PATRICIA ADAIR GOWATY and STEPHEN P. HUBBELL

The switch point theorem (SPT) is the quantitative statement of the hypothesis that stochastic effects on survival, mate encounter, and latency affect individuals’ time available for mating, the mean and variance in fitness, and thus, originally favored the evolution of individuals able to make adaptively flexible reproductive decisions. The SPT says that demographic stochasticity acting through variation in (*i*) individual survival probability, *s*; (*ii*) individual encounter probability, *e*; (*iii*) latency, *l*; (*iv*) the number of potential mates in the population, *n*; and (*v*) the distribution of fitness conferred, the *w* distribution, together affect average lifetime fitness, and induce adaptive switches in individual reproductive decisions. The switch point is the rank of potential mates at which focal individuals switch from accepting to rejecting potential mates, a decision rule that the SPT proves maximizes the average lifetime fitness of a focal individual under given values of ecological constraints on time. The SPT makes many predictions, including that the shape of the distribution of fitness conferred affects individual switch points. All else equal, higher probabilities of individual survival and encounter decrease the fraction of acceptable potential mates, such that focal individuals achieve higher average lifetime fitness by rejecting more potential mates. The primary prediction of the SPT is that each decision a focal

Department of Ecology and Evolutionary Biology and Institute of the Environment, University of California, Los Angeles, CA 90095 and Smithsonian Tropical Research Institute, APO, AA 34002.

Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 213

11
Reproductive Decisions
Under Ecological Constraints:
It’s About Time
PATriCiA ADAir GoWATy and sTePhen P. hUBBell
The switch point theorem (SPT) is the quantitative statement of
the hypothesis that stochastic effects on survival, mate encoun-
ter, and latency affect individuals’ time available for mating, the
mean and variance in fitness, and thus, originally favored the
evolution of individuals able to make adaptively flexible repro-
ductive decisions. The SPT says that demographic stochasticity
acting through variation in (i) individual survival probability, s; (ii)
individual encounter probability, e; (iii) latency, l; (iv) the number
of potential mates in the population, n; and (v) the distribution of
fitness conferred, the w distribution, together affect average life-
time fitness, and induce adaptive switches in individual reproduc-
tive decisions. The switch point is the rank of potential mates at
which focal individuals switch from accepting to rejecting potential
mates, a decision rule that the SPT proves maximizes the aver-
age lifetime fitness of a focal individual under given values of
ecological constraints on time. The SPT makes many predictions,
including that the shape of the distribution of fitness conferred
affects individual switch points. All else equal, higher probabili-
ties of individual survival and encounter decrease the fraction of
acceptable potential mates, such that focal individuals achieve
higher average lifetime fitness by rejecting more potential mates.
The primary prediction of the SPT is that each decision a focal
Department of ecology and evolutionary Biology and institute of the environment, Uni-
versity of California, los Angeles, CA 90095 and smithsonian Tropical research institute,
APo, AA 34002.

OCR for page 213

4 / Patricia Adair Gowaty and Stephen P. Hubbell
individual makes is determined jointly by e, s, l, n, and the w
distribution.
W
hen Darwin’s critics said that natural selection (Darwin, 1859)
could not explain the evolution of traits such as the outrageous
tails of peacocks that reduce their bearers’ survival probabili-
ties, he countered with sexual selection (Darwin, 1871). he argued that
costly traits would evolve if they also increased males’ abilities to attract
females or to win behavioral contests over access to females. in “Principles
of sexual selection,” the first chapter of Part ii of The Descent of Man, and
Selection in Relation to Sex (Darwin, 1871), Darwin defined sexual selection
as that type of selection that “depends on the advantage which certain
individuals have over other individuals of the same sex and species, in
exclusive relation to reproduction” (1871, p. 256). Darwin distinguished
sexual selection from natural selection as selection that arises from some
individuals having a reproductive advantage over other same-sex, conspe-
cific individuals, not from different “habits of life,” but from reproductive
competition with rivals. Darwin’s discussion focused overwhelmingly
on traits in males that could be explained by 2 mechanisms of sexual
selection—male-male competitive interactions and female choice—each
of which could result in variation among males in fitness and thereby
favor traits that helped males win fights and attract females (see Jones
and ratterman, Chapter 9, this volume). Most modern discussions of
typical sex roles begin with Darwin’s 1871 volume, and statements about
choosy females and profligate, competitive males. however, Darwin prob-
ably suspected that male choice was common; he argued in Part i of the
1871 book that men’s choice of mates was seemingly more common than
women’s at least in “civilized societies.” he even argued that the beauty
of women was due to male choice. in Part ii he also described cases of
male domestic and companion animals refusing to copulate with some
females. he was aware too of gaudy, pugnacious, and competitive females
in some bird species. Controversy over whether females had the esthetic
capability for discrimination dogged Darwin and his followers into the
20th century.
After most people finally agreed that females had the sensibilities to
choose, focus narrowed so that modern students of sexual selection sim-
ply assumed that males were competitive and indiscriminate and females
“coy,” passive, and discriminating. For example, when Bateman (1948)
studied sex differences in fitness variances in Drosophila melanogaster, he
attributed the larger variances of males to their “undiscriminating eager-
ness” and the “discriminating passivity” of the females (p. 367), even
though he did not watch behavior (Dewsbury, 2005). Bateman’s study

