1. Under the hypothesis of environmental buffering, populations are expected to minimize the variance of the most influential vital rates; however, this may not be a universal principle. Species with a life span <1 year may be less likely to exhibit buffering because of temporal or seasonal variability in vital rate sensitivities. Further, plasticity in vital rates may be adaptive for species in a variable environment with reliable cues.
2. We tested for environmental buffering and plasticity in vital rates using stage-structured matrix models from long-term data sets in four species of grassland rodents. We used periodic matrices to estimate stochastic elasticity for each vital rate and then tested for correlations with a standardized coefficient of variation for each rate.
3. We calculated stochastic elasticities for individual months to test for an association between increased reproduction and the influence of reproduction, relative to survival, on the population growth rate.
4. All species showed some evidence of buffering. The elasticity of vital rates of Peromyscus leucopus (Rafinesque, 1818), Sigmodon hispidus Say & Ord, 1825 and Microtus ochrogaster (Wagner, 1842) was negatively related to vital rate CV. Elasticity and vital rate CV were negatively related in Peromyscus maniculatus (Wagner, 1845), but the relationship was not statistically significant. Peromyscus leucopus and M. ochrogaster showed plasticity in vital rates; reproduction was higher following months where elasticity for reproduction exceeded that of survival.
5. Our results suggest that buffering is common in species with fast life histories; however, some populations that exhibit buffering are capable of responding to short-term variability in environmental conditions through reproductive plasticity.
Variation in the population growth rate generally reduces realized population growth relative to the expected value without variation (Lewontin & Cohen 1969). Variation in population growth rate is the result of variation in the underlying vital rates (e.g. survival and fecundity; Tuljapukar 1982); however, the population growth rate is differentially sensitive to variation in these vital rates (Caswell 1978; Caswell & Trevisan 1994). Minimizing the variance associated with the most influential vital rates is expected to increase realized population growth rate by reducing the impact of environmental variability. This has led to the hypothesis of environmental buffering (or canalization): vital rates with the highest sensitivities will be those under the greatest selection pressure and in turn are expected to have the lowest variability (Pfister 1998; Gaillard & Yoccoz 2003).
Previous studies of buffering have focused on taxa with relatively long generation time (>1 year) and modest annual reproduction (Pfister 1998; Gaillard & Yoccoz 2003; Rotella et al. 2012); however, few empirical tests of this hypothesis have been conducted for rodents with fast life histories. Species with short generation times are more sensitive to climate-induced variability in vital rates (Morris et al. 2008; Dalgleish, Koons & Adler 2010), which suggests that buffering environmental variability would be likely. Further, for animals with short generation times, the temporal scale of the analysis may impact the results. A temporal scale of a single month would be appropriate for rodents as this approximately corresponds to a single breeding attempt and would be the temporal scale at which reproductive decisions would be made. In periodic environments, vital rates interact such that the relative influence of a vital rate in a single time-step (i.e. a month) may differ from that estimated for a longer time period (i.e. a season, Caswell & Trevisan 1994).
Environmental buffering and vital rate plasticity often are presented as opposing life-history strategies (Reed et al. 2010), but this is not necessarily the case (Koons et al. 2009). Environmental variability may result in interannual variation in vital rate sensitivities (Chirakkal & Gerber 2010). The most influential vital rate in an individual month may differ from that estimated from the long-term mean of the vital rate. Therefore, the best strategy in that month may differ from the usual strategy. For example, the environmental conditions of a specific month may indicate that the most benefit would be gained from survival even though, on average, reproduction has the most influence on the population. Interannual variation in vital rate sensitivity could favour adaptive plasticity in vital rates (Reed et al. 2010). Buffering of vital rates may suggest a lack of plasticity, particularly in vital rates that represent a life-history trade-off (e.g. survival and reproduction).
The ability of populations to tolerate environmental variability has received considerable attention because of projected changes in regional climate (Boyce et al. 2006). Species with short generation times are expected to be more sensitive to changes in climate (Morris et al. 2008) but also may be less likely to exhibit buffering (Jongejans et al. 2010). Further, buffering may indicate a lack of vital rate plasticity (Reed et al. 2010) which could be maladaptive in a changing environment. We use a long-term data set on four species of small mammals – prairie vole [Microtus ochrogaster (Wagner, 1842)], cotton rat (Sigmodon hispidus Say & Ord, 1825), white-footed mouse [Peromyscus leucopus (Rafinesque, 1818)] and deer mouse [Peromyscus maniculatus (Wagner, 1845)] – to assess the prevalence of buffering and plasticity in the vital rates of mammals with a short generation time. We expect to find evidence of buffering in our species; however, S. hispidus may be the least likely to exhibit buffering as it has only recently (c. 80 years) moved into our study region (Cameron & Spencer 1981). The taxon is native to subtropical regions, and species that have expanded their range are likely to exhibit demographic lability (Koons et al. 2009). We expect that any species that exhibit vital rate buffering will exhibit less plasticity in reproductive effort.
