The Evolution of Paternal Care with Overlapping Broods

 
vol. 164, no. 4      the american naturalist           october 2004                                                                     

         The Evolution of Paternal Care with Overlapping Broods

Andrea Manica* and Rufus A. Johnstone†

Department of Zoology, University of Cambridge, Cambridge                    their mating frequency. Male care, on the other hand, is
CB2 3EJ, United Kingdom                                                      predicted to be much rarer, since males can greatly increase
                                                                             their reproductive success by finding more mates. How-
Submitted January 15, 2004; Accepted May 11, 2004;
Electronically published September 1, 2004                                   ever, male care is predominant in fish and some groups
                                                                             of insects such as assassin bugs and giant water bugs
Online enhancement: appendix.                                                (Clutton-Brock 1991).
                                                                                Empiricists attempting to measure the costs of care have
                                                                             outlined three major categories of costs: energetic, survival,
                                                                             and mating costs (Gross and Sargent 1985; Clutton-Brock
abstract: Most attempts to model the evolution of parental care              1991). Parental care is often energetically expensive (en-
assume that caring and mating are mutually exclusive activities (i.e.,       ergetic costs) as a result of reduced feeding opportunities
individuals acquire and guard broods “sequentially”). However, in            or of direct calorific expense while caring, thus leading to
most fish and certain insects, males can keep mating and collecting          reduced future fertility (e.g., Clutton-Brock et al. 1989;
additional eggs while continuing to guard broods obtained earlier            Crowl and Alexander 1989). For example, in the Galilee
(i.e., males guard “overlapping” broods). We present a model of
                                                                             St. Peter’s fish, a biparental mouthbrooding cichlid,
parental care with overlapping broods in which males can mate and
guard simultaneously, even though there is a trade-off between these         mouthbrooding parents lose more weight than parents that
two activities. Within this framework, we show that male care is             have their brood removed after only 1 day of care
favored by short female processing times and high population den-            (Balshine-Earn 1995). Parental survival can also be affected
sities, which minimize the mating cost of care. Relatively low mor-          by care (survival costs) through the increased visibility of
tality while guarding is also important for the stability of male care.      caring adults to predators (e.g., Lack 1966; Shine 1980).
Female care, on the other hand, is favored by long female processing
                                                                             In lumpsuckers, a fish with adhesive demersal eggs and
times and low populations densities, which lead to longer intermating
intervals. Biparental care is stable only when the cost to benefit ratio     paternal care, males suffer very high levels of predation
of care was not biased toward either sex. We derive quantitative             by otters while guarding, whereas females of the same
estimates of fitness for different strategies for two species of assassin    species are rarely caught by this predator (H. Kruuk, un-
bugs with male and female uniparental care and show that the model           published data, in Clutton-Brock 1991). Finally, while car-
predicts the correct form of care for both species. We believe our           ing, individuals may suffer reduced access to mates (mat-
model might help explain the prevalence of male uniparental care
                                                                             ing costs), a potentially important issue for males whose
in certain taxa, such as fish.
                                                                             reproductive success is commonly limited by access to
Keywords: parental care, population density, natural selection, sexual       their mates (Bateman 1948).
selection.                                                                      The magnitude of mating costs probably differs between
                                                                             homeotherms, which have to keep their young warm and
                                                                             thus can look after only a small number of offspring
Which sex should care for the offspring? The evolution of                    (Clutton-Brock 1991), and ectotherms, which can simul-
any form of parental care depends on the cost to benefit                     taneously guard a large number of young (e.g., Blumer
ratio of this strategy to each parent. Because of anisogamy,                 1979; Sargent 1988). For the latter, providing parental care
females often end up caring for the young, since they have                   and acquiring mates are not mutually exclusive activities
a high investment in each of the gametes and cannot in-                      (i.e., they can obtain “overlapping” broods), and the mat-
crease their reproductive success significantly by increasing                ing cost of care is therefore reduced (Gross and Shine 1981;
                                                                             Gross and Sargent 1985). However, because of the asym-
* E-mail: am315@cam.ac.uk.                                                   metry between the sexes in the cost of gamete production,
†
    E-mail: raj1003@hermes.cam.ac.uk.                                        only males are likely to be able to take full advantage of
Am. Nat. 2004. Vol. 164, pp. 517–530. 䉷 2004 by The University of Chicago.   this opportunity, since the time needed by females to pro-
0003-0147/2004/16404-40280$15.00. All rights reserved.                       duce more eggs is likely to be longer than the period of
518   The American Naturalist