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
also led many to infer that female multiple mating was unlikely to be very
common as it was unlikely to enhance female fitness.
resistance to such “narrow-sense sexual selection” was afoot in Dar-
win’s century [see citations in Gowaty (2007)], and accelerated with the
flush of empirical papers that followed Parker et al. (1972) and Trivers
(1972). Parker argued that the sexes are what they are because of the size
of the gametes they carry: females having large, relatively immobile,
resource-accruing gametes and males having smaller, mobile gametes that
competed for access to the larger ones. Trivers argued, echoing Williams
(1966), that females were usually the choosy sex because in most species
females bore the greater cost of reproduction. Challenges to the generaliza-
tions of parental investment theory included hrdy’s (1981) book about the
near ubiquity of competitiveness of primate females and their anything
but “coy” and “passive” sexual behavior; discovery that females fight
females to defend “genetic maternity” in birds (Gowaty, 1981; Gowaty
and Wagner, 1987), documentation of multiple mating by wild-living
female Drosophila pseudoobscura (Anderson, 1974), Sialia sialis and other
bird species (Gowaty and Karlin, 1984; Gowaty, 1985); and the first study
of male mate choice in beetles with typical female-biased parental invest-
ment (Johnson and hubbell, 1984). importantly, sutherland (1985a,b, 1987)
showed theoretically that the sex differences in fitness variances could
arise in the absence of mate choice and intramale competition and could
be due entirely to chance. Building on sutherland’s insights, hubbell and
Johnson (1987) showed that the variation in lifetime mating success results
from chance and selection, so that measures of selection should rely only
on the residual variance that cannot be ascribed to known and quantifi-
able nongenetic life history variation. hrdy and Williams (1983) and hrdy
(1986) offered an explanation for why, in the face of so much evidence,
so many biologists seem invested in the “myth of the coy female.” in
response to the failures of the simpler versions of parental investment
theory, new theory to explain reproductive decisions appeared (hubbell
and Johnson, 1987; Crowley et al., 1991; Clutton-Brock and Parker, 1992)
predicting that variation in encounters, latencies, survival, and their more
complex proxies (relative reproductive rate, the operational sex ratio, and
density) favored shifts in mean behavior of the sexes, and as a result more
nuanced reports of ecologically induced variation in sex-typical behavior
appeared [e.g., Magnhagen (1991), Forsgren (1992), shelly and Bailey
(1992), Berglund and rosenqvist (1993), hedrick and Dill (1993), Berglund
(1994, 1995), Poulin (1994), simmons (1995), Dill et al. (1999), Grand and
Dill (1999), itzkowitz and haley (1999), Gowaty et al. (2002), Jiggins (2002),
Drickamer et al. (2003)]. Currently, few investigators think that sex role
behavior is entirely fixed for either sex, particularly in females in species
with female-biased parental investment in which many observations of

OCR for page 213

/ Patricia Adair Gowaty and Stephen P. Hubbell
changes in mating behavior exist (Magnhagen, 1991; Gong, 1997; Grand
and Dill, 1999). nevertheless, we know little about how male mate choice
behavior varies under ecological constraints, because few investigators
study male mate choice in species with female-biased parental investment.
The possibility of sex role flexibility has seldom been simultaneously
tested in both sexes (Gowaty et al., 2003b).
Most modern theories of sex roles begin with Trivers’s (1972), and
Parker et al.’s (1972) ideas about sex differences to predict further sex
differences. The derivative theories assume that the origin theories are an
accurate reflection of past selection, which is an intuitive place to begin
in refinement of theory—until one seriously considers the explicit chal-
lenges to the embedded assumptions about sex differences in fitness vari-
ances. Consider what hubbell and Johnson (1987) proved theoretically:
that demographic stochasticity gives rise to chance variances in lifetime
reproductive success. They proved that nonheritable environmental varia-
tion in an individual’s lifetime number of mates could have favored the
evolution of mate assessment in the first place. They showed that vari-
ances in number of mates similar or identical to those usually attributed
to sexual selection could arise without mate choice and other sources of
intrasexual competition. in other words, the arrow of causation linking
classic mechanisms of sexual selection to fitness variances can go in either
direction (Gowaty and hubbell, 2005). This means that fitness variances
can arise from demographic stochasticity, acting through chance effects on
individual encounter probabilities with potential mates, individual sur-
vival probabilities, and latencies, which are variables that can then induce
individual behavior (Fig. 11.1). This conclusion is turned around from
the usual conclusion that mate choice and intrasexual competition cause
fitness variances. This observation was profound, just as sutherland’s
(1985a) earlier one was, because it showed that the usual linkage between
FiGUre 11.1 scheme showing a scenario of the evolution of switches in repro-
ductive decisions.
soUrCe: Modified from Gowaty and hubbell (2005).

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
mechanisms of sexual selection and means and variances in number of
mates is best interpreted as correlation rather than causation until one
partitions the deterministic and stochastic components of means and vari-
ances in number of mates.
Most recent sex roles research has focused on female mate choice for
fancy male traits, much of which was inspired by hamilton and Zuk’s
(1982) hypothesis that fancy male traits indicate good genes for offspring
viability, a compelling hypothesis given red Queen dynamics between
hosts and pathogens. later quantitative genetics theory suggested that
indirect fitness effects are less likely than direct fitness effects to favor the
evolution of such traits (Kirkpatrick, 1985; Wolf and Wade, 2001). Addi-
tionally, until relatively recently, few data existed in support of the good
genes hypothesis of mate choice. however, female and male choice stud-
ies in flies (Anderson et al., 2007), cockroaches (Moore et al., 2001, 2003),
ducks (Bluhm and Gowaty, 2004a,b), and mice (Drickamer et al., 2000,
2003; Gowaty et al., 2003a) have demonstrated that offspring viability
was significantly higher when choosers were mated with discriminatees
they preferred, as was productivity (the number of offspring surviving to
reproductive age). in contrast, fecundity (the numbers of eggs laid or off-
spring born) was almost always lower and sometimes significantly lower
when choosers were experimentally paired with partners they preferred
compared with partners they did not prefer (Gowaty et al., 2007). These
findings are consistent with the hypothesis of reproductive compensation
(Gowaty, 2008) and inconsistent with theories predicting that mate choice
favors enhanced fecundity.
Why were these studies (Drickamer et al., 2000, 2003; Moore et al.,
2001, 2003; Bluhm and Gowaty, 2004a,b; Anderson et al., 2007; Gowaty et
al., 2007) successful in showing the predicted associations between mate
choice and offspring viability, when others were not? There are at least 3
reasons. First, the studies had controls that eliminated the effects of intra-
sexual behavioral contests and intersexual coercion that often confound
mate choice studies (Kingett et al., 1981). second, these studies were not
designed to understand the evolution of traits mediating preferences,
but to test the effects of mate choice independent of their effect on the
evolution of discriminatee traits. The investigators picked choosers and
discriminatees at random with respect to their phenotypes, so these stud-
ies are silent about the traits in the discriminatees that mediated choos-
ers’ preferences. Questions about the evolution of fancy traits are really 2
separate questions: one about the fitness payouts of preferences, and the
other about the evolution of the traits that mediate the preferences. Mul-
tiple traits may mediate preferences (Candolin, 2003). however, few have
evaluated the hypothesis that fancy traits may exploit preexisting sensory
biases that could manipulate choosers in ways that could decrease rather

