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The effect of density dependence and weather on population size of a polyvoltine species.


Ecologists continue to debate what factors determine population size. Populations can be affected by simple density-dependent factors (Nicholson 1933, Lack 1954; see references in Dennis and Taper 1994), density-independent factors (Andrewartha and Birch 1954, see references in Martinat 1987), and time-delayed or nonlinear density-dependent forces (sometimes leading to chaotic fluctuations; May 1976, Hastings et al. 1993, Ellner and Turchin 1995). How these factors affect populations has broad theoretical and practical implications because the strength and form of density dependence influences the survival time of natural populations (Stacey and Taper 1992) and the structure of communities (Diamond and Case 1986). Also, populations strongly affected by weather are less accurately forecast (Hastings et al. 1993) and less manageable from the standpoints of conservation biology (Ginzburg et al. 1990, Stacey and Taper 1992, Allen et al. 1993) and pest management (Berryman 1991a, Turchin 1991).

The relative importance of density-dependent and -independent factors varies among and within species, depending on life history strategy (Hanski and Woiwod 1991, Holyoak and Lawton 1992), geographical location (e.g., Turchin and Hanski 1997), habitat type (e.g., Stubs 1977), season (e.g., Kaufman et al. 1995, Reid et al. 1997), and community structure (e.g., predation pressure: Turchin and Hanski 1997). However, detection of which factors drive population fluctuations can be affected by the scale of the study. For example, populations studied on a local spatial scale may not detect regulation that occurs at a regional scale (see references in Gilpin and Hanski 1991, and for a review of spatial scale in ecology see Levin 1992). The length of the study also affects the detection of density dependence, where studies with fewer than 20-30 annual censuses do not consistently detect density dependence in univoltine species (Solow and Steele 1990, Woiwod and Hanski 1992). In addition, when density dependence affects a developmental stage of a univoltine (i.e., 1 generation per year) species, studies using annual census data often cannot detect any density dependence (Hassell 1985, 1986, 1987, Hassell et al. 1987). Finally, density dependence is more detectable in species with long life-spans than in those with short life-spans when censuses are taken at less than generation intervals (Holyoak and Baillie 1996).

The importance of census interval to population studies of polyvoltine (i.e., [greater than]1 generation per year) species is particularly important to discern, because most populations are censused once annually (for lists of data sets see Turchin 1990, Turchin and Taylor 1992, Wolda and Dennis 1993, Ellher and Turchin 1995), regardless of the life history of the species. When a population is censused at only every fourth generation, for example, it is unclear whether sufficient detail exists to identify what factors drive population fluctuations and thus to create accurate forecasts. For polyvoltine species, annual density dependence represents the effect of the total number of individuals at a snapshot in time that resulted from birth, death, immigration, and emigration over multiple generations in 1 yr, on the same total number next year. Although annual density dependence in polyvoltine species is not classical density dependence (Wolda and Dennis 1993), this annual relationship may be mediated by resources, competitors, or predators that vary on an annual scale. In the only study so far to evaluate how census interval affects the identification of the key determinants of population size, Holyoak (1994) found that the time delay for density dependence was often misidentified when densities were smoothed over multiple generations. In addition, most annual studies including Holyoak misinterpret annual density dependence in polyvoltine species as direct instead of delayed density dependence.

Density-independent factors such as weather also may vary on an annual scale, creating an annual pattern in population size of polyvoltine species. However, polyvoltine species may respond only to variability from the annual weather cycle. For example, if animals respond to daily or monthly changes in weather, peak annual densities could be caused by the accumulation of short-term interactions between weather and monthly rate of population growth. In addition, if weather affects a specific point in the life cycle (e.g., breeding, lactation, pupation), annual weather data representing averages or sums may not describe potential multiple intra-annual effects on specific life stages in polyvoltine species. Finally, which factors have the strongest influence on a population of polyvoltine species may vary by season, where, for example, density-independent death from environmental conditions may be very important in winter but unimportant in summer.

In this paper, we determined what factors affected the population dynamics of the white-footed mouse (Peromyscus leucopus), a generally nonoutbreak, polyvoltine species. Density data were collected monthly over 23 yr in an isolated, 2-ha, Ohio woodlot. Monthly censuses represent generation to within-generation intervals because P. leucopus require 2 mo from conception until the subadult can be detected and because conception occurs every 1-2 mo depending on postpartum estrus. We used a multifactorial approach (Hilborn and Stearns 1982, Lidicker 1988, Gaines et al. 1991) by concurrently studying the effect of density dependence (linear, nonlinear, and time-delayed) and weather on the monthly rate of population growth over 257 mo. We used this model to forecast future densities. In addition to this comprehensive population model, we evaluated the importance of time scale in the detection of which factors affected the population by creating additional models using subsets of the data. We evaluated whether long time spans (22 yr) of annual data or short time spans (2-4 yr) of monthly data were better able to detect factors affecting the population. Although monthly data contain finer detail than annual data and thus may be able to detect effects on single generations and individual life stages, annual data span substantially more time and thus incorporate more interannual variability.

We had three goals for this study: (1) to identify what factors determined intra- and interannual variability in a population of polyvoltine mice by studying time-dependent relationships between monthly rate of population growth, density dependence, and weather; (2) to evaluate the accuracy of forecasts of population dynamics; and (3) to determine what type of data is best able to address the above two issues: long time spans of annual data or short time spans of generational data.


Field site

Carter Woods is an isolated 2-ha woodlot located in northwest Ohio (Section 16, Center Township, Wood County). It is dominated by secondary growth of shag-bark hickory (Carya ovata) and red oak (Quercus rubrum), and commonly has standing water in spring. It is surrounded by cropland planted in corn (Zea mays), soybeans (Glycine max), or wheat (Triticum aestivum), and the fields often are plowed in autumn. A drainage ditch, [approximately]1 m deep, runs along the eastern edge of the woodlot and is paralleled by a paved road. The nearest woodlot is 1.5 km away.

Field techniques

In 1973, a 13 X 26 trap grid was established through the entire woodlot with 7.6 m between stations and one trap per station. Beginning in January 1988, the very eastern edge of the woodlot was also trapped, expanding the grid to 15 X 26 stations. Sherman and Leathers live traps (typically, Sherman Traps, Tallahassee, Florida, USA, model number LNATDG, size [equivalent to]7.5 X 7.5 X 23 cm; Leathers live traps of similar size [company and model unknown]) were baited with peanut butter and oatmeal and supplied with cotton nestlets for insulation during winter. From April 1973 to March 1977, April 1978 to June 1978, and December 1982 to April 1983, the woodlot was divided into four sections, and two sections were trapped in a calendar week with each section trapped for two consecutive days, yielding a 2-wk trap session. For all other trap dates, the entire woodlot was trapped simultaneously for two consecutive days, yielding a 2-d trap session. During the study period (1973-1995), the entire woodlot was trapped once or twice per month, typically from April to November because high trap mortality and reduced manpower limited winter trapping. From 1979 to 1982 and 1989 to 1993, animals also were captured in nestboxes arranged at approximately every fourth trap station (Jacquot and Vessey 1995). Nestboxes were checked twice monthly in winter and spring, but never during a trap session to ensure consistent trappability. Mice were toe-clipped or ear-tagged at the time of capture, and information was gathered regarding sex, age, reproductive status, and location.

During the 23 yr, there were 363 558 trap nights, 22 950 nestbox nights, and 5543 individual P. leucopus were caught in 33 295 trap captures. In addition, there were 307 short-tailed shrew (Blarina brevicauda) and 8 meadow vole (Microtus pennsylvanicus) captures. Thus, there was little interspecific competition with P. leucopus for food or shelter (S. H. Vessey, personal observation). Few predators (e.g., owls) were ever observed near the woodlot, except during very high mouse densities (e.g., late summer 1980; S. H. Vessey, personal observation). Finally, because Carter Woods is isolated, it was poorly connected with other P. leucopus populations through immigration or emigration (S. H. Vessey, personal observation).

Mouse density data

We estimated P. leucopus densities using "minimum number known alive" (MNA; Krebs 1966), a direct enumeration technique [ILLUSTRATION FOR FIGURE 1 OMITTED]. MNA estimates density by tallying individuals from two counts: (1) individuals caught during a particular trap session, and (2) individuals caught at any time both before and after, but not during the trap session. The second count includes animals in the density estimate that are known to be alive but that were not captured during the trap session. When trap data were not available, nestbox data were used. To make MNA estimates of nestboxes and traps consistent with each other and with previous research, we counted the entire trappable population in trap estimates, and we included in nestbox estimates only those animals that weighed at least 8 g (the smallest trappable mouse) and that had been trapped at some point in their life. All densities represent population size per 2 ha, the size of the woodlot.