care (e.g., fish: Wootton 1984; insects: Thomas and Man-       whereas pmc is the processing time for caring males (i.e.,
ica, in press). This asymmetry might explain the wide-         the time taken by a male to mate and subsequently re-
spread occurrence of paternal care in fish, where males        plenish his gametes).
can mate with a large number of females and simulta-              In both females and males, care takes the form of egg
neously guard all the eggs they acquire. It might also ex-     guarding. The time required for fertilized eggs to hatch is
plain the association between male care and territoriality     denoted h, and the probability of offspring survival de-
(Krebs and Davies), since males that mate on their ter-        pends on the proportion of this time period during which
ritories are likely to be able to continue doing so while      either or both parents are present to protect the eggs.
protecting the already acquired offspring. Male stickle-       Specifically, we assume that unguarded eggs suffer a con-
backs, for example, keep attracting females while guarding     stant instantaneous mortality rate m. Guarding by the male
up to 10 broods simultaneously (Wootton 1984). This pos-       reduces egg mortality to a fraction M mc of the unguarded
sibility has received little attention by theoreticians, who   mortality m, whereas female guarding reduces it to M fc.
generally assume that parents can care for only one brood      When both parents guard the eggs simultaneously, mor-
at a time (e.g., Maynard Smith 1977; Balshine-Earn and         tality is reduced to a fraction equal to the product of
Earn 1997; Wade and Shuster 2002).                             M mc and M fc. We also assume that care is nondepreciable
   In this article, we develop a model of parental care in     (sensu Clutton-Brock 1991); that is, its effectiveness does
which males are able to simultaneously care and mate. We       not depend on the number of offspring that receive it.
show that, given the possibility, male care is favored in         Parental care has three major costs (Gross and Sargent
species found at high densities, whereas female care evolves   1985; Clutton-Brock 1991): energetic, mating, and survival
more easily at low densities. We then test our model using     costs. All these three costs are built into the model. Guard-
data from two species of assassin bugs: Rhinocoris tristis     ing individuals incur an energetic cost when they are not
(Stål), which shows paternal care, and Rhinocoris carmelita   able to meet the energetic demands of guarding by foraging
Stål, which shows maternal care. These two species are        while looking after the brood. In order to offset this en-
sympatric and have very similar morphology and behavior        ergetic cost, they will take a longer time to replenish their
(Thomas 1994; Thomas and Manica 2003 and in press),            gametes than individuals that do not care, thus leading to
thus providing an ideal model system to investigate sex        pc 1 pnc. Because of anisogamy, care is likely to affect the
differences in the costs and benefits of parental care.        processing time of females disproportionately more than
                                                               that of males.
                                                                  In our model, females can produce and guard only one
  A Model of Parental Care with Overlapping Broods
                                                               brood at a time, and thus they pay a high mating cost of
We consider mating and offspring production in a large,        care since they cannot mate while caring. When females
stable, continuously breeding population. A proportion a       guard a single brood until hatching, time out is equal to
of newly born individuals are males and 1 ⫺ a are females.     the sum of h and pfc. On the other hand, we assume that
Individuals can be in either of two states: searching for      caring males guard a territory and can continue to acquire
mates (time in) or not available for mating (time out).        broods while guarding, although the rate at which they
While searching, an individual randomly encounters will-       encounter females will be reduced by some proportion g
ing members of the opposite sex at a rate proportional to      (because of the males’ restricted ability to search out fe-
their density d, with the constant of proportionality being    males while remaining on their territory). This scenario
denoted k. Since encounters are random, the time until         is common: overlapping care by territorial individuals is
the next encounter follows a negative exponential prob-        the predominant form of male care in fish (Gross and
ability distribution so that its mean (average search time     Sargent 1985) and some insects (e.g., assassin bugs: Tho-
s) is simply the inverse of encounter frequency l. In the      mas and Manica, in press), whereas female carers are al-
absence of any form of parental care, time out is equal to     ways sequential brooders (e.g., intertidal fish from the fam-
the processing time p, defined as the time needed to mate      ily Stichaeidae: Coleman 1992; assassin bugs: Thomas and
and to replenish egg or sperm supplies after a mating.         Manica, in press).
Because of anisogamy, processing as well as search time           Parents guarding eggs or territory are likely to expose
are likely to differ between the sexes. Subscripts “m” and     themselves to predation, thus incurring survival costs. The
“f” will therefore be used to denote parameters for males      instantaneous mortality rates for guarding individuals of
and females, respectively. Subscripts “c” and “nc” denote      either sex, m mc and m fc, are thus assumed to be higher (or
parameters for caring and noncaring individuals, respec-       equal, in the case of no cost) to the mortality rates of
tively. Following this notation, pfnc is the processing time   nonguarding individuals, m mnc and m fnc, respectively. For
for females in the absence of care (i.e., the time taken by    caring males, we assume the higher mortality to be as-
a female to mate and subsequently replenish her gametes),      sociated with their limited movements while holding onto
Paternal Care with Overlapping Broods          519

