Population models are increasingly being applied for ecotoxicological risk assessments to predict population-level effects after application of pesticides or the impact of agricultural practices (Pastorok et al. 2002, 2003; Dalkvist et al. 2009; Forbes et al. 2009; Topping et al. 2009; Wang and Grimm 2010). Despite various attempts to establish population modeling in ecotoxicological risk assessment the acceptance of such models is still limited in Europe. The main criticism is that the realism of such models is mostly not well demonstrated: in a recent review, Schmolke et al. (2010) have shown that in the current literature only 3% of the models were validated using independent data sets. This is astonishing because an earlier review 10 years ago by Grimm (1999) had already pointed out that validation of population models is hardly done and needs further investigation. Hence validation of population models may be a requirement that is still not sufficiently included in current models and that may be a major obstacle for using such models for practical applications, such as the risk assessment of pesticides. However, Barnthouse (1992) pointed out that models used in risk assessment do not have to be fully validated to be useful.
True validation in a strict sense is never possible (Barlas and Carpenter 1990; Oreskes et al. 1994), i.e., it can never be shown that a model produces correct results for all possible situations. Therefore, here the term “validation” is used to describe a method that shows that a model produces a realistic prediction for a variety of situations by comparing model results with observational data (Conroy et al. 1995; Rykiel 1996). Validation includes a systematic comparison of real data, usually obtained from field studies, and results from a population model. This comparison of model and field data includes both variability and uncertainty. Although natural variability describes the variability in nature, e.g., standard deviation of mean litter size, uncertainty refers to measurement errors, sample size effects, or lack of knowledge of the distribution of individuals in the population. Although the need for estimating uncertainty and variability was recognized early in the advent of population ecology (Andrewartha and Birch 1954; Strong 1986) these factors are not commonly addressed in population models (for exceptions see Wiegand et al. 2004 and Clark 2003). However, this is partly due to the fact that in field studies uncertainty can often not easily be distinguished from natural variability, whereas in a population model it is often possible to reduce uncertainty of the model output by increasing the number of simulations or individuals (see also Wang 2012 for a comparison of uncertainties of a model and of field data).
For the analysis of the realism of a population model, all key parameters need to be validated that may have an affect on the output of the model. Because the purpose of a model used for risk assessments is to realistically reproduce the population dynamics of the species under concern, a great number of parameters have to be validated, which affect population dynamics, including reproduction, survival, spatial behavior and the amount of population fluctuations, and possibly population cycles. However, validation of population models with independent data is not a standard approach. Although some authors have validated their models by qualitatively comparing the model output with laboratory data (Rinke and Vijverberg 2005) other authors have tried to compare values or ranges from several field observations with model predictions (Wang and Grimm 2007). In practice, the method chosen for validation strongly depends on the availability of data and on the methodology with which these data were obtained.
In the present study, we developed an individual-based model for the common vole, Microtus arvalis, one of the standard species considered in risk assessments needed for pesticide registration in Europe. The purpose of the model is to realistically reproduce population dynamics and to predict recovery after potential impacts on the populations by pesticides. All key parameters of the model are systematically validated based on independent field data. We show how a structurally adequate and well-parameterized model can result in a very realistic model, realistically reproducing natural spatial behavior, habitat preference, reproduction, survival and population dynamics of the common vole. Unexpectedly, the model also reproduced population cycles that are a typical feature of vole populations. These cycles were not explicitly included in the model but emerged as a result of home range behavior.
The common vole (M. arvalis) is a common herbivore living in most parts of central Europe. Compared to other small mammals, common voles can reproduce rapidly (Niethammer and Krapp 1982) reaching high densities. Common voles are also known as pests in agricultural areas, because they can cause considerable damage to crops at mass occurrences (Tertil 1977; Truszkowski 1982). Mass occurrences are related to a very fast reproduction, characterized by large litter size and early maturity (Niethammer and Krapp 1982). Females can already conceive while they are still lactating (Niethammer and Krapp 1982; Flowerdew 1987). Feeding primarily on green plant material or roots (Niethammer and Krapp 1982), which is usually abundant compared to seeds or invertebrate food, common voles are less limited by food resources than other small mammal species, such as granivorous mice or insectivorous shrews, allowing common voles to live in rather small home ranges (Jacob 2000; Briner et al. 2005). Evidence from the literature is still controversial regarding territoriality. Although several authors observed territoriality and no overlap of female home ranges in common voles (Frank 1954a, 1954b; Reichstein 1960; Corbet and Harris 1991) others reported that females may breed together (Boyce and Boyce 1988a, 1988b, 1988c). However, Boyce and Boyce (1988a, 1988b, 1988c) only found 2 groups of females in their study. Frank (1954a, 1954b) observed that resident females do defend their home ranges. Corbet and Harris (1991) reported that home ranges are used rather exclusively, and Niethammer and Krapp (1982) found that home ranges of adult females usually do not overlap whereas those of males do. Territoriality in female small herbivores is usually interpreted as a measure to ensure sufficient resources for female reproduction (Ostfeld 1990). It can therefore be assumed that some degree of territorial behavior exists in females. Male and subadult home ranges often overlap indicating that territoriality in males is limited, although aggressive behavior between males was observed (Niethammer and Krapp 1982).