Monthly rate of population growth ([R.sub.t]) was calculated over a calendar month as: [R.sub.t] = ln([N.sub.t]/[N.sub.t-1]), where [TABULAR DATA FOR TABLE 1 OMITTED] t is in months, using densities ([N.sub.t]) that had been interpolated to the last day of the month from the nearest two trap sessions. We used the last day of the month so that both the rate of population growth and weather variables were calculated over the same calendar month.

Weather data

We obtained 12 monthly weather variables from a National Weather Service station located 3 km southwest of Carter Woods. Data included eight temperature and four precipitation variables of three types: monthly averages, extreme values of the month, and monthly accumulations (Table 1). Because P. leucopus at this latitude have a seasonal breeding pattern, monthly densities were closely correlated with average monthly temperatures. To identify proximate rather than ultimate relationships with weather, we subtracted the monthly mean from all weather variables, yielding variables that represented deviations from each month's average weather. All weather data used throughout this paper are in deviations from this seasonal mean. Threshold (i.e., binary) variables and extreme weather events were identified graphically from these continuous weather variables (see Model development: Threshold weather effects).



We modeled the effect of density dependence (linear, nonlinear, and time delayed) and weather factors (using continuous and binary variables) on monthly rate of population growth ([R.sub.t]) in P. leucopus, using 257 mo of data. We used this "comprehensive population model" to evaluate the relative importance of density dependence and independence, to forecast future densities, and to act as a standard to which the short-term monthly models and annual model were compared.

Statistical analyses. - We modeled time-dependent relationships and autocorrelated residuals using transfer function analysis (Box and Jenkins 1976) for continuous independent variables and intervention analysis (Box and Jenkins 1976) for qualitative independent variables and outliers. These multivariate time series analyses can be thought of as regression analyses between two or more time series data sets. They identify the start-up delay between the occurrence of an independent variable and its effect on monthly [R.sub.t], the number of months the independent variable has an effect, and a possible decay in the effect over time. Uncovering the entire time-delayed relationship is critical because parameter estimates are statistically inconsistent if significant time lags are excluded or irrelevant ones are included (Pankratz 1991). In addition, multivariate time series analyses remove autocorrelation from the residuals, leaving random independent residuals, which are required for valid t and F statistics.

Because they identify patterns that are not easily seen graphically, multivariate time series analyses can identify potentially important interactions in poorly understood ecological systems (Poole 1978, Rastetter 1987). These methods have been used to study the effect of weather on densities in fish (Fogarty 1988, Stocker and Noakes 1988, Tsai et al. 1991, Tsai and Chai 1992) and insects (Hacker et al. 1975, Hacker 1978, Poole 1979), the effect of weather on physiology in plants (Kanninen 1985, Ford et al. 1987, Keller 1987, McLaughlin et al. 1987), community interactions (Rastetter 1987), and energy flows in an ecosystem (Birkett 1994).

Structure and estimation of multivariate time series models.--A simple transfer function model can be estimated as a regression model. For example, the rate of population change in P. leucopus ([R.sub.t]) can be correlated with precipitation ([P.sub.t]) at numerous time lags, such that

[R.sub.t] = C + ([v.sub.0][P.sub.t] + [v.sub.1][P.sub.t-1] + [v.sub.2][P.sub.t-2] + ...)+[A.sub.t] (1)

where C is a constant, v are the parameters to estimate, and [A.sub.t] are the residuals or "disturbance" to the system at month t. Transfer function models are written in backshift notation to ease understanding and estimation of more complex models, where Eq. 1 is rewritten as

[R.sub.t] = C + ([v.sub.0] + [v.sub.1]B + [v.sub.2][B.sup.2] + ...)[P.sub.t] + [A.sub.t] (2)


[R.sub.t] = C + v(B)[P.sub.t] + [A.sub.t] (3)

where the backshift operator (B) moves its associated variable ([P.sub.t]) back in time by L months where [B.sup.L][P.sub.t] = [P.sub.t-L].

When numerous time-lagged correlations are significant, as in Eq. 1, there can be a pattern in the terms. For example, the correlation between the rate of population change and precipitation may decay as the time lag increases. This model can be parameterized more parsimoniously by estimating the decay with an initial peak value ([Omega]) and decay rate ([Delta]), yielding more accurate parameter estimates. Continuing the example, transfer function analysis also identifies a possible time lag before precipitation affects the rate of change. This more complex model is written as

[R.sub.t] = C + [Omega](B)/(1 - [Delta]B) [P.sub.t-b] + [A.sub.t] (4)


[Omega](B) = [[Omega].sub.0] - [[Omega].sub.1]B - [[Omega].sub.2][B.sup.2] - ... - [[Omega].sub.s][B.sup.s].

Here, b is the number of months before precipitation starts to affect the rate of population change, and s in the number of months that precipitation affects rate of change.

When the independent variable does not explain all autocorrelation in the dependent variable, the residuals of the model are autocorrelated. Residual autocorrelation is removed by adding additional terms to the model that relate observations of the independent variable to observations at time intervals in the past (i.e., autoregressive terms [[Phi](B)[R.sub.t]] and differencing terms [[R.sub.t] - [R.sub.t-d]] or [1(B)[R.sub.t]]) and to past values of the residual term (i.e., moving average terms [[Theta](B)[a.sub.t]]), such that

[R.sub.t] = C + [Omega](B)/(1 - [Delta]B) [P.sub.t-b] + [Theta](B)/[Phi](B) [a.sub.t]


[Phi](B)[R.sub.t] = C + [Omega](B)/(1 - [Delta]B) [P.sub.t-b] + [Theta](B)[a.sub.t] (5)


[Theta](B) = -[[Theta].sub.1]B - [[Theta].sub.2][B.sup.2] - ... - [[Theta].sub.q][B.sup.q]

[Phi](B) = -[[Phi].sub.1]B - [[Phi].sub.2][B.sup.2] - ... - [[Phi].sub.p][B.sup.p]

and where [a.sub.t] are random, independent residuals; [[Phi].sub.p] are autoregressive terms with a time lag of p; and [[Theta].sub.q] are moving average terms with a time lag of q. [Theta](B) and [Phi] (B) are often extended to include seasonal autocorrelation (Box and Jenkins 1976). To make our models more accessible, we write them without the backshift operator where possible (i.e., when there is no decay in the time-lagged correlations and no autocorrelation in the disturbance).

When qualitative independent variables are included, the same model structure is used but the method is called intervention analysis (Box and Jenkins 1976). Qualitative variables only estimate changes in the mean response of the dependent variable, and these interventions can have temporary or permanent effects on the dependent variable. For example, a severely cold day may have a temporary effect on rate of population change, and the construction of a drainage ditch may have a permanent effect. Such an event is best represented with a binary variable identifying its timing and duration (i.e., single or multiple months). As with continuous variables, the time-dependent relationship between a qualitative independent variable and the dependent variable can have a start-up delay, multiple-month effect, and decay in the effect over time. Simple intervention models can be estimated with regression with indicator variables (Neter et al. 1989), but more complex time relationships require intervention analysis. By creating interactions with continuous variables, one can model changes in the slope of the regression line of the continuous variable, depending on the level of the qualitative variable (Box and Tiao 1975) in a manner similar to regression with indicator variables (Neter et al. 1989). We use this method to model the effect of continuous variables (e.g., density dependence and weather) by season.

In the past, transfer function and intervention models were restricted to a single independent variable because of limitations in the cross-correlation identification method introduced by Box and Jenkins (1976). However, a newly available method, the linear identification method, allows these models to be extended to multiple independent variables (Liu and Hanssen 1982, Pankratz 1991). The construction procedures for the transfer function and intervention models followed the three-step process of identification, estimation, and diagnostic checking (Box and Jenkins 1976, Vandaele 1983, Pankratz 1991, and references therein). During the development phase of the model, we used surrogates for missing data that were estimated according to the univariate time series process of mouse densities (Chen and Liu 1993). During final parameter estimation, we used a newly available technique to omit missing data during model estimation (Scientific Computing Associates 1995; R. H. Lewellen and S. H. Vessey, unpublished manuscript). All parameters were estimated concurrently using SeA statistical software (Scientific Computing Associates 1995). We reserved the last 12 data points (November 1994-October 1995) to evaluate forecasts, and we estimated the model with a data set that spanned 257 mo (June 1973-October 1994) and contained 217 observations of monthly [R.sub.t].

