Polyandry and protandry in butterflies

Bulletin of MathematicalBiology Vol.54, No. 6, pp. 957-976, 1992. Printedin Great Britain. 0092 8240/9255.00+0.00 PergamonPressLtd 9 1992Societyfor MathematicalBiology POLYANDRY AND PROTANDRY IN BUTTERFLIES C. ZONNEVELD Vrije Universiteit, De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands (E.mail: cor@bio.vu.nl) Current models on protandry in butterflies assume that females mate only once, yet for many species this assumption is not realistic. In this paper a model is formulated to study how polyandry, i.e. repeated mating of females, affects protandry. Moreover, the model is elaborated to describe the probability distribution of the number of matings per female. Field data on this distribution are well described by the model, which supports the use of the law of mass action to describe the encounter rate between males and females. Finally, a weight factor is derived, taking into account the decline in oviposition rate with age, as well as the chance that a female is remated. In comparison with the situation that all matings contribute equally to a male's reproductive success, the application of the weight factor enhances protandry. This suggests that mate competition is not the sole cause of protandry. 1. Introduction. In many butterfly species that appear in discrete generations, the males on average emerge before the females, a phenomenon referred to as protandry. Various mathematical models have been developed to explain protandry (Bulmer, 1983; Fagerstr6m and Wiklund, 1982; Iwasa et al., 1983; Wiklund and Fagerstr6m, 1977; Zonneveld and Metz, 1991). These models have one key assumption in common, namely that females mate only once--a condition called monandry, whereas males are capable of multiple matings. All the models indicate that males should indeed emerge some time before females, if they are to maximize their expected reproductive success. Although experimental evidence shows that monandry does occur in a number of species, it is far from ubiquitous. Wiklund and Forsberg (1991) showed that in the subfamily Pierinae, polyandry, i.e. repeated mating of females, prevails. Several species belonging to other families are also polyandric (Burns, 1968; Drummond, 1984; Ehrlich and Ehrlich, 1978; Sv~ird and Wiklund, 1989). These observations show that the assumption ofmonandry is not generally valid. Wiklund and Fagerstr6m (1977) suggested that polyandry will relax the selection for protandry. Theoretical analysis of this idea is hampered by the assumption that females are mated instantaneously upon eclosion, an assumption shared by most models. Admittedly, the fact that virgin females are seldom captured in the field (Burns, 1968; Wiklund and Forsberg, 1991) suggests that this assumption is realistic. Mathematically, however, it poses a problem. The 957 958 C. Z O N N E V E L D assumption amounts to an infinitely large encounter rate between males and virgin females. Therefore, if mating with non-virgin females would be allowed in these models, the number of matings would invariably become infinitely large. Thus, this assumption cumbers the analysis of polyandry. Zonneveld and Metz (1991) formulated a model in which the mating rate depends on both male and female density, in agreement with the law of mass action. Hence, in their model the encounter rate between males and females is finite. The first aim of this paper is to present an extension of the model of Zonneveld and Metz (1991), which shows how polyandry affects the evolutionarily stable amount of protandry (ESP). The second aim of this paper is to test the applicability of the law of mass action, used to describe the encounter rate between males and females. For this purpose, the model is further elaborated, in order to describe the frequency distribution of the number of matings per female. Subsequently, the elaborated model is tested against field data. This also yields rough estimates of two parameters, which facilitates the evaluation of the model results. Until now, models on protandry assumed that all matings contribute equally to a male's reproductive success. For monandric species this seems to be a plausible assumption. However, for two reasons this assumption is less likely to be realistic in polyandric species. First, since eggs are generally fertilized by sperm received during the latest mating (Drummond, 1984), a female only contributes to a male's reproductive success as long as she does not mate again. Thus, the expected value of a mating depends on male presence, which changes in time. Second, the oviposition rate of butterflies declines with age (e.g. Boggs, 1986; Wickman and Karlsson, 1987). Matings early in the flight period are therefore likely to be more important than later ones, since, in view of the increase in mean age with time, the expected oviposition rate decreases. The final aim of this paper, then, is to derive a time-dependent weight factor taking these facts into account, and to show how this factor affects the evolutionarily stable amount of protandry. 