OCR for page 213

/ Patricia Adair Gowaty and Stephen P. Hubbell
than increase their fitness. Third, the investigators (Gowaty et al., 2007)
were motivated to study the effects of constraints on the free expression
of mate preferences, so their methodologies were designed to get unam-
biguous pretouching behavioral indicators that choosers preferred one
discriminatee more than the other. once the investigators knew whom
the choosers liked and did not like, they randomly assigned the choosers
to breed with one of the discriminatees. in that way, they captured the
effects of constraints on the fitness payouts for choosers in unconstrained
and constrained partnerships.
Why are these studies (Drickamer et al., 2000, 2003; Moore et al., 2001,
2003; Gowaty et al., 2003a, 2007; Bluhm and Gowaty, 2004a,b; Anderson et
al., 2007) of interest in an article on sex roles? (i) They provide powerful
empirical counterpoint to theory that argues that indirect fitness effects
are unlikely to accrue from mate preferences. (ii) They demonstrate trade-
offs in components of fitness for choosers breeding under constraints, thus
suggesting that to understand selection from mate choice, investigators
may profit from knowing about as many components of fitness as possible,
including fecundity, productivity, and offspring viability, when individu-
als breed under constraints. (iii) They show that individuals—both males
and females—can and do make pretouching assessments of likely fitness
payouts before mating. They made clear that males, not just females,
are able to modify their behavior and physiology when breeding under
constraints (Gowaty et al., 2007), observations inconsistent with typi-
cal ideas about sex role variation in species with female-biased parental
investment.
What is needed now is a theory that will allow investigators to parse
differential effects on fitness and behavior, and the direction of their
effects on each other (behavior to fitness/fitness to behavior) on both
females and males. such a theory will allow investigators to attribute
reproductive decisions and behavior to 3 causal factors: (i) chance varia-
tion in ecological contingencies that may induce flexible and adaptive
individual reproductive decisions; (ii) competitive forces including natural
and sexual selection; and (iii) fixed sex differences. The model we present
facilitates these goals.
To describe the model, which we call the switch point theorem (sPT),
we (i) discuss individual reproductive time budgets, which encapsulate
some of the most important ecological constraints on reproduction. (ii)
We introduce the concept of the fitness conferred by alternative potential
mates. (iii) We verbally describe the steps in model construction (the
mathematical description is in the Appendix to this chapter). (iv) We show
some of the results of the model. in the discussion, we (v) list what the sPT
does and does not do, and (vi) describe several possible empirical tests
and applications of the sPT.

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
CONSTRAINTS ON INDIVIDUAL
REPRODUCTIVE TIME BUDGETS
For all mortal individuals, their time is finite. Time available for mat-
ing affects means and variances in number of mates (Fig. 11.1) (Gowaty
and hubbell, 2005), and chance effects on life history that affect time
available for mating can have strong effects on fitness means and vari-
ances (sutherland, 1985a; hubbell and Johnson, 1987). The simplest set of
parameters to affect lifetime variance in numbers of mates is based on sto-
chastic effects on an individual’s survival and the individual’s encounters
with potential mates, and, if the individual is a nonvirgin (i.e., a remating
individual), its latency from one copulation to receptivity for the next
(sutherland, 1985a; hubbell and Johnson, 1987; Gowaty and hubbell,
2005). indeed, in order for a receptive individual to mate, it must encoun -
ter a potentially mating opposite-sex individual; thus an individual’s
encounter probability (e) with potential mates affects how much time the
focal individual spends searching for mates. likewise, for already-mated
individuals in iteroparous species, the time they spend “handling” the
reproductive consequences of mating, during which they are in latency (l)
and unavailable for further mating, affects the time remaining for future
matings and the opportunity cost of the past mating (sutherland, 1985a,b,
1987; Gowaty and hubbell, 2005). importantly, individuals vary in repro-
ductive life span, a function of their survival probability (s). All else equal,
when search time and latencies are short, and life span is long, individuals
have more time for reproduction than when search time and latencies are
long, and life span is short. individuals with short search times and long
lives have more opportunities to mate than individuals whose search
time is long and whose probability of future life is short. intuitively (Fig.
11.2), it is easy to see that when opportunities vary, the costs and benefits
of accepting or rejecting potential mates also vary. This means that for
individuals with many opportunities, the costs of rejecting more potential
mates are smaller, whereas, when focal individuals have fewer opportuni-
ties, the costs of rejecting potential mates are greater. however, relative
opportunities predict only means and variances in number of mates; to
predict adaptive, flexible reproductive decisions of an individual, one
must also know the fitness that would be conferred if a focal individual
were to mate with any given alternative potential mate.
FITNESS DISTRIBUTIONS
Just as most models of mate preferences do (Andersson, 1994), the
sPT assumes that individuals assess the fitness that would be conferred
(w) by potential mates before making reproductive decisions. We further
assume that individuals obtain information during their development of

