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Environmental Variability and Semelparity vs. Iteroparity as Life Histories
J. theor. Biol. (2002) 217, 391–396 doi:10.1006/yjtbi.3029, available online at http://www.idealibrary.com on
Environmental Variability and Semelparity vs. Iteroparity as Life Histories Esa Ranta*w, David Tesarz and Veijo Kaitalaz wDepartment of Ecology and Systematics, Integrative Ecology Unit, Division of Population Biology, University of Helsinki, P.O. Box 65 (Viikinkaari 1), FIN-00014, Finland and zDepartment of Biological and Environmental Science, University of Jyva¨skyla¨, P.O. Box 35, FIN-40351 Jyva¨skyla¨, Finland (Received on 8 May 2001, Accepted in revised form on 11 February 2002)
Research on the evolution of life histories addresses the topic of fitness trade-offs between semelparity (reproducing once in a lifetime) and iteroparity (repeated reproductive bouts per lifetime). Bulmer (1994) derived the relationship v þ PA o1 (PA is the adult survival; vbS and bS are the offspring numbers for iteroparous and semelparous breeding strategies, respectively), under which a resident semelparous population cannot be invaded by an iteroparous mutant when the underlying population dynamics are stable. We took Bulmer’s population dynamics, and added noise in juvenile and adult survival and in offspring numbers. Long-term coexistence of the two strategies is possible in much of the parameter region of v þ PA o1 when noise occurs simultaneously in all three components, or (more restricted) when it affects juvenile and adult survival or adult survival and offspring numbers. Iteroparity cannot persist when the environmental variability involves juvenile survival and offspring numbers, or when the noise acts on the three components separately. r 2002 Elsevier Science Ltd. All rights reserved.
Introduction Reproducing once (semelparity) or several times (iteroparity) during a lifetime is one of the central questions in the evolution of life histories (Cole, 1954; Roff, 1992; Stearns, 1992). The major conclusion of previous research, using simple population models, can be summarized by noting that in many scenarios semelparity is a superior life history over iteroparity. In a striking contrast with this theoretical outcome, numerous plant and animal species reproduce iteroparously over their lifespan, which often extends over several reproductive seasons (e.g. Roff, 1992). The disagreement between theory and nature, also known as Cole’s paradox, has n Corresponding author: Fax: +352-9-191-287-01. E-mail address: csa.ranta@helsinki.fi (E. Ranta).
0022-5193/02/$35.00/0
puzzled evolutionary ecologists for the past five decades. Major features of previous research are summarized by Roff (1992), Stearns (1992) and Charlesworth (1994). In his review, Charlesworth (1980, 1994) concluded that iteroparity is favoured over semelparity in fluctuating environments, in which either offspring number per female or juvenile survival varies (Murphy, 1968; Stearns, 1976, 1977). On the other hand, Bulmer (1985, 1994) was the first to note that much of the research of relative evolutionary merits of semelparity over iteroparity had ignored the relevance of density dependence in affecting vital parameters of population renewal. Thus, Charlesworth’s (1994) discussion on semelparity and iteroparity also largely revolves around the issue of the evolution of the two contrasting life histories in density-independent environments. r 2002 Elsevier Science Ltd. All rights reserved.
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E. RANTA ET AL.
Our task here is to take Bulmer’s (1985, 1994) model with density dependence affecting dynamics of the populations reproducing either semelparously or iteroparously. Within this framework, we then address the question, to what extent environmental variability acting on survival and fecundity favours either of the two reproductive strategies? Density-dependent Dynamics and Environmental Variability Bulmer (1994), following Charnov & Schaffer (1973), proposed the following scheme to account for the temporal dynamics of semelparous (S), and iteroparous (I) reproducers (notation from Ranta et al., 2000b): NS ðk þ 1Þ ¼ lS NS ðkÞ; NI ðk þ 1Þ ¼ lI NI ðkÞ;
ð1Þ
where lS ¼ PJ bS and lI ¼ PJ vbS þ PA ; N ðkÞ refers to population size in different years, k; and l is the strategy-specific geometric growth rate, and b is the offspring number. The number of offspring produced by semelparous breeders is scaled by a factor of v to obtain the offspring number of iteroparous breeders, bI ¼ vbS : In principle, the scaling parameter can assume any value v40; but often 0ovo1: The proportion of offspring surviving the first year is PJ ; and PA is the adult survival rate of iteroparous breeders (0oPJ o1 and 0oPA o1; PJ oPA ). Equation (1) ignores the significance of density dependence on the dynamics of the populations, but it can be modified to account for density dependence via juvenile survival (Bulmer 1985, 1994) by writing PJ ¼ PJ expfa½bS NS ðkÞ þ vbS NI ðkÞ g:
ð2Þ