2. The Evolutionarily Stable Amount of Protandry 2.1. Model formulation. The model is based on the following assumptions: (i) generations do not overlap; (ii) the time to emergence of males and females is logistically distributed (Zonneveld, 1991), with similar variances and only the mean date of eclosion is under genetic control (Bulmer, 1983); (iii) males and females have equal and constant death rates (Scott, 1973); (iv) mating occurs as soon as a male encounters a female. (By definition, an encounter is an interaction between a male and female that results in a mating.) The duration of a mating is negligibly small in comparison with the total life expectancy; (v) the encounter rate follows the law of mass action (see Section 3); (vi) the individualspecific encounter rate for non-virgin females is less than or equal to that of POLYANDRY AND PROTANDRY IN BUTTERFLIES 959 virgin females (see discussion); (vii) each mating contributes equally to the reproductive success of a male (see section 4 for a weakening of this assumption). Changes in densities of males (M*), virgin females (V*) and non-virgin females (F*) are given by the following system of differential equations [from (iii)-(v)]: dM* - 6Nm(t*) - c~M* dt* (1) dV* - (1-6)Nv(t*)- (o~+ ~9M*)V* dt* (2) dF* dt* (3) - 0M* V* - aF*. Parameters in these equations are: N, the total number of adults that appear throughout the flight period, in a given area; ~, the individual-specific death rate; 6, the fraction of males among the total number of adults; and ~, the individual-specific encounter rate for virgin females. (Variables and parameters with dimensions and interpretation are summarized in Table 1.) The quantities re(t*) and v(t*) denote the male and female emergence frequency, which follow from (ii): m(t*) = exp {(t*-pm)/fl} fl(1 +exp {(t* _#,,)/fl})2 (4) v(t*) = exp{(t* --#v)/fi} fl(1 + exp{(t* -- tt~)/fi})2" (5) A measure for the reproductive success of a male is the number of matings with virgin and non-virgin females, q~. The expected reproductive success of an arbitrary male emerging at time t* is given by [from (iii)-(vii)] ~b(t*)= e -~t*-'*)~V* d t * + e e -~(** t*)~OF*dt*. (6) Here, e is the ratio of the encounter rates for non-virgin and virgin females. According to assumption (vi), 0 ~<E~<1. Equation (6) implies that all matings contribute equally to a male's reproductive success. Below (Section 4) a possibly more realistic approach will be discussed. To simplify equations (1)-(3) and (6), the dimensionless variables M = M*/N, V = V*/N, F = F*/N, and t = (t* -#v)/fi are used. After insertion of equations (4) and (5), the equations describing changes in male, virgin female and female density become: 960 C. ZONNEVELD Table 1. Variables and parameters. No., T, and L denote the dimensions number, time and length Parameter Dimension #m,/~ T 1 T /3 Individual-specific death rate Mean time of male and female emergence Dispersion measure of emergence times Individual-specific encounter rate between males and virgin females Total number of butterflies that emerge throughout the flight period, per unit area Proportion of males among the total number of adults Scaled death rate: 2 = ~fl Scaled encounter rate: 9 = N0/3 Scaled protandry: ~ = (My- #m)//3 Ratio of encounter rates for non-virgin and virgin females r L2/No. T N No./L 2 6 2 "C 8 Variable Dimension M*, V*,F* N o . L 2 t* T ~(t*) M,V,F Interpretation -- t r Interpretation Male, virgin female, and non-virgin female density Time Expected reproductive success of a male emerging at rime't* Scaled densities of males, virgin females and non-virgin females: M = M * / N V= V*/N, F = F * / N Scaled time: t = (t*-/t~)//3 Scaled reproductive success of a male emerging at time te dM e t+~ - 6 +~)2 dt (1 + e t dV dt dF dt - (1-6) - ~MV-- 2M et (1 + eZ) 2 (7) (fl + ~ M ) V 2F (8) (9) where 2 = cq~ is the scaled death rate, 9 = NOfl the scaled encounter rate, and = (#v-I%)/fi the scaled a m o u n t of protandry. The reproductive success of a male emerging at t e, expressed in dimensionless variables, is given by: ~(te; "C)= exp { - 2 ( t - t e ) } [ ~ V ( t ; r)+e~F(t; r)] dt. e Since the presence of virgin and non-virgin females depends on male presence, which in turn depends on the protandry, q) is a function of