OCR for page 213

0 / Patricia Adair Gowaty and Stephen P. Hubbell
FiGUre 11.2 Four stick models of idealized reproductive careers of 4 individuals.
on each stick, the gray bars represent the stage, search. The ovals attached to black
bars represent the time used when an individual encounters and accepts for mat-
ing a potential mate, and enters a latency, a period during which the individual
is unreceptive to further mating. The gray vertical ovals attached to gray bars
represent encountering and rejecting a potential mate, after which the individual
reenters search. some sticks are longer than others, indicating that some individu-
als die before others, when they enter the absorbing state, death.
zpq9990980790002.g.tif
20the distribution of fitness that would be conferred (the w distribution) by
p6
potential mates in their population. To conceptualize the meaning of w
distributions, imagine an experiment with no carryover effects in which
every female in a population is mated with every male and every male
with every female. Fill in every cell of a matrix with the fitness that would
result from each pairwise mating. This matrix specifies what we mean by
“fitness conferred by alternative potential mates.” We assume that focal
individuals have information about the distribution, which is a key com-
ponent of expected mean reproductive success. The fitness components
one might consider empirically or theoretically include: fecundity (the
number of eggs laid or offspring born), productivity (the number of their
offspring that survive to reproductive age), or offspring viability (the
proportion of eggs laid or offspring born that survive to reproductive
age). on completion, one would have a matrix of values representing
“fitness conferred by alternative potential mates, w.” one can imagine
a variety of scenarios for how fitnesses are distributed in such a fitness
matrix. For example, one might observe a situation in which w is as an
absolute, meaning that all focal individuals rank a potential mate the same
way, as predicted by many theories of sexual selection (Andersson, 1994).
Alternatively, w might often differ from one focal individual to another,
as we think likely, and be an interaction effect and self-referential, mean-
ing that each focal individual will not rank specific potential mates in the

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
same way. such self-referential mate preferences occur in mice and other
organisms (ryan and Altmann, 2001; ryan and lacy, 2003). The entries
in the fitness matrix have some statistical distribution, which we call the
w distribution.
in the sPT, we assume the w distributions are beta distributions.
The beta distribution is convenient because a beta random variate takes
values from 0 to 1, as fitness does, and a beta distribution can assume a
very large diversity of shapes depending on its 2 parameters, nu ( ν) and
omega (ω), from flat to strongly unimodal, left or right skewed, and even
bimodal. We illustrate several possible fitness distributions in the analyses
including beta (1, 1), which gives a uniform probability density from 0 to
1; beta (8, 3), which is skewed to high values; beta (3, 8), which is skewed
to low values; and beta (5, 5), which has a strong central tendency. note
that although we use a beta probability density function in the sPT, there
is no necessity for the w distribution to be beta for the sPT to be valid.
THE SWITCH POINT THEOREM
The sPT arises from a Markov chain state transition model. The sPT is
a generalization of an earlier model (hubbell and Johnson, 1987; Gowaty
and hubbell, 2005) in which focal individuals made mating decisions
when potential mates occurred in only 2 qualities. in the current model,
focal individuals make mating decisions when potential mates may occur
in up to n qualities, where n is the number of potential mates in the popu-
lation, so that there is a state for mating each potential mate. in Markov
state transition models, individuals move from one state to another with
some probability. The model is an absorbing Markov chain, so individuals
continue to move through states until death (a terminal state individuals
cannot leave). The solution to the model is a theorem that specifies the
mean and the variance of the number of times the individual enters each
state. The solution to the sPT (below) is f *, the number of potential mates
a focal individual finds acceptable that maximizes relative lifetime fit-
ness. The sPT says that focal individuals should accept all potential mates
whose rank (with 1 being the highest fitness rank) is less than or equal to
the rank at which average lifetime fitness would be maximized (f *); and
reject all whose fitness rank is above f *.
The sPT assumes that individuals encounter potential mates at ran-
dom with respect to their rank, and shows that focal individuals who
follow the mating decision rule to accept any potential mate i for which
wi > wf *, and reject any potential mate j for which wj < wf *, will maximize
their average lifetime reproductive fitness. note that any given focal
individual in this stochastic ensemble may not actually mate with all
individuals having wi > wf *, in their lifetime. however, the rule that maxi-

OCR for page 213

/ Patricia Adair Gowaty and Stephen P. Hubbell
mizes lifetime reproductive success is to find acceptable any individual
encountered whose w is greater than wf *, the fitness of the potential mate
at rank f * (Fig. 11.3).
DERIVATION OF THE SPT
The sPT proof is provided in the Appendix. here, we describe the
steps that allowed us to analytically solve for the number of potential
mates that a focal individual should find acceptable to maximize the
focal’s average lifetime fitness.
in step 1, we constructed a series of absorbing Markov chain mod -
els for each decision rule associated with n, under specified ecological
constraints affecting values of e, s, l, and the w distribution. if n = 3, for
example, there are 3 decision rules, and 3 matrices are required to deduce
the rule that would maximize average lifetime fitness: (i) accept the poten-
tial mate ranked 1 and reject potential mates ranked 2 and 3; (ii) accept
potential mates ranked 1 and 2 and reject the potential mate ranked 3; and
(iii) accept all 3 potential mates. if n = 100, there are 100 decision rules
and, without the sPT proof by induction, 100 matrices would have been
required to deduce the rule that would maximize the focal individual’s
FiGUre 11.3 An example of a switch point graph.
zpq9990980790003.g.tif

