A damped precipitation‐driven, bottom‐up model for deer mouse population abundance in the northwestern United States

Abstract Small‐mammal population densities can be regulated by bottom‐up (food availability) and top‐down (predation) forces. In 1993, an El Niño Southern Oscillation event was followed by a cluster of human hantavirus with pulmonary syndrome in the southwestern United States. An upward trophic cascade hypothesis was proposed as an explanation for the outbreak: Increased plant productivity as a consequence of El Niño precipitations led to an unusual increase in distribution and abundance of deer mice (Peromyscus maniculatus; reservoir host of Sin Nombre virus). Could such drastic events occur in mesic habitats, where plant productivity in response to climate conditions is likely to be much less dramatic? In this work, we investigate to what extent deer mouse populations follow a precipitation‐driven, bottom‐up model in central and western Montana and discuss important conditions for such a model to be possible. We found positive correlations between deer mouse abundance and on‐the‐ground measured plant productivity with a several‐month lag in three of six study sites. This effect was weaker when deer mouse populations were more abundant, indicating density‐dependent effects. Dispersal resulting from territoriality may be important in attenuating local density increments in spite of high food availability. In addition, there is evidence that population abundance in the study area could respond to other abiotic factors. In particular, precipitation in the form of snow may reduce deer mice survival, thus compensating the benefits of improved plant productivity. Deer mouse populations in Montana study sites follow complex dynamics determined by multiple limiting factors, leading to a damped precipitation‐driven bottom‐up regulation. This prevents dramatic changes in rodent abundances after sudden increments of food availability, such as those observed in other regions.