Table 1: List of all the parameters in the model                  of factors such as population density on the stability of
Parameter                        Definition                       male versus female care.
                                                                     A pair of male and female strategies is said to be stable
c            Clutch size
                                                                  if the male strategy maximizes the lifetime reproductive
m            Instantaneous egg mortality
M            Fraction of unguarded egg mortality suffered by
                                                                  success for males given the female strategy and vice versa.
               guarded eggs                                       A population in which a stable pair of strategies is adopted
s            Search time                                          cannot be invaded by a mutant male or female who adopts
p            Processing time (time taken to mate and              some other strategy, since any such mutant would suffer
               replenish gametes)                                 a lower lifetime reproductive success. Here, we consider
h            Hatching time of eggs                                the ancestral state of no care by either sex and investigate
m            Instantaneous adult mortality                        the stability of this system against invasion by caring in-
g            Proportional decrease in search efficiency of        dividuals from either sex. A detailed treatment of the sta-
               guarding males                                     bility of male-only, female-only, and biparental care can
l            Encounter rate                                       be found in the appendix in the online edition of the
d            Population density
                                                                  American Naturalist. Mixed equilibria (where both strat-
k            Constant of proportionality between density
               and encounter rate
                                                                  egies coexist) are possible where a mutant strategy can
a            Primary sex ratio                                    invade but is not able to become fully fixed within the
                                                                  population. For the sake of simplicity, we do not analyze
                                                                  in detail the relative frequencies of different strategies
their territory and thus not be influenced by whether eggs        within mixed equilibria but simply indicate the range
are present in the territory.                                     within which these equilibria occur.
  The adult sex ratio will differ from the primary ratio,
depending on the mortality associated with the strategy
adopted by each sex. Assuming a stable population, the              Stability of No Care against the Invasion of Male Care
proportion of adult males is given by
                                                                  We begin by looking at the stability of a system without
                                am fy                             care against invasion by caring males. The mean payoff
               a mxfy p                       ,                   (vmnc) from a single batch of eggs in our noncaring pop-
                        a(m fy ⫺ m mx) ⫹ m mx
                                                                  ulation is equal to the number of eggs c multiplied by their
where “x” and “y” represent the strategy (caring or non-          survival probability when unguarded (e⫺hm). To obtain a
caring) adopted by males and females, respectively. Table         brood, a male will have first to survive the search phase
1 provides a summary of all the parameters used in the            (with probability (1/s m)/(1/s m ⫹ m mnc ) p 1/(1 ⫹ s m m mnc )),
model.                                                            after which it will obtain a payoff vmnc. Subsequently, the
                                                                  male will be able to reenter the search phase only if he
                                                                  survives the processing phase, with probability e⫺p mnc m mnc.
                          Analysis                                Writing wmnc for the lifetime reproductive success of a
                                                                  nonguarding male, we thus have
We will assume in the following analysis that females can
produce and guard only one brood at a time while males
                                                                                         1
can acquire several overlapping broods and guard them                      wmnc p                (v ⫹ e⫺p mnc m mnc wmnc ).
simultaneously. For both males and females, several care                            1 ⫹ s m m mnc mnc
strategies are theoretically possible. A behavioral “rule”
covering all situations would be very complex, and the set        Substituting vmnc and solving for wmnc, we obtain
of all possible strategies of this kind would be very large,
especially for males that can care simultaneously for several                                      ce⫺hm
                                                                                wmnc p                                 .        (1)
broods of different ages. Here, for the sake of simplicity,                              1 ⫹ s m m mnc ⫺ e⫺m mnc p mnc
we consider only two strategies for each sex: no care, in
which individuals immediately abandon a new clutch to                The lifetime reproductive success of a guarding male
seek out more mates, and full care, in which females guard        mutant that arises in the population can be calculated
eggs until hatching and males remain permanently in one           using the same approach. The mean payoff (vmc) from a
location, guarding any clutch they may acquire and waiting        single batch of c eggs depends on the amount of time the
for females to locate them (although we still refer to this       male remains alive to guard them. If the male guarded the
period of waiting as search time). Although simplistic, this      brood up to hatching, the survival probability of the eggs
limited strategy set still allows us to investigate the effects   until hatching would be e⫺hmMmc. However, if we take into
Paternal Care with Overlapping Broods               521

account that the male might die at some time t ! h, the                                     for mates or replenishing their sperm stores. We define
average payoff is                                                                           the proportion of searching males within the population
                                                                                            as Ps mnc and the proportion of males that are replenishing

                  冕                                       冕
                                                          ⬁
                                                                                
                     h
                                                                                            their sperm stores as Pc mnc. At any given moment in time,
        vmc p c              ⫺m mct ⫺mMmct ⫺(h⫺t)m
                          m mce   e       e           ⫹         m mce⫺m mcte⫺hmMmc         a proportion of males moves from the searching pool into
                   tp0                                   tph                              the replenishing pool at a rate equal to their average en-
                c[e⫺h(mMmc⫹m mc)(mM mc ⫺ m) ⫹ e⫺hmm mc ]                                    counter rate with females (l mnc p 1/s m). On the other
            p                                            .                                  hand, in a stable population, males will move out of the
                           m mc ⫹ mM mc ⫺ m
                                                                                            search pool either because they die (at rate m mnc) or because
   A guarding male will also suffer a mating cost, leading                                  they have completed the whole replenishment period (with
to longer search times (s m /(1 ⫺ g)) compared with the rest                                probability e⫺p mnc m mnc). Also, in a stable population, the rate
of the population. Writing wmc for the lifetime reproductive                                of change in Pc mnc is equal to 0 (i.e., no change in the
success of a guarding mutant, we thus have                                                  proportion of replenishing males), and it can be written
                                                                                            as
                           1
       wmc p                            (v ⫹ e⫺p mc m mc wmc ).
                 1 ⫹ [m mc s m /(1 ⫺ g)] mc                                                      dPc mnc
                                                                                                         p lPs mnc ⫺ m mnc Pc mnc ⫺ lPs mnce⫺p mnc m mnc p 0.
Substituting vmc and solving for wmc, we obtain                                                   dt

                                      wmc p
                                                                                            Since Pc mnc p (1 ⫺ Ps mnc ), we can rearrange the above equa-
           c[e⫺h(mMmc⫹m mc )(mM mc ⫺ m) ⫹ e⫺hmm mc ]                                        tion as
                                                              .
    (m mc ⫹ mM mc ⫺ m){1 ⫹ [m mc s m /(1 ⫺ g)] ⫺ e⫺p mc m mc}
                                                                                      (2)                                           1
                                                                                                         Ps mnc p                                          .
                                                                                                                    1 ⫹ (l mnc /m mnc )(1 ⫺ e⫺p mnc m mnc)
  For stability, we require that wmnc 1 wmc, that is, that the
nonguarding males that make up the population should
have a higher lifetime reproductive success than the guard-                                 Similarly,
ing mutant. Writing out wmnc and wmc, we obtain