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
average lifetime fitness under specified values of e, s, l, n, and w distribu-
tion. The 100 rules take the following form: accept up to rank f * (where f
equals the highest acceptable rank), reject all from rank n − f *. The prob-
abilities associated with e, s, and l determine the mean number of times
the focal will pass through each cell in each matrix. it is possible to solve
this algebraically without specifying numerical values of e, s, l, or n.
in step 2, we computed for each decision rule the mean number of
times that an individual passed through the mating state with each accept-
able potential mate and multiplied this mean by the fitness, w, that would
be conferred if the focal actually mated with the acceptable mate. The w
for each potential mate comes from the specified beta distribution.
in step 3, we summed up these products from each decision rule for a
focal with specified e, s, l, n, and w distribution to find f *, the decision rule
that would maximize average lifetime fitness, if the focal actually mated
with each acceptable mate.
Because of social and ecological constraints (Gowaty et al., 2007), it is
unlikely that focal individuals actually mate with all potential mates who
are acceptable to them. Therefore, we characterize the rule as the “switch-
point” at which focal individuals switch from accepting potential mates to
rejecting them. That is, it is important to keep in mind that the sPT does
not state that individuals actually mate up to f *, only that they will accept
any potential mate they encounter whose rank is 1 to f *. Thus, the sPT is
only a decision rule, not a statement of how many mates a focal individual
will have. The sPT specifies whether to accept or reject a potential mate
under chance effects from demographic and environmental stochasticity.
The information embodied in the sPT is about future opportunities, and,
if one actually mates with a potential mate, the opportunity costs associ-
ated with having mated with a particular potential mate. We hypothesize
that individuals use this information to adjust their reproductive decisions
as their ecological circumstances change. The “two body” problem that
occurs when 2 individuals meet and one accepts but the other rejects can-
not be studied in the current Markov chain model.
The analytical solution for a switch point set at f, for which the deri-
vation is the si, is:
)
(
es ∑ i = 1 wi / n
f
2
{∑ }
M SA = (1)
W
( 1 − s ) + es[1 − l+1
− g / n] '
f
fs
where g = n − f.
The solution is the decision rule f * that maximizes eq. 1. note that e,
s, l, n, and the w distribution could be functions of 1 or more of the other
parameters. There is nothing in the theorem that precludes such functional
interactions between the parameters. We have presented the version with-
out interaction for heuristic simplicity.

OCR for page 213

/ Patricia Adair Gowaty and Stephen P. Hubbell
rejecting potential mates—could simultaneously reject one hypothesis and
provide support for the alternative.
The SPT is an Alternative Hypothesis to Parental Investment Theory
For species in which parental investment is biased toward one sex,
investigators could compete the predictions of the sPT with parental
investment theory. Controlling for s, e, n, l, and w distribution for experi-
mental subjects, the predicted behavior of individuals of different sexes
would be the same under the sPT. Parental investment theory, by contrast,
predicts that in a species with female-biased parental investment, females
would reject more and males would accept more potential mates, whereas,
in a species with male-biased parental investment, females would accept
more and males reject more potential mates. Another valuable test would
be of virgins of both sexes, for whom l = 0, in species with female-biased
parental investment and in species with male-biased parental investment.
As with anisogamy theory, these alternative predictions of the sPT and
parental investment theory could be tested with a crucial experiment.
Almost Nothing Is Known Empirically About w Distributions
The w distribution has only been characterized for a few populations
(unpublished data), and no one to our knowledge has tested the effects
of w distributions on individual reproductive decisions. For laboratory
populations of flies and other organisms with short generation times and
no sex biases in dispersal, it is relatively easy to estimate the shape of the
w distribution, measuring fecundity, productivity, and offspring viability
from a sample of random pairs breeding under enforced monogamy. An
experiment that we plan to do will begin with flies cultured under inbreed-
ing and outbreeding, which may produce w distributions with different
shapes, and then to test the predictions (Fig. 11.7) for virgins (l = 0) when
e, s, and n are held constant, using pretouching arenas.
Implications for Experimental Studies of Mate Preferences
for Fancy Male Traits
The sPT is not a hypothesis for the evolution of fancy male traits, nor
does it predict the evolution of traits mediating preferences. nevertheless,
the sPT suggests that selection should favor traits that increase a focal indi-
vidual’s encounters with potential mates. enhanced encounters increase
reproductive opportunities, thereby reducing the opportunity costs of
accepting potential mates who would confer low w. Traits, such as bizarre
or easily seen plumage, loud calls, songs, or pheromones that travel over

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
large distances may attract more potential mates to the focal individual.
in contrast to classic sexual selection ideas, however, the sPT predicts a
different payout, not necessarily more mates, but in more reproductive
opportunity. The sPT thus predicts that attractive focal individuals reject
more potential mates than other, less attractive same-sex conspecifics, all
else equal (i.e., if s, l, n, and w distribution are equal for attractive and
unattractive focals). This prediction has not been made by any hypothesis
of classic sexual selection.
The sPT suggests that many previous failures to associate fitness
rewards for female mate choice for fancy traits in males might be explained
by experimentally uncontrolled variation in e, s, l, n, or w distribution. The
sPT predicts that a focal individual exposed only to a small number of
potential mates would fail to show a preference. it is also possible that
investigators have exposed females to males of nearly equivalent fitness.
if this were the case, adaptively flexible females would more often accept
potential mates, because the differences in w between potential mates may
be very small, thus reducing the opportunity costs from any particular
mating. Thus, the sPT may inform previous ambiguities in mate prefer-
ence studies.
The Ecology of Sex Roles
Thinking about time constraints on reproductive decisions suggests
a new, powerful framework for sex differences research. if individuals of
different sexes have different feeding niches and/or different exposure
to pathogens or predators, there are likely to be associated differences in
e, s, and l experienced by individuals of different sexes and, as the sPT
predicts, differences in the typical reproductive decisions of individuals of
different sexes. or, if species in which one sex disperses, but the other does
not, individuals of different sexes may see different w distributions and
therefore express differences in the fraction of potential mates acceptable.
Thus, it is possible that sex differences in typical reproductive decisions
may have more to do with what Darwin called “habits of life,” rather than
with fixed sex differences. Data on how different habits of life affect e, s, l,
n, and w distribution of individuals are not yet systematically collected to
our knowledge, and may prove interesting indeed, perhaps providing a
path toward resolution of the some of the controversies that have dogged
Darwin and his followers up to the current day.
ACKNOWLEDGMENTS
We thank John Avise and Francisco Ayala for inviting P.A.G. to con-
tribute to the Arthur M. sackler Colloquium of “in the light of evolution

