prey. Similarly, studies of topi leks report that hyenas hunt disproportionately more frequently at leks than at solitary territories (Gosling and Petrie, 1990). However, observational and experimental data from another Uganda kob population suggest that males pay attention to predation risk while establishing territories and prefer areas with greater visibility (Deutsch and Weeks, 1992). Predation risk could also potentially affect female mating decisions. Our model, which is focused on male mating decisions, did not explore this influence.
In our model, predation risk had interesting effects when it interacted with female mating bias. In general, combining a female bias for larger clusters with an increase (even small) in survival in larger clusters increased the proportion of males adopting more extreme clustering options. Furthermore, female mating bias functions with low levels of nonlinearity that did not generate clustering when acting alone, produced clustering in combination with predation risk. Thus, our model emphasizes the importance of studying interactions among factors when testing hypotheses about lek evolution.
Male harassment
We modeled male harassment of estrous females because it has been highlighted in recent studies as a primary factor in lek evolution (Clutton-Brock et al., 1993; Nefdt and Thirgood, 1997; Stillman et al., 1996). It has been proposed as the factor most likely to cause female movement between territories, this movement in turn is necessary to produce clustering in the black hole model (Stillman et al., 1993). We modeled male harassment of estrous females as the probability that a female entering a male's territory is not driven away by harassing intruders but remains to mate with him. We were especially interested in assessing whether male harassment of estrous females could favor clustering in the absence of any kind of a female mating bias for clustering. In our model, harassment did not produce clustering of territories. This is because of the way in which the two parameters, the probability of an intrusion and the probability of at least one territorial male successfully driving away the intruder, scaled with cluster size. These two components of male harassment increased and decreased proportionately with cluster size. A disproportionate increase in the chance of an intruder being successively driven away in larger clusters may have produced clustering. However, we did not have any a priori expectation of such a disproportionate increase.
One mechanism that might generate such a disproportionate increase in intruder repulsion is cooperation between males in driving away intruders, so that two males in a cluster acting together are more likely to drive away an intruder than two males individually. Such cooperation between territorial males has not been systematically investigated in ungulate leks. However, studies of other territorial systems report that territory owners are often less aggressive toward neighboring territorial males than toward strangers (the "dear enemy" phenomenon; e.g., Beletsky and Orians, 1989). These studies suggest that males may favor neighbors over strangers because remaining among neighbors whose strengths have already been assessed may minimize the costs of aggression and territory defense. Observations of a lekking antelope suggest that the dear enemy phenomenon may also operate on ungulate leks (Isvaran K, unpublished data). During prolonged fights between a territorial male blackbuck and an intruder, the fighting males tend to move out of the owner's territory and often cross into neighboring territories. When such a fighting pair continues to remain in a neighboring territory, the neighboring owner usually attacks the intruder, and most often, the intruder ends the interaction and leaves the area. Male harassment of estrous females could also produce clustering if fewer intruders appeared at larger clusters. Data from antelope leks do not support this pattern because intrusions reportedly increase with cluster size (Nefdt and Thirgood, 1997). Clustering is most likely, according to our model, when intruder pressure decreases with lek size and the probability of driving intruders away increases disproportionately with lek size, an unlikely combination. Thus, we conclude that male harassment of estrous females is unlikely to be the primary factor favoring the evolution of leks because it is unlikely to show a strongly nonlinear relationship with cluster size. However, the black hole model does not rely on this factor in its argument about lek evolution; male harassment is only one of several factors that can facilitate the black hole process by which females are retained better in larger clusters (Stillman et al., 1993, 1996).
Sensitivity analyses
We manipulated the basic parameters of the model to assess the sensitivity of model predictions to these parameters. None of the manipulations we performed produced large qualitative changes in model outcomes; e.g., none of them produced clustering when clustering was never the optimal tactic in the original model result. Similarly, clustering never disappeared after manipulations, although the level and extent of clustering sometimes changed. These results suggest that the parameters manipulated are unlikely to be primary factors underlying lek evolution. Our sensitivity analyses also suggest that our main model results are robust and do not rely on a restricted set of parameter assumptions.
Two particular manipulations resulted in significant changes in model outcomes. Reducing the difference between tactics in foraging yield (i.e., reducing the cost of clustering) resulted in a large increase in clustering. This suggests that males in habitats in which forage is abundant and accessible are more likely to defend territories in large clusters than are males with less access to forage. Lekking males usually forage close to their territory cluster and likely compete with other males in the cluster for forage. Hence, resources may constrain clustering. A study of lekking capercaillie (Tetrao urogallus) found that lek size and spacing were regulated by the quality of the habitat around each lek (Wegge and Rolstad, 1986).
The second manipulation that had a relatively large effect on the predictions of the model was a reduction in estrus synchrony. When females were distributed more uniformly across the breeding season, extreme clustering increased and fewer males adopted solitary territories. One explanation for this pattern in our model is that a reduction in synchrony was associated with an increase in time intervals with a large number of females (Figure 1). Thus, the mating benefits to males adopting join 5 were high enough to offset the large costs of extreme clustering in many more time intervals, rendering extreme clustering an optimal tactic in more time intervals. Previous models of mating system evolution (e.g., Emlen and Oring, 1977) have also predicted that in systems in which males do not defend either females or important resources used by females, a decrease in estrus synchrony should favor leks.
Conclusions
Our model produced distinct sets of predictions for the three selective factors that we evaluated. Female mating bias generated the greatest levels of territory clustering. Furthermore, empirical evidence suggests that it may naturally have the forms and strengths of nonlinear relationships with cluster size that were essential to generate clustering in our model. More generally, one of the most striking results of our model exploration is that leklike clustering was consistently produced when there were certain nonlinear increases in the benefits associated with clustering. From this, we conclude that any factor that generates such a relationship can potentially explain the evolution of lekking. This model also highlights the importance of studying multiple factors and their interactions. Factors in interaction predicted male mating decisions that were often different from predictions generated by the same factors acting alone.
Although our model is based on lekking ungulates, its results are applicable to other taxa, especially those that show similar lekking behavior (e.g., birds). Further, the structure of the basic model is general, and the only factors that are based on data from ungulates are the costs to clustering. Similar increases in costs with cluster size have been reported in lekking birds (Alatalo et al., 1992; Höglund et al., 1993). Of the three selection pressures evaluated, harassment of estrous females is the only one that is considered less important in other species (e.g., highly mobile taxa like birds; Höglund and Alatalo, 1995).
Nevertheless, our model is a simplification of natural lekking systems. Clustering is a phenomenon in which a male's behavior depends on that of other males in a population (e.g., even if extreme clustering is optimal for a male of a given state, he may be unable to adopt it if there are no large clusters available for him to join). This is a system in which a game approach potentially describes the outcomes of male decision-making processes more realistically. However, adding a game component to this model greatly increases the complexity of the model and of resulting inferences. With this straightforward dynamic model, which included some major costs and benefits, we have been able to produce patterns in clustering similar to those seen in natural populations. We have also been able to evaluate the potential role of different factors in lek evolution and produce predictions that vary both qualitatively and quantitatively.
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ACKNOWLEDGEMENTS |
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We are grateful to Jane Brockmann, Rebecca Hale, Kai Lindström, Ben Miner, Suhel Quader, Nat Seavy, Laura Sirot, Ron Ydenberg, and two anonymous reviewers for useful discussions and valuable comments on previous versions of the article. We thank Craig Osenberg and Vinayaka Pandit for helpful suggestions regarding the model. This model was motivated by a course in dynamic state variable modeling taught at the Department of Zoology, University of Florida, Gainesville, Florida, USA.
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