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Ecology, 91(1), 2010, pp. 3–6
Ó 2010 by the Ecological Society of America
Parasite virulence, host life history, and the costs and benefits of sex
CURTIS M. LIVELY1
Department of Biology, Indiana University, Bloomington, Indiana 47405 USA
Abstract. The widespread existence of sexual reproduction is widely considered to be one
of the most pressing anomalies for evolutionary theory. One possible solution is that
coevolution between hosts and parasites might favor sexual over asexual reproduction (the
Red Queen hypothesis), provided infection is genotype specific and highly virulent. This
requirement for high virulence has been seen as a limitation of the theory. In the present study,
I solve for the cost of sex per reproductive time step of the host, as well as the minimum
virulence required to select for sex. The results show that the cost of sex per time step increases
with increases in the host’s mortality rate, reaching twofold in annual host species. The results
also show that high virulence is not required to select for sexual reproduction, especially in
long-lived organisms. These findings might help to explain the paucity of parthenogenesis in
organisms having long generation times.
Key words: evolution of sex; host; logistic population growth; parasite virulence; Red Queen hypothesis.
INTRODUCTION
1980, Lloyd 1980, Hamilton et al. 1990). The intuitive
appeal of the hypothesis is that coevolving parasites
would seem to be a likely source of selection to provide
the kind of rapid response against common host
genotypes that would seem necessary to prevent the
elimination of sexual populations by asexual clones. On
the other hand, based on computer simulations, the
theory would seem to require that infections are both
common and highly virulent in order to select for
obligate sexual reproduction (May and Anderson 1983,
Howard and Lively 1994, Otto and Nuismer 2004),
which could limit the general applicability of the
hypothesis. The simulation models, however, make a
number of important assumptions regarding the genetic
interface for parasite evasion, the number of parasite
exposures per host, and the host’s life history. Generally,
in these models, the hosts are assumed to be semelparous
annuals. In the present study, I solve analytically for the
minimum virulence required for parasite-mediated
selection to favor sexual reproduction in iteroparous as
well as semelparous hosts.
The minimum strength of selection required to
prevent the fixation of an asexual clone would clearly
depend on the cost of producing males. A previous study
suggested that this cost is reduced in species with very
high birth rates or very low death rates (Doncaster et al.
2000). The rationale for this suggestion was based on
analytical solutions that showed that (1) the carrying
capacities of asexual populations is higher than that for
Almost 40 years after Maynard Smith asked, ‘‘What
use is sex?’’ (Maynard Smith 1971), the persistence of
obligate sexual reproduction in the face of competition
with obligately asexual clones remains mysterious. The
mystery stems from the simple fact that sexual
populations produce males, which do not directly
produce offspring, leading to a lower per capita birth
rate than that expected for asexual populations (the
‘‘cost of males,’’ Maynard Smith 1978). This leads to the
expectation that, all else equal, asexual reproduction
should dominate. Yet the reverse is true. Sexual
reproduction is vastly more common in eukaryotes than
asexual reproduction (Bell 1982).
Many different ideas have been put forth to explain
the long-term persistence of sexual reproduction (Williams 1975, Maynard Smith 1978, Bell 1982, Kondrashov 1993). One idea that has withstood empirical
scrutiny, at least in some systems, is the Red Queen
Hypothesis (Lively 1987, Busch et al. 2004, Decaestecker
et al. 2007, Jokela et al. 2009, King et al. 2009, Wolinska
and Spaak 2009). The gist of the Red Queen Hypothesis
is that parasites should select against common genotypes, giving an advantage to the production of
genetically variable offspring (Jaenike 1978, Hamilton
Manuscript received 28 June 2009; revised 28 July 2009;
accepted 29 July 2009. Corresponding Editor: L. M. Wolfe.
1
E-mail: [email protected]
3
4
CURTIS M. LIVELY
sexual populations, and (2) that the difference in
carrying capacities becomes smaller as birth rates
increase relative to death rates. I recently found a
similar result using a different formulation (Lively
2009). In the present study, I solve for the rate of
population growth per reproductive time step for an
asexual population immediately following its introduction into a sexual population. This then gives the cost of
sex per time step when the asexual clone is rare. This
estimate for the cost of sex is then used to determine the
minimum virulence required to prevent the fixation of an
asexual clone.
Ecology, Vol. 91, No. 1
WasexðUÞ Wasexð1Þ
Csex 1
.
:
WasexðUÞ
f Csex g
Note that the left-hand side of the equation is equal to
virulence (V ), as defined above in Eq. 1. Hence selection
for sex requires that, at some point during the spread of
an asexual clone,
V . Vmin ¼
Csex 1
f Csex g
V . Vmin ¼
Reports
Minimum virulence
V¼
WðUÞ WðIÞ
:
WðUÞ
ð1Þ
The definition is conceptually identical to that used for
inbreeding depression by plant population biologists
(Lloyd 1979, Charlesworth and Charlesworth 1987).