Outlier analysis. - Outlier detection in time series is recommended as a standard procedure in all multivariate time series analyses (Liu and Chen 1991, Chen and Liu 1993). Outliers can strongly affect parameter estimation and model specification (Guttman and Tiao 1978, Tsay 1986, Chang et al. 1988), which in turn affects forecast accuracy (Ledolter 1989). These effects can be particularly strong in time series because data are autocorrelated, causing a single outlier to affect multiple observations by shifting the mean or changing residual autocorrelation (Pankratz 1991). We subjectively chose 3.4 standard deviation units (1 in 1667 observations at [Alpha] = 0.05) to identify observations not successfully accommodated by the comprehensive population model. In addition to improving parameter estimates, external events detected by outlier analysis often have identifiable causes, which can lead to a better historical understanding of the data. Forecasts can be created that are conditional on the event happening again (Pankratz 1991).

Model development. - 1. Density-dependent effects.Density-dependent factors describe feedback in the system and include the interaction of resources, competitors, and predators with P. leucopus. Although we did not have data on other species in the community, we indirectly detected their influence by studying the effect of lagged densities on current monthly [R.sub.t]. For example, future [R.sub.t] is affected by current densities ([N.sub.t]) through competition and affected by past densities mediated through predation, because predator numbers can have a time-delayed response to prey numbers (Turchin 1991). By using time-delayed densities, we described the dynamics of a multivariate system using only univariate time series data (for mathematical proofs see Packard et al. 1980, Takens 1981, Casdagli 1992). For this seasonally breeding species, direct (nonlagged) density dependence was represented by 1- and 2-mo lags because there were [approximately]2 mo between conception and the detectable increase in population size. Delayed density dependence was represented by a time lag of [greater than or equal to]3 mo, and nonlinear density dependence (e.g., Allee effects) was represented by Box-Cox power transformations (Box and Cox 1964) of monthly densities (Turchin and Taylor 1992).

Because P. leucopus is a seasonal breeder, we expected that the intrinsic growth rate and possibly density dependence would vary throughout the year. We did not have enough data to accurately estimate these parameters during each month. Instead, we identified biological seasons for P. leucopus by creating a preliminary density-dependent model for each month (12 models), and then using similarities in parameter estimates to group months into seasons. In these 12 preliminary models, monthly [R.sub.t] was correlated with mouse densities from the prior 2 mo ([N.sub.t-1], [N.sub.t-2]), and the monthly mean response represented the intrinsic growth rate. To increase the degrees of freedom, we estimated all 12 models concurrently using 11 binary variables ([S.sub.i]). The preliminary density-dependent model (written without the backshift operator) was

[Mathematical Expression Omitted] (6)

where [S.sub.1] = 1, February; 0, otherwise. The other binary variables represented the other 10 mo. [[Omega].sub.i] are the parameters to estimate; [e.sub.t] is random, independent error; and t is in months.

January parameters were arbitrarily chosen as the standards to which the indicator variables were compared (Neter et al. 1989), where for January, [[Omega].sub.0] describes intrinsic growth rate, and [[Omega].sub.12] and [[Omega].sub.24] describe density-dependent effects with 1- and 2-mo lags, respectively. All other parameters were estimated as deviations from January parameters. These standards were for estimation purposes only (Neter et al. 1989: 350-351) and had no biological relevance. To understand the meaning of the coefficients, Eq. 6 can be solved for each binary variable. For example, in the February relationship, [S.sub.1] = 1 and [S.sub.2] to [S.sub.11] = 0, and Eq. 6 reduces to:

[R.sub.t] = ([[Omega].sub.0] + [[Omega].sub.1]) + ([[Omega].sub.12] + [[Omega].sub.13])[N.sub.t-1] + ([[Omega].sub.24] + [[Omega].sub.25])[N.sub.t-1] + [e.sub.t]. (7)

Each month has a similar equation.

By comparing parameters among months, we identified three biological seasons for the preliminary model. The growth phase (summer) occurred from May through August; the decline phase (autumn) occurred from September to December; and the stagnant phase (winter) occurred from January to April. The model did not have constant variance, so we transformed [N.sub.t-1] and [N.sub.t-2] to the natural log scale and re-estimated the model using three seasons. We checked for additional nonlinearity by trying other transformations in the Box-Cox family (Box and Cox 1964), and we searched for delayed density dependence by including additional time delays in the model.

2. Continuous weather effects. - Because there were many weather variables to consider, we did a preliminary analysis using principal components (Manly 1990) to identify the general weather effects on monthly [R.sub.t] (Lewellen and Vessey, 1998). We used three principal components (not shown) to describe the maximum amount of variability in the 12 strongly collinear weather variables (Table 1; SAS Institute 1987). The preliminary analysis suggested that we study TotalPrecip, TotalSnow, and MinTemp [less than] 0 in autumn and winter; and AvgMaxTemp and MaxTemp [greater than] 32 in summer (see Table 1 for abbreviation descriptions). We included these continuous weather variables in the model by season and studied their time-delayed relationship with monthly [R.sub.t]. We looked for start-up lags, extension in time, and potential decays in the effect. We also considered variable sums over 2, 6, and 12 mo.

3. Threshold weather effects. - We hypothesized that mice might respond to thresholds in weather, which would not be captured by a polynomial relationship with continuous variables. We identified five potential thresholds by considering uncommon highs or lows in the 12 weather variables (in deviations from monthly means) and their 2-, 6-, and 12-mo sums, and we weighed their biological relevance. First, HotSummer temperatures may have caused the documented midsummer breeding hiatus in P. leucopus (Rintamaa et al. 1976). We described this effect using a daily threshold variable (MaxTemp [greater than or equal to] 32) summed over 2 mo. Using these criteria, there were six extremely hot summers during the 22 yr [ILLUSTRATION FOR FIGURE 2A OMITTED]. Second, WarmWinter temperatures and normal winter precipitation may have caused winter breeding. We used a daily threshold variable (MaxTemp [less than or equal to] 0) summed over the month to describe temperatures, and excluded winters with far below average total precipitation [ILLUSTRATION FOR FIGURE 3 OMITTED]. Using these criteria we identified four WarmWinters during the study period [ILLUSTRATION FOR FIGURE 2B OMITTED]. We also examined ExtremeCold, HeavyPrecip, and HeavySnow (Table 1), because prior studies have documented reduced survival in the laboratory in temperatures below -15 [degrees] C for 15 h (Sealander 1951), reduced litter size in the field after exposure to heavy precipitation during early pregnancy (Myers et al. 1985), and an insulating effect of heavy snowfall (Merritt 1984). For each of these five weather thresholds, we created a binary variable, where the observations had a value of 1 when the threshold was crossed, and a value of 0 at all other time periods. When thresholds occurred in the winter, only time periods with mouse data were included in the binary variables.

4. Extreme weather events. - A few weather events were drastically different from normal conditions and occurred only once during the study period. They had an historical effect on monthly [R.sub.t], which could help explain densities but not forecast them. The 1974 autumn drought and 1988 summer drought both had far below average precipitation for numerous months in a row (Table 1, [ILLUSTRATION FOR FIGURE 3 OMITTED]). These two droughts were not considered together as a single threshold variable because how each drought affected monthly [R.sub.t] varied, depending on its season and duration. We were unable to identify the time-dependent relationship (i.e., startup lag, length of effect, and decay in effect) with monthly [R.sub.t] using standard intervention analysis because the relationship depended on the subjective definition of the drought period, the timing and level of which could not exactly match a mouse's perspective. Instead, we used visual inspection of the residuals of the density-dependent model to identify when the reduction in monthly [R.sub.t] occurred, and created a binary variable to represent this low [R.sub.t]. In contrast, for the other two extreme weather events, the 1977 hot May and 1977/1978 blizzard (Table 1), we created a binary variable to describe the weather event and used standard intervention analysis to identify the time-delayed effect of monthly [R.sub.t].

5. Density-data collection. - There were two departures from standard data collection practices (i.e., trap session lasting 2 d) at Carter Woods: trapping the woodlot in four parts over a period of 2 wk, and capturing mice in nestboxes. We created a binary variable for each nonstandard data collection practice to determine if either practice consistently over- or underestimated densities compared to the standard 2-d trap session.