both the time of eclosion of an individual male, and the protandry of the population. The POLYANDRY AND PROTANDRY IN BUTTERFLIES 961 expected value of the reproductive success of a male emerging on the average time units before the females is given by: [Eqb](r)= M[V(t; r)+eF(t;-r)] dt. -00 (1 +et+~) 2 @(t; z) dt = ~ - To answer the question whether there is an evolutionarily stable strategy with respect to protandry, let us consider a population with protandry z, and a mutant male with protandry ?. Mutant males are considered to be so rare that they do not affect the density of virgin females. The expected reproductive success of this mutant, E~, depends on both 9 and f, and is given by: M(t; f)[V(t; z)+eF(t; "c)] dt lEVI(f; "c)=~- (10) oo where the density of mutant males is denoted by M(t; f). Let us define fm to be the value of ~ for which E~, at a given r, is maximized. Generally fm depends on z, so we can write fro(Z) (see Fig. 1). The evolutionarily stable amount of: protandry (ESP), z*, is given by the solution of: ~m(~) = (11) T. 0.8 4J ~o 0.6 L E 0.4 0 L fl_ E 0.2 O_ 0 0 I 0 I I 0.4 I I O.B I i I .2 Population I ~ 1 .6 i I 2 Protandry Figure l. Optimal protandry for a mutant, given a fixed level of protandry for the population. The parameters 6 and ~ are taken 0.5 for all curves. The relationship is given for the following parameter combinations: ( ~ ) : 2=0.1, ~ = 1; ( x ): 2=0.1, u?=25; (O): 2=1.5, ~ = 2 5 ; (~,<1): 2=1.5, ~ = 1 . The straight line indicates equality of population protandry and optimal mutant protandry. The intersection of the line with a curve yields the ESP for that parameter combination. 962 C. ZONNEVELD 2.2. Numerical methods. I approximated the solution of equation (11) using a Newton-Raphson procedure (Burden and Faires, 1985). For each iteration, fm was obtained by equating the (numerically evaluated) derivative of E r to zero, where I again used the N e w t o n - R a p h s o n method. I used a fifth order Adams Predictor Corrector method (Burden and Faires, 1985) to evaluate E ~ along with the differential equations (7)-(9). Although the differential equations (7)-(9) can be solved, the solutions contain integrals that have to be evaluated numerically. For the problem at hand, numerical solution of the differential equations is more convenient. If e = 1, the ESP equals zero (see below). Numerical approximations to this solution were accurate to about 10-5; hence I expect that numerical inaccuracies are negligibly small. 2.3. Results. After a short remark on the uniqueness of the solution of equation (11), I will first evaluate the behaviour of the ESP for the most extreme parameter values, i.e. for e = 0 or 1,2-+0 or oo and ~ 0 or oo. Next I will show how protandry depends on polyandry for realistic values for the death and encounter rate. Finally, I will show how the expected number of matings per female depends on the parameter values. Figure 1 shows that equation (11) has one solution for a given combination of parameter values. This strongly suggests that, for a particular set of parameter values, there is one evolutionarily stable value for z. The different limits for monandric species (i.e. e = 0) have been studied by Zonneveld and Metz (1991). The results are briefly restated here. If the death rate approaches zero, the ESP will become infinitely large. If the encounter rate tends to zero, or the death rate becomes infinitely high, the ESP will approach zero. If the encounter rate becomes infinitely large, the ESP will approach a finite limit. If a species is completely polyandric (i.e. e = 1), males should maximize the chance to encounter just any female. Let W = V+ F denote all females. Apart from a translation of z units along the abscis, M and W have identical curves. Since W does not depend on male presence, the problem of finding the ESP reduces to optimizing ~MW dr, which is achieved for z =0. Hence the ESP equals zero, a result that is independent of the other parameters. The following arguments hold if a species is neither monandric nor completely polyandric, i.e. if 0 < e < 1. If ~--+0, or 2 ~ , the contribution of non-virgin females to a male's reproductive success becomes negligibly small. Since for very small encounter rates or high death rates virgin female presence no longer depends on male presence, the problem of finding the ESP reduces to optimizing ~MVdt. Renewed application of the argument outlined in the previous paragraph shows that this, again, is achieved for 9=0. If ~ , or 2--+0, matings with virgin females contribute relatively less and less to a male's reproductive success. Thus a male should optimize the chance to encounter POLYANDRY AND PROTANDRY IN BUTTERFLIES 963 non-virgin females, i.e. a male should maximize ~MFdt, which, once more, is achieved for z = 0. Figures 2 and 3 show the solution of equation (11) for different values of the scaled death and encounter rates. The depth axis shows the degree of polyandry, e. F o r monandric species (i.e., e = 0 ) , the E S P decreases with increasing death rate, and increases with increasing encounter rate (cf. Zonneveld and Metz, 1991). F o r e > 0, these monotonicities break down. F o r /\ \ 0 / 0"5 0.2 0 . 