Materials and methods
Data used in this analysis were collected from 1975 to 2003 at the Nelson Environmental Study Area, located c. 14 km NE of Lawrence, Kansas, MO, USA. Small mammals were trapped monthly on a 2·5 ha grid that consisted of 99 trap stations set at 15-m intervals; data on sex, mass and reproductive condition were recorded for each individual (Swihart & Slade 1990). Peromyscus leucopus and P. maniculatus were not individually marked until 1989. Therefore, our data set for P. leucopus includes 1989–2003 and for P. maniculatus 1989–1997 after which the species became uncommon on our study area. We classified individuals into size classes [3 for S. hispidus (1: ≤60 g; 2: 61 g–110 g; 3: >110 g) and M. ochrogaster (1: ≤20 g; 2: 21–32 g; 3: >32 g) and 2 for P. maniculatus and P. leucopus (1: ≤15 g; 2: >15 g)] based on body mass (Reed & Slade 2006a,b, 2007) that roughly correspond to juvenile, subadult and adult animals. We used a multi-state model in program MARK to estimate survival and transition among size classes in each month (Buckland, Burnham & Augustin 1997; White & Burnham 1999; Lebreton et al. 2009). We estimated the probability of an individual being pregnant using external reproductive data collected in the field (Reed & Slade 2008). To estimate litter size in S. hispidus, we used the mass of each pregnant female and a published regression equation (Campbell & Slade 1995) to arrive at an estimate of litter size for each female and then calculated the mean litter size for each month. We used published estimates of mean litter size in M. ochrogaster (Rose & Gaines 1978), P. leucopus (Lackey, Huckaby & Ormiston 1985) and P. maniculatus (Meyers & Master 1983). We used reverse capture histories and a multi-strata model to estimate immigration of stage 2 and 3 animals into the populations (Nichols et al. 2000). We estimated the number of immigrants per adult female by multiplying the proportional recruitment obtained from the MARK output by the number of individuals in that stage and then dividing the estimated total number of recruits by the number of adult females (Reed & Slade 2008). Previous simulations have indicated that including immigration is necessary to achieve realistic population growth rates.
We included correlation among vital rates in models using the methods of Morris & Doak (2002). We first estimated correlation coefficients for each pair of vital rates. We then assembled a correlation matrix of these values and checked that the matrix was positive semi-definite by calculating the eigenvalues of each matrix. If one or more of the eigenvalues was negative, we set the negative eigenvalues to 0 and multiplied a diagonal matrix (D) of eigenvalues by a matrix of the right eigenvectors of the correlation matrix (W) and its inverse: Cm = W*D*W−1. This covariance matrix then was modified to a correlation matrix by:
We used these correlation matrices to draw correlated normal values for our vital rates and then used an incomplete beta function to convert these values to beta variates. Because we had no empirical estimates of litter size, we used the mean value for each species in all models. We combined the selected vital rates to parameterize monthly, single-sex (female), stage-classified matrices. Values for vital rates were combined into a stage-structured 3 × 3 matrix for S. hispidus and M. ochrogaster and a 2 × 2 matrix for P. maniculatus and P. leucopus (Table 1; Tables S1 and S2, Supporting information). Because the analytical estimation of stochastic elasticities is sensitive to high levels of variation, we estimated vital rate elasticities using a simulation method (Morris & Doak 2004) and assessed the relationship between vital rate elasticity and variance based on periodic matrices (Caswell & Trevisan 1994).
Table 1. Formulae for calculating transition probabilities from vital rates
aTransitions are: F, fecundity; S, survival and remaining in the same stage; G, survival and growth into the next stage. Subscripts indicate stage. bVital rates are indicated by the following symbols: σ survival; γ growth into a larger stage; β probability of being pregnant; ι per-capita immigration; and ψ estimated litter size.