Term pJ is the maximum juvenile survival in the absence of competition from other juveniles. Parameter a is a scaling coefficient affecting the maximum values N can achieve (throughout, we assume a ¼ 0:1). By replacing PJ from eqn (2) into eqn (1), a representation of the effect of density dependence is achieved. Bulmer (1994), by using invasion analysis, gives an answer to the question whether iteroparous breeders F when initially rare F will be able to invade a population of semelparous
breeders: he showed that invasion is impossible when the following inequality holds: bS >
vbS 1 PA
or v þ PA o1:
ð3Þ
The same inequality is also the condition for a semelparous mutant strategy to invade a population of iteroparous reproducers. For this to hold, the population dynamics has to be stable (Bulmer, 1994). Equation (1), extended by eqn (2), is a modification of the Ricker equation with r ¼ lnðlÞ: It is well known that the Ricker model yields stable dynamics when 1orp2 (May, 1974, 1975, 1976). Thus, in what follows, we shall take ro2; in order to remain within the range of stable population dynamics. We also assume that inequality (3) holds. Ranta et al. (2000a) have shown that the dynamics of semelparous breeders becomes unstable far sooner than that of iteroparous breeders. Thus, we ensured that rS o2: On top of this system, we shall add noise in order to address whether, under density-dependent dynamics, environmental variability enables long-term persistence of the two life history strategies. In our treatment, the noise can affect the number of offsprings produced or survival of juveniles and adults. The noise, wðkÞ; with a desired interval [MIN y MAX], is generated by a first-order autoregressive process, AR (1). We first generate a noise function as follows (Ripa & Lundberg, 1996): qffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ð4Þ w ðkÞ ¼ bw ðk 1Þ þ s 1 b2 : Here b is the autocorrelation parameter (21obo1), where negative and positive values of b represent negatively and positively autocorrelated time series, respectively. The term s is a normally distributed random variable with mean zero and variance one. Next, time series w0 is scaled as wðkÞ ¼ ðMIN þ MAX Þ=2 þ w0 ðkÞðMAX MIN Þ=5:1516;
ð5Þ
which results in 99% of values fitting into the assigned interval [MIN y MAX]. The values of the time series that do not fit into the interval are
393
LIFE HISTORY EVOLUTION UNDER UNCERTAINTY
replaced by the corresponding boundary values. The final noise time series produced, w; has the desired autocorrelation structure with mean and variance equal to 0.5 and 0.0236, respectively. qffiffiffiffiffiffiffiffiffiffiffiffiffi The term
1 b2 scales the variance of the time
series, so that its true variance is independent of b and the length of the time series (Heino et al., 000). Equation (4) is initiated with a random number drawn from a normal distribution with mean zero and variance one. The effect of noise on juvenile survival rates was implemented as P* J ðkÞ ¼
exp ln PJ ðkÞ=½1 PJ ðkÞ þ eðkÞ ; 1 þ exp ln PJ ðkÞ=½1 PJ ðkÞ þ eðkÞ ð6Þ
where e is drawn from normally distributed random numbers with mean mðkÞ and variance s: For s; we used two values 1.0 and 0.25, referred to as wide and narrow noise around mðkÞ: The variable mðkÞ was generated by eqns (4) and (5) with range 1 to +1. A similar procedure was also applied to adult survival [replace subscript J in equation (6) by A]. To superimpose environmental variability on offspring numbers, we generated random numbers wðkÞ using eqns (4) and (5) from the range 12c to 1 þ c; having two values for c; 0.5 (wide) and 0.1 (narrow). For each time unit k; the offspring numbers were multiplied with this number, wðkÞbS ; wðkÞbI : To evaluate whether long-term joint persistence (also referred to as coexistence) of the two life history strategies is possible or not with environmental noise, we proceeded as follows. First, a semelparous and an iteroparous breeding lineage were initiated by drawing uniform random numbers between 0 and 1. The lineages were allowed to compete for the resources according to eqns (1) and (2) for 5000 generations. The next 1000 generations were then used to sample for the possible persistence of the two life histories in the population. To tally longterm coexistence, we calculated the average population sizes over the 1000 generations. As the Ricker dynamics allows presence of infinitesimally small population densities, we set the criterion for persistence such that the long-term average had to exceed 1 103. As this is an
arbitrary limit, we repeated the process 1000 times, and report here the long-term persistence of the two life history strategies. To us, it is natural to assume that if the population dynamics, as described by eqns (1) and (2), are affected by environmental noise, the noise affects all the different components ( juvenile survival, adult survival, offspring numbers for semelparous and iteroparous life histories) in concert (Benton & Grant, 1999). The Bulmer inequality, eqn (3), is the benchmark against which the noise effect in all components (i.e., three-factor interaction) should be compared. However, for historical reasons (Charlesworth, 1994), we also used the tedious method of assessing persistence scenarios for the two life histories when the environmental variability