Density-dependent effects. - The monthly rate of population growth ([R.sub.t]) in P. leucopus was correlated with densities (transformed to the natural log scale) with 1- and 2-mo time lags. We found no evidence of longer time lags (i.e., lags 3-12), and little evidence of nonlinear density dependence. In contrast to the preliminary model, the final model suggested only two biological seasons, where summer density-dependent effects varied from winter, but autumn effects did not (Table 2). Any difference between autumn and winter was undetectable because of substantial missing data during winter. The intrinsic growth rate was almost three times higher in summer than autumn/winter (Table 2), as expected in this seasonally breeding species.

Continuous weather effects. - AvgMaxTemp in summer negatively affected monthly [R.sub.t] with a 2-mo lag (Table 2). Thus, when temperatures were higher than normal in March through June, densities declined 2 mo later [ILLUSTRATION FOR FIGURE 4 OMITTED]. Also, TotalPrecip in autumn positively affected monthly [R.sub.t] in the same month (Table 2), indicating that densities declined more slowly during wet autumns (September to December; [ILLUSTRATION FOR FIGURE 4 OMITTED]). We found no evidence of multiple months of significance or of decays in the effect over time for either variable. Continuous weather effects were weak compared to the other components in the model (Table 2). We found no effect at any time lag or in any season of MaxTemp [greater than] 32, MinTemp [less than] 0, or TotalSnow.

Threshold weather effects. - During WarmWinter months, monthly [R.sub.t] was higher than normal (Table 2, [ILLUSTRATION FOR FIGURE 4 OMITTED]). Residual analysis indicated that all years with WarmWinters (i.e., 1983, 1987, 1988, and 1992) were represented equally well with the single parameter; however, for all four years, residuals in November, December, and April were positive, while residuals in January through March were negative. These differences indicated that WarmWinter months might have had a stronger positive effect on monthly [R.sub.t] in late autumn and late winter than in early winter. These differences suggested that the WarmWinter effect might be better represented by a trinary parameter or with a decayed effect. During HotSummers, monthly [R.sub.t] declined in the same month (Table 2, [ILLUSTRATION FOR FIGURE 4 OMITTED]). This threshold effect was in addition to the continuous weather effect of AvgMaxTemp with a 2-mo lag. We found no detectable effect at any time lag of ExtremeCold, HeavyPrecip, or HeavySnow. However, we could not test ExtremeCold at 0- or 1-mo lag because of lack of density data.

Extreme weather events. - The 1988 summer drought had a more severe and longer lasting effect on monthly [R.sub.t] than did the 1974 autumn drought, and it had only a 1-mo time delay before the resulting reduction in monthly [R.sub.t], compared to the 5-mo delay of the 1974 drought [ILLUSTRATION FOR FIGURE 4 OMITTED]. For the 1988 drought, residual analysis indicated that even though the worst of the drought occurred in June, the lower precipitation in March and April [ILLUSTRATION FOR FIGURE 5B OMITTED] caused the densities in April and May to decline [ILLUSTRATION FOR FIGURE 5A OMITTED]. The severely hot May in 1977 had the strongest effect on monthly Re of any factor in this study (Table 2). It caused a severe decline in densities during May and June [ILLUSTRATION FOR FIGURE 4 OMITTED] However, two other unusual weather events also may have contributed to this population decline: a severely cold January (i.e., MinTcrop [less than] -18 was [greater than] 3[Sigma]) or a very hot April (second hottest on record). There was no way to determine which temperature extreme or combination caused the decline. We found no detectable effect of the 1977/1978 blizzard. However, immediate or short-term delayed effects were not studied because of the lack of mouse data that winter.

Density data collection and outliers. - We found no detectable difference between trap sessions lasting 2 d and those lasting 2 wk, but densities estimated from nestbox censuses were significantly higher than from trap censuses (Table 2). The highest winter densities recorded (winter 1991/1992, [ILLUSTRATION FOR FIGURE 1 OMITTED]) were estimated with nestbox data and were affected by a WarmWinter. Outlier analysis identified three single-month outliers [ILLUSTRATION FOR FIGURE 4 OMITTED] that were adjacent to missing winter data, were at a transition between nestbox and trap data, or occurred during extremely low density estimates (6-7 animals). These outliers were interpreted as probable density-estimation errors because MNA would include few to no "span" animals, those caught both before and after but not during the trap session.

Outlier analysis also detected three regions of the [TABULAR DATA FOR TABLE 2 OMITTED] data where an external event with unknown cause probably affected the population (Table 2, [ILLUSTRATION FOR FIGURE 4 OMITTED]). Two months in summer 1980 had rapid population growth, yielding the highest peak density in 23 yr [ILLUSTRATION FOR FIGURE 1 OMITTED]. The second highest peak density was caused by the high population growth in October 1991, a month that typically showed negative growth. Finally, two months in autumn 1981 experienced rapid decline in densities, yielding the second lowest densities on record [ILLUSTRATION FOR FIGURE 1 OMITTED].

Mathematical description. - The final comprehensive population model correlated monthly rate of population growth ([R.sub.t]) with seasonal intrinsic growth rate (constants); seasonal density dependence ([N.sub.t]); continuous ([C.sub.t]), threshold ([T.sub.t]), and extreme ([E.sub.t]) weather variables; and nestbox censuses ([D.sub.t]). The final equation is written (without the backshift operator) as

[R.sub.t] = [[Omega].sub.0] + [[Omega].sub.0,s][S.sub.t] + ([[Omega].sub.0,N] + [[Omega].sub.1,N]) [N.sub.t-1]

+ ([[Omega].sub.0,[N.sub.S]] + [[Omega].sub.1],[N.sub.S]])[S.sub.t][N.sub.t-1]+[[Omega].sub.0,[C.sub.1][S.sub.t][C.sub.t-2,1] + [[Omega].sub.0,[C.sub.2]][F.sub.t][C.sub.t,2]

+ [[Omega].sub.0, [T.sub.1]][T.sub.t,1] + [[Omega].sub.0,[T.sub.2]][T.sub.t,2]+ [[Omega].sub.0,[E.sub.1]][E.sub.t,1] + [[Omega].sub.0,[E.sub.2][E.sub.t,2]

+([[Omega].sub.0,[E.sub.3]] + [[Omega].sub.1, [E.sub.3]])[E.sub.t,3] + [[Omega].sub.0,D][D.sub.t] + [e.sub.t] (8)


[S.sub.t] = 1, "summer" months May-August; 0, otherwise

[F.sub.t] = 1, "autumn" months September-December; 0, otherwise

[N.sub.t] = In mouse densities (continuous variable)

[C.sub.t1] = AvgMaxTemp (continuous variable)

[C.sub.12] = TotalPrecip (continuous variable)

[T.sub.t1] = 1, WarmWinters; 0, otherwise

[T.sub.12] = 1, HotSummers; 0, otherwise

[E.sub.t1] = 1, January-May 1975, drought's effect on monthly [R.sub.t]; 0, otherwise (see note below on time lag)

[E.sub.12] = 1, April-December 1988, drought's effect on monthly [R.sub.t]; 0, otherwise (see note below on time lag)

[E.sub.t3] = 1, May 1977, extremely hot temperature; 0, otherwise

[D.sub.t] = 1, Nestbox data; 0, otherwise.

[B.sup.g] indicates a start-up time lag of g months; [[Omega].sub.g] is a parameter with time-delayed effect of g months; [e.sub.t] is random, independent error; and t is in months. The timing of the effect of 1974 and 1988 droughts on monthly [R.sub.t] was determined by graphical analysis, and the two drought binary variables represent the timing of the lowered monthly [R.sub.t] rather than the drought period. We detected no decay in an effect over time in any variable. The model was re-estimated with outliers removed (Table 2).

The final comprehensive population model (Table 2, Fig. 4) explained 81% of the variability in monthly [R.sub.t] over the 22-yr study period (May 1973-October 1994). Intrinsic monthly [R.sub.t] and density-dependent effects explained 51% of the variability and described the seasonal cycle found throughout the data set. Deviations from this seasonal cycle were explained by weather effects, variations in how density data were collected, and external events with unknown causes (i.e., multimonth outliers). Annual peak densities were caused by an accumulation of the effects of weather on monthly [R.sub.t] over the year. With the exception of droughts, no density-dependent or weather variable affected monthly R, with a delay of [greater than]2 mo.