(1 + ι2) σ2β2ψ2
(1 + ι3) σ3β3ψ3
σ1 (1−γ12) (1−γ13)
σ1 (γ12) (1−γ13)
(1 + ι2) σ2 (1−γ23)
σ1 (γ13) (1−γ12)
(1 + ι2) σ2γ23
(1 + ι3) σ3
All of our simulations were performed with matlab® (MathWorks, Natick, MA, USA) using code modified from Morris & Doak (2002). We calculated monthly means for stage-specific survival, transition between stages and reproductive probability and then corrected for sampling variance using White’s (2000) variance discounting method. We estimated elasticities for each vital rate by simulating the population and making small, proportional changes to each vital rate within a month. We projected the population through a periodic matrix (Ai) that was the product of the transition matrix of the current and 11 subsequent months. We generated the matrices by drawing correlated values for all vital rates where our correlation included serial correlation in vital rates among months. We synthesized transition matrices for each month and post-multiplied the twelve matrices together:
We projected the initial population vector through this matrix and repeated the process for 1000 years.
We first estimated a baseline stochastic growth rate by generating a series of 1000 years and projected an initial population vector, the stable stage vector, through the series of matrices without changing the mean for any vital rate. We then estimated the stochastic growth rate as the mean growth rate over the 1000-year simulation. This was repeated for every month and resulted in a baseline stochastic growth rate for each month. To estimate elasticity for a vital rate, we increased the mean of the rate in a single month by 0·0001, drew correlated values for all vital rates and synthesized the 12 monthly matrices from these values and calculated Ai. We projected the population by post-multiplying a population vector by the periodic transition matrix. The resulting vector was then used to project the population through the next periodic projection matrix. The population was projected through a series of 1000 years of randomly generated matrices, and we estimated the stochastic growth rate as the mean growth rate for the simulation. We calculated the vital rate sensitivity as:
and converted to elasticity by:
where Δλs is the difference between the stochastic growth rate between the perturbed matrices and the baseline stochastic growth rate, Δxi is the change in the vital rate, and xi is the mean value of a vital rate. This process was repeated for each vital rate other than litter size for each month. Thus, we produced estimates of vital rate elasticity for each vital rate (10 for M. ochrogaster and S hispidus, 5 for Peromyscus spp.) for each of the twelve months. We calculated the CV for each vital rate from our estimate of process variance and divided this by the maximum possible CV; this measures the percentage of possible variation that is present in the population and corrects for spurious correlations between Ei and variance (Morris & Doak 2004).
We tested for relationships between Ei and the CV of vital rates using common correlation coefficients (Miller et al. 2011). For each species, we first calculated correlation coefficients for each month and then tested whether the correlation coefficients differed among months. If we found no evidence of a monthly difference, we calculated a common correlation coefficient for each species and tested whether this value was significantly different from zero using a randomization test where we randomly paired values of Ei and CV without replacement to generate a test statistic based on 10 000 randomizations.
Interannual variation in monthly elasticity
For each month of the breeding season (March–November), we tested for an association between reproductive effort and the relative influence of reproduction on the population growth rate. We used estimates of survival and reproduction for individual months (e.g. June 1979) to synthesize matrices for each month of the study for which we had estimates of all vital rates (M. ochrogaster n = 142; S. hispidus n = 102; P. leucopus n = 87; P. maniculatus n = 45). Survival estimates were from our MARK analysis and the variance from the variance components procedure within MARK. We used the same correlation matrices from the mean matrices and assumed that patterns of correlation were consistent across the duration of the study. We estimated Ei as above but used vital rate and process variance estimates from a single month rather than the mean matrix. This produced estimates of vital rate elasticity for each individual month of our study. For each month, we compared the rank-order Ei for large-adult survival and reproduction. Ei values for reproduction being higher than survival imply that population growth would be maximized by increasing reproductive effort. We then determined whether the proportion of reproductive females in the following month was greater than the average for that month. We used contingency tables to test for an association between the rank of Ei for reproduction relative to that of survival and increased reproduction in the subsequent month. We tested for this association for all months combined and separated by seasons and used Fisher’s exact test for any test with expected cell counts ≤5. We then calculated odds ratios for all associations as an estimate of effect size. A statistically significant result with an odds ratio >1 indicates that reproductive effort was greater than average following months with high relative elasticity for reproduction.
The pattern of Ei was relatively consistent for M. ochrogaster and P. leucopus in all months of the year (Fig. 1). Survival of large-adult M. ochrogaster potentially was the most influential rate in all 12 months, followed by survival of the smallest stage and reproduction of the largest stage. The pattern of elasticity was similar in P. leucopus. Survival of the largest stage had the highest elasticity followed by survival of the smaller stage with little variability in the rank order of other Ei. Survival of adult P. maniculatus was the most influential vital rate in all months, but the elasticity for reproduction also was high in most months. Conversely, the rank order of Ei for S. hispidus (Fig. 2) and P. maniculatus showed considerable variation among months. Elasticity for survival of large adults was highest among vital rates for S. hispidus in spring and early summer, but was relatively low in autumn and winter. Elasticity for survival of small adults (stage 2) was highest among vital rates in autumn and winter.