involves one component only (juvenile survival, adult survival, offspring numbers), or two components in various combinations (PJ and PA ; PJ and offspring number, PA and offspring number b). Because of the relationship bI ¼ vbS we shall, however, assume that environmental variability simultaneously affects offspring numbers of the two life histories. It should be noted that the life histories, as treated here, are pure strategies (Maynard Smith, 1982). Results A straightforward finding is that when the environmental noise is in one component only F either in adult survival rate, juvenile survival rate or in offspring number F the two life history strategies, obeying eqns (1) and (2) in population renewal, cannot coexist. Thus, semelparity is an ESS in the range of parameter values that satisfy the Bulmer (1994) inequality v þ PA o1; independent of the fact that environmental variability affects either b or PJ or PA : The outcome changes radically when the noise simultaneously affects two of the three factors, or all three of them. The parameter space (v and PA ) enabling long-term coexistence is rather wide with noise in adult and juvenile survival and offspring numbers [Fig. 1(a)], as well as with noise in adult survival and offspring numbers [Fig. 1(c)]. When noise affects juvenile survival and offspring numbers, semelparity is an ESS over the entire range of v and PA values
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Fig. 1. The effect of environmental variability on longterm persistence of the two life history strategies. A white area below the line v þ PA ¼ 1 indicates that semelparity is an ESS, a black area that iteroparity is an ESS, and a grey area that coexistence of the two strategies is possible. The jaggedness in the graphed areas is due to the fact that we used 0.05 as the step in the search both for v and PA (for both from 0.05 to 0.95). (a) Noise in adult and juvenile survival and in offsprings numbers; (b) Noise in adult and juvenile survival; (c) Noise in adult survival and in offspring numbers; (d) Noise in juvenie survival and in offsprings numbers.
[Fig. 1(d)]. When the noise affects both adult and juvenile survival three differing parameter regions emerge. The largest region gives semelparity as an ESS, but there is also a narrow range for evolutionarily stable iteroparity. Between these areas, there is an area of v and PA enabling long-term coexistence of the two life history strategies [Fig. 1(b)]. Population sizes in the coexistence range are very rarely high for both semelparity and iteroparity (Fig. 2). Somewhat unexpectedly, the colour of the noise, that is, autocorrelation structure of the noise signal, had only marginal effect on the results (Fig. 2). The ‘‘colour’’ relates to the autocorrelation structure such that the time series with red and blue colour are positively and negatively autocorrelated, respectively, whereas in white time series successive data points are independent of each other
Fig. 2. Long-term averages of population sizes with adult survival PA ¼ 0:05 and examining the offspring number scaling parameter (bI ¼ vbS ) across all possible values of v: The population data are displayed for the three different environmental variability types, white, red and blue noise (here wide noise was used). The different panels (a)–(d) correspond to panels (a)–(d) in Fig. 1. (a) Noise in adult and juvenile survival and in offsprings numbers; (b) Noise in adult and juvenile survival; (c) Noise in adult survival and in offspring numbers; (d) Noise in juvenie survival and in offsprings numbers.
(e.g. Ripa & Lundberg, 1996; Kaitala et al., 1997). We finally investigate the extent to which the noise range affects the outcome. For this purpose, we used two values of s; 1.0 and 0.25 (survival), and two values for c; 0.5 and 0.1 (offspring numbers). With two values for both parameters, we have four combinations, thus we restricted ourselves to one noise colour only: white. The results are as expected: reducing the
LIFE HISTORY EVOLUTION UNDER UNCERTAINTY
noise regime also reduces the possibilities for long-term coexistence of the two life histories (Fig. 3). Experimentation shows that changing the colour of the noise does not have much influence on the outcome.
Discussion In recent papers, Orzack & Tuljapurkar (1989) and Benton & Grant (1999) demonstrate that variability does not select towards semelparity or iteroparity alone. Selection towards one or the other reproductive strategy also depends on factors such as density dependence, and the vital rates that are affected by the environmental variation, as is also observed in our research.
Fig. 3. The effect of the range of environmental variability (white noise only; four different combinations of s and c values) on long-term coexistence and population size of the two life history strategies. The results for noise both in juvenile mortality and in offspring numbers are not displayed, as semelparity is an ESS and iteroparity cannot invade in the range of noises implemented. (a) Noise in adult and juvenile survival and in offsprings numbers; (b) Noise in adult and juvenile survival; (c) Noise in adult survival and in offspring numbers.
