Does the trade-off between growth and reproduction select for female-biased sexual allocation in cosexual plants?
However, these previous sexual allocation models assumed that a common pool of resources is instantaneously allocated to male and female functions. However, plants generally have a phenological component to their reproduction. For example, an annual plant grows vegetatively from a seed, produces flowers, and finally produces fruits. The resources produced by its vegetative parts during the flower and the fruit stages cannot be used for the prior vegetative growth stage, and the resources produced during the fruit stage cannot be used for flower production. This phenological aspect seems to prevent equal resource allocation to male and female functions because resources cannot be allocated to these functions freely. In this context, Burd and Head (1992) theorized that female-biased sexual allocation occurs because flower initiation imposes an opportunity cost on fruit production by diverting resources from further vegetative growth that could have fueled later fruit production. Seger and Eckhart (1996) also developed a sexual-allocation model that takes the trade-off between growth and reproduction into account.
In this paper, however, we reanalyze sexual allocation taking the phenological aspect into consideration, and we propose that this trade-off does not select for female-biased sexual allocation.
We consider the resource allocation to the male and the female functions and to vegetative growth by an annual or a perennial plant within a growing season. Let T be the length of a growing season and g(t) be the amount of resources instantaneously available at t in the growing season; g(t) derives from photosynthates produced by the vegetative parts and/or resources recovered from the reserves. The plant allocates a fraction 1 - r(t) of g(t) to its vegetative parts and/or to reservation, and the remaining fraction r(t) is allocated to reproduction. Vegetative growth and reproduction simultaneously occur if 1 [greater than] r(t) [greater than] 0. g(t) is a function of the size of the vegetative part, which is a function of the allocation schedule, r(t), before t. Of the r(t)g(t) resources, fractions m(t) and f(t) (m[t] + f[t] = 1) are allocated to the male and the female functions, respectively. m(t), f(t) [greater than] 0 during the period when both the male and the female functions exist (flowers exist), and m(t) = 0 and f(t) [greater than] 0 during the period when only the female functions exist (only fruits exist). Here, flower and fruit production can overlap; m(t) may decrease and f(t) may increase with t as the number of flowers decreases and that of fruits increases with t. Early flower onset (r[t] [greater than] 0 where t is small) may carry the opportunity cost because the vegetative parts cannot grow large in this case. For a given schedule of growth and reproduction, r(t), the total resource allocation to reproduction within the growing season, R, is
R = [integral of] r(t)g(t) dt between limits T and 0, (1)
and the resource allocations to the male and the female functions within the growing season, M and F, are
M = [integral of] m(t)r(t)g(t) dt between limits T and 0, (2a)
F = [integral of] f(t)r(t)g(t) dt between limits T and 0, (2b)
respectively, (R = M + F).
To emphasize our arguments, we assume that the fitness gains through the male and the female functions are proportional to M and F, respectively (i.e., other factors that are known to select for female-biased sexual allocation are removed). Then, to obtain the evolutionarily stable strategy (ESS) sexual allocation schedule, [m.sup.*](t) and [f.sup.*](t), we employ the product theorem (Charnov 1982), which states that the ESS sexual allocation is that which maximizes the product of the male and the female reproductive success, M and F. However, it is clear that mutants that can allocate more resources to reproduction, or to the male and/or the female functions, than the wild type can invade into the population, and hence R must be maximized (or optimized in perennial plants) at the ESS. Thus, the conditions for the ESS sexual allocation are FM [approaches] max and R [approaches] max.
RESULTS AND DISCUSSION
To obtain [m.sup.*](t) and [f.sup.*](t),
FM = [(F + M).sup.2] - [(F - M).sup.2]/4 = [R.sup.2] - [(F - M).sup.2]/4. (3)
Here, R does not depend on F nor M because the value of R is determined by r(t), not by [f.sup.*](t) and [m.sup.*](t). Thus, FM is maximized when F = M for a given r(t). However, natural selection also acts on r(t), which results in changes in R, to maximize the product of F and M. As long as F and M are equalized, the product of F and M is an increasing function of R. So, if maximizing R by changing r(t) and equalizing F and M by changing f(t) and m(t) are possible at the same time, equal resource allocation to the male and the female functions is the ESS even if phenological aspects are taken into account.
What are the reasons for this difference from the result of Burd and Head (1992)? First, theirs is not a game theory model, though equal sexual allocation is a result of a mate-competitive game. Second, they assumed that the duration of the fruit stage is fixed and the resource allocation ratio to the male and the female functions in flowers is also fixed. In other words, Burd and Head assumed that only the schedule of allocation to reproduction is allowed to vary, whereas we assume that both this schedule and allocation to the male and the female functions are allowed to vary. Thus, if the latter two constraints are assumed,
M = [integral of] [p.sub.m] r(t)g(t) dt between limits [t.sub.2] and 0, (4a)
F = [integral of] (1 - [p.sub.m]) r(t)g(t) dt between limits [t.sub.2] and 0, + [integral of] r(t)g(t) dt between limits T and [t.sub.2], (4b)
where [t.sub.2] is the beginning of the fruit stage (flower and fruit production do not overlap in their model) and [p.sub.m] is the resource allocation ratio to the male functions in the flowers. Because both [t.sub.2] and [p.sub.m] are constants, there is only one parameter, r(t), that can change F and M. However, a change in r(t) inevitably results in a change in R because r(t) in a given time affects the future photosynthetic rates. Thus, the maximization of R (Burd and Head  concentrated on the reproductive success in one growing season) and the equalization of F and M are incompatible, and hence equal resource allocation is not evolutionarily stable in this case.