OCR for page 213

4 / Patricia Adair Gowaty and Stephen P. Hubbell
iii: Two Centuries of Darwin” and J. P. Drury, Brant Faircloth, Graham
Pyke, steve shuster, and elliott sober for comments on a previous ver-
sion of the manuscript. This work was supported by national science
Foundation grant “Beyond Bateman,” which has allowed us and our col-
laborators, W.W. Anderson and y.K. Kim, to study variances in number of
mates and reproductive success in 3 species of flies with different gamete
size asymmetries.
APPENDIX
in this Appendix, we derive eq. 1 for the switch point theorem given
in the paper, and we also give the equations used in the sensitivity analy-
sis of lifetime fitness to changes in each of the model parameters. We also
present some graphical results.
Derivation of the Switch Point Theorem
The switch point theorem is an analytical solution to the question
of how many potential mates a focal individual should accept or reject
to maximize lifetime fitness. Consider a focal individual with a finite
reproductive life span, during which the individual searches for potential
mates, accepting some and rejecting others, and having to take time out
from searching and mating every time it mates and reproduces, a latency
period from the onset of one copulation to receptivity to remating. We can
represent movement of the individual through its reproductive lifetime
as a series of states, each lasting 1 time unit, such as searching but no
potential mate is encountered in a given time step, searching and potential
mate i is encountered, mating with potential mate i, and in time period j
of a latency period of length l.
let e be the probability that a potential mate is encountered per unit
time. let there be n potential mates, and let pi be the probability that the
encountered potential mate is individual i, where ∑ pi = 1. The relative
fitness conferred on the focal individual by mating i with potential mate
i is a beta-distributed random variable, b(w,n) where w and n determine
the shape of the fitness distribution. let s be the probability of survival of
the focal individual over 1 unit of time. The reproductive life span of the
focal individual is the time from the onset of reproductive maturity until
the individual’s death. in terms of the probability of survival per unit time,
the expected reproductive life span of the focal individual is 1/(1 – s). note
that small changes in the probability of survival when s is near unity can
cause large changes in life span. Thus, for example, survival probabilities
of 0.9, 0.99, and 0.999 correspond to mean life span of 10, 100, and 1000
time units, respectively. in a Markov chain there are discrete time steps.
When s = 0, the individual lives one time step and dies. When s = 1, the

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
ratio of 1/1 – s is infinite and the reproductive life span is infinite, the
individual lives forever. Thus, s = 1 is not a biologically meaningful value.
values of s > 0 and s < 1 are the biologically meaningful values and these
are the ones we use in our sensitivity analyses.
The relative fitness of potential mates influences whether the focal
individual should or should not mate a particular potential mate. To illus-
trate the Markovian process, we analyze a case of 3 potential mates ( n = 3).
in this simple case, we can distinguish 3 possible decisions. A focal indi -
vidual could accept all 3 potential mates as encountered. or an individual
might accept 2 and reject 1 of the 3 potential mates. or a focal individual
might accept only 1 of 3, rejecting the other 2 potential mates.
Consider the “accept all 3” case first. The state of the focal individual
can be receptive and searching for a mate, but it does not encounter a
potential mate in the current time step. label this state sA. or the focal
individual may encounter a potential mate in the current time step, in
which case there are 3 possible states: encountering potential mate number
1 (state S1), encountering potential mate number 2 (state S2), or encounter-
ing potential mate number 3 (state S3). Because the focal individual accepts
all 3, it will then mate with the potential mates as encountered (i.e., enter
mating states M1, M2, and M3, respectively). let l be the length of time that
the focal individual is in latency before returning to the pool of receptive
individuals. Measure the duration of the latency period in the same time
units used for measuring s and e. in this example, let the latency l = 1. let
the state of latency be labeled L1, and the absorbing state of death D. The
matrix of transition probabilities for the case “accept all 3” is thus:
To: D M1 M2 M3 S1 S2 S3 SA L1
From:
1 000 0 0 0 0 0
D
1s000 0 0 0 0 s
M1
1s000 0 0 0 0 s
M2
1s000 0 0 0 0 s
M3
H= 1ss00 0 0 0 0 0 [1]
S1
1s0s0 0 0 0 0 0
S2
1s00s 0 0 0 0 0
S3
1 es 0
1 s 0 0 0 esp1 esp2 esp3
SA
1 s 0 0 0 esp1 1 es 0
esp2 esp3
L1
We list “to” and “from” states along the top and left side of matrix
H, respectively. The column state (“to”) is the next state reached after
the states (“from”) listed on the rows. note that the focal individual has
a probability 1 – s of dying during every time step, and moving to state
D (death). if death occurs, this is the terminating point of the absorbing