0 /I / I/ 12,5 1 25 ~ 0 25 n v r2 ] 1 Encounter rate Figure 2. ESP as a function of the encounter rate and the degree of polyandry, for two different values of the death rate. (a) 2 = 0.1; (b) 2 = 1.5. 964 C. ZONNEVELD (a) 1.8- o ~- 1.2- 0.6- 0 O0.1 0 5 (b) 3 i i Death rate 1.5 Figure 3. E S P as a function of the death rate and the degree of p o l y a n d r y , for t w o different values of the encounter rate. (a) q~ = 1; (b) qJ = 50. increasing values of the encounter rate W, the ESP first increases up to a maximum, and then decreases. With increasing death rate, the ESP first increases, reaches a maximum, and subsequently decreases. (Notice that in Fig. 3a the maximum is attained for some 2 e [ 0 , 0 . 1 ] . ) The ESP decreases POLYANDRY AND P R O T A N D R Y IN BUTTERFLIES 965 monotonically with increasing degree of polyandry. Polyandry affects the ESP most strongly at low death rates and high encounter rates. Zonneveld and Metz (1991) showed that in monandric species protandry should increase with an increasing fraction of males among adults (5). For polyandric species (i.e. e > 0) the relationship between protandry and sex ratio is more complicated. For low encounter rates the pattern remains identical to that of monandric species. For high encounter rates, however, it changes: ESP is maximal for some intermediate value of the parameter 6, and decreases at both lower and higher values of 3. The mean number of copulations per female is generally used as a measure for the degree of polyandry. Figure 4 shows the mean number of matings per female, which is given by (6/1 - 6 ) E ~ , for various parameter combinations at the evolutionarily stable amount of protandry. For e > 0, the mean number of matings increases almost linearly with the encounter rate ~ . For larger values of the scaled death rate, 2 > 0.5, the mean number of copulations increases only slowly with increasing encounter rate. For higher death rates, the mean number of copulations is also relatively insensitive to the degree of polyandry. 3. Testing the Law of Mass Action. The law of mass action is a cornerstone of the model presented in Section 2. I will elaborate this model, to describe the probability distribution of the number of matings per female, and timedependent changes therein. This elaborated model will then be tested against field data. The probability distribution is obtained as follows. Denote females that have mated i times by F~. The differential equations describing changes in the densities of the F~'s are given by: dV dt - (1 - dF, dt dt 6)v(t)- - ugMV- (2 + - eqZMF~_ 1 -- (2 + (12) ~M)V (13) eUgM)F1 (2 + e~M)F/ for i> 1 (14) where M is given by the solution of equation (7). Solutions of these equations yield the densities of females that have never mated (the virgins), have mated once, twice and so on. These densities are normalized by dividing them by: V(t) + 5(t) i=1 966 C. ZONNEVELD (a) c 1 O.B o u 0.6 o L 0.4 E ~ 0.2 0 l l ~ 0 l l i 10 ] l 20 l l l 30 40 ~ncounter 50 Fate (b) m c o 6 ~ 5 3 o u 4 o ~ 3 N 3 E 2 E 0 1 0 1 1 1 10 1 1 20 1 1 1 30 Encounter 1 1 40 50 re~e Figure 4. Mean number of copulations per ~male. Left: e = 0 ; right: e=0.1. From top downwards, 2 = 0.1, 0.5, 1, and 1.5. the result being the probability distribution of the number of matings per female. To test this model, I compared it to data on the number ofmatings per female during one generation of the green-veined white, Pieris napi, presented by Forsberg and Wiklund (1989). To do this, I proceeded as follows. First I obtained an estimate of the scaled death rate, by fitting data on the abundance of Pieris rapae and P. napi, collected in the Netherlands in 1978 (Zonneveld, 1991), to the solution of equation (1). These two species have equal flight periods and life histories, and are hard to distinguish in flight. I therefore combined the data of the two species. Figure 5 shows that the time course of POLYANDRY AND PROTANDRY IN BUTTERFLIES 967 abundance is well described by equation (1). The scaled death rate equals 0.27. Next, 1 calculated the ESP, assuming 2 = 0.27, for different values of qJ and e, and solved equations (12)-(14) for the different parameter combinations with accompanying ESP. Finally, I calculated expected frequencies and compared these with the observed. To allow this comparison, the time points of the observations have to be divided by the scaling parameter, ft. A good description of the data was obtained with fi = 1.875 days, q~ = 75, and e -- 0.025; the result is shown in Fig. 6. 