| INTRODUCTION
How small-mammal population densities are controlled has been widely argued for a long time. Most works have focused on the effects of food availability and predation, which are referred as bottom-up and top-down forces, respectively. Support for both is found early in the literature. For example, Lack (1954) showed that vegetable food was the major factor controlling various rodent populations' density which in turn led to predator-prey cycles. On the other hand, Pearson (1966Pearson ( , 1971 observed Microtus californicus populations were limited by carnivorous predation, which determined the amplitude and synchronization of abundance cycles. More recently, Prevedello, Dickman, Vieira, and Vieira (2013) conducted a meta-analysis of food supplementation studies and concluded that both bottom-up and top-down forces are important for the regulation of populations. However, intrinsic factors such as intraspecific competition could also be relevant for controlling population densities (Conley, 1976). Socially intolerant individuals tend to disperse when density increases, keeping populations locally stable (Krebs, Gaines, Keller, Myers, & Tamarin, 1973;Myers & Krebs, 1971). In situations of extreme densities, even survival and reproduction can be diminished by physiologically driven behavioral changes (Christian, 1950(Christian, , 1978David, 1978). Altogether, the complexity of population regulation makes hard to predict how climatic events and other abiotic factors can impact on rodent abundance and dispersal.
Human health risks associated with disease-bearing species are likely to be triggered by sudden changes in rodent populations.
In 1993, a cluster of Hantavirus pulmonary syndrome (HPS) cases caused by Sin Nombre virus (SNV) occurred in the southwestern United States. It was hypothesized that precipitation associated with an El Niño Southern Oscillation event produced increased plant productivity and, consequently, deer mouse (Peromyscus maniculatus, the reservoir host of SNV) abundance increased after a several-month lag in locations where they previously were absent or rare and where plant productivity had previously been extremely low (Dearing & Dizney, 2010;Glass et al., 2002). The concept that the increased abundance and wider distribution of deer mice in the southwestern United States occurred in response to increased plant productivity was called the trophic cascade hypothesis (Parmenter, Brunt, Moore, & Ernest, 1993;Yates, Mills, Parmenter, & Ksiazek, 2002). This increase in the deer mouse population may have increased rodent-to-rodent transmission of SNV that ultimately spilled over to humans (Mills, Ksiazek, Peters, & Childs, 1999).
This bottom-up precipitation-driven process used to explain HPS cases is still discussed in relation to variability among habitats and climatic regimens Mills, 2005;Loehman et al., 2012). Deer mice are absent or not abundant in many areas of the normally arid US southwest during typical dry years. In this region, dramatic changes in plant productivity after an El Niño event may produce habitats more appropriate for deer mouse populations, at least temporarily.
Studies in the southwestern United States demonstrated the relationship between vegetation growth and deer mouse abundance 1 year later (Engelthaler, 1999;Glass et al., 2002Glass et al., , 2006Glass, Shields, Cai, Yates, & Parmenter, 2007). In addition, evidence of a delayed relationship between precipitation and SNV prevalence in deer mouse populations, likely associated with increased plant productivity, has been found in the Channel Islands in California (Orrock, Allan, & Drost, 2011).
In contrast, in the mesic parts of the US west, deer mouse populations tend to be nearly ubiquitous, although at varying abundance regardless of climatic changes Douglass et al., 2001). Mesic habitats (coniferous forests, grasslands, and sagebrush) present situations where plant productivity in response to climate conditions is likely to be much less dramatic than in the arid US southwest.
Consequently, the response of deer mouse populations to changing climatic conditions in mesic areas is likely to be less pronounced than those seen in the arid southwest. In addition, predator richness has also been identified as a factor regulating rodent populations independently of precipitation, even in habitats where productivity is strongly affected by precipitation (Orrock et al., 2011). Therefore, the bottom-up model cannot be generalized straightforwardly to other contexts.
Because the emergence of SNV is linked to changes in climate (Carver et al., 2015), it is important to clarify the link between deer mouse population abundance and plant productivity in the northwestern US where deer mouse populations are persistent. In a study on the effects of climate on deer mouse populations, Luis, Douglass, Mills, and Bjørnstad (2010) showed that deer mouse population dynamics at one location in western Montana were correlated with precipitation, time of precipitation, and temperature after 0-to 5-month lags.
However, similar correlations were not found at another location in western Montana (Luis pers. com.). Moreover, Loehman et al. (2012) found that remotely sensed plant productivity provided limited predictive information regarding deer mouse abundance on two sampling grids on which Luis et al. (2010) found climate effects in Montana.
These contradictory observations could indicate that remotely sensed plant productivity may not accurately account for available biomass at the scale of single 100 × 100-m sampling grids.
Our primary objective was to assess whether deer mouse populations follow a precipitation-driven bottom-up model in central and western Montana. In so doing, we identify important assumptions of the hypothesis and provide potential explanations for variable results among sampling sites. For this purpose, we evaluated the relationship between deer mouse abundance and various environmental characteristics. In particular, we focus on the response to on-the-ground measured plant productivity after various time lags, and we investigate density-dependent effects.

| Study area and sampling design
We used deer mouse trapping data based on 850,000 trap nights and environmental data collected at six sites in central and western grassland, sagebrush, meadow, and subalpine fir. For a detailed habitat description, see Douglass et al. (2001). We sampled three grids per site over 12-17 years (Table 1). Except for high-altitude grids (>1,590 m), deer mice were present during all sampling periods at all locations .
Trapping and animal handling followed Douglass et al. (1996), according to Mills et al. (1999), and approved by the University of Montana Animal Use Committee, approval #011-04RDTECH-021304.
We livetrapped for three nights in each sampling period. All animals were marked with ear tags, and sex, breeding condition, weight, and presence of scars were recorded. Blood samples were collected from deer mice at two of the three grids at each site, with the third grid acting as a control grid to determine the effect of blood collection on deer mice (Douglass, Kuenzi, Wilson, & Van Horne, 2000). We released all animals back to the grid on which they were captured.