OCR for page 213

/ Patricia Adair Gowaty and Stephen P. Hubbell
Markov chain. in the “accept all 3” case, whenever the focal individual
encounters a potential mate, it mates, with a probability simply equal to
the survival probability (because the only behavior the individual exhibits
next is to mate) (e. g., Pr{M1|S1} = Pr{M2|S2} = Pr{M3|S3} = s). During
search, the focal individual encounters a potential mate of quality i with
probability Pr{SAS|S1} = espi. since the encounter probability is e, the prob-
ability that the search fails in the current time step and the focal individual
has to continue searching in the next time step is: Pr{SA|SA} = (1 – e)s.
We can solve eq. (1) for the expected number of times the focal indi-
vidual passes through each state before death, by computing E = (I–H)–1.
Matrix E always exists and its element ei,j is the expected number of times
over its lifetime that the focal individual is in column state j given that the
focal individual starts in row state i. By convention we assume a newly
mature focal individual begins the reproductive portion of its life in state
SA, searching for a mate. From eq. (1), the expected number of matings
of a focal individual with potential mate i is
es 2 pi
. [2]
E Mi SA
es 1 s 2 p1
1s p2 p3
Therefore, the total lifetime mating success of the focal individual over
all potential mates is
es 2 p p2 p31 es 2
. [3]
E M1 M2 M3 SA 2
1 s es 1 s 2
1s es 1 s p1 p2 p3
Generalizing this expression to a latency of l time units yields a total
lifetime mating success of
D M 1 M 2 M 3 S1 S2 S3 SA L1 L2
To:
From:
D1 000 0 0 0 0 00
M1 1 s s0
000 0 0 0 0
M2 1 s s0
000 0 0 0 0
M3 1 s s
0 0 0 0 0 0 0 0
S1 1 s s 0 0 0 0 0 0 0 0
H= [4]
S2 1 s s
0 0 0 0 0 0 0 0
S3 1 s s
0 0 0 0 0 0 0 0
SA 1 s 0 esp1 esp2 esp3 1 es 0 0
0 0
L1 1 s 0 0 0 0s
0 0 0 0
L2 1 s 0 0 0 esp1 esp2 esp3 1 es 0 0
es 2 p p2 p31 es 2
. [5]
E M1 M2 M3 SA 3
1 s es 1 s 3
1s es 1 s p1 p2 p3

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time /
The total lifetime mating success for a latency of two time units (l = 2)
is therefore
es 2 p p2 p31 es 2
es 2 p p2 p31 es 2 . [5]
EM M M SA
1 s es 1 s 3 . [5]
E M1 M2 M3 SA 1 s 3 p1 p2
1 s es p3
1 2 3
1 s 3 p1 p2 1 s es 1 s 3
1 s es p3
Generalizing this expression to a latency of l time units yields a total
lifetime mating success of
es 2 p p p es 2
es 2 p p2 p31 es 2 . [6]
E M1 M2 M3 SA 2 31
. [6]
E M1 M2 M3 SA es 1 s ll 1 p1 p2 1
es 1 s ll
1s 1s
p3
es 1 s 1 p1 p2 1
1s 1s es 1 s
p3
now, consider when a focal individual mates with 2 of the 3 potential
mates but not the third. in this case, state M3 no longer exists (although
potential mate 3 is still encountered, which is state S3), so the transition
matrix becomes:
To: D M 1 M2 S1 S2 S3 SA L1
From:
1
00 0 0 0 0 0
D
1s00 s
0 0 0 0
M1
1s00 s
M2 0 0 0 0
1ss0
S1 0 0 0 0 0
H= [7]
s 0s
S2 1 0 0 0 0 0
esp2 esp3 1 es 0
s 0 0 esp1
S3 1
s 0 0 esp1 esp2 esp3 1 es 0
SA 1
s 0 0 esp1 esp2 esp3 1 es 0
L1 1
note that when the focal individual encounters potential mate 1 (S1)
it moves to the state of mating with individual 1 (M1) and likewise when
the focal individual encounters potential mate 2 (S2), it moves to mating
that individual (M2). however, note that when potential mate 3 is encoun-
tered, the focal individual does not mate with it (there is no state M3), but
it resumes search, enters states S1, S2, S3, and SA, depending on what the
search results are.
For a focal individual with a latency of 1, this decision rule yields a
total lifetime mating success of:
es 2 p p2
[8]
E M1 M2 SA 2
1s es 1 s p1 p2 p3
es 2 p p21
. [9]
E M1 M2 SA l1
1s es 1 s p p p

OCR for page 213

es 2 p p2
[8]
E M1 M2 SA 2
1s es 1 s p1 p2 p3
/ Patricia Adair Gowaty and Stephen P. Hubbell
and for a focal individual with a latency of l time units:
es 2 p p21
. [9]
E M1 M2 SA l1
1s es 1 s p1 p2 p3
in the case in which the focal individual mates with only 1 of the 3
possible potential mates, 1 but not 2 or 3, the transition matrix is
D M1 S1 S2 S3 SA L1
To:
From
1
0 0 0 0 0 0
D
1s0 s
0 0 0 0
M1
1s s
S1 0 0 0 0 0
s esp1 esp2 esp3 e s
H = S2 1 0 1 0 [10]
s esp1 esp2 esp3 e s
S3 1 0 1 0
s esp1 esp2 esp3 e s
SA 1 0 1 0
s esp1 esp2 esp3 e s
L1 1 0 1 0
Focal individuals with a latency, l = 1, with this decision rule gain a
lifetime mating success of
es 2 p1
. [11]
E M1 SA es 2 p
es 1es 2s 2 p1
1s p2 p3 . [11]
E M1 SA 1
1
. [11]
E M1 SA es 1 s 2 p1
1s p2 p3
2
1s es 1 s p1 p2 p3
For the generalization of this decision rule to focal individuals with
latency of l time units, expected lifetime mating success becomes
es 2 p1
. [12]
E M1 SA es 2 p
11
es 1 ess l
2
1s pp p2 p3 . [12]
E M1 SA 1
11 . [12]
E M1 SA es 1 s ll
1s 1 p1 p2 p3
1s es 1 s p1 p2 p3
one can find by induction the general solution for the lifetime mat-
ing success of a decision rule in which the focal individual mates with
f individuals out of a total of n total potential mates, where f ≤ n. The
solution is f
2
es p
if 1 i
es 2 . [13]
p
E M SA f
es 2 p
i1 if n
f
pi . [13]
E M SA 1
es 1 s l
1s p
i1 i
pi . [13]
E f M SA if 1 i n
if1
1
es 1 s ll
1s p
f n
f 1
1s es 1 s p pi
i1 i if1
i1 i if1