We found no evidence that correlations differed among months for any species (M. ochrogaster U = 3·6; P > 0·9; S. hispidus U = ·7; P > 0·9; P. leucopus U = 3·3; P > 0·9; P. maniculatus U = 1·0, P > 0·9). Therefore, we calculated common correlations for each species across months. We found significant negative relationships between elasticity and CV for M. ochrogaster ( = −0·17; P = 0·05), S. hispidus ( = −0·19; P = 0·04) and P. leucopus ( = −0·34; P < 0·01). The relationship between elasticity and CV for P. maniculatus was negative, but the common correlation coefficient was not statistically significant ( = −0·15; P = 0·2).
The proportion of pregnant female M. ochrogaster was greater following months where reproduction had a higher elasticity than survival in summer (P = 0·04), autumn (P = 0·003) and annually (P = 0·002; Table 2). Reproduction in P. leucopus increased when the elasticity of reproduction was higher than that of survival in spring (P = 0·04), summer (P = 0·009) and annually (P = 0·001). Reproductive effort was not related to relative rank of elasticity of reproduction in S. hispidus and P. maniculatus.
Table 2. Results of tests of association between the probability of reproduction and the influence of reproduction, relative to survival, on the population growth rate. Entries in bold indicate statistically significant (P<0·05) associations. Cells with only a P-value indicate the use of a Fisher’s exact test because of expected cell counts <5
aOR = odds ratio, an odds ratio >1 indicates reproduction was higher than average in the month following a high reproductive, relative to adult survival, elasticity. bIndicates too few samples (expected cell counts = 0) to test association.
χ2 = 11·1; P = 0·001
χ2 = 0·02; P = 0·96
χ2 = 12·9; P < 0·001
P = 0·42
ORa = 5·2
OR = 1·02
OR = 7·2
OR = 0·45
P = 0·94
P = 0·61
P = 0·039
OR = 1·1
OR = 1·02
OR = 7·0
P = 0·04
P = 0·9
P = 0·009
OR = 6·8
OR = 0·88
OR = 20·0
χ2 = 8·9; P = 0·003
χ2 = 0·13; P = 0·7
P = 0·34
ORa = 16·8
OR = 0·8
OR = 3·1
Life-history theory predicts that species with short generation times are most sensitive to changes in vital rates associated with reproduction (Yoccoz et al. 1998; Oli & Dobson 2003; Gaillard et al. 2005) and that the variance associated with these vital rates should be minimized (Pfister 1998). Therefore, we predicted that we would find the highest vital rate elasticities and lowest variance for reproduction. Although there were some seasonal differences among taxa, we found that our populations were most sensitive to changes in survival. Previous work at the site found a similar pattern of sensitivity in S. hispidus; sensitivities for survival and somatic growth were higher than for reproduction (Sauer & Slade 1985). We found significant negative relationships between vital rate elasticity and the CV of the vital rate in three populations, thus supporting the hypothesis of environmental buffering for the majority of our species. Although reproduction was less influential than survival, two of the populations showed increased reproductive effort when reproduction had more influence on the population than survival suggesting some plasticity even in populations that exhibit environmental buffering.
Our analyses were conducted at a monthly scale, whereas many previous studies of vital rate elasticities on rodents have been conducted on a single matrix generated from seasonal or annual data (Gaillard & Yoccoz 2003; Gaillard et al. 2005). Analysis at the seasonal scale would include multiple reproductive attempts for each individual, resulting in larger values for fecundity elements. Survival values also would be lower as few individuals survive for more than a few months. These changes in vital rate values likely would alter the rank order of elasticities and may produce higher elasticity values for reproduction and lower elasticity values for survival. Elasticity is related positively to the value of the vital rate; because adult survival probabilities would be very small, and reproduction very large, when estimated over a 3-month period, one would expect elasticity of reproduction to be greater than that of survival.