                                    ce⫺hm                                                                                             1
                                                                                                          Ps fnc p                                          .
                                                        1                                                            1 ⫹ (l fnc /m fnc )(1 ⫺ e⫺p fnc m fnc)
                          1 ⫹ s m m mnc ⫺ e⫺m mnc p mnc

           c[e⫺h(mMmc⫹m mc )(mM mc ⫺ m) ⫹ e⫺hmm mc ]
                                                              .                                The rate at which individuals move from the search
    (m mc ⫹ mM mc ⫺ m){1 ⫹ [m mc s m /(1 ⫺ g)] ⫺ e⫺p mc m mc}
                                                                                            phase to the processing phase (i.e., the encounter rate) is
                                                                                      (3)   proportional to the product of the population density (d),
                                                                                            the proportion of females within the population (1 ⫺
  To solve this inequality, we need an explicit formula for                                 a mncfnc), and the proportion of individuals of the other sex
male search time (s m) in a population of nonguarding                                       available for mating (i.e., searching; Ps fnc). Thus, we can
individuals. Males can be in either of two states: searching                                write

Figure 1: Effect of population density and relative female processing time on the ability of male care to invade a noncaring population under
different costs scenarios: (A) low (g p 0.3 ) and (B) high (g p 0.8 ) mating cost, (C) low (mmc p 0.015 ) and (D) high (mmc p 0.03) survival cost,
and (E) low (pmc p 1.5) and (F) high (pmc p 3) energetic cost. Successively lighter shaded areas represent zones of stability for decreasing values
of Mmc (i.e., higher benefit of care). When males suffered mating or mortality costs, the invasion of care was favored at high densities and low
female processing times, whereas energetic costs favored the invasion of care at low densities and high female processing times. In each of the
scenarios, the costs that were not being investigated were set to 0 (i.e., g p 0 , pmc p pmnc , mmc p mmnc , respectively). Each graph shows the range
of densities and values of pfnc (relative to pmnc ) over which male care could invade for three different values of Mmc , the proportion of unguarded
egg mortality suffered when guarding (A, 0.90, 0.91, 0.92; B, 0.55, 0.60, 0.65; C, 0.8560, 0.8561, 0.8562; D, 0.5950, 0.5955, 0.5960; E, 0.96, 0.97, 0.98;
F, 0.7, 0.8, 0.9; unguarded egg mortality m was set to 0.3). Other parameter values were a p 0.5 (an even sex ratio), pmnc p 1 , k p 1, h p 10,
mfnc p mmnc p 0.01 (no sex difference in mortality). Note that the units of density are undefined; the graph simply shows the effects of variation
over a 10-fold range to invade a nonguarding population.
Paternal Care with Overlapping Broods                     523

                                                               1                   nonguarding females is analogous to that of males de-
                 l mnc p kd(1 ⫺ a mncfnc )Ps fnc p               ,                 scribed in equation (1):
                                                              sm
                                                      1                                                                  ce⫺hm
                     l fnc p kda mncfnc Ps mnc p         .                                            wfnc p                                  .               (5)
                                                      sf                                                        1 ⫹ m fnc s f ⫺ e⫺p fnc m mnc

Substituting the above in the equations for Ps mnc and Ps fnc                      On the other hand, a guarding female mutant needs to
and solving for s m and s f, we obtain                                             survive both the guarding and the processing time before
                                                                                   being able to produce another batch of eggs:
                              (1 ⫹ m fnc s f ⫺ e⫺p fnc m fnc)
                      sm p                                    ,                                              1
                                (1 ⫺ a mcfnc )dkm fnc s f                                         wfc p               [v ⫹ e⫺(h⫹p fc )m fc wfc ],
                                                                                                          1 ⫹ s f m fc fc
                              (1 ⫹ m mc s m ⫺ e⫺p mc m mc)
                       sf p                                .
                                  a mcfncdkm mc s m                                where vfc can be computed by adapting the formula for
                                                                                   vmc, giving
This pair of equations can be solved for s m:
                                                                                                    c[e⫺h(mMfc⫹m fc )(mM fc ⫺ m) ⫹ e⫺hmm fc ]
                                                                                       wfc p                                                          .       (6)
                                                                                                  (mM fc ⫺ m ⫹ m fc )[1 ⫹ m fc s f ⫺ e⫺m fc(h⫹p fc )]
      sm p A ⫺  ([       1 ⫺ e⫺p mnc m mnc a mncfnc(1 ⫺ e⫺p fnc m fnc)
                             m mnc
                                          ⫹
                                             m fnc(1 ⫺ a mncfnc )        ]           For the nonguarding population to be stable against the
                                                                                   invasion of a guarding mutant, we need wfnc 1 wfc. Writing
                                                                     2
             1 ⫺ e⫺p mnc m mnc a mncfnc(1 ⫺ e⫺p fnc m fnc)                         out wfnc and wfc, we obtain
           {[
        ⫹ A⫺
                 m mnc
                              ⫹
                                 m fnc(1 ⫺ a mncfnc )                ]                                             ce⫺hm
                                                                                                                                        1
                                                                                                          1 ⫹ m fnc s f ⫺ e⫺p fnc m mnc