395
Orzack & Tuljapurkar (1989) compare a set of different iteroparous life histories assuming density independence and equal growth rates for different types. In their study, reproduction is either concentrated in early life, or spread over the whole lifetime, or concentrated in the end of life. They conclude that, for small environmental variation, a life history with early reproduction and a short life has the highest stochastic growth rate. An advantage to a life history under intermediate environmental variation will depend on the correlation of the survival and fecundity values. For extreme environmental variation, the life history with long life is favoured. If the survival and fecundity values are positively correlated, concentrated reproduction towards the end of life history with the largest stochastic growth rates is favoured. The potential for coexistence between life history strategies in variable environments, as presented here, is actually much larger than anticipated by Orzack & Tuljapurkar (1989). Benton & Grant (1999) incorporate density dependence into their study of reproductive strategies. They assume a trade-off between fecundity and survival. They conclude that environmentally driven variations in vital rates have a substantial impact on the outcome of the optimal life history. Of interest in the context of our study is the fact that the size and direction of a change in reproductive effort depends on which vital rate is sensitive to the environmental variation and how the vital rates interact under density dependence. When fecundity varies more than survival, the optimal reproductive effort tends to increase while the opposite is true if survival varies more than fecundity (i.e. there is selection towards iteroparity). Our results support these conclusions. Both Orzack & Tuljapurkar (1989) and Benton & Grant (1999) ignore the effects of temporal structure in the environments. The autocorrelation structure of environmental time series has been shown to have a distinct effect on the intrinsic growth rate (Kaitala et al., 1997). Also, noise colour in dispersal variability enabled coexistence of two different reproductive strategies in space (Tesar et al., 2002). In the present explorations, however, autocorrelation structure in the environmental noise seemed to
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have a minuscule effect on the coexistence of the two life histories. The model of Caswell & Cohen (1995), as compared to ours, differed mainly in assumptions concerning space and noise. While their environmental variation is an on/off process, our environmental fluctuations also took values between the two extremes. Thus, noise, as implemented by us, refers to temporal variability of some climatic variable, like temperature or humidity. The abrupt character of the environmental fluctuations in the model by Caswell & Cohen (1995)Fprobably untypical of most environmental variationFmay have increased the autocorrelation effect of their environmental time series and caused a larger difference between the different colour treatments. Nevertheless, Caswell & Cohen (1995) note that the persistence of the competitively inferior type was hindered in very red environmental fluctuations, that is, blue environments better promote coexistence. From an investigation on the coexistence of a semelparous and an iteroparous population with spatial structure, we came to a similar conclusion (Tesar et al., 2002). However, based on our present results, we suggest that the effect of noise colour in environmental fluctuations is smaller than anticipated for the coexistence of iteroparity and semelparity. This suggests that the conclusions drawn by Orzack & Tuljapurkar (1989) for density-independent systems, as well as Benton & Grant (1999) for density-dependent systems, despite ignoring the autocorrelation problem, may generally be valid when space does not play a role (Ranta et al. 2000b). We stress, however, that, within the parameter range defined by the Bulmer inequality, the coexistence of iteroparity and semelparity is greatly enhanced when noise affects life history traits. REFERENCES Benton, T. G. & Grant, A. (1999). Optimal reproductive effort in stochastic, density-dependent environments. Evolution 53, 677–688. Bulmer, M. G. (1985). Selection of iteroparity in a variable environment. Am. Nat. 126, 63–71.
Bulmer, M. G. (1994). Theoretical Evolution and Ecology. Sunderland, MA: Sinauer Associates. Caswell, H. & Cohen, J. E. (1995). Red, white and blue: environmental variance spectra and coexistence in metapopulations. J. theor. Biol. 176, 301–316. Charlesworth, B. (1980). Evolution in age-structured populations, 1st Edn. Cambridge, U.K.: Cambridge University Press. Charlesworth, B. (1994). Evolution in age-structured populations, 2nd Edn. Cambridge, U.K.: Cambridge University Press. Charnov, E. L. & Schaffer, W. M. (1973). Life-history consequences of natural selection: Cole’s result revisited. Am. Nat. 107, 791–793. Cole, L. C. (1954) The population consequences of life history phenomena. Q. Rev. Biol. 29, 103–137. Heino, M., Ripa, J. & Kaitala, V. (2000). Extinction risk under coloured environmental noise. Ecography 23, 