Environmental buffering is expected to be widespread because of the negative effect of variability on population growth rate, a proxy for fitness (Pfister 1998; Gaillard & Yoccoz 2003; Rotella et al. 2012; but see Jongejans et al. 2010). Our results support this as three of four populations showed evidence of environmental buffering. We did observe considerable variation in CV for parameters with low elasticity. This does not refute the buffering hypothesis as vital rates that are less influential likely are not under considerable selection pressure as they have little effect on the population. Further, one would expect some vital rates to have low variability especially in relatively constant environments (Gaillard & Yoccoz 2003). Rather, if environmental buffering is present, we only expect to see low variance of vital rates with high elasticities, i.e. no values in the upper right quadrant of our graphs.
We included immigration in our models as previous simulations had indicated that ignoring immigration produced unrealistically low projected population growth rates. Our sampling was conducted on a 2·5-ha grid in open oldfield habitat. Consequently, movement of adults onto the study site is relatively common. Selection could not act directly on immigration making interpretation of immigration in relation to buffering difficult. We interpret immigration as another measure of survival of reproductive aged females, but the per-capita rate of immigration is so small (∼0·05) that elasticity values were low for the variable. However, including immigration in the current analyses did not affect our results as the relationships between vital rate elasticity and CV were qualitatively the same if immigration was removed from the correlation analysis (A.W. Reed and N.A. Slade, unpublished data).
We found limited evidence for buffering in P. maniculatus because of the high elasticity values of reproduction which were among the most variable vital rates. The overall trend for the species was negative; however, the relationship was not statistically significant. Vital rate buffering is a bet-hedging strategy (Gaillard & Yoccoz 2003) that should minimize the variance in population growth rate. In the simplest case, a bet-hedging organism uses a conservative strategy to maximize its long-term fitness (Philippi & Seger 1989). Survival and reproduction can be antagonistic vital rates; increasing reproduction may result in decreased survival (Hamel et al. 2010). Therefore, we might expect a bet-hedging strategy to include lower rates of reproduction to provide a better chance of survival. It is possible that our population of P. maniculatus was too conservative in balancing the cost of reproduction. By 1998, the taxon had become rare on our study area; in the final 6 years of the study (data not used in this analysis), we captured only 31 individual P. maniculatus.
Environmental buffering and plasticity are often presented as opposing life-history strategies (e.g. Reed et al. 2010). Our results suggest that this is not the case, that a population can be plastic in some vital rates while other vital rates are buffered. Simulations suggest that plasticity in vital rates, particularly reproduction, is beneficial (Reed et al. 2010; Bårdsen et al. 2011). Interannual variation in vital rate sensitivity also suggests that some plasticity is adaptive. The most sensitive vital rates in some months differed from those estimated from the mean matrix. Individuals that could adjust to the conditions and reproduce when the sensitivity of reproduction was higher than that of survival likely would have higher fitness.
The mechanism responsible for environmental buffering is unknown (Meiklejohn & Hartl 2002) and likely differs among populations and species. Striking an appropriate balance between survival and reproduction could be a mechanism to reduce variability in survival. Reproductive rodents are expected to have lower survival because of the energetic and physiological stresses of reproduction (Hamel et al. 2010). This cost can be offset by reproducing under only the most environmentally and physiologically favourable conditions. Rodents are considered income breeders; using available, rather than stored, resources to offset the energetic demands of reproduction (Jönsson 1997); therefore, reproductive success is dependent on current environmental conditions (e.g. food availability). Limiting reproduction only to times when conditions are favourable would maximize this ability and therefore minimize the cost of reproduction. Similarly, reproducing when it is most influential on the population likely would provide the highest benefit to the individual, even if a survival cost was present.
Buffering is expected to make populations less vulnerable to the increased environmental variability associated with anthropogenic climate change (Boyce et al. 2006). Sensitive vital rates that are less affected by environmental variability should result in less variability in population growth rate even in the face of climate change. The plasticity apparent in reproductive rates also suggests that these species should be relatively unaffected by climate change as the species can respond appropriately to change in environmental conditions. However, the effect of buffering of vital rates on vulnerability to climate variability has not been tested. It is possible that a change in climate would alter the correlation among vital rates or the rank order of vital rate elasticities making buffering less adaptive. For plasticity to be adaptive environmental cues must provide reliable information; changes in climate could reduce the reliability of these cues making plasticity less adaptive (Reed et al. 2010). Further research that explores the roles of buffering and plasticity in vital rates is necessary to draw broad inference on the demographic effects of increased environmental variability.
We thank all the individuals who helped collect small mammal data for this long-term data set and provided logistical support at the Kansas Ecological Reserves. Fieldwork was supported by grants from the University of Kansas General Research Fund. J.M. Gaillard, D.A. Miller and an anonymous reviewer provided helpful suggestions on a previous version of this manuscript.