We forecast the population size of P. leucopus in Carter Woods using the comprehensive population model, which had been developed with 22 yr of monthly data. Forecast accuracy is strongly affected by whether or not the data being forecast were used to develop the model, by whether or not data on independent variables during the forecast period are used, and by how far into the future data are being forecast. Throughout this paper the following definitions apply. "Forecasts" predict data that were not used to develop the model. "Simulations" predict data that were used to develop the model, and inaccuracies were not corrected before additional predictions were made. "Fitted data" predict data that were used to develop the model, but inaccuracies were corrected before additional predictions were made. We created five types of predictions.

First, fitted values of the population model were calculated using known densities and weather that were updated before each month was predicted (called "fitted data"). Any inaccuracies in predicted monthly [R.sub.t] were corrected before the next prediction was made. Second, we simulated the entire data set using known weather during the 22-yr simulation period, unknown densities during the period, and no error (called "simulated densities"). Inaccurate monthly [R.sub.t] simulations were never corrected during the 22-yr simulation period, which caused a single error in simulated [R.sub.t] to be propagated according to the time-dependent relationships between variables. Third, to evaluate model accuracy, we forecast monthly [R.sub.t] during 1995 (November 1994-October 1995) using unknown weather during the forecast period and using densities calculated from forecast monthly [R.sub.t] (called "true forecasts"). Data from 1995 were not used during model development. Fourth, we forecast 1995 with known weather during the forecast period (called "forecasts with known weather"). We identified threshold weather effects using previously developed definitions (Table 1), and we looked for potential extreme weather events. Finally, we determined the sensitivity of monthly [R.sub.t] to small changes in the model, by adjusting parameters that described the relationship between weather and [R.sub.t] until 1995 forecasts matched observed densities exactly (called "sensitivity analysis").


"Fitted data" indicated an excellent fit of the population model throughout the 22-yr study period [ILLUSTRATION FOR FIGURE 6 OMITTED] and were within 95% confidence intervals. They also suggested that one-month-ahead forecasts would be very accurate.

"Simulated densities" indicated that the comprehensive population model accurately described the seasonal cycle of P. leucopus and accurately predicted the variability in autumn and winter densities [ILLUSTRATION FOR FIGURE 7 OMITTED]. In addition, the model predicted average and below-average peaks in summer densities (except 1982 and 1986; Fig. 7), but was unable to predict peak densities in years with high monthly [R.sub.t] in the summer [ILLUSTRATION FOR FIGURE 7 OMITTED]. However, because monthly densities are autocorrelated, a single inaccurate estimate of monthly [R.sub.t] strongly affected future density estimates, even though future monthly [R.sub.t] estimates may have been accurate. For example, a single inaccurate density estimate in April 1978 was propagated for 5 mo [ILLUSTRATION FOR FIGURE 8 OMITTED]. In addition, March and April breeding in 1979 was not represented by the model, which allowed only for summer intrinsic growth rates from May to August. This deviation was propagated for 4 mo and caused a substantial difference in observed and simulated peak densities because of the importance of May densities to peak density levels [ILLUSTRATION FOR FIGURE 8 OMITTED]. Thus, because inaccuracies in simulated data were never corrected during the 22-yr period, deviations between observed and simulated data seen in Fig. 7 are often caused by the propagation of a single month error from many months earlier.

"True forecasts" predicted densities double to triple their observed levels [ILLUSTRATION FOR FIGURE 9 OMITTED], and predicted a peak density of 93 animals, compared to the 36 observed. These forecasts represented the average seasonal cycle of the mice, given a starting point of 68 animals in October. They also represented the best forecasts available if weather data were not used. Forecasts using unknown weather data during the forecast period were equivalent to forecasts created only with density-dependent relationships because weather effects did not extend beyond 2 mo and we forecast 12 mo in the future.

For "forecasts with known weather," we identified no threshold weather effects [ILLUSTRATION FOR FIGURE 2 OMITTED] but did identify a 1994 autumn drought that could affect the 1995 forecasts [ILLUSTRATION FOR FIGURE 3 OMITTED]. Because we did not know the strength or duration of its effect on monthly [R.sub.t], we modeled its effect using parameters and time delays identical to the 1974 drought (Table 2), even though there was no reason to assume that the effect of the two droughts would be identical. Using parameters identical to the 1974 drought, we forecast densities within three animals up to 8 mo in the future, and predicted a peak density (10 mo in the future) in August of 64 animals compared to the 36 observed [ILLUSTRATION FOR FIGURE 9 OMITTED]. For the "sensitivity analysis," we produced the observed peak 1995 densities by altering the model's parameters to make the negative effect of the 1994 drought on [R.sub.t] last 2 mo longer than for the 1974 drought.



We evaluated whether short-term monthly data or long-term annual data were better able to identify factors affecting the population of P. leucopus, and to create accurate forecasts.

Short-term monthly models. - We looked for density-dependent and weather effects in short segments of the 23-yr data set. We used the four longest continuous segments of the data: June 1973-December 1975, June 1982-January 1985, April 1986-January 1989, and May 1989-December 1992. These segments contained a range of dynamics, including years without peaks, years with high winter estimates because of nestbox censuses, variability in peak height from 50 to 183 animals, and variability in trough height from 3 to 50 animals. For each short-term monthly model, we estimated all density-dependent and weather parameters detected by the comprehensive population model, and excluded those threshold weather variables and extreme weather events that did not occur during each time span. Only the 1989-1992 model spanned any outliers detected by the comprehensive model, and we included these outliers during parameter estimation but did not use them during simulations or forecasts. We estimated all models using transfer function analysis and intervention analysis, as in the comprehensive model.

We evaluated how well each model fit its respective data segment using percent variability explained and the accuracy of simulated data. For each short-term monthly model, we simulated the time span that was used to develop the model. As with the comprehensive population model, these simulations used known weather data during the time span, unknown densities, and no error (see "simulated densities" under Forecasting population size: Methods). Time-delayed densities used in the model were calculated from simulated monthly [R.sub.t]. In addition, we forecast all 23 yr with each of the four short-term monthly models, to evaluate the ability of the monthly models to forecast dynamics that did not occur during the respective data segments used in model development. For each model and year, we provided July and August densities of the prior year and forecast 18 mo into the future. We used known weather during this period, but calculated densities from the forecasted monthly [R.sub.t].

Annual model. - We identified the effect of density dependence and weather on annual rate of population growth using 22 yr of annual data. Annual [R.sub.t] was calculated using August densities [ILLUSTRATION FOR FIGURE 10 OMITTED] according to: [R.sub.t] = ln([N.sub.t]/[N.sub.t-1]), where t is in years [ILLUSTRATION FOR FIGURE 10 OMITTED]. Although P. leucopus densities peak from June to August (except for the peak in October 1991), we used August densities so that annual [R.sub.t] spanned a consistent time period. Thus, these August densities did not always represent the peak population size during the year [ILLUSTRATION FOR FIGURE 10 OMITTED]. Weather variables used in the annual model were created by averaging or summing monthly variables over 1 yr and over individual seasons. Annual continuous weather data included summed deviations from AvgMaxTemp from March to June and summed TotalPrecip from April to August, from September to December, and from September to August. Annual binary variables included HotSummers, WarmWinters, and a single variable for both droughts.

We looked for linear, nonlinear, and time-delayed density dependence using response surface methodology (RSM; Turchin and Taylor 1992). First, we calculated autocorrelation functions (ACF; Box and Jenkins 1976) to identify potential periodicity [ILLUSTRATION FOR FIGURE 10 OMITTED]. ACFs are calculated as the correlation coefficient between observations separated by lag g (i.e., [N.sub.t-g] and [N.sub.t]). RSM looks for complex density dependence by studying 2-yr time delays between densities ([N.sub.t-2]) and the annual [R.sub.t] and by considering multiple types of nonlinear transformations ([[Theta].sub.1]). We estimated the following model for all combinations of [[Theta].sub.1] and [[Theta].sub.2] equaling (-1, -0.5, 0, . . . , 2.5, 3), where values farther from 1 represent stronger nonlinear transformations:

[Mathematical Expression Omitted] (9)

where [a.sub.i] are the parameters to estimate, and [e.sub.t] is random, independent error. The model with the smallest residual sum of squares best represented the dynamics of the system. We then iterated the model to identify predicted dynamics, including stable equilibrium, limit cycle, quasiperiodic dynamics, and chaotic dynamics. Populations with complex dynamics would be best described by a model with time-lagged and/or nonlinear relationships. We modified the best fit RSM model to exclude parameters not significantly different from zero. Then, we regressed annual [R.sub.t] on the density-dependent parameters and on annual weather variables. We used percent variability explained to evaluate model fit.

