Abstract
In principle, the intensity of sexual conflict is best measured as a loss of fitness associated with the expression of conflict-related traits. But because the relevant traits may be difficult to manipulate and fitness difficult to assess, proxy variables linked to conflict intensity may provide important tools for empirical measurement. Here we identify two common types of sexual conflict—one within mating pairs over the less expensive male role, and one between mating pairs and intruders seeking to obtain fertilizations—and consider how they vary in intensity along gradients of population density and predation risk. To do this, we develop and analyze a model of mating dynamics in the chalk bass, an egg-trading simultaneous hermaphrodite that lives on Caribbean coral reefs. In this species, within-pair sexual conflict leads each female-role partner to provide in each mating episode only a subset (parcel) of its egg clutch to its mate for fertilization. Pair-intruder sexual conflict (i.e., sperm competition) increases the proportion of the gonad allocated to male function. In the model, more parceling and greater male allocation both resulted in lower fitness at the ESS, our measure of conflict intensity. Male allocation increased along the density gradient but decreased along the predation-risk gradient, reflecting shifts in intrusion frequency. Parcel number sharply increased and then decreased more gradually along a gradient of increasing local density, initially responding to increased availability of alternative mates across low densities and then to diminishing clutch size toward higher densities. Parcel number decreased with predation risk as each mating episode became more dangerous. Conflict intensities were usually greatest at intermediate positions along the two environmental gradients, and each conflict ameliorated the intensity of the other. Overall, parceling and sex allocation may be good though imperfect proxies for intensities of within-pair and pair-intruder sexual conflicts among chalk bass.
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Acknowledgments
We thank Eric Fischer, Yoriko Saeki, Craig Sargent, Crowley lab members, Chuck Fox, and Nico Michiels for discussions of these and closely related ideas. We especially appreciate comments and suggestions on the manuscript from Chris Petersen, Bob Warner, Dave Westneat, Gisela García-Ramos, Derik Castillo Guajardo, Lukas Schärer, Cristina Lorenzi, Gabriella Sella, Sara Helms Cahan and three anonymous reviewers. M. K. H. acknowledges UK Graduate School and Smithsonian Tropical Research Institution (STRI), UK Biology Department Gertrude Flora Ribble Fund, UK Association of Emeriti Faculty, the PADI Foundation, Animal Behavior Society, and Sigma Xi for fellowships and research grants. Work cited as unpublished or in preparation is part of the dissertation project of MKH.
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Appendices
Appendix 1: End game solutions
To keep things tractable and generally consistent with observed behavior, we assume that a pair separates only when one of the individuals has no parcels remaining. It seems likely—though, to our knowledge, unproven—that an individual has some chance b of detecting that its partner has no remaining parcels, in which case seeking another partner to exchange any of its own remaining parcels for male mating opportunities would be advantageous. Regardless of mechanism, we let b be the chance of breakup following any mating after which one partner has no parcels left. Moreover, also in the interest of tractability, we assume that a focal individual separating from its partner with parcels remaining, if successful in finding a new partner, finds one with the same number of parcels remaining as the focal has. Other justifications for this assumption include (1) that size-assortative mating of the chalk bass suggests a tendency of mate choice to equalize the egg-providing potential of the partners, which could extend to these additional mating opportunities, and (2) that mates available after some rounds of parcel exchange during a mating interval may tend to be similarly depleted of eggs.
Our approach to solving for ESS allocations and parcel numbers (see the text) means that we must consider several possible “end games”, starting (1) when both the focal partner and the other partner both have one remaining parcel, (2) when the focal has one parcel and the other has two, or (3) when the focal has two parcels and the other has one. Recall that we are only concerned here with what can be gained in the male role, since all parcels produced by all individuals are assumed to be released and fertilized. We let γ xw be the expected net fitness gain to the focal partner (mutant or wild type) over the end game, where x here indicates the number of parcels remaining to the focal partner and w indicates the number of parcels remaining to the other partner, which must be wild type. Let α i represent the chance of finding the ith mate and β i be the cost of seeking the ith mate. Recall that g x is the expected gain by the focal from a single mating as male (see Eq. 3). Since the focal is female first in a mating sequence with probability 0.5 and male first with probability 0.5, we have that
where \( \gamma_{11}^{\prime } \) is the same as γ 11 except that it applies to a second or subsequent partner.
Equation 12 indicates that when female first (first additive term) the focal individual gains another mating if there is no breakup following the fertilization of the focal’s last parcel. When male first (second additive term), the focal gains the benefit of fertilizing the partner’s last parcel and may then leave, seeking and possibly finding a second mate (which we assume also has one parcel). Finding the second mate would then repeat the same scenario, except that the cost of seeking and the chance of finding a third partner following a second breakup (α 3 and β 3) may differ from α 2 and β 2 in Eq. 12. To preserve tractability, we assume that for any j > 3, α j = α 3 and β j = β 3. This means that
which can be substituted into Eq. 12 to solve for γ 11. By equivalent logic, we find
Here, the focal has no opportunity to seek an additional mate, regardless of whether it is first “male” or first “female”, because it will be the first to expend all of its parcels.