OCR for page 213

f
es 2 pi
i1
. [13]
E M SA f n
f 1
es 1 s l
1s p p
1i 1i
i if
Reproductive Decisions Under Ecological Constraints: It’s About Time /
We are now in a position to evaluate the cumulative lifetime fitness
for focal individuals exhibiting this decision rule. if the fitness of the focal
individual of mating with potential mate i is wi, then the focal individual’s
cumulative lifetime fitness is simply its mating success with potential mate
i multiplied by wi, summed over all of the focal individuals’ mates:
f
es 2 pi wi
i1
. [14]
W M SA f n
f 1
es 1 s l
1s p p
1i 1i
i if
The expression in eq. (14) is the analytically derived mean of cumula-
tive lifetime fitness computed over an stochastic ensemble of focal indi-
viduals, each of whom assesses the fitness distribution of potential mates
in the same way, experiences the same values of e, s, and l, and has the
same mating decision rule. eq. (14) takes on different values as we change
f, the number of acceptable mates, out of a total of n potential mates. The
objective is to find the value of f, call it f, which maximizes eq. (14); f is
the switch point. Given values of e, s, and l and a distribution of fitnesses
wi across a set of n potential mates, the recipe for maximizing eq. (14) is
as follows: (1) rank the n potential mates in fitness conferred on the focal
individual from high to low. (2) next, compute the average focal individu-
al’s expected cumulative lifetime reproductive fitness for these parameters
and fitness distribution, assuming that the focal individual mates with
only one potential mate, the highest fitness-conferring individual, and
rejects the remaining n – 1 individuals. This corresponds to f = 1 in the eq
(14). (3) now, repeat step 2, but assume that the focal individual mates
with the top two potential mates, so that f = 2 in eq .(14). (4) Continue this
process, adding 1 more mate at a time and computing the expression in
eq. (14) until f = n. (5). Plot the curve of reproductive fitness as a function
of f, the number of ranked potential mates that are accepted, and find
that value of f, f, that produces the highest average reproductive fitness.
This is the switch point, and f/n is the fraction of acceptable mates that
maximizes the cumulative lifetime fitness.
if the potential mates are equally likely to be encountered, then we
can further simplify eq. (14)
f
es 2 wi
i1
, where g = n-f . [15]
W M SA 1
es 1 fs l
1s g
f
This is the expression given by eq. (1) in the text of the paper.
M SA /de
dW f*

OCR for page 213

40 / Patricia Adair Gowaty and Stephen P. Hubbell
f
es 2 wi
i1
, where g = n-f . [15]
W M SA
Sensitivity Analysis 1
es 1 fs l
1s g
f
f
2
es w
We studied the sensitivity of the fitness function to changes in the
i1 i
, where g = n-f . [15]
W M SA
encounter probability f e, the survival probability s, and the latency l, by
1 s es 1 fs l 1 g
computing the derivatives of eq. (14) with respect to each of the param-
eters. We evaluated the derivatives at the switch point, f. The sensitivity
of the fitness function at f to changes in encounter probability e is given
by
M SA /de
dW f*
f* f*
M SA /de
dW s2 1 s l 1P Q es 3
pi wi pwii
f*
, [16]
i1 i1
2
es f *1 s l 1P Q
1s 1 s es 1 s l 1P* Q
f
s 2 pi wi 1 s l 1P Q es 3 pwii
, [16]
i1 i1
f* n 2
sl 1
pi a1 sQ es 1 pi . P Q es 1 s l 1P Q
1s
where P nd
i1 i f* 1
f* n
where P pi and Q p.
The sensitivity of ithe1 fitness function at f to changes in the survival
i
i1 f*
probability s is
M SA /ds
dW f*
f* f*
es 2 1 e (2 l )es l P eQ
2 es p i wi pi wi
M SA /ds i 1
dW
. [17]
i1
f*
2
1 s es f1 s l 1P Q 1 s es 1 s l 1P Q
* f*
es 2 1 e (2 l )es l P eQ
2 es p i wi pi wi
. [17]
i1 i1
2
l1
1s es 1 s P Q es 1 s l 1P Q
1s
Finally, the sensitivity of the fitness function at f to changes in latency
l is
f*
e 2 s l 4 ln s P pi wi
M SA /dl . [18]
dW i1
l1
1s es 1 s P Q
f*
Results
Fig. 11.6 shows the effect of varying latency l and the population size
of acceptable mates n on the fraction of mates acceptable, for a w distribu-
tion of β(1,1). The fraction of acceptable mates declines with population

OCR for page 213

Reproductive Decisions Under Ecological Constraints: It’s About Time / 4
size, and with increasing latency, but the effect of latency is minimal for
long latency times. Fig. 11.5 shows how the mean fraction of acceptable
mates and its standard deviation change as a function of the w distribution
and of each of the model parameters. The left-most column of graphs ( a–d)
are for a uniform w distribution of β(1,1). The middle column of graphs
(e–h) are for a w distribution, β(3,8), skewed to low fitness values, and the
right-most columns of graphs (i – l) are for a w distribution, β(8,3), skewed
to high fitness values. note that a higher fraction of potential mates are
acceptable when the fitness distribution is skewed high than when it is
skewed low or is uniform. Fig. 11.8 presents a graphical representation of
the results of the sensitivity analysis. The top row of panels ( a, d, g) rep-
resents how the sensitivity of lifetime fitness to survival rate s is affected
by variation in s, e, and l. The middle row of panels (b, e, h) show how the
sensitivity of lifetime fitness to encounter rate e is affected by variation in
s, e, and l. The bottom row of panels (c, f, i) show how the sensitivity of
lifetime fitness to latency l is affected by variation in s, e, and l. note that
in general, the sensitivity of lifetime fitness to survival is much greater
than sensitivity to encounter rate, which in turn is much greater than the
sensitivity to latency. note also that the parameters interact in complex
nonlinear ways in their impact on the sensitivity.

OCR for page 213