177–184. Kaitala, V., Lundberg, P., Ripa, J. & Ylikarjula, J. (1997). Red, blue and green: dyeing population dynamics. Ann. Zool. Fenn. 34, 217–228. May, R. M. (1974). Biological populations with nonoverlapping generations: stable points, stable cycles and chaos. Science 186, 645–647. May, R. M. (1975). Biological populations obeying difference equations: stable points, stable cycles and chaos. J. theor. Biol. 51, 511–524. May, R. M. (1976). Theoretical Ecology. Principles and Applications. Oxford, UK: Blackwell Scientific. Maynard Smith, J. (1982). Evolution and the theory of games. Cambridge, UK: Cambridge University Press. Murphy, G. I. (1968). Pattern in life history and the environment. Am. Nat. 102, 391–403. Orzack, S. H. & Tuljapurkar, S. (1989). Population dynamics in variable environments. VII The demography and evolution of iteroparity. Am. Nat. 133, 901–923. Ranta, E., Kaitala, V., Alaja, S. & Tesar D. (2000a). Nonlinear dynamics and the evolution of semelparous and iteroparous reproductive strategies. Am. Nat. 155, 294–300. Ranta, E., Tesar, D., Alaja, S. & Kaitala, V. (2000b). Does evolution of iteroparous and semelparous reproduction call for spatially structured systems? Evolution 54, 145–150. Ripa, J. & Lundberg, P. (1996). Noise colour and the risk of population extinctions. Proc. R. Soc. London, Ser. B 263, 1751–1753. Roff, D. A. (1992) The Evolution of Life Histories: Theory and Analysis. New York: Chapman & Hall. Stearns, S. C. (1976). Life history tactics: a review of the ideas. Q. Rev. Biol. 51, 3–47. Stearns, S. C. (1977). The Evolution of life history traits: a critique of the theory and a review of the data. Ann. Rev. Ecol. Syst. 8, 145–171. Stearns, S. C. (1992). The Evolution of Life Histories. Oxford, UK: Oxford University Press. Tesar, D., Kaitala, V. & Ranta, E. (2002). Stochasticity and spatial coexistence of semelparity and iteroparity as life-histories. Evol. Ecol. (accepted).
Environmental Variability and Semelparity vs. Iteroparity as Life Histories Esa Ranta*w, David Tesarz and Veijo Kaitalaz wDepartment of Ecology and Systematics, Integrative Ecology Unit, Division of Population Biology, University of Helsinki, P.O. Box 65 (Viikinkaari 1), FIN-00014, Finland and zDepartment of Biological and Environmental Science, University of Jyva¨skyla¨, P.O. Box 35, FIN-40351 Jyva¨skyla¨, Finland (Received on 8 May 2001, Accepted in revised form on 11 February 2002)
Research on the evolution of life histories addresses the topic of fitness trade-offs between semelparity (reproducing once in a lifetime) and iteroparity (repeated reproductive bouts per lifetime). Bulmer (1994) derived the relationship v þ PA o1 (PA is the adult survival; vbS and bS are the offspring numbers for iteroparous and semelparous breeding strategies, respectively), under which a resident semelparous population cannot be invaded by an iteroparous mutant when the underlying population dynamics are stable. We took Bulmer’s population dynamics, and added noise in juvenile and adult survival and in offspring numbers. Long-term coexistence of the two strategies is possible in much of the parameter region of v þ PA o1 when noise occurs simultaneously in all three components, or (more restricted) when it affects juvenile and adult survival or adult survival and offspring numbers. Iteroparity cannot persist when the environmental variability involves juvenile survival and offspring numbers, or when the noise acts on the three components separately. r 2002 Elsevier Science Ltd. All rights reserved.
Introduction Reproducing once (semelparity) or several times (iteroparity) during a lifetime is one of the central questions in the evolution of life histories (Cole, 1954; Roff, 1992; Stearns, 1992). The major conclusion of previous research, using simple population models, can be summarized by noting that in many scenarios semelparity is a superior life history over iteroparity. In a striking contrast with this theoretical outcome, numerous plant and animal species reproduce iteroparously over their lifespan, which often extends over several reproductive seasons (e.g. Roff, 1992). The disagreement between theory and nature, also known as Cole’s paradox, has n Corresponding author: Fax: +352-9-191-287-01. E-mail address: csa.ranta@helsinki.fi (E. Ranta).
0022-5193/02/$35.00/0
puzzled evolutionary ecologists for the past five decades. Major features of previous research are summarized by Roff (1992), Stearns (1992) and Charlesworth (1994). In his review, Charlesworth (1980, 1994) concluded that iteroparity is favoured over semelparity in fluctuating environments, in which either offspring number per female or juvenile survival varies (Murphy, 1968; Stearns, 1976, 1977). On the other hand, Bulmer (1985, 1994) was the first to note that much of the research of relative evolutionary merits of semelparity over iteroparity had ignored the relevance of density dependence in affecting vital parameters of population renewal. Thus, Charlesworth’s (1994) discussion on semelparity and iteroparity also largely revolves around the issue of the evolution of the two contrasting life histories in density-independent environments. r 2002 Elsevier Science Ltd. All rights reserved.