| Vegetation sampling
Each September, after seed ripening, we measured plant cover at 30 randomly selected plots on each grid. We used a one-half-meter point frame with 10 rods and recorded the contacts with bare ground, rock, mosses, lichens, duff litter, grasses, forbs, and shrubs. We clipped and placed all herbaceous matter in individual paper bags from each 0.1 m 2 plot for drying. We also recorded the maximum height of shrubs contacting or overhanging the frame. Herbaceous matter was dried and weighed to determine productivity, beginning in 2002.

| Statistical methods
The vegetation variables were determined for each grid as the average number of contacts for bare ground (bg), rock (ro), moss (mo), lichens (lich), duff litter (dl) , grass (gra), forbs (fo), and shrubs (shr) and the average of maximum shrub heights (avshr). Biomass (biom) was calculated as the dry weight per sampled area unit. All the statistical methods described below were performed using the software R (R Core Team, 2016).
Correlation among variables may be underestimated if their distributions are too different in shape (Goodwin & Leach 2006). Therefore, vegetation variables were either logarithmically or square-root transformed to obtain more symmetrical distributions. Pearson correlation coefficients among transformed variables were <0.33, except between shr and avshr, for which it was 0.51. Collinearity between variables is undesired as it can lead to larger standard errors in parameter estimates. Therefore, the variable avshr was fitted on a linear model in terms of shr, and the residuals were used instead of the original values (i.e., the uncorrelated part of the variable). The remaining variables were centered by subtracting their corresponding mean values after the transformation.
We were interested mainly in the effect of productivity, measured by biomass, on deer mouse abundance, estimated as minimum number alive (MNA). MNA estimates of population size at each sampling period were calculated as the sum of all animals captured during that period, plus the number of individuals that were captured during at least one previous and one subsequent sampling period, but not during the current period (Chitty & Phipps, 1966). Because biomass was not measured from 1994 to 2001, we fitted a linear regression model to extrapolate biomass from point-frame cover values. We used all habitat cover measures except biom as explanatory variables. We ranked the full model and all its nested models based on the Akaike information criterion corrected for finite sample size (AICc; Burnham & Anderson, 2002). The best model (lowest AICc) included the variables: bg, fo, and gra (Table 2) and was significantly better than any other model (ΔAICc ≥ 2). Consequently, the model using cover of bg, fo, and gra was used to extrapolate biomass. To determine the error in biomass extrapolation, we made a leave-one-out cross validation.
This procedure simulates the extrapolation on known data, providing an estimate of the expected extrapolation error (Burnham, 1983). We also considered the uncertainty in regression coefficients. Therefore, extrapolated biomass errors were calculated as (σ 2 + δ 2 ) 1/2 , where σ is the regression error and δ is the cross-validation error.
To evaluate the relationship between MNA and vegetation variables, we constructed log-linked Poisson generalized linear mixedeffect models (GLMMs). We used MNA as the response variable, seven habitat variables as explanatory variables (ro, mo, lich, dl, shr, avshr, and biomass [biom]), the grid location (site among the six sites listed under "study area" above) as fixed effect, and grid as a random factor. To evaluate whether such a relationship may have a delayed effect on MNA, various data sets were created by shifting the abundances with respect to the explanatory variables. Each sampling session where vegetation data were available was assigned the MNA measured a given number of months later (lag). Incomplete entries were discarded. For each lag T A B L E 1 Geographic characteristics and sampling periods for the six study sites in Montana