3a o o o o 20 % 10 0 I L 0 I 20 I I dO I ~ NO I L I BO Time, t I 100 d Figure 5. Time course of abundance of the two whites, Pieris rapae and Pieris napi, with the solution of equation (1) fitted to these data. I assume that the observations at given time are Poisson distributed. Parameter estimates (with sd): N = 50.7 (6.8), /~= 17.5 (6.8) days,/7= 6.05 (0.95) days, and ~=0.0445 (0.0061) day-1. To test the goodness of fit I used the test statistic ~ ' = 1(Oi-ei)2/ei, where oi and e i denote observed and expected frequencies. Simulations gave an upper tail probability of 0.41. Thus, deviations from the model predictions could well have arisen from sampling errors. Since the fitted model heavily relies on the law of mass action, the good fit considerably supports the applicability of the law. 4. Weighing Matings. The results presented so far are based on the assumption that all matings contribute equally to a male's reproductive success. I will relax the "equal contribution assumption" in two different ways. The first one is very simple: suppose that matings with virgin females contribute more to a male's reproductive success than matings with non-virgin females, but that both contributions are still constant. If so, in equation (10) the 968 C. ZONNEVELD 1 J o.s / t L ~ 5 10 o 0 1 2 NUmber of COpulation3s 4 Figure 6. Relative frequencies of the females that have mated 0, 1,2, 3 or 4 times, as a function of time. See text for details. term eMFshould be multiplied by a constant, say co, representing the value of a mating with a non-virgin female relative to that with a virgin female. This gives rise to a new constant, e* = eco, which no longer has the interpretation of the degree of polyandry. Mathematically, however, one can treat E* as if it where e. Hence the results presented so far remain applicable. The second approach to relax the equal contribution assumption is possibly more realistic. I will weigh matings with a time-dependent weight factor, say e)(t), that takes into account the possibility that females are remated, and the decrease of the oviposition rate with age. This factor represents the expected number of eggs that a female will contribute to a male's reproductive success, if they mate at time t. Butterflies often show last male sperm precedence, i.e. the eggs are fertilized by the sperm received during the latest mating (Drummond, 1984). Thus, after mating a female contributes to a male's reproductive success as long as she does not mate again, and, naturally, stays alive. The oviposition rate of butterflies decreases with age (e.g. Boggs, 1986; Wickman and Karlsson, 1987). Little is known of the relationship between the age and the oviposition rate. In view of the desire to keep the weight factor as simple as possible, I assume that the oviposition rate decreases exponentially with age, with a rate parameter equal to the death rate. For the small heath butterfly, Coenonymphapamphilus, this assumption seems to be realistic: data on the decrease of oviposition rate are well described by an exponential POLYANDRY AND PROTANDRY IN BUTTERFLIES 969 decrease (Wickman and Karlsson, 1987), with a rate parameter equal to 0.12 day -x, which is close to the death rate as found by Zonneveld (1991). Assuming complete last male sperm precedence, and an exponential decrease in oviposition rate, one can derive the weight factor co(t) as follows. The expected oviposition rate, 6, at time t is given by: 6(0 = J o o(a)A(a, t) da, where o(a) denotes the oviposition rate as a function of age, and distribution at time t. The age distribution is given by: A(a, t)= A(a, t) the age v(t-a)S(a) f : v(t--u)S(u) du where S(a) denotes the fraction of animals surviving up to age a, and v the emergence curve, given by equation (5). The oviposition rate as a function of age is given by: o(a) = r exp{--2a}, where r denotes the oviposition rate at zero age. Since the expected reproductive success [see equation (15)] is linear in r, the ESP does not depend on r. Figure 7 shows the expected oviposition rate for several death rates. Before the peak of female emergence (at t = 0), the expected oviposition rate is nearly constant. Just before the female emergence peak the expected oviposition rate starts to decline. This is due to the fact that the rate at