Burd and Head (1992) argued that the duration of the fruit stage may be fixed because there is a minimum time required for fruits to mature. However, even if [t.sub.2] is fixed, F = M can be realized if m(t) ([p.sub.m] in the above equations) is free to vary under the constraints of f(t) + m(t) = 1 and f(t), m(t) [greater than or equal to] 0. Moreover, even if both [t.sub.2] and m(t) are fixed, sexual allocation is either male-biased or female-biased depending on the parameter values. To see this, we reanalyze their model with correcting it to a game theory model and assuming that both male and female successes are linear functions of M and F. With the above two constraints, male-biased sexual allocation evolves if
[k.sub.v] (T - [t.sub.2]) [less than] 1/2[.sub.m] - 1,(5)
whereas female-biased sexual allocation evolves if the left-hand side is larger than the right-hand side, where [k.sub.v] (constant) is the relative growth rate during the vegetative growth stage. Male-biased sexual allocation is the ESS if the duration of the fruit stage, T - [t.sub.2], and/or [k.sub.v] are small and/or [P.sub.m] is large [ILLUSTRATION FOR FIGURE 1 OMITTED]. Thus, female-biased sexual allocation postulated by Burd and Head (1992) is due to their assumption that both [t.sub.2] and [P.sub.m] are fixed and to the parameter values used in their analyses, but not due to the trade-off between growth and reproduction.
Does the constraint on r(t), which results in the constraint on R, select for biased sexual allocation? The answer is no, and the reason is clear in light of the product theorem: namely, for a given value of R, whether or not this value is constrained, FM is maximized at F = M.
However, Seger and Eckhart (1996) developed models in which growth and reproduction overlap and flowers and fruits are produced continuously during periods 0, 1, and 2. Pollen produced in a period i fertilizes ovules produced in the same period, but these ovules will mature using resources available in the next period of i + 1. Using this model, Seger and Eckhart showed that female-biased sexual allocation is the ESS if there is a trade-off between growth and reproduction. However, it can be easily shown that this trade-off does not select for biased sexual allocation in a constant environment if ovules fertilized in a period i mature using resources available in same period i or those in the period i and in later periods. We thus think that their assumption as to the time of ovule maturation, not the trade-off between growth and reproduction, is responsible for their findings of biased sexual allocation.
Zhang and Wang (1994) and Zhang et al. (1996) developed sexual allocation models for perennial plants assuming the trade-off between current reproduction and postbreeding survival and obtained the condition that justifies the separate study of reproductive and sexual allocation. They showed that if female fitness gain increases linearly with resource investment, the ESS reproductive effort is immune from the effect of sexual allocation, and if male fitness gain is a power function of investment, the ESS sexual allocation does not depend on total reproductive effort. Zhang and colleagues studied the independence of sexual allocation and reproductive allocation between breeding periods, and we studied the one during a single breeding period. In this sense, their result and ours are complementary to each other on the significance of life-history consideration in the study of sexual allocation.
In conclusion, there are two problems in sexual allocation: optimizing the amount of resources allocated to reproduction in a growing season and equalizing resources allocated to the male and the female functions. If these two are possible at the same time, equal resource allocation to the male and the female functions is the ESS (given that the fitness gains through the male and the female functions are proportional to M and F). Biased sexual allocation only occurs when constraints make it impossible to simultaneously optimize allocation to reproduction and allocation to male and female functions. Even if female-biased sexual allocation occurs due to the addition of other factors (e.g., constraints on resource allocation within flowers and fruit development duration as in Burd and Head , time lag of fruit development as in Seger and Eckhart , self-fertilization, and local mate competition), the trade-off between growth and reproduction itself is not an important factor that selects for female-biased sexual allocation.
We would like to thank N. Anten, K. Kikuzawa, Y. Iwasa, and K. Matsui for their critical comments on the manuscript.
BURD, M., AND G. HEAD. 1992. Phenological aspects of male and female function in hermaphroditic plants. Am. Nat. 140:305-324.
CHARLESWORTH, D., AND B. CHARLESWORTH. 1981. Allocation of resources to male and female functions in hermaphrodites. Bot. J. Linn. Soc. 15:57-74.
CHARNOV, E. L. 1979. The genetical evolution of patterns of sexuality: Darwinian fitness. Am. Nat. 113:465-480.
-----. 1982. The theory of sex allocation. Princeton Univ. Press, Princeton, NJ.
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|Author:||Sakai, Satoki; Harada, Yasushi|
|Date:||Aug 1, 1998|
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