Range of elevation (in meters)
General between 0 and 16 months, GLMMs including all the variables and all nested models were fitted and averaged using AIC weights with a correction for finite sample sizes (AICc). Model fitting and averaging were conducted using R packages lme4 (Bolker, 2013) and MuMIn (Barton, 2013), respectively. Only models with ΔAICc < 10 were included in the average (Burnham & Anderson, 2002). No assessment of significance other than model selection was made at this stage.
To account for the error in the extrapolation of biomass, the analysis described above was repeated following a randomization procedure. For each replicate, a new random variable was generated for each entry for which biomass was extrapolated, drawing its value from a normal distribution with the mean equal to the extrapolated value and standard deviation equal to the extrapolation error. The randomization-fitting cycle was repeated 350 times. This number of replicates was decided upon a preliminary analysis, so that the standard error of averaged coefficients would be smaller than their corresponding errors in each replicate. Final errors in the replicate-averaged model were calculated considering single-replicate errors and the dispersion due to randomizing extrapolated biomass. The error of each coefficient α was estimated as α i is their mean value, and SE(α i ) is their standard error from each replicate. Effects for a given time lag were considered significant when the corresponding 95% confidence intervals (α±1.96E α ) did not include zero. In order to test for possible density-dependent effects, we investigated the combined effects of previous abundances and lagged values of biomass on deer mouse populations. The use of autoregressive models (i.e., models for which each observation of the response variable is modeled in terms of other observations of the same variable) has proved useful for understanding important correlations-both temporal and spatial-in ecology (Vieira et al. 2008, Ives et al. 2010. Applying these models to the present data is not straightforward as trapping sessions were not always evenly spaced. However, trapping sessions were conducted often enough so that characteristic times of population dynamics comprised multiple sessions. Therefore, we adopted a coarser approach: For each sampling session, we calculated the log-transformed (i.e., log[1 + x]) mean abundances of three previous 6-month periods (short term: 1-6 months, midterm: 7-12 months, and long term: 13-18 months prior to current session). These three averages, sampling grid, and the mean biomass for the three time lags which showed stronger effects (8-10 months prior to current session, see Results) were considered as covariates in a log-linked Poisson GLM. Two-and threefold interaction terms among biomass, previous abundances, and site were also included in the full model (but no interactions among averaged abundances). We grouped interactions per site (instead of per grid) to avoid having too many parameters to estimate. The full model and all nested models were fitted and averaged based on their AICc (Burnham & Anderson, 2002), using package MuMIn for R (Barton, 2013). Relative importance (RI) of each term was calculated as the sum of Akaike weights of all models having that term. For this analysis, data from site Wisdom were excluded due to consistently too low capture rates.

| RESULTS
Measured biomass ranged from 29 to 1,666 kg/ha.
The results of models tested to determine best overall for extrapolating biomass based on their Akaike information criterion corrected for small sample size (AICc)

| DISCUSSION
The bottom-up regulation model assumes that energy (i.e., food) is the only factor limiting populations, so their densities should increase continuously with greater food availability. Plant material (mostly as seeds but some vegetative parts) directly provides energy and supports many insect populations which are also a source of energy for deer mouse populations (Pearson & Callaway, 2006).
Therefore, rodent populations should expand after periods of warm temperatures and abundant precipitation due to the subsequent increase in plant productivity (Hansson, 1979). Considering all the stages in a bottom-up model, the maximum positive effect on rodent abundance can be expected about a year after warm and rainy weather for several reasons (Heisler, Somers, & Poulin, 2014): It takes a growing season between precipitation and the expression of productivity. Once mice receive adequate biomass, it takes time for the population to respond through survival and reproduction.
The same applies for insect populations before they represent an increased source of energy for mice.
When the conditions for a bottom-up regulation are met, a sudden increase in food availability may unleash a population explosion.
Examples of such situations outside the US southwest are found worldwide: In temperate Europe, bank voles populations increase after mast years (Johnson, Moraes Figueiredo, & Vapalahti, 2010); in western Patagonia, infrequent flowering of colihue cane is followed by a drastic growth of granivorous rodent populations (Jaksic & Lima, 2003;Piudo, Monteverde, González Capria, Padula, & Carmanch, 2004). However, the same species subject to bottom-up regulation in one region may show completely different dynamics in another context, such as the contrasting predation-driven top-down regulation of bank voles in northern Europe (Johnson et al., 2010).
While there is sound evidence of a precipitation-driven bottom-up process ruling deer mouse population dynamics in the arid southwestern United States and the Channel Islands in California (Orrock et al., 2011), the mechanism is not so clear in the western mesic habitats.
Below, we analyze whether the conditions for a precipitation-driven bottom-up regulation are met in Montana study sites.