392
E. RANTA ET AL.
Our task here is to take Bulmer’s (1985, 1994) model with density dependence affecting dynamics of the populations reproducing either semelparously or iteroparously. Within this framework, we then address the question, to what extent environmental variability acting on survival and fecundity favours either of the two reproductive strategies? Density-dependent Dynamics and Environmental Variability Bulmer (1994), following Charnov & Schaffer (1973), proposed the following scheme to account for the temporal dynamics of semelparous (S), and iteroparous (I) reproducers (notation from Ranta et al., 2000b): NS ðk þ 1Þ ¼ lS NS ðkÞ; NI ðk þ 1Þ ¼ lI NI ðkÞ;
ð1Þ
where lS ¼ PJ bS and lI ¼ PJ vbS þ PA ; N ðkÞ refers to population size in different years, k; and l is the strategy-specific geometric growth rate, and b is the offspring number. The number of offspring produced by semelparous breeders is scaled by a factor of v to obtain the offspring number of iteroparous breeders, bI ¼ vbS : In principle, the scaling parameter can assume any value v40; but often 0ovo1: The proportion of offspring surviving the first year is PJ ; and PA is the adult survival rate of iteroparous breeders (0oPJ o1 and 0oPA o1; PJ oPA ). Equation (1) ignores the significance of density dependence on the dynamics of the populations, but it can be modified to account for density dependence via juvenile survival (Bulmer 1985, 1994) by writing PJ ¼ PJ expfa½bS NS ðkÞ þ vbS NI ðkÞ g:
ð2Þ
Term pJ is the maximum juvenile survival in the absence of competition from other juveniles. Parameter a is a scaling coefficient affecting the maximum values N can achieve (throughout, we assume a ¼ 0:1). By replacing PJ from eqn (2) into eqn (1), a representation of the effect of density dependence is achieved. Bulmer (1994), by using invasion analysis, gives an answer to the question whether iteroparous breeders F when initially rare F will be able to invade a population of semelparous
breeders: he showed that invasion is impossible when the following inequality holds: bS >
vbS 1 PA
or v þ PA o1:
ð3Þ
The same inequality is also the condition for a semelparous mutant strategy to invade a population of iteroparous reproducers. For this to hold, the population dynamics has to be stable (Bulmer, 1994). Equation (1), extended by eqn (2), is a modification of the Ricker equation with r ¼ lnðlÞ: It is well known that the Ricker model yields stable dynamics when 1orp2 (May, 1974, 1975, 1976). Thus, in what follows, we shall take ro2; in order to remain within the range of stable population dynamics. We also assume that inequality (3) holds. Ranta et al. (2000a) have shown that the dynamics of semelparous breeders becomes unstable far sooner than that of iteroparous breeders. Thus, we ensured that rS o2: On top of this system, we shall add noise in order to address whether, under density-dependent dynamics, environmental variability enables long-term persistence of the two life history strategies. In our treatment, the noise can affect the number of offsprings produced or survival of juveniles and adults. The noise, wðkÞ; with a desired interval [MIN y MAX], is generated by a first-order autoregressive process, AR (1). We first generate a noise function as follows (Ripa & Lundberg, 1996): qffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ð4Þ w ðkÞ ¼ bw ðk 1Þ þ s 1 b2 : Here b is the autocorrelation parameter (21obo1), where negative and positive values of b represent negatively and positively autocorrelated time series, respectively. The term s is a normally distributed random variable with mean zero and variance one. Next, time series w0 is scaled as wðkÞ ¼ ðMIN þ MAX Þ=2 þ w0 ðkÞðMAX MIN Þ=5:1516;
ð5Þ
which results in 99% of values fitting into the assigned interval [MIN y MAX]. The values of the time series that do not fit into the interval are
393
LIFE HISTORY EVOLUTION UNDER UNCERTAINTY
replaced by the corresponding boundary values. The final noise time series produced, w; has the desired autocorrelation structure with mean and variance equal to 0.5 and 0.0236, respectively. qffiffiffiffiffiffiffiffiffiffiffiffiffi The term
1 b2 scales the variance of the time
series, so that its true variance is independent of b and the length of the time series (Heino et al., 000). Equation (4) is initiated with a random number drawn from a normal distribution with mean zero and variance one. The effect of noise on juvenile survival rates was implemented as P* J ðkÞ ¼
exp ln PJ ðkÞ=½1 PJ ðkÞ þ eðkÞ ; 1 þ exp ln PJ ðkÞ=½1 PJ ðkÞ þ eðkÞ ð6Þ