which fresh females emerge increases less and less fast, so that the average age starts to increase. In addition, Fig. 7 shows that as time proceeds, the expected oviposition rate for high death rates exceeds that for low ones. This can be understood as follows. At a high death rate, 2 > 1, the youngest females always predominate, so mean age remains low, whereas at low death rates, 2 < 1, the mean age increases after the peak of female emergence. Since young females have higher oviposition rates, the mean oviposition rate will remain high if mean age remains low. However, the mean oviposition rate will continue to decline if mean age increases. Let us now consider a female that has been mated at time t. The chance that this female dies or mates again at any time z ~>t, i.e. the hazard rate h(z), is given by: h(z) = e~M(z) + 2. 970 C. ZONNEVELD oJ ~3 L O.B 4~ 0.6 o (3_ 0.4 u ~ 0.2 0 I i -10 i t 0 1 I 10 I i 20 l 3O Time Figure 7. Expected oviposition rate as a function of time, for four different death rates. The oviposition rate at zero age is put equal to 1. (<>): 2=0.1; ( x ): 2=0.5; (O): 2=1; (+): 2=1.5. Using the relationship between the hazard rate and the survivor function, h = - S'/S, one can evaluate the chance that our female is still alive and did not mate again after 0 time units. It is given by: exp{-e~ftt+~ dz-20}. This chance has to be multiplied by the expected oviposition rate at time t + 0 to obtain the expected rate at which a female adds to a male's reproductive success. The resulting product should then be integrated over all possible values of 0. The expected oviposition rate is given by 6(0 e x p { - 2 0 } . The weight factor is thus given by: dO. In solving equation (11), equation (10) is replaced by: [E~] (f; z) = ~- co(t; v)M(t; f) IV(t; "c)+ eF(t; ~)] dt. (15) oo Figure 8 shows the weight factor for some parameter combinations. Quantitatively, the weight factor depends on all parameters. More important, however, is the qualitative result that the weight factor decreases, though not necessarily monotonically, in time. This non-monotonic decrease can be POLYANDRY AND PROTANDRY IN BUTTERFLIES 971 understood as the result of two counteracting factors, namely male presence and the expected oviposition rate. Male presence is highest around t = 0. As males disappear the weight factor tends to increase, since the chance for a female to become remated decreases. However, since female age increases, the expected oviposition rate decreases. As a result, the weight factor starts to decrease again. In the end, nearly all males have disappeared, and the weight factor is determined solely by the life expectancy and the oviposition rate. (a) 01 i i i -10 i 0 i i 18 l I i 20 3B Time (b) i-, 0.2 4~ .E 03 0.16 0.12 0.0B I -10 I I I 0 I i 10 i I 20 / 30 Time Figure 8. Weight factor as a function of time, for several parameter combinations. (a) 2=0.1, ~=3; (b) 2=1.5, ~=0.5. For both figures, ug=25. (~): e=0; (x): s=0.05; ( 9 ~=0.1. 972 C. ZONNEVELD Figure 9 compares the ESP obtained with unweighed matings to that obtained with weighed matings. The intuitively clear result is that protandry is enhanced if matings are weighed, a result which is independent of any specific parameter. An important implication of this result is that protandry is evolutionarily stable even if a species is completely polyandric, i.e. the encounter rates for virgin and non-virgin females are identical (~ = 1). O.G .~ 8.5 Q_~ (/1 U O.4 0.3 0.2 0.1 0 0 0.1 0.2 ESP, 8.~ unwe ighed Figure 9. Comparison between ESP's with equal or varying weight factor. The straight line indicates equality of both ESP's. Parameter values: 2 = 1.5. From top downwards e =0, 0.25, 0.5, 0.75 and 1; points with equal values o f t are connected. Different values of the encounter rate are indicated by different symbols. From left to right, ~ takes the values 1, 5, 10 and 25. 5. Discussion. In the present model, the degree of polyandry is characterized by e, the ratio of encounter rates for non-virgin and virgin females. Various behavioural aspects bring about differences in these encounter rates. In the small heath butterfly, Coenonympha pamphilus, females behave differently before and after mating. Virgin females search out male territories and show a lengthy "solicitation" flight, whereas mated females avoid being detected in male territories (Wickman, 1986). The ringlet butterfly, Aphantopus hyperanthus, shows a similar shift in female behaviour after mating (Wiklund, 1982). In the pierid family, unreceptive females have the ability to reject male courtship by alighting in the vegetation, spreading their wings and raising the abdomen straight up in the air (e.g. Wiklund and Forsberg, 1985, and refs therein). Chemical cues may also be involved. In Heliconius erato, females use antiaphrodisiac pheromones that are transferred at mating to reject subsequently courting males (Gilbert, 1976). Forsberg and Wiklund (1989) argue that in the POLYANDRY AND PROTANDRY IN BUTTERFLIES 973 (polyandric) green-veined white, Pieris napi, female attractiveness is also chemically mediated by pheromones emitted by the female. As outlined in Section 2.3, the ESP's of monandric and polyandric species differ in their limit behaviour if 2--. 