| Food as the only limiting factor
Liebig's law of the minimum states that population growth will always be controlled by the scarcest essential resource (Salisbury, 1992). In this context, energy must be the only limiting factor. Other requirements such as nest sites and escape cover cannot be more limiting than energy. Food effects have been tested with several species of rodents. An increase in food results in various demographic changes (Duquette & Millar, 1995), but neither increased food (Duquette & Millar, 1995;Wolff, 1985) nor natural seed production (Kaufman et al. 1995;Elkington et al., 1996) necessarily increased population density.
Increased mast yield does increase deer mouse population density (Ostfeld, Jones, & Wolff, 1996;Wolff, 1985;Schnurr et al. 2002), but there were no mast-producing plants on our study sites. Thus, it is doubtful that energy is always the only limiting factor for our study populations.
The Other habitat features, including rocks, duff litter, moss, and lichens, had a negative association with abundance only for some time lags, mostly around 7-10 months. This is coincident with the strongest positive association with biomass, suggesting that there is a connection among all effects. Rocky environments affect nesting habitats (Wolff & Sherman, 2008) while moss and lichens may be indicators of recent climatic conditions such as humidity and temperature or habitat quality. It is possible that these abiotic factors also affected deer mouse survival, and due to the characteristic times of their population dynamics, all the effects on MNA become apparent after about the same time lag.

| Density-independent behavior
In order for populations to grow as long as additional food becomes available, intraspecific interactions must remain constant through all population densities. If deer mice were territorial, their numbers may be limited by social behavior before resources become limiting (Krebs et al., 1973).
In control buildings, mouse populations remained stable and were comprised of the same individuals for the duration of the study.
Our results offer evidence that such territoriality may indeed constrain the effects of forage (biomass). For Cascade and Gold Creek study sites (both with highest abundances out of the three sites where we found a positive effect of biomass on MNA), the interaction term between lagged biomass and short-term previous MNA was negative.
This appears as a slight saturation in the fitted effect of lagged biomass on MNA (Figure 4). The meaning of this saturation is that population growth resulting from increased plant productivity becomes less pronounced in moments of higher abundance, thus supporting the existence of a density-dependent social limit.

| No interspecific competition or predation
Competing species may interfere with deer mice using available energy, thus reducing the impact of changes in plant productivity on mice populations. The most abundant other small-mammal species at the study sites were voles (Microtus sp.), which were only present sporadically on the grids. Small-mammal communities at our study sites were relatively simple compared to studies conducted in the US southwest . Therefore, although competition (either by aggressive interference or by simply getting to the food first) may have occurred at some point on some of our grids, we can expect that it was not a strong factor determining deer mouse abundance at our study sites.
On the other hand, increased survival or recruitment as a consequence of increased food availability could be countered by increased predation, leading to a mixture of top-down and bottom-up processes (Prevedello et al., 2013). The predators' coyote (Canis latrans), ermine (Mustela erminia), and rattlesnake (Crotalus viridus) were occasionally observed or trapped on or near various grids. We do not have data on the effect of these predators on deer mouse abundance in our study sites. However, in the Channel Islands in California, predator richness has been associated with lower hantavirus prevalence, likely as result of reduced deer mouse density (Orrock et al., 2011). Kotler (1984 documented predation on deer mice by owls in the Great Basin Desert. Later, Kotler (1985) described avoidance of open areas and foraging in bushes as antipredation strategies, which eventually determined microhabitat use. Reduced foraging activity of deer mice in response to artificial light was also observed in experiments (Clarke 1983). If density-dependent behavior forces some individuals to forage in open areas due to increased density, predation risk also increases at higher densities. Thus, predation may limit population growth as a consequence of the social limit caused by intraspecific strife. However, this compensatory effect is expected to be secondary to that of plant productivity (Mutshinda, O'Hara, & Woiwod, 2009;Ostfeld & Holt, 2004).