where e is drawn from normally distributed random numbers with mean mðkÞ and variance s: For s; we used two values 1.0 and 0.25, referred to as wide and narrow noise around mðkÞ: The variable mðkÞ was generated by eqns (4) and (5) with range 1 to +1. A similar procedure was also applied to adult survival [replace subscript J in equation (6) by A]. To superimpose environmental variability on offspring numbers, we generated random numbers wðkÞ using eqns (4) and (5) from the range 12c to 1 þ c; having two values for c; 0.5 (wide) and 0.1 (narrow). For each time unit k; the offspring numbers were multiplied with this number, wðkÞbS ; wðkÞbI : To evaluate whether long-term joint persistence (also referred to as coexistence) of the two life history strategies is possible or not with environmental noise, we proceeded as follows. First, a semelparous and an iteroparous breeding lineage were initiated by drawing uniform random numbers between 0 and 1. The lineages were allowed to compete for the resources according to eqns (1) and (2) for 5000 generations. The next 1000 generations were then used to sample for the possible persistence of the two life histories in the population. To tally longterm coexistence, we calculated the average population sizes over the 1000 generations. As the Ricker dynamics allows presence of infinitesimally small population densities, we set the criterion for persistence such that the long-term average had to exceed 1 103. As this is an
arbitrary limit, we repeated the process 1000 times, and report here the long-term persistence of the two life history strategies. To us, it is natural to assume that if the population dynamics, as described by eqns (1) and (2), are affected by environmental noise, the noise affects all the different components ( juvenile survival, adult survival, offspring numbers for semelparous and iteroparous life histories) in concert (Benton & Grant, 1999). The Bulmer inequality, eqn (3), is the benchmark against which the noise effect in all components (i.e., three-factor interaction) should be compared. However, for historical reasons (Charlesworth, 1994), we also used the tedious method of assessing persistence scenarios for the two life histories when the environmental variability involves one component only (juvenile survival, adult survival, offspring numbers), or two components in various combinations (PJ and PA ; PJ and offspring number, PA and offspring number b). Because of the relationship bI ¼ vbS we shall, however, assume that environmental variability simultaneously affects offspring numbers of the two life histories. It should be noted that the life histories, as treated here, are pure strategies (Maynard Smith, 1982). Results A straightforward finding is that when the environmental noise is in one component only F either in adult survival rate, juvenile survival rate or in offspring number F the two life history strategies, obeying eqns (1) and (2) in population renewal, cannot coexist. Thus, semelparity is an ESS in the range of parameter values that satisfy the Bulmer (1994) inequality v þ PA o1; independent of the fact that environmental variability affects either b or PJ or PA : The outcome changes radically when the noise simultaneously affects two of the three factors, or all three of them. The parameter space (v and PA ) enabling long-term coexistence is rather wide with noise in adult and juvenile survival and offspring numbers [Fig. 1(a)], as well as with noise in adult survival and offspring numbers [Fig. 1(c)]. When noise affects juvenile survival and offspring numbers, semelparity is an ESS over the entire range of v and PA values
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Fig. 1. The effect of environmental variability on longterm persistence of the two life history strategies. A white area below the line v þ PA ¼ 1 indicates that semelparity is an ESS, a black area that iteroparity is an ESS, and a grey area that coexistence of the two strategies is possible. The jaggedness in the graphed areas is due to the fact that we used 0.05 as the step in the search both for v and PA (for both from 0.05 to 0.95). (a) Noise in adult and juvenile survival and in offsprings numbers; (b) Noise in adult and juvenile survival; (c) Noise in adult survival and in offspring numbers; (d) Noise in juvenie survival and in offsprings numbers.
[Fig. 1(d)]. When the noise affects both adult and juvenile survival three differing parameter regions emerge. The largest region gives semelparity as an ESS, but there is also a narrow range for evolutionarily stable iteroparity. Between these areas, there is an area of v and PA enabling long-term coexistence of the two life history strategies [Fig. 1(b)]. Population sizes in the coexistence range are very rarely high for both semelparity and iteroparity (Fig. 2). Somewhat unexpectedly, the colour of the noise, that is, autocorrelation structure of the noise signal, had only marginal effect on the results (Fig. 2). The ‘‘colour’’ relates to the autocorrelation structure such that the time series with red and blue colour are positively and negatively autocorrelated, respectively, whereas in white time series successive data points are independent of each other
Fig. 2. Long-term averages of population sizes with adult survival PA ¼ 0:05 and examining the offspring number scaling parameter (bI ¼ vbS ) across all possible values of v: The population data are displayed for the three different environmental variability types, white, red and blue noise (here wide noise was used). The different panels (a)–(d) correspond to panels (a)–(d) in Fig. 1. (a) Noise in adult and juvenile survival and in offsprings numbers; (b) Noise in adult and juvenile survival; (c) Noise in adult survival and in offspring numbers; (d) Noise in juvenie survival and in offsprings numbers.