                                                     U
                                               1/2

                 ⫹4
                    A(1 ⫺ e⫺p mnc m mnc)
                         m mnc                })      2,                                        c[e⫺h(mMfc⫹m fc )(mM fc ⫺ m) ⫹ e⫺hmm fc ]
                                                                                              (mM fc ⫺ m ⫹ m fc )[1 ⫹ m fc s f ⫺ e⫺m fc(h⫹p fc )]
                                                                                                                                                  .           (7)
                                                                             (4)
                                                                                   Using the approach described for equation (4), we can
where                                                                              obtain an expression for female search time:

                                         1
                           Ap
                                  (1 ⫺ a mncfnc )kd
                                                    .

Substituting the above equation into equation (3), we can
                                                                                     sf p A ⫺([      1 ⫺ e⫺p fnc m fnc (1 ⫺ a mncfnc )(1 ⫺ e⫺p mnc m mnc)
                                                                                                        m fnc
                                                                                                                      ⫹
                                                                                                                               m mnca mncfnc
                                                                                                                                                          2
                                                                                                                                                              ]
                                                                                           1 ⫺ e⫺p fnc m fnc (1 ⫺ a mncfnc )(1 ⫺ e⫺p mnc m mnc)
thus predict when a noncaring population will be stable
against the invasion of a caring male mutant.                                           {[
                                                                                      ⫹ A⫺
                                                                                              m fnc
                                                                                                            ⫹
                                                                                                                     m mnca mncfnc                        ]
                                                                                                                              U
                                                                                                                        1/2
 Stability of No Care against the Invasion of Female Care
Let us now consider the likelihood of the invasion of a
                                                                                              ⫹4
                                                                                                 A(1 ⫺ e⫺p fnc m fnc)
                                                                                                      m fnc            })      2,

caring female mutant. Lifetime reproductive success for                                                                                                       (8)

Figure 2: Effect of population density and relative female processing time on the ability of female care to invade a noncaring population under
different costs scenarios: (A) high (k p 2 ) and (B) low (k p 0.1 ) encounter rates, (C) low (mfc p 0.015) and (D) high (mfc p 0.03) survival cost,
and (E) low (pfc p 1.5) and (F) high (pfc p 3 ) energetic cost. Successively lighter shaded areas represent zones of stability for decreasing values of
Mfc (i.e., higher benefit of care). The invasion of care was favored at low densities and high female processing times. Each graph shows the range
of densities and values of pmnc (relative to pfnc ) over which female care can invade for three different values of Mfc , the proportion of unguarded
egg mortality suffered when guarding (A, 0.7, 0.8, 0.9; B, 0.5, 0.6, 0.7; C, 0.2, 0.3, 0.4; D, 0.01, 0.05, 0.10; E, 0.4, 0.5, 0.6; F, 0.3, 0.4, 0.5; unguarded
egg mortality m was set to 0.5). Other parameter values are a p 0.5 (an even sex ratio), pfnc p 1 , k p 1 , h p 5 , mfnc p mmnc p 0.01 (no sex difference
in mortality). Note that the units of density are undefined; the graph simply shows the effects of variation over a 10-fold range on the ability of
female care to invade a nonguarding population.
524     The American Naturalist

Figure 3: Effect of the benefit of male and female care on the stability of biparental care under three cost scenarios: (A) equal costs of care for the
two sexes (pmc p pfc p 2.4, mmc p mfc p 0.012), (B) higher costs for males (pmc p 4 , mmc p 0.02), and (C) higher costs for females (pfc p 4,
mfc p 0.02). Larger benefits of care are represented by smaller values of Mmc and Mfc . BC, Area of stability of biparental care; MC, area where a
noncaring female mutant can invade, leading to male care; FC, area where a noncaring mutant can invade; U, unstable areas where both mutants
can invade. Other parameter values are a p 0.5 (an even sex ratio), pmnc p pfnc p 2, mmnc p mfnc p 0.01, k p 1, d p 5, h p 10, m p 0.7.

where                                                                        M fc, in lighter gray). Also, it is intuitively obvious that
                                                                             smaller values of M mc and of M fc (i.e., large benefit of care)
                                      1                                      are required for caring mutants to invade when care is
                           Ap                  .
                                  a mncfnckd                                 more costly (figs. 1, 2). Biparental care is stable only when
                                                                             the balance between the costs and benefits of care for the
                                                                             two sexes is comparable (fig. 3). A sex bias in terms of
                                                                             costs has to be counterbalanced by a bias in the benefits
                                Results                                      of care (fig. 3B, 3C). Any sex bias in the cost to benefit
It is clear that when the benefit of care is high (represented               ratio of care favors uniparental care by the sex with the
in figs. 1 and 2 by a small M mc and M fc, respectively, in                  most advantageous ratio. However, a balance in the cost
darker gray), uniparental care is likely to become estab-                    to benefit ratio in the two sexes would not be necessary
lished in a noncaring population over a wider range of                       for the evolution of biparental care if the benefit of two
conditions than when the benefit is low (large M mc and                      parents guarding the offspring was disproportionately
Paternal Care with Overlapping Broods                   525