0 or 9 ~ oo. These differences are caused by the additional term in the reproductive success measure of polyandric species, i.e. the matings with non-virgin females. The relative contribution of this term to the reproductive success becomes small if the death rate becomes high or the encounter rate low. If so, the limit for the ESP is identical for monandric and polyandric species. However, the relative contribution of matings with nonvirgin females approaches unity if the death rate approaches zero or the encounter rate becomes infinitely large. In that case, the limit for the ESP for monandric species differs from that of polyandric species. In addition to the data on the number of matings per female, Forsberg and Wiklund (1989) did not provide information on the abundance of males and females. Hence, to fit the model outlined in Section 3 to their data, some crude approximations had to be made. Nevertheless, the result is promising. Not only are the data well described by the model, but, as far as I can judge, the parameter values are realistic too. For the scaling factor needed to compare the data with the model solutions (i.e. the dispersion measure fi) I found 1.875 days. This agrees well with other known values, which range from 0.9 to 5.5 days (Zonneveld, 1991). Given that 2 = 0.27, this results in a death rate of 0.14 day-1, which is quite realistic for many butterfly species. Since protandry strongly depends on the parameter values, a rough knowledge of which parameter values are realistic is necessary to evaluate the model results. Despite the uncertainties, the values ~ = 7 5 and e=0.025 correctly indicate at least the order of magnitude. Thus the encounter rate for non-virgin females is much smaller than that for virgin females. For such small values of e, protandry is diminished but still close to that of monandric species. The present study shows that with increasing degree of polyandry, protandry decreases, a result which is independent of any specific parameter. This confirms the suggestion of Wiklund and Fagerstr6m (1977) that polyandry will relax the selection for protandry. On the other hand, the degree of polyandry in one of the most polyandric species, Pieris napi, is still so low (e ~ 0.025) that it should hardly affect protandry. The same probably is true for two other whites, P. rapae and P. brassicae. Thus the fact that these species are scantily protandric, if at all, can not be explained by the degree of polyandry. Among others, Wiklund and Forsberg (1991) argued that the absence of protandry in polyandric species is best explained by sperm competition. In butterflies, the mass of the ejaculate delivered by a male is possitively correlated with male size, and large ejaculates induce a longer period of female unreceptivity. This results in sexual selection favouring male size in species where females mate repeatedly. This hypothesis will be the subject of a future study. 974 C. Z O N N E V E L D Many studies present data on the number of copulations per female (for a review, see D r u m m o n d , 1984), tacitly assuming this as a measure of polyandry (or explicitly so, Wiklund and Forsberg, 1991). However, the mean number of copulations per female not only depends on the degree of polyandry (as measured by e), but also on the encounter rate, death rate and population density. Therefore this variable is not a suitable measure of the degree of polyandry. Since e, the ratio ofencouter rates for non-virgin and virgin females, is independent of any other parameter, it is a more appropriate choice to measure polyandry. Data on the number of matings per female are sometimes presented as a function of female age (e.g. Ehrlich and Ehrlich, 1978; Pliske, 1973). However, age per se has no effect, apart from the fact that older females have had a larger chance to encounter a male than younger ones. Age together with male presence determines the number of matings of a female. Therefore, such a presentation is only meaningful if data on male presence are given too. The weight factor introduced in this paper is based on quite