(e.g. Ripa & Lundberg, 1996; Kaitala et al., 1997). We finally investigate the extent to which the noise range affects the outcome. For this purpose, we used two values of s; 1.0 and 0.25 (survival), and two values for c; 0.5 and 0.1 (offspring numbers). With two values for both parameters, we have four combinations, thus we restricted ourselves to one noise colour only: white. The results are as expected: reducing the
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noise regime also reduces the possibilities for long-term coexistence of the two life histories (Fig. 3). Experimentation shows that changing the colour of the noise does not have much influence on the outcome.
Discussion In recent papers, Orzack & Tuljapurkar (1989) and Benton & Grant (1999) demonstrate that variability does not select towards semelparity or iteroparity alone. Selection towards one or the other reproductive strategy also depends on factors such as density dependence, and the vital rates that are affected by the environmental variation, as is also observed in our research.
Fig. 3. The effect of the range of environmental variability (white noise only; four different combinations of s and c values) on long-term coexistence and population size of the two life history strategies. The results for noise both in juvenile mortality and in offspring numbers are not displayed, as semelparity is an ESS and iteroparity cannot invade in the range of noises implemented. (a) Noise in adult and juvenile survival and in offsprings numbers; (b) Noise in adult and juvenile survival; (c) Noise in adult survival and in offspring numbers.
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Orzack & Tuljapurkar (1989) compare a set of different iteroparous life histories assuming density independence and equal growth rates for different types. In their study, reproduction is either concentrated in early life, or spread over the whole lifetime, or concentrated in the end of life. They conclude that, for small environmental variation, a life history with early reproduction and a short life has the highest stochastic growth rate. An advantage to a life history under intermediate environmental variation will depend on the correlation of the survival and fecundity values. For extreme environmental variation, the life history with long life is favoured. If the survival and fecundity values are positively correlated, concentrated reproduction towards the end of life history with the largest stochastic growth rates is favoured. The potential for coexistence between life history strategies in variable environments, as presented here, is actually much larger than anticipated by Orzack & Tuljapurkar (1989). Benton & Grant (1999) incorporate density dependence into their study of reproductive strategies. They assume a trade-off between fecundity and survival. They conclude that environmentally driven variations in vital rates have a substantial impact on the outcome of the optimal life history. Of interest in the context of our study is the fact that the size and direction of a change in reproductive effort depends on which vital rate is sensitive to the environmental variation and how the vital rates interact under density dependence. When fecundity varies more than survival, the optimal reproductive effort tends to increase while the opposite is true if survival varies more than fecundity (i.e. there is selection towards iteroparity). Our results support these conclusions. Both Orzack & Tuljapurkar (1989) and Benton & Grant (1999) ignore the effects of temporal structure in the environments. The autocorrelation structure of environmental time series has been shown to have a distinct effect on the intrinsic growth rate (Kaitala et al., 1997). Also, noise colour in dispersal variability enabled coexistence of two different reproductive strategies in space (Tesar et al., 2002). In the present explorations, however, autocorrelation structure in the environmental noise seemed to
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have a minuscule effect on the coexistence of the two life histories. The model of Caswell & Cohen (1995), as compared to ours, differed mainly in assumptions concerning space and noise. While their environmental variation is an on/off process, our environmental fluctuations also took values between the two extremes. Thus, noise, as implemented by us, refers to temporal variability of some climatic variable, like temperature or humidity. The abrupt character of the environmental fluctuations in the model by Caswell & Cohen (1995)Fprobably untypical of most environmental variationFmay have increased the autocorrelation effect of their environmental time series and caused a larger difference between the different colour treatments. Nevertheless, Caswell & Cohen (1995) note that the persistence of the competitively inferior type was hindered in very red environmental fluctuations, that is, blue environments better promote coexistence. From an investigation on the coexistence of a semelparous and an iteroparous population with spatial structure, we came to a similar conclusion (Tesar et al., 2002). However, based on our present results, we suggest that the effect of noise colour in environmental fluctuations is smaller than anticipated for the coexistence of iteroparity and semelparity. This suggests that the conclusions drawn by Orzack & Tuljapurkar (1989) for density-independent systems, as well as Benton & Grant (1999) for density-dependent systems, despite ignoring the autocorrelation problem, may generally be valid when space does not play a role (Ranta et al. 2000b). We stress, however, that, within the parameter range defined by the Bulmer inequality, the coexistence of iteroparity and semelparity is greatly enhanced when noise affects life history traits. REFERENCES Benton, T. G. & Grant, A. (1999). Optimal reproductive effort in stochastic, density-dependent environments. Evolution 53, 677–688. Bulmer, M. G. (1985). Selection of iteroparity in a variable environment. Am. Nat. 126, 63–71.
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