We predicted all 23 yr of data by using August density from the prior year and known weather to predict the following August density. Because the annual model was developed with data from 1973 to 1994, these predictions were fitted data, and 1995 was a forecast.


Short-term monthly models. - All short-term monthly models detected higher intrinsic growth rate in the summer than autumn/winter (Table 3). However, the parameter values varied from -0.188 to 1.085 in the autumn/winter (Table 3), which represents more than a threefold difference in the intrinsic growth rate between the models. A similar difference was detected in the summer intrinsic rates (0.503 to 1.632; Table 3). In addition, each model detected only one of the four density-dependent factors identified by the comprehensive population model (Table 3).

The ability of the short-term monthly models to detect weather effects varied depending on the data segment used during model development. AvgMaxTemp and TotalPrecip both had a significant effect in only two of four models (Table 3). When they occurred during [TABULAR DATA FOR TABLE 3 OMITTED] the study period, the effect of the two threshold variables, WarmWinters and HotSummers, were detected in all but one case (Table 3). As there were only two droughts during the 22-yr period, only two of the four short-term models included droughts, and both detected a strong negative effect. The 1986-1988 model detected all the weather effects identified by the comprehensive model. No model studied the period during the 1977 hot May.

The short-term models had high [R.sup.2] values (67-88%), two of which were higher than the comprehensive model. Simulations indicated an excellent fit for all short-term monthly models [ILLUSTRATION FOR FIGURE 11 OMITTED], and suggested that a single density-dependent parameter and seasonal growth rate were sufficient to describe the intraannual cycle for each data segment. Excellent model fit was caused by targeting the parameter estimates directly to the data segment under study.

Annual model. - Annual [R.sub.t] was negatively correlated with August densities of the prior year ([N.sub.t-1]). We found no evidence of time-delayed annual density dependence ([N.sub.t-2]) or substantial nonlinearity in this relationship. The best model had transformations of [[Theta].sub.1] = 0.5 for [N.sub.t-1] and [[Theta].sub.2] = 3 for [N.sub.t-2]; however, none of the parameters for [N.sub.t-2] were significantly different from zero. Simulations using the model converged on a stable equilibrium. We detected no significant effects of any continuous or threshold weather variable on annual [R.sub.t]. However, drought years (1974, 1988) had significantly lower annual [R.sub.t] than the rest of the data. The best annual model was: [R.sub.t] = 1.28 - 0.0012[N.sub.t-1] [1.039D.sub.t] + [e.sub.t], where [D.sub.t] is a binary variable representing drought years; and [e.sub.t] is random, independent error. This model explained 48.4% of the variability in annual [R.sub.t].

Comparison of short-term monthly models and the annual model. - We compared the effectiveness of the data types using two criteria: ability to detect factors affecting the population, and forecast accuracy. We found that three of the four short-term monthly models were better able to detect weather factors affecting the population than the annual model, and the 1973-1975 model was equally effective (Table 3). In addition, the annual model detected an annual density-dependent relationship that was not suggested by the comprehensive population model.

We found that monthly models developed on 2-4 yr of data created better forecasts than a 22-yr annual model. Although neither the annual model nor any of the monthly models could predict high-density years during the 23-yr data set, all other years (i.e., average to low-density years) were generally forecast better by the short-term monthly models than the annual model. The average deviation from observed peak values for the nine average to low-density years (i.e., 1974, 1976-1978, 1982, 1986, 1987, 1989, 1992) was 56% for the 1973-1975 model, 36% for the 1982-1984 model, 39% for the 1986-1988 model, 85% for the 1989-1992 model, and 80% for the annual model. The three drought years (i.e., 1974, 1988, 1995) were forecast better by the 1973-1975 ([+ or -] 13%) and 1986-1988 models (-18%) than the annual model ([+ or -]20%), but poorly by the 1982-1984 (+148%) and 1989-1992 (+202%) monthly models. The 1982-1984 and 1989-1992 models did not span a drought year and were unable to predict the effect of these severe events. These comparisons were biased in favor of the annual model because the annual model, but not the monthly models, predicted densities for years that had been used during model development.

We graphed three example years to describe forecast accuracy of the five models [ILLUSTRATION FOR FIGURE 12 OMITTED]. First, although none of the models directly detected the effect of the 1977 hot May, the monthly models used other weather variables (i.e., AvgMaxTemp and HotSummers) to improve their estimates of this crash [ILLUSTRATION FOR FIGURE 12A OMITTED]. As the annual model did not detect any effect of temperature on [R.sub.t], it was unable to predict the reduction in densities. Second, the 1986 WarmWinter was associated with a higher than average peak density. Three of the four models were able to detect an increase in spring [R.sub.t] caused by a WarmWinter, which was correlated with the increased peak density [ILLUSTRATION FOR FIGURE 12B OMITTED]. By not detecting a WarmWinter effect, the annual model did not predict increased [R.sub.t] and the resulting higher peak densities. Instead, it predicted a peak density of average height, and by August, densities had declined exactly to this level. This contrast is a good example of the difficulties of selecting an appropriate standard to compare models, especially when the models use different time scales. Third, the 1995 drought was accurately forecast by the annual model and the two short-term monthly models that used a data segment that included a drought [ILLUSTRATION FOR FIGURE 12C OMITTED].


Factors determining population size

All natural populations are affected to some degree by density dependence and weather. We determined the relative importance of each by season for the polyvoltine white-looted mouse. By using a 23-yr monthly data set, we studied both the detailed intra-annual relationships and long-term population dynamics. This comprehensive population model indicated that interannual variability was driven by the accumulative effect of monthly weather that directly or indirectly influenced monthly rate of population growth ([R.sub.t]) with short time lags (i.e., 0-2 mo). The intra-annual cycle typical of this species at this latitude was caused by intrinsic monthly [R.sub.t] and short-term (i.e., 1-2 mo) density-dependent effects that varied seasonally. Two droughts during the 22-yr model were the only factors that had long-lasting effects (i.e., 9 mo) on the population. The analysis did not detect any relationship between peak densities in one year and peak densities of the prior year. Overall, the comprehensive population model indicated that population size of P. leucopus in Carter Woods was driven almost exclusively by short-term intrinsic and extrinsic factors.

Interpretation of the comprehensive population model. - Although we did not include biological mechanisms directly in the comprehensive population model, correlations between monthly [R.sub.t] and the independent variables can be interpreted biologically using time-delayed relationships. Density dependence and weather could have directly affected survival, physiology, or behavior of each age class, or indirectly affected individuals through food, refuges, or disease.

Density dependence detected in the population was probably caused by intraspecific competition for resources, because Carter Woods does not support substantial predator or competitor species (S. H. Vessey, personal observation). In autumn and winter, high densities were correlated with an increase in monthly [R.sub.t] with a 1-mo lag and a decline in monthly [R.sub.t] with a 2-mo lag. The negative 2-mo correlation suggests that fewer animals bred when densities were high due to competition for resources (Gilbert and Krebs 1981, Taitt 1981). A reduction in breeding would not be detectable in the trappable population (typically adults) for 2 mo, because gestation and weaning in P. leucopus each require [approximately]1 mo. The positive 1-mo correlation probably was caused by more animals creating more offspring, because we used a monthly scale to study a species with a typical generation length of 2 mo. This relationship could also be caused by sampling error in density estimates (Wolda and Dennis 1993; see below in Time scale of population studies). In contrast, in summer, [R.sub.t] was negatively correlated with densities at 1- and 2-mo lags. This difference between seasons suggests that in summer, high densities negatively impacted in utero and pup survival, causing [R.sub.t] to be reduced at a 1-mo lag. Resource limitation often negatively affects lactating females and thus juvenile survival (Briggs 1986). Finally, as expected in this nonoutbreak species (but see Sexton et al. 1982), we found little evidence of time-delayed or nonlinear density dependence.