Table 2: Parameter estimates obtained from the literature
                                                                                                                                  Rhinocoris      Rhinocoris
Parameter                                                             Source                                                        tristis       carmelita
m                    Estimated in the wild by daily monitoring broods from which the guarding parent                                  .0892           .2345
                        was removed.a
Mmc                  Estimated in the wild by monitoring broods of R. tristis daily. Laboratory experi-                               .3037           .7252
                        ments determined that some of the mortality was due to filial cannibalism.a,b We
                        assumed R. carmelita males would also cannibalize if they were to guard eggs.
                        Since there is hardly any sexual dimorphism, we assumed both sexes to be
                        equally capable of defending eggs. An estimate of egg mortality when R. carme-
                        lita males provided protection was obtained by subtracting mortality due to can-
                        nibalism from the estimate from females of the same species (see below).
Mfc                  Estimated in the wild by daily monitoring broods of R. carmelita.a In R. tristis, we                             .0793           .3439
                        used the estimate obtained for males (see above) but discounted mortality due to
                        filial cannibalism.
pmnc                 The fastest intermating time recorded in the laboratory.c                                                        .3333           .3333
pmc                  The fastest intermating time for an R. tristis male that was allowed to keep his                                 .3333           .3333
                        eggs.c Male R. tristis cannibalize some of their own eggs, minimizing the cost of
                        care. We assumed R. carmelita males would be able to do the same if they were
                        to guard eggs.
pfnc                 The fastest interlaying time for R. tristis in the laboratory and for R. carmelita from                         2               2
                        which the brood was artificially removed.c
pfc                  Laboratory estimate for R. carmelita.c We assumed R. tristis female reproductive                              15.6             15.6
                        physiology to be similar to that of R. carmelita (supported by the identical pfnc
                        values).
h                    Field estimates from demographic data.a                                                                       16.5             12
sm                   Field estimates from demographic data.a                                                                        4.3               …
sf                   Field estimates from demographic data.a                                                                         …               5.8
mmc p mmnc p
   mfc p mfnc        Estimated from demographic data for R. tristis; there was no difference between                                  .0286           .0286
                        sexes or between guarding and nonguarding individuals.a No data exist for R.
                        carmelita, but we assumed mortality to be equal to R. tristis since the two species
                        are sympatric and morphologically similar.
g                    Derived from gas equation, assuming males and females move at the same speed                                     .2929           .2929
                        when searching for mates. Males were assumed to be stationary when guarding.
a                    Field estimates from demographic data.a                                                                         .5               .5
d                    Field estimates from demographic data.a                                                                        5.4              1.2
c                    Field estimates from demographic data.a                                                                       19.63            35.82
k                    Estimated using the appropriate equation and the parameters detailed in this table.                             .2099            .2913
   Note: Figures in bold were estimated directly either through demographic studies or through manipulative experiments. Values in italic represent indirect
estimates either from first principle or by assuming the biology of one species to be similar to that of the other species for which information was available.
   a
     Data obtained from Thomas (1994).
   b
     Data obtained from Thomas and Manica (2003).
   c
     Data obtained from Thomas and Manica (in press).

greater than the combined benefits arising from two in-                          time lead to an increase in encounter rates and thus a
dividual carers (as opposed to our assumption that                               reduction in the proportion of time that males spend
M bc p M mc # M fc). For simplicity’s sake, we do not in-                        searching/waiting for mates. A short search time mini-
vestigate this possibility in detail, even though the model                      mizes the mating cost and decreases the probability that
could be easily modified to do so.                                               a guarding male will die because of the increased mortality
   When males suffer a mating (fig. 1A, 1B) or a survival                        while guarding. Comparison of figure 1A and 1B reveals
cost (fig. 1C, 1D), male care is more likely to become                           that density is more important compared with processing
established when density is high and when female pro-                            time when mating costs are higher (so that encounter rates
cessing time is low relative to male processing time. This                       for caring males are lower). This is because an increased
is because both high density and low female processing                           encounter rate for caring males means that even at low
526    The American Naturalist

Figure 4: Sensitivity analysis of the stability of care in (top) Rhinocoris tristis and (bottom) Rhinocoris carmelita. Plausible transitions (when the
lifetime reproductive success of a mutant is larger than that of the rest of the population) are shown by solid black arrows, whereas dashed arrows
show implausible transitions. Note that sex-specific traits can affect transitions that refer to the other sex (e.g., the transition between biparental
care and male-only care depends on the lifetime reproductive success of caring and noncaring females, but these are strongly influenced by male
mortality, which determines the likelihood that a caring male will survive long enough to guard the brood for a significant amount of time). The
ratio between the lifetime reproductive success of the mutant and the rest of the population is given together with a 95% confidence interval
obtained from simulations in which all parameters were allowed to vary simultaneously. Parameters that destabilize a transition during simulations,
changing only one parameter at a time (with a coefficient of variation p 0.1), are presented next to the transition they affected.

density, females mate very soon after they become avail-                     density is the most important factor determining the abil-
able, so that female processing has the main influence on                    ity of male care to invade a noncaring population.
the proportion of time that males spend searching/waiting                       When females guard the eggs, on the other hand, the
for a mate. When encounter rates are low, by contrast,                       effect of male processing time and density is different. For
Paternal Care with Overlapping Broods                527