A typical feature of the population ecology of voles are population cycles, which have been observed in various vole or lemming species, including M. arvalis (Krebs and Myers 1974; Ostfeld et al. 1993; Hanski et al. 2001; Lambin et al. 2000; Inchausti et al. 2009). In the common vole population cycles have been reported for Scandinavia and central Europe and usually show peak densities every 3 to 4 years (Mackin-Rogalska and Nabaglo 1990; Tkadlec and Stenseth 2001; Hanski et al. 2001).
The general model design was based on a previously published model for the common shrew (Sorex aranaeus) (Wang and Grimm 2007). The model is an individual-based model, in which the behavior of each single individual is simulated to reveal the development of the entire population. The model description follows the overview, design concepts, and details (ODD) protocol for describing individual- and agent-based models (Grimm et al. 2006).
The purpose of the model is to realistically reproduce population dynamics for the common vole to predict recovery after potential impacts on the populations by pesticides. For this purpose, the dynamic spatial arrangement of home ranges in different habitats, effects on density dependent population regulation, and population cycles are included.
State variables and scales
The entities of the model are the landscape, the individuals, and the home ranges. The landscape consists of hexagonal cells (5 m diameter), each characterized by habitat type (e.g., grassland) and amount of food resources on a given calendar date. Individuals are characterized by the state variables: age, gender, developmental stage (lactating offspring, subadult, adult), reproductive status (breeding or nonbreeding), fertility (fertile, infertile—applies to females only), pregnancy, and identity of the home range occupied. Home ranges are represented by a number of cells of the landscape used by an individual. All cells of a home range are connected, i.e., no isolated cells are allowed. Animals without a home range are dispersers (except for unweaned juveniles). This means that dispersers are identified by a home range size of zero cells. The time step in the model is 1 day; simulations were usually run over 1 to 50 years. Model landscapes usually covered between 5 and 25 ha.
Process overview and scheduling
The following pseudocode gives an overview of the processes and their scheduling run each day for every individual:
If individual does not survive
Delete individual from population
Update developmental stage
If in breeding season
Update reproductive state (receptive, gestating)
If home range size is larger than zero
Update home range
Disperse (and try to acquire a home range)
If individual is adult and if breeding season
Changes in state variables caused by the model processes are updated immediately.
Population dynamics (including intra- and interannual cycles) and the spatial arrangement of individuals and home ranges emerge from the model (e.g., the size and location of home ranges is driven by home range optimization; population dynamics are regulated by the number of reproducing individuals, which themselves depend on population density and the amount of food resources, see Update of home ranges). Life cycle, reproduction, and survival rates are represented by empirical rules and parameters.
Individuals sense the amount of food, vegetation cover, and the presence of other individuals in cells of the habitat, which are adjacent to their home range or which they pass during dispersal.
Individuals try to optimize their home range by preferentially selecting cells with high-food resources and by avoiding cells occupied by other individuals.
Fitness consequences of the home range behavior are indirectly included by allowing only females with a home range to reproduce. Dispersers do not reproduce.
Individuals interact when they try to occupy the same cell. Adult females can expel other adult females from 1 or more cells with a probability of 0.5.
Values of almost all parameters are sampled from probability distributions (Table 1) to reflect the natural variability of the parameter values, e.g., litter sizes. Parameters are sampled from the respective distribution at the time when they are used.
Table 1. Parameters and values of the common vole model
Max = Maximum; Min = Minimum; SD = standard deviation.
1. Food resources in a cell
Data series for vegetation height obtained from Jacob (2000)
For model testing, the behavior of single individuals and of the whole population was observed. More specifically, reproduction (litter number, percentage of pregnant/lactating females), home range size distributions, habitat selection, dispersal, population density and growth rate, age structure, and population cycles were analyzed.
The model is initialized with a given number of individuals. The animals are randomly distributed in the landscape. The initial age distribution of the animals corresponds to the age distribution observed after 1-year simulations (a γ distribution with α = 7.554 and β = 19.2879). The home range of each individual initially contains a single cell of the landscape. A model prerun of 14 days allows the animals to extend their home ranges or acquire new ones.
The only environmental conditions considered are the amount of food resources and vegetation cover in a habitat. Weather conditions are not included because the literature provides no information about the impact of winter temperatures or other climatic factors on survival or other parameters. Food resources were estimated as by vegetation height reported by (Jacob 2000) for several habitats over a period of 3 years. Jacob and Hempel (2003) have shown that home range size correlates linearly with vegetation height indicating that vegetation height is an adequate measure for food resources.
Mortality is interpreted as the probability of survival on a given day. Mortality rates depend on the season, habitat, and on the developmental stage of the individuals. For adults, survival rates were obtained from Adamczewska-Andrezejewska and Nabaglo (1977) who observed common voles in arable land over a period of 2 years. These survival rates can be considered conservative, because animals not found (for 3 months) were assumed to have died (but could either just not have been trapped or might have dispersed). For juveniles up to an age of 20 days, survival rates were obtained from survivorship curves from Boyce and Boyce (1988a). Habitat-specific survival was included based on data from Jacob (2000), who observed differences in survival arable land and other habitats. Mortality is simulated by sampling a random number between 0 and 1 for each animal and day. If this number is smaller than the daily mortality rate, the individual is considered to die.
Update developmental stage
The update of the developmental stage is based on the length