We identified numerous weather variables that affected monthly [R.sub.t], and interpreted them as follows. Higher AvgMaxTemp than normal in March to June caused reduced breeding in those months, creating lower monthly [R.sub.t] 2 mo later. This result suggested that the timing of the midsummer breeding hiatus documented in P. leucopus (Rintamaa et al. 1976, Cornish and Bradshaw 1978, Wolff 1985) may vary depending on temperature. This relationship was not necessarily direct and could have been mediated through food resources. Numerous other mechanisms have been suggested for the midsummer breeding hiatus including: food quality or quantity (Wolff 1986), sampling error caused by immigration (Cornish and Bradshaw 1978), density (Terman 1993), reproductive inhibition (Terman 1987), and water availability (Nelson 1993). There is general agreement that the cause of initiation and cessation of reproductive behavior in Peromyscus is not well understood (Wolff 1985, Terman 1993).

Lower TotalPrecip than normal in September through December caused an immediate decline in densities, resulting from either death or permanent emigration of adults or subadults. This effect would be too late to have resulted from abortion of the seed crop (Waller 1988). However, it may have reduced other food resources for P. leucopus, including vegetation, insect adults, and larvae. Increased trapping success on rainy nights (Drickamer and Capone 1977, Vickery and Bider 1981) would not be reflected in density estimates because of the MNA span. Previous work on rodents has identified much longer time lags (e.g., 12 months) between precipitation and resulting densities (Smith et al. 1974, Grant 1976, Pinter 1988). However, these studies did not estimate weather- and density-dependent effects concurrently and did not include all potentially important time lags in the relationship. When included time lags are correlated with significant excluded time lags (which occurs when data are autocorrelated), parameter estimates are biased and inconsistent (Pankratz 1991), and the analysis can show a significant relationship at the hypothesized lag where no significant relationship exists at that lag (for an example see Pankratz 1991: 183).

HotSummer temperatures had a negative effect on densities. Because the HotSummer variable represented two consecutive hot months, HotSummers probably reduced in utero and pup survival, which together resulted in a decline in [R.sub.t]. This effect may have been mediated through reduced plant productivity. We do not believe that HotSummer months reduced survival in adults. Although P. leucopus have been documented to die within 2 d when held in the laboratory at temperatures [greater than]35 [degrees] C and given no food, water, or shelter (Sealander 1951), animals would never suffer these conditions in the field.

WarmWinter temperatures had an immediate positive effect on densities, suggesting that warmer temperatures increased survival directly or indirectly through augmented food resources. The lack of a time delay suggested that breeding was not affected, and trap censuses documented winter breeding in only one of the four WarmWinters. Finally, we could not identify any effect of harsh winter weather because of the lack of census data during those time periods.

The 1977 hot May was associated with a density crash in May and June, which suggested abortion of litters and strong reduction in adult and juvenile survival, possibly enhanced by acclimation to cool spring temperatures (Sealander 1951). This effect was not captured by either the AvgMaxTemp in summer or HotSummers. Although cool spring temperatures may eventually reduce the seed crop in the autumn (Sork et al. 1993), this effect was not detected in our model.

Two droughts strongly reduced monthly [R.sub.t] for numerous months. The difference in the timing of the droughts (i.e., 1- vs. 5-mo start-up lag) may have been related to season, severity, or length of each drought. The 1988 drought, which began in March, may have reduced breeding and recruitment throughout the summer. In contrast, the 1974 drought, which began in July, may have affected survival and emigration. The effect of both droughts lasted longer than any other continuous or threshold weather effect, reducing densities for 5 and 9 mo for the 1974 and 1988 droughts, respectively. The length of the effect and its extension beyond the drought period suggested that food resources were strongly affected and recovered slowly. P. leucopus in winter feed extensively on seeds, and the autumn crop in a number of species is correlated with summer precipitation (Sork et al. 1993). During the 1988 and 1994 droughts, the seed crop contained only 20% of the nuts that were available during the five nondrought years that were sampled (S. H. Vessey, personal observation; seed data were unavailable in 1974). We did not detect any relationship between high summer/autumn precipitation and high spring densities.

Our model supports recent studies suggesting that seed failure is correlated with low spring densities but that heavy seed production is not necessarily correlated with high spring densities (Kaufman et al. 1995, Elkinton et al. 1996). Other studies have found that seed crop is correlated with peak density (Ostfeld et al. 1996, Wolff 1996). We found that although a relationship with peak summer densities existed, droughts only reduced autumn/winter [R.sub.t] but not summer [R.sub.t]. Thus, low peak summer density was a result of the small number of animals breeding in May, and was not a result of continued negative effects of the drought. Seed production may affect densities by influencing winter survival (Hansen and Batzli 1979), conditions for winter breeding (Gashwiler 1979, Wolff 1996), or initiation of breeding (Hansen and Batzli 1978). Our model suggests that an early initiation of breeding may have the strongest effect on the resulting peak densities [ILLUSTRATION FOR FIGURE 8 OMITTED].

Three external events with unknown causes were identified with outlier analysis. Rapid summer growth in April and May of 1980 may have been caused by a large seed crop. As we did not study the effect of seed crop directly, our analysis would not detect an increase in spring densities if masting were not directly correlated with precipitation. The rapid autumn decline in November and December of 1981 may have been caused by the stochastic appearance of owls. Owls have occasionally hunted in Carter Woods during peak density in early autumn, and can have a strong negative effect on density in a small woodlot. Finally, the rapid summer growth during September 1991 was particularly unusual, as September typically has negative growth.

Limitations of the comprehensive population model. - The comprehensive population model did not adequately identify factors causing the occasional large population growth in early summer (e.g., 1980, 1981). This effect typically lasted for a single month between April and June. These high monthly [R.sub.t] were probably not caused by immigration, because the nearest woodlot is 1.5 km away and only two mice have been documented dispersing that far during the 23-yr study. High monthly [R.sub.t] in April was particularly difficult to include in the model, even if weather effects were known, because April was defined as a winter month with an intrinsic growth rate at one-third of summer levels. Because of this design of our model, we could not detect if weather (or indirectly reduced seed crop) affected the initiation of breeding in spring. Although high April [R.sub.t] could have been included in the model using a binary variable, the inclusion would have to be subjective and historically based because the cause of initiation of reproductive behavior in Peromyscus is not well understood (Terman 1993). In addition, because mice acclimate to temperature, rapidly changing temperature on a daily scale may be more stressful than a month of constant, far below average temperatures (Myers et al. 1985).

We believe that the remaining unexplained variability in monthly [R.sub.t] is accounted for by inaccuracies in density data. Few density-dependent or large weather effects should remain because time-delayed densities represented all density-dependent effects and because few multiple-month deviations in the residuals (representing large weather effects) remained (Fig. 6). A number of factors could have caused inaccuracies in density estimates. First, monthly [R.sub.t] estimates were based on interpolations of densities to the end of the month, which assumed that consecutive density estimates (2-6 wk apart) were linearly related in time. Also, density estimates from nestbox data (used only during winter) were significantly higher than from trap data, and because densities in an isolated woodlot could not be overestimated with a direct enumeration technique, trap data must have underestimated winter densities. Nestbox captures also indicated that some individual mice were never caught in traps. Finally, although outlier analysis removed three significant density-estimation errors in the data, we expect that there were other, less severe errors that were not beyond the cutoff of 3.4 standard deviation units. As we discuss below, these potential sampling errors may have increased our chance of detecting density dependence (Wolda and Dennis 1993).

Forecasting population size

Although the model explained 81% of variability in monthly [R.sub.t], it created poor forecasts of 1995 data. These "true forecasts" did not use weather data during the forecast period, and could not be improved by forecasting weather because we used only the stochastic variability in the model. When we used weather data during the forecast period, forecasts were substantially improved, but were still 77% higher than observed peak density. In these forecasts, the effect of the drought on [R.sub.t] was described using parameters identical to a prior drought, even though our analysis indicated that individual droughts have substantially different effects on [R.sub.t] (Table 2). A sensitivity analysis indicated that observed peak density could be produced by extending the duration of the 1994 drought's effect on [R.sub.t] by two additional months. This extended effect could not be predicted because drought effects appear to be highly variable, and could never be confirmed because other factors could have caused the low densities. In general, we expect that forecasts during extreme weather events would usually be poor because extreme events occur rarely, and consequently, prior data does not exist to suggest how an event would affect densities. Thus, forecasts during an extreme event could be inaccurate even if data on the event were available.