| Only productivity-mediated effects of precipitation affect deer mouse populations
In the precipitation-driven bottom-up model, precipitation effects populations. In northern Europe, snow has been found to provide shelter, reducing predation risks during winter (Hansson 1985).
In contrast,  reported populations reached annual lows on all grids at the end of winter, with the exception of mild winters (no midwinter snow accumulation) when deer mouse population numbers were higher. This suggests a negative relationship between snow accumulation and overwinter survival. It is not clear whether this is actually a consequence of snow or it is due to more general weather conditions correlated with snow accumulation (e.g., lower temperatures). In either case, should plant productivity be increased after snowy winters, reduced overwinter survival will limit the benefits of subsequently increased food availability.

| A population regulation model for deer mouse populations in the Northern Great Plains
Periods of low food availability acting as a limiting factor for deer mouse were observed at three study sites, but population fluctuations at two other sites could not be explained in terms of biomass availability. Moreover, at the three study sites where biomass was related to increased MNA, the food limit would not be much lower than the social limit due to intraspecific competition and density-dependent behavior. Therefore, increased food availability likely enhances survival and leads to population growth, but individuals soon leave crowded areas. Dispersal thus attenuates the local density increment below the higher food limit ( Figure 5).
Increased predation may occur as a result of higher rodent density, but most likely after abundance is already limited by density-dependent interactions. For this reason, this increment in predation would not be a crossover from bottom-up to top-down regulation as more prey becomes available for predators (Orrock et al., 2011;Prevedello et al., 2013). Instead, populations are limited by a combination of energy availability and social behavior, leading to a damped bottom-up process. Other abiotic factors, such as snow accumulation and availability of nesting sites, may also contribute to compensate beneficial effects of increased plant productivity in response to precipitation.
Still, the proposed damped precipitation-driven bottom-up model adequately explains the observed dynamics only in the three study sites where consistent fluctuations in response to measured biomass were observed. What energy source replaces biomass at the study sites with lower measured plant productivity and whether it acts as a limiting factor remain to be explored.

| CONCLUSION
In the arid US southwest, deer mouse populations expand after El Niño events that produce widespread plant growth where typically little growth occurs during dry years (Parmenter et al., 1993;Yates et al., 2002). Similar strong associations between precipitation and rodent density, mediated by increased plant productivity, were observed in the Channel Islands in California (Orrock et al., 2011). In contrast, the conditions required for a strictly precipitation-driven bottom-up regulation to occur are only partly met by persistent deer mouse populations in Montana. Although we found positive correlations between deer mouse abundance and plant productivity with a several-month lag, as required to fit the hypothesized upward trophic cascade model, the effect was neither particularly strong nor universal over 18 livetrapping grids in Montana. Predation and interspecific competition appear to be of little F I G U R E 5 . Schematic description of the damped trophic cascade timeline in Montana (green dashed line represents the population density theoretically allowed by the food supply alone; red dotted line (social limit) is the population density allowed by density-dependent factors). In periods of low food availability, rodent survival (represented by brown/gray mice ratio) may be limited by food. After warm and rainy periods which increase plant productivity, higher food availability may enhance survival, leading to population growth. However, once the population density (blue line) approaches a social limit, mice disperse despite surplus food availability. Local abundance thus increases, but not as much as expected if food were the only limiting f a c t o r importance in regulating deer mouse populations in Montana study sites, but social and abiotic factors may play roles not observed in desert habitats of the US southwest.
It is clear that deer mouse populations in northwestern Montana display complex dynamics which requires consideration of multiple potential limiting factors (Heisler et al., 2014). Thus, a combination of factors prevents dramatic changes in rodent abundances after sudden increments of food availability, such as those observed in other regions.