For sex to be favored, the following must be true, at
least at some point before the asexual population
eliminates the sexual population:
W̄sex . W̄asex
ð2Þ
where W̄sex is the mean fitness in the sexual population,
and W̄asex is the mean fitness in the asexual population.
If we let g be the frequency of infection in the sexual
population, and f be the frequency of infection in the
asexual population, the inequality above becomes
where the subscripts I and U refer to infected and
uninfected, respectively. Assuming that Wsex(I) is equal to
Wasex(I)/Csex, and Wsex(U) is equal to Wasex(U)/Csex, we get
gWasexðIÞ þ ð1 gÞWasexðUÞ
. f WasexðIÞ þ ð1 f ÞWasexðUÞ
Csex
ð4Þ
where Csex is the cost of sexual reproduction per
reproductive time step for the host. If f – (g/Csex) . 0,
the result can be rearranged to give the following as the
condition for which sex is favored:
1
:
2f g
ð7Þ
If all the clonal individuals are infected ( f ¼ 1), and none
of the sexual individuals are infected (g ¼ 0), the
condition reduces to
1
V . Vmin ¼ :
2
ð8Þ
Hence, as a limiting case, virulence must be greater than
one-half, assuming a twofold cost of sex (Csex ¼ 2). This
result is consistent with computer simulations that
showed that sexual reproduction would be evolutionarily stable, or that sexuals and asexuals would coexist
for V . 1/2 (see Fig. 1 in Howard and Lively 1994).
Clearly, the minimum virulence to select for sex would
be greater than one half whenever f is less than one, and
g is greater than zero. For example, given sterilizing
parasites (V ¼ 1), and a twofold cost of sex, the
minimum frequency of infection in the asexual population required to select for sexual reproduction must
become greater than (1 þ g)/2. Thus if 10% of the sexual
population is infected (g ¼ 0.10), then more that 55% of
the asexual clone must be infected in order to favor
sexual reproduction. On the whole, f would be expected
to increase, and g to decrease as the clone becomes
common in the population (e.g., Lively 2009). The exact
values would be expected to depend on the genetic basis
for infection, and the number of exposures to parasite
propagules.
The cost of males revisited
gWsexðIÞ þ ð1 gÞWsexðUÞ . f WasexðIÞ þ ð1 f ÞWasexðUÞ
ð3Þ
ð6Þ
where Vmin is the minimum virulence required to select
for sex. Thus, for a twofold cost of sex (Csex ¼ 2), the
condition becomes
MODELS
Virulence is normally defined in the theoretical
literature as the increase in host death rate caused by
infection (e.g., Anderson and May 1982, Levin 1996).
One difficulty with this definition is that it does not
consider the effects of infection on total host fitness as
mediated through fecundity. Here I use a different
definition that allows for infection-mediated reductions
in fecundity and/or survival. Specifically, virulence (V )
is defined as the difference between the fitness of
uninfected, W(U), and infected individuals, W(I), divided
by the fitness of uninfected individuals (Lively 2006):
ð5Þ
An asexual lineage will increase when rare when
Csex ¼
Atþ1 1 ½D þ yð1 DÞ þ Bð1 xÞ
¼
.1
Stþ1
1 D þ ð1 sÞB
ð9Þ
where the numerator (Atþ1) gives the per capita growth
rate for an asexual clone per reproductive time step; and
the denominator (Stþ1) gives the per capita growth rate
for a sexual population per reproductive time step. The
variable B is the birth rate for both sexual and asexual
females, and D is the death rate, which is constrained to
be greater than zero, and less than or equal to one (0 ,
D 1). The variable s is the frequency of males in the
sexual population; x is the birth-rate cost of asexual
January 2010
PARASITE VIRULENCE AND THE COST OF SEX
reproduction; and y(1 – D) gives the increase in the
death rate in the asexual population. The birth and/or
death rates would be expected to be functions of total
host density.
At carrying capacity for the sexual population, the
death rate (D) is equal to the per capita birth rate [(1 –
s)B]; hence: B ¼ D/(1 – s). Substituting for B in Eq. 9, the
condition for the spread of an asexual clone into a sexual
population at carrying capacity is given by
yð1 DÞ
:
ð10Þ
ð1 xÞ .ð1 sÞ 1 þ
D
Csex ¼ 1 þ D:
ð11Þ
Hence, the cost of sex per time step increases linearly
with increases in D, reaching twofold in annual
populations (D ¼ 1). Note that there is no cost of sex
per time step when Csex is equal to one, which is the case
for an immortal sexual population (D ¼ 0).
Eq. 11 gives the cost of sex per reproductive time
step. What about the cost of sex per generation? The
intrinsic advantage of asexual reproduction per host
1=D
generation is simply Csex ¼ (1 þ D)1/D, where 1/D gives
the average number of time steps per generation
(assuming the birth and death rates are constants).