specific assumptions. However, the qualitative result, namely that matings early in the flight period are more important than later ones, is probably robust. Since protandry is an ESS even in the absence of mate competition (i.e. if e = 1; then mating with a female does not reduce the opportunity for other males to mate, hence there is no mate competition), an important conclusion from the present study is that, at least in polyandric species, mate competition is not the sole cause ofprotandry. Though most models treat protandry as an exclusively male strategy, Fagerstr6m and Wiklund (1982) and Zonneveld and Metz (1991) showed that if females aim to minimize the time between emergence and mating, protandry will also be an evolutionarily stable strategy. Quantitatively, there are only slight differences between the male and female ESP's. In polyandric species, it is also likely that females aim to minimize the time between emergence and the first mating. Hence, the female ESP will be independent of the degree of polyandry. Since the male ESP decreases with the degree of polyandry, there will be a conflict of interests between males and females, which becomes more severe with the degree of polyandry. However, since in most species the degree of polyandry is probably low (e < 0.025), I expect the conflict of interests still to be too small to be of importance. The various mathematical models all predict protandry, suggesting a robustness with respect to specific assumptions. The key assumption the models have in c o m m o n is that there is competition for virgin females. Since mate competition and protandry are both c o m m o n phenomena, their simultaneous occurrence only weakly confirms the theory. If mate competition were indeed the main cause of protandry, then its absence should imply the absence of protandry. This would provide a more rigorous test of the theory. POLYANDRY AND PROTANDRY IN BUTTERFLIES 975 For instance, in the brimstone, Gonepteryx rhamni, their is no mate competition during the emergence period, since adults emerge during the summer, but mate after hibernation. Thus, the theory predicts absence of protandry in the brimstone. This prediction still awaits experimental (dis-) confirmation. The comma butterfly, Polygonia c-album, offers another possibility to test the theory. In this species, the adults hibernate, and mate in spring. Their offspring consists of two morphs, a light and a dark one. Butterflies of the dark morph enter hibernation before they mate. The light morph, however, produces a second brood in the same year. In view of the possibility to alter the proportion of light and dark morphs experimentally (Nylin, 1989), this species provides a nice opportunity to test the theory. Since in the light morph mate competition is present, one expects this morph to be protandric. In the dark morph, on the other hand, there is no mate competition, and protandry should be absent. Experimental evidence confirms these predictions (Nylin, 1991), which provides strong evidence favouring the theory. In biological research, one can choose between the experimental and the comparative method. Wiklund and Forsberg (1991 ) maintained that questions about patterns, such as the association between polyandry and protandry, can only be studied by the comparative method. The present study shows that there is a third method, namely heuristic modelling, which suits the study of pattern. In the comparative method, species compared will differ in many more aspects than the one under study. This can obscure the pattern one is looking for. In heuristic modelling, one only varies the characteristics of interest. Hence, the answers obtained are not obscured by differences between species that are irrelevant for the problem at hand. The present study, for instance, shows beyond doubt that the decrease of protandry with polyandry cannot explain the absence of protandry even in the most polyandric species. The reason is that the observed degree of polyandry in these species is still too low to affect protandry appreciably. Although Wiklund and Forsberg (1991) arrived at the same conclusion, they had to cope with much variation, which makes their conclusion less powerful. I am indebted to Hans Metz, who suggested the possibility to weigh matings. I gratefully acknowledge (in alphabetical order) Jacques Bedaux, Hugo van den Berg, Ger Ernsting, Bas Kooijman, Wim van der Steen and Carla Zijlstra for comments on draft versions of the manuscript. I am also grateful to Oren Hasson, whose remarks improved the clarity of presentation. LITERATURE Boggs, C. L. 1986. 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