To obtain γ 21, we must allow for the possibility that the focal is in the male role first and breaks up with its partner when the partner provides its last parcel, leaving the focal with two parcels to trade if it finds another mate. Under the assumption that the new mate has the same number of parcels (i.e., 2) as the focal, we now need to know γ 22. But here each partner provides a parcel before either partner can expend its last, which, for the focal partner, means that
and therefore
The final variable μ x in text Eq. 1 is defined in text Eq. 8, except for the expected mating frequency per daily mating interval, q x . We address q x here because of the link to the break-up of mating pairs.
We can express q x as
where the first additive term indicates the number of matings in the female role, the second term is the expected number of matings in the paired-male role, the third term is the expected number of streaker matings, and the final term is the expected number of matings with unpaired females. Here θ ij is the expected number of “end game” matings in the paired male role:
where i is the number of parcels retained by the focal individual immediately before a mating in which one of the partners could expend its last parcel, and j is the number of parcels retained by the other partner as this end game begins. These equations are derived by the same logic used to determine G x and I x .
Appendix 2: Parameter values as functions of density and predation risk
Here we formulate the dependence of the model’s parameter values on density and predation risk using simple mathematical functions. Data generally consistent with the relationships described here are already published (Hart et al. 2010), as noted below, or are to be presented elsewhere (M.K. Hart, in preparation). To avoid confusion, we describe how parameter magnitudes would respond to increases in the magnitude of each variable; decreases in the variables would yield opposite responses. We present and justify each functional relationship between variable and parameter for each ecological variable. In all cases we chose the simplest linear or hyperbolic functions with fewest parameters that would capture the basic expected trends, and we then adjusted the descriptive parameter magnitudes so that the function would pass through the default parameter magnitude at the default value of the variable. (Using other two-parameter functions with the same general shape, such as exponential functions, with the same linear-to-asymptotic shape as the hyperbolic functions, would not qualitatively influence the results.) Since the model itself is expressed in terms of the parameters, the relationships described here can be used numerically to link the ecological variables density and predation risk to the key response variables \( \hat{a} \), \( \hat{p} \), and \( \hat{s} \). Solutions and graphical output were obtained using computer programs written in MATLAB® (The MathWorks, Inc. 2009).
For the two primary causal factors of interest here, density and predation risk, we first consider population density S, where S = 3.93 fish m−2 at default parameter magnitudes, an intermediate density in nature [e.g., chalk bass densities on nine Panamanian reefs ranged from 2.33 to 6.20 fish m−2, with a mean of 3.93 fish m−2—Hart et al. (2010)]. Increasing population density is assumed to reduce body size and maximal clutch fitness B, perhaps through reduced growth rates resulting from local exploitation of planktonic prey, suggesting a function of the form B = −xS + y. We obtained the parameters from a regression of body weight on density (Hart et al. 2010, based on transects from reefs in Panama), dividing both sides by the body weight at the default density 3.93. This yields \( B = - 0.043S + 1.169 \), where B = 1 at the default density. This is the basic approach used to specify all of the relationships between model parameters and primary causal factors specified below. It is important to emphasize that these functions (e.g., B(S)) and descriptive parameters (e.g., x and y) were chosen a priori—independent of the model’s output.
Based on access to mating pairs, we expect streaking rate to increase with density, and data are consistent with σ = 0.397S (Hart et al. 2010). The chance of finding a partner (α i ) and the break-up probability (b) may increase with density via greater partner availability at high densities but must approach an upper limit asymptotically [b = S/(3.93 + S); α 1 = S/(0.437 + S); α 2 = α 3 = 0.555α 1]. And the cost of seeking a partner should decrease as density increases and more potential mates become available, but with a positive lower limit, since there should always be some cost of searching for a partner [β 1 = 0.03 + 0.0459/(0.655 + S); β 1 = β 2 = β 3]. Other parameters—namely relative fertilization efficiency of streakers ε, the background mortality rate μ 0, the cost of a unit of female allocation c, and the cost of each mating event k—we considered to be independent of population density and held constant (see Table 1). For simplicity, we also take S and R to be independent of each other.
Next we address predation risk R, which we express as a per-capita mortality rate; R = 0.006 day−1 at default parameter magnitudes. Increasing risk should dissuade streakers, with σ declining asymptotically to zero [σ = 0.0187/(0.006 + R)]. Both the chance of break-up b and of finding a first partner α 1 should decline with increasing risk, approaching zero asymptotically [b = 0.009/(0.012 + R); α 1 = 0.054/(0.054 + R), and α 2 = α 3 = 0.555α 1]; both are expressions of the increased difficulty of finding partners when potential partners are attempting to remain inconspicuous to predators. The incremental mortality cost of each individual mating and unit of sex allocation should be approximately proportional to predation risk; we set k = 0.0167R and c = 0.167R. Other parameters (B, μ 0, β 1, β 2, β 3, and ε) were assumed to be independent of R and kept at default values. (Note: d = 0 regardless of S and R, under the assumption of negligible male allocation cost.)
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Hart, M.K., Shenoy, K. & Crowley, P.H. Sexual conflicts along gradients of density and predation risk: insights from an egg-trading fish. Evol Ecol 25, 1081–1105 (2011). https://doi.org/10.1007/s10682-010-9459-1
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DOI: https://doi.org/10.1007/s10682-010-9459-1