Figure 5: Effect of density and male mating cost on the stability of parental care in (A) Rhinocoris tristis and (B) Rhinocoris carmelita. The asterisks
denote the estimates of these two parameters obtained from the literature. MC, Area of stability of male care; FC, area of female care; BC, biparental
care; NC, no care. Areas where both male and female care could become stable, depending on which one evolves first, are denoted by MC or FC.
Small zones of instability (i.e., where two strategies would coexist) around the boundaries between areas of stability are not shown for the sake of
simplicity. All these zones are very narrow. Parameter values are given in table 2.

any given value of M fc, female care is more likely to invade                                An Empirical Test of the Model
when density is low and when female processing time is
high. This is because both low density and high female                       The ability of the model to predict the evolutionarily stable
processing time lead to an increase in female intermating                    form of care was tested on two species of assassin bugs:
interval, with the result that care involves a proportionately               Rhinocoris tristis, which exhibits paternal care, and Rhino-
smaller decrease in mating rate. Contrary to what was                        coris carmelita, which exhibits maternal care. Despite
observed for male care, stability was negatively affected by                 showing different patterns of care, the two species are
short search time even with a survival cost. This is the                     sympatric and morphologically very similar, thus provid-
result of guarding time acting in a similar fashion to a                     ing an ideal test for our model. We have detailed infor-
large energetic cost of care for males, forcing the female                   mation on several population parameters (such as en-
to have a longer time out. Stability would be favored by                     counter rates and population density) and on the impact
a relatively short search time only if hatching time was                     of care on both adult and egg survival as well as gamete
extremely short relative to the processing time of non-                      replenishment times for both species (Thomas 1994; Tho-
guarding females. It is also apparent, comparing figure 2A                   mas and Manica 2003 and in press). The parameters of
and 2B, that the relative importance of female processing                    the model were estimated from either demographic data
time compared with density increases when encounter                          obtained by daily monitoring of individuals in the field or
rates are high. This is because an increased encounter rate                  simple manipulations (table 2 and references therein). For
means that even at low density, females mate very soon                       each species, it was obviously not possible to estimate di-
after they become available. Consequently, it is female pro-                 rectly the impact of care for the noncaring sex (e.g., we
cessing time rather than density that has the main influ-                    cannot directly measure how long a mutant caring R. tristis
ence on the intermating interval. When encounter rates                       female would need to replenish her gametes after guarding
are low, by contrast, density is the critical factor. A similar              the eggs, since females of this species never guard). Since
effect is seen when males pay an energetic cost. Under this                  the two species have very similar ecology and morphology,
scenario, when processing time for the mutant guarding                       we assumed the impact of care on the noncaring sex in
males is extremely long relative to the nonguarding males,                   each species to be equal to the impact measured in the
a relatively long search time (given by high density and                     other species (e.g., the impact of care on processing time
long female processing time) minimizes the advantage that                    for a mutant caring R. tristis female was assumed to be
nonguarding males have in processing broods at a faster                      the same as measured in caring R. carmelita females). The
rate.                                                                        only parameter that could not be estimated directly from
528     The American Naturalist

 Table 3: Predictions on the stability of care                                             Discussion
 Type of care                      Favored by                                       Insights from the Model
 Male            High population density
                                                                  Our model shows that, when males can guard overlapping
                 Short female processing time
                 Low mortality for caring males
                                                                  broods, the evolution of exclusive paternal care is not an
 Female          Low population density                           unlikely event as suggested by models that assume that
                 Long female processing time                      guarding and mating are mutually exclusive (e.g., Maynard
 Biparental      Small sex bias in the cost to benefit ratio of   Smith 1977; Balshine-Earn and Earn 1997; Wade and Shus-
                   care                                           ter 2002). When males are able to care for overlapping
                                                                  broods, their cost of care is drastically reduced compared
                                                                  with caring males that can only guard broods sequentially.
                                                                  Unless the benefit of biparental care is disproportionately
empirical data for either species was the mating cost of
                                                                  greater than the combination of the benefits provided by
guarding males. Caring males are stationary and have to
                                                                  each parent individually, an asymmetry in the costs and
wait for a female to find them, whereas noncaring indi-
                                                                  benefits of care to the two sexes has the obvious effect of
viduals (both males and females) can freely move searching
                                                                  favoring uniparental care by the sex with the best balance;
for mates. We assume the latter to move randomly through
                                                                  the ability to care for overlapping broods together with
space, with both sexes being equally efficient at searching
                                                                  anisogamy makes male care relatively more likely to evolve
for mates. Under this scenario, the movement of individ-
                                                                  than female care.
uals can be modeled using the gas equation (e.g., Dusen-
                                                                     Out of the factors we investigated, male mortality had
bery 2000), and the decrease in encounter rates for sta-
                                                                  the strongest influence on the stability of paternal care (see
tionary males can be estimated as g p 1 ⫺ 1/2(1/2).
                                                                  fig. 1C, 1D, where very small steps in M mc lead to sharp
   Using the parameter estimates detailed in table 2, the         shifts in the ability of care to invade, and fig. 4 [bottom],
model predicted the correct form of care in both species,         where the stability of care in Rhinocoris carmelita was
with paternal care being the only stable strategy for R.          mostly influenced by this parameter). This result is a con-
tristis and maternal care for R. carmelita (fig. 4). The sen-     sequence of the ability of males to continuously add new
sitivity of these predictions was investigated through sim-       broods, thus leading to potentially very high payoffs for
ulations. Initially, we allowed all parameter values to vary      increased longevity. This factor is particularly important
simultaneously. Each parameter value was randomly sam-            in R. carmelita, where the benefit of care is relatively low
pled from a normal distribution with the estimated value          when compared with Rhinocoris tristis. Since caring for
from table 2 as the mean and a coefficient of variation           each brood gives only a limited payoff to caring males,
equal to 0.05. Confidence intervals from 1,000 simulations        their ability to process a large number of broods when
are given in figure 4. Subsequently, we investigated the          caring is fundamental in determining the stability of this
effect of each individual parameter by keeping all other          strategy.
parameters equal to the estimates given in table 2. For this         Density was shown to be an important ecological factor
second set of simulations, the coefficient of variation for       that could influence the stability of care. The effect of
the parameter being investigated was raised to 0.1. Param-        density depends on the relative magnitude of the three
eters that affected the stability of care in more than 5%         costs (promiscuity, mortality, and energetic). In general,
of 1,000 simulations are given in figure 4. The predictions       we would expect male care to be most stable in high-
for R. tristis are very robust with respect to the exact pa-      density populations, since short search times minimize
rameter estimates, whereas the predictions for R. carmelita       both the mating cost of care and the probability of dying
are more sensitive to the actual values of the benefit of         without having obtained many matings. In most species,
care (m, M mc) and male mortality (m mnc and m mc; fig. 4).       this effect is unlikely to be counteracted by the energetic
   We further investigated the sensitivity of our results with    cost of care, which is often relatively small for males and
respect to the reduction in mating efficiency g, since the        can be further minimized through filial cannibalism (Man-
mating cost was not estimated from field data, and pop-           ica 2002). On the other hand, low densities are more likely
ulation density d, since this parameter is likely to vary         to favor female care, since females tend to pay a high
widely across populations of the same species. The stability      energetic cost of care because of their expensive gametes
of care in the two species shows similar patterns (fig. 5).       and thus have a relatively long time out when guarding
Female care is stable only at relatively low densities,           because of their inability to guard multiple broods si-
whereas male care becomes stable at higher densities. The         multaneously. Similarly, we would expect short female pro-
effect of the promiscuity parameter g decreases slightly          cessing times to favor paternal care and long female pro-
with increasing population density.                               cessing times to favor female care.
Paternal Care with Overlapping Broods       529