Critiques of ecology have argued that models should be judged by their predictive power (Peters 1991). Population ecology has focused on predicting population size because of the applied concerns of managing populations of endangered species, agricultural pests, disease vectors, and exotic species. In addition, much of the research on chaos in population dynamics has been directed toward making more predictive models. If population fluctuations are driven by deterministic factors, such as density dependence, then even wild (i.e., chaotic) fluctuations would be predictable over short time scales (Hastings et al. 1993), often considered to be 2 yr for annual analyses. However, when fluctuations are driven by stochastic events, such as weather, then prediction and management are substantially more difficult (Hastings et al. 1993). Our work suggests that when studying polyvoltine species driven by weather, predictions may be accurate for only a few weeks or months in the future, because animals respond to weather on a short time scale (e.g., 1-2 mo). Future research will need to identify how species respond to fluctuating environments (Jillson 1980) and at what time scales.

Time scale of population studies

The question of scale is critical to all ecological studies. Ecological systems typically show variability on a range of spatial and temporal scales, and it is often difficult to determine the scale best capable of identifying the mechanisms underlying observed patterns (Levin 1992). The population dynamics of polyvoltine species, in particular, often contain intra- and interannual patterns, but the time scale at which intrinsic and extrinsic factors affect the population is often unknown. We found that for a nonoutbreak polyvoltine mouse, models using data collected on a generational scale (i.e., monthly) better described the population dynamics and created better forecasts than a model using annual data, even though the monthly models spanned only a fraction of the years used by the annual model.

On average, short-term monthly models detected half of the weather factors affecting the population (as determined by the comprehensive population model), whereas the annual model detected no weather effects except droughts, and only when a subjectively identified binary variable was supplied. The annual analysis also could not detect the effect of AvgMaxTemp or HotSummers on annual peak densities, even through these weather variables affected the growth phase of the population and occurred just 2 mo before peak densities. Thus, although short-term monthly models could not detect all weather effects because of the short data segment used, they still were more effective than an annual model spanning 22 yr.

Although the short-term monthly models' parameters were tailored to 3-yr data segments, they still represented density dependence on a generational scale, whereas the annual model detected an annual relationship. This relationship indicated that annual rate of population change was correlated with peak density of the prior year. In a polyvoltine species, annual density dependence represents delayed density dependence (Holyoak 1994), which was not detected by the comprehensive population model. The comprehensive population model may have missed an annual effect because the density-dependent relationships were studied by season. However, because the comprehensive model had greater statistical power, and because of the statistical difficulties associated with detecting annual density dependence in polyvoltine species, we believe that the annual model detected a spurious relationship.

Annual density dependence may have been inaccurately identified by using a statistical test (i.e., response surface methodology) not designed for polyvoltine species. All currently available tests to detect annual density dependence assume that the data set is a first-order Markovian time series, which requires that species are univoltine and semelparous (Wolda and Dennis 1993, Wolda et al. 1994). It is currently unclear how violating this assumption affects the outcome of the test (Holyoak and Lawton 1993). In addition, independent, identically distributed random variables can cause annual density-dependent tests to be fooled (den Boer 1990, Wolda and Dennis 1993, Wolda et al. 1994). Simulations have demonstrated that a statistically significant return tendency is often detected in independent, identically distributed variables (Wolda et al. 1994), regardless of their subject matter. For example, although a return tendency has been detected in precipitation data (Wolda and Dennis 1993), this relationship is not equivalent to ecological population "regulation." Also, error in density estimates can cause density dependence to be erroneously identified, because sampling error is bounded where small sampling errors are likely to be followed by large sampling errors (Holyoak and Lawton 1993, Wolda and Dennis 1993, Dennis and Taper 1994). Sampling error increases the chance of detecting density dependence on an annual or monthly scale. Other reasons that a spurious annual relationship can be detected include "constraint" (Hunski et al. 1993), "spreading the risk" (den Boer 1968), autocorrelated noise (Royama 1992), and autocorrelation in an exogenous factor (Williams and Liebhold 1995).

Studies of density dependence attempt to demonstrate that populations have growth rates regulated by their own densities. A polyvoltine species in a seasonal environment has a varying equilibrium - a seasonally varying carrying capacity, which is often driven by seasonally varying food resources. Each month's density estimate is either above or below this varying equilibrium, and thus, density-dependent factors affect the population in relation to a varying equilibrium. (For a discussion of the importance of varying equilibrium to density-dependent theory see Wolda 1989, 1991, Berryman 1991b). Because of this seasonal equilibrium, the population of P. leucopus at Carter Woods has a return tendency that occurs within months rather than over a year. The immediacy of this return explains why peak population sizes are not correlated with each other in the Carter Woods population. By considering only August densities in a polyvoltine species, only a single point in this varying equilibrium is studied, and the comprehensive model suggests that the population does not take 12 mo to return to the equilibrium.

Although our study and others (Kesner and Linzey 1997) indicated that P. leucopus should be studied on a generational scale, this result cannot be indiscriminately applied to all polyvoltine species. When polyvoltine population dynamics are driven by factors that have an annual or multiannual pattern, such as resources, competitors, predators, or disease, polyvoltine dynamics can have a strong multiannual pattern (Hornfeldt 1994; for a review of hypothesized causes of cycles see Akcakaya 1992, Krebs 1996), which may be effectively studied with annual data (Turchin and Hanski 1997). Multiannual patterns are more common in voles than in Peromyscus, in part because voles have a much higher intrinsic rate of increase (see references in Turchin and Ostfeld 1997) than Peromyscus (Millar 1989), and because vole populations are often driven by specialist predators (Akcakaya 1992, Turchin and Hahski 1997). However, those vole populations that do not contain multiannual cycles may need to be modeled with a more detailed data set. In addition, not all P. leucopus populations necessarily need to be studied with generational data. When a population has multiannual fluctuations caused by specialist predators (Drost and Fellers 1991), annual data might be effective at identifying factors driving the population.

Our results suggest that before censusing begins, the species and the community in which it resides need to be evaluated to identify the time scale at which factors affecting the population may occur. Census interval should be based on (1) generation interval, interval between life stages, and length of breeding season of the species under study; (2) population pattern or generation interval of resources, competitors, and predators that may be affecting the population; and (3) the strength of the annual vs. monthly weather patterns and variability to which individuals may respond. These factors determine the length of the return tendency, and thus the appropriate time scale at which to census the population to yield an accurate understanding of which factors are affecting the population.

Currently, most published long-term population data sets ([greater than]20 yr) represent annual censuses, regardless of the life history of the species. (For lists of data sets see Turchin 1990, Turchin and Taylor 1992, Wolda and Dennis 1993, Ellner and Turchin 1995). Although some polyvoltine species can be effectively modeled with annual data (e.g., some vole populations), this prevalence of annual data will limit our understanding of the population dynamics of certain polyvoltine species until data on an appropriate time scale can be collected. In addition, our work indicates that where data are available, analyses of polyvoltine species using annual data (e.g., Holyoak and Lawton 1992, Woiwod and Hahski 1992, Fryxell et al. 1998) should be reanalyzed on a time scale appropriate to their life histories and community interactions.


This study yielded three main conclusions. First, by analyzing data on a generational time scale using multivariate time series analysis, we identified which factors drive the population fluctuations of the polyvoltine white-footed mouse - density dependent or independent. Contrary to much of the Peromyscus literature, our analysis indicated that, with the exception of droughts, P. leucopus respond to density-dependent factors and weather only on a 1- or 2-mo time scale. Second, even with extensive data at an appropriate time scale for analysis and with an excellent explanatory model, populations driven by weather may not be accurately forecastable on a time scale useful to management and conservation concerns. Finally, we found that annual censuses were incapable of detecting most factors affecting the population and possibly identified spurious relationships. These results suggest that population studies on an inappropriate time scale may yield inaccurate results. Future studies need to identify the most appropriate time scale of the study system before data collection is initiated.


We thank B. Dennis, J. Fryxell, R. Ostfeld, and one anonymous reviewer for their insightful suggestions that improved the research this article describes. We also thank J. Bruseo, T. Carpenter, J. Miner, A. Porter, and H. Wolda for their critical reading of earlier drafts of the manuscript. We are grateful for the aid of a large number of graduate and undergraduate students who trapped Carter Woods over a period of 23 yr including J. Cummings, D. Fleming, T. Goundie, J. Jacquot, A. Korytko, P. Mazur, R. McKenna, D. Rintamaa, D. Sanchez, and M. Schug. This work was supported in part by a Whitehall Foundation grant to S. Vessey and Doctoral Dissertation Improvement grant number DEB9423424 from the National Science Foundation.


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