Thus, taking the limit, the intrinsic advantage of
asexual reproduction per generation converges on the
exponential (2.72) as the death rate approaches zero.
Thus the cost of sex per generation may be greater than
2, owing to the ‘‘compound interest’’ accrued from
offspring produced in the early time steps. The cost of
sex per generation decreases to twofold in an annual
population (D ¼ 1).
Assuming that parasites can infect during each time
step for the host, the relevant cost of sex may be the
expected cost per time step, rather than the cost per
generation. If parasites are virulent enough to counter
the cost of sex per time step, they could prevent the
short-term fixation of asexual clones. Substituting the
result for the cost of sex per time step (Eq. 11) into Eq. 6,
the minimum virulence required to select for sex
becomes the following:
D
:
f ð1 þ DÞ g
ð12Þ
For f ¼ 1 and g ¼ 0, the equality simplifies to
Vmin ¼
D
:
1þD
ð13Þ
Thus, the limiting case for virulence becomes V . 1/2
only if the death rate is equal to unity (D ¼ 1). For lower
values of D, the condition becomes less stringent. For
example for D ¼ 1/2, the limiting case becomes V . 1/3.
As such, the maintenance of sex in long-lived species
may be less problematical.
DISCUSSION
The results show that the cost of producing males
depends on the death rate in the sexual population. In
the extreme, there is no cost of sex per time step in
immortal sexual populations. The cost of sex per time
step increases linearly to twofold when the death rate is
equal to one, as in annual populations. The minimum
virulence to select for sexual reproduction, therefore,
deceases with increases in the death rate. The minimum
virulence to select for sex is also reduced as (1) the
fraction of infected asexual individuals increases, and/or
(2) the fraction of infected sexual individuals decreases,
both of which happen rapidly in simulation models as
the clone becomes common (Lively 2009).
Doncaster et al. (2000) were the first to suggest that
the cost of sex might be affected by the birth and death
rates in the sexual population. They showed that the
carrying capacities of asexual populations can be higher
than that of sexual populations, and that this difference
decreases as the intrinsic birth rate increases relative to
the death rate (see also Lomnicki 2001, Olofsson and
Lundberg 2007). Based on this result, Doncaster et al.
(2000) suggested that ‘‘males are less costly to species
with high growth capacities.’’ In the present paper, I
solved for the cost of sex at carrying capacity for the
sexual population. Here the cost of sex does not depend
on the intrinsic birth or death rates per se, but rather on
the equilibrium values, which vary from zero to one. The
results nonetheless suggest that Doncaster et al. (2000)
were correct in suggesting that the cost of sex depends
on the life histories of the sexual and asexual populations.
Reducing the cost of sex reduces the strength of
selection required to maintain it. Previous studies have
focused on resource competition between sexual and
asexual populations, and have suggested that coexistence is easier in long-lived species (Doncaster et al.
2000, Olofsson and Lundberg 2007, Scheu and Drossel
2007). The results of the present study similarly suggest
that the minimum parasite virulence required to prevent
the fixation of an asexual clone decreases with the host’s
death rate. Thus it may be that highly virulent parasites
are only required for the evolutionary persistence of sex
in short-lived host species. If so, these results may help
Reports
Hence the condition for the spread of the asexual
population depends strongly on the death rate. If, for
example, the death rate is one, as in an annual
population, then the condition for spread of the clone
is (1 – x) . (1 – s). As the death rate approaches zero,
the condition becomes y , 0, suggesting that a clone
would not spread into an immortal sexual population.
The intrinsic advantage of asexual reproduction, in
the absence of any costs (x ¼ y ¼ 0), can be solved for by
substituting 0 for x and y in equation (9). At carrying
capacity for the sexual population [B ¼ D/(1 – s)], and
assuming an equal sex ratio (s ¼ 1/2), the advantage of
asexual reproduction per reproductive time step is
simply
Vmin ¼
5
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CURTIS M. LIVELY
to explain the paucity of obligate asexual reproduction
in large organisms having long generation times (Bell
1982).
CONCLUSIONS
Decreasing the death rate of uninfected individuals
has two effects that are favorable to the Red Queen
hypothesis. First, it decreases the cost of sex per time
step, thereby reducing the minimum virulence to favor
sex. Second, it reduces the rate of spread of a clone into
a sexual population, which increases the time (in
generations) for parasites to respond to the clone before
it goes to fixation. Taken together, these factors suggest
that asexual reproduction should be more common in
annuals, and that sexual reproduction should be more
common in long-lived iteroparous species. Sex might
also be expected to be more common in stable, resourcelimited populations.
ACKNOWLEDGMENTS
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I thank Aneil Agrawal, Lynda Delph, Hadas Hawlena,
Jukka Jokela, Kayla King, Maurine Neiman, and anonymous
reviewers for helpful suggestions. This study was supported by
the U.S. National Science Foundation (DEB-0515832, DEB0640639).
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