   Density might play an important role in the evolution         location where females can still locate mates, whereas non-
of paternal care in assassin bugs. Whereas R. carmelita is       predatory species are more likely to be unable to effectively
found at low densities and thus is prone to potentially          guard more eggs than can be covered by their body (e.g.,
high costs for guarding males, R. tristis is found at very       Mappes and Kaitala 1994).
high densities, possibly because of its specialized associ-         Sexual selection has been recently suggested as a po-
ation with the legume Stylosanthes (Thomas 1994). Inter-         tentially important factor in the evolution of uniparental
estingly, the other three known paternal species of assassin     male care, but only verbal models have been used to infer
bugs, Rhinocoris albopilosus, Rhinocoris albopunctatus, and      the effect of density and processing times on the evolution
Zelus sp., are also found at relatively high densities as well   of care (Tallamy 2000, 2001). Our model could be used
as being specialized for hunting on the flowers of certain       to investigate the effect of sexual selection by allowing the
species of legumes (Odhiambo 1959; Ralston 1977; Tho-            mating cost to be mitigated or even to be turned into a
mas 1994).                                                       benefit (such that s mc ! s mnc), if females prefer guarding
                                                                 males and actively search for them.
                 Applicability of the Model
                                                                                         Conclusions
The model accurately predicted the caring sex in assassin
bugs. However, its most obvious application would be to          Our model makes a set of testable predictions for when
investigate why paternal care is so common in fish. Trivers      parental care should evolve and which sex should look
(1972) suggested that this association might be a conse-         after the offspring in species where males can obtain over-
quence of gametic proximity. He speculated that the sex          lapping broods (see table 3 for a summary). We believe
that is last associated with the eggs is most likely to care,    our framework to be general enough to fit most species,
because it is faced with a “cruel bind.” Since fish mostly       and the parameters should be relatively easy to measure
show external fertilization (Gross and Sargent 1985), males      empirically. The model successfully predicted the stable
would usually be left behind to care for the eggs. However,      form of care in two species of assassin bugs with uniparen-
this hypothesis does not explain why several fish species        tal care, but further testing in other species (especially in
with internal fertilization show paternal care (Clutton-         fish) would be very informative.
Brock 1991) nor why paternal care in internally fertilizing
insects such as assassin bugs strongly resembles care shown                          Acknowledgments
by fish (Thomas and Manica 2003 and in press). The ability
                                                                 We are indebted to L. Thomas, who was the first to suggest
to look after a large number of offspring (i.e., multiple
                                                                 that high population densities could favor paternal care
broods) is likely to be a key factor in explaining paternal
                                                                 with overlapping broods and encouraged us to look at this
care in fish and insects, which both have relatively small
                                                                 issue, providing advice and data on assassin bugs. We also
eggs that need to be guarded only until hatching, with no
                                                                 thank T. Clutton-Brock, N. Davies, H. Kokko, and A.
or little provisioning for the larvae.
                                                                 Young for providing useful comments on the manuscript.
   The major difficulty in applying our model to fish is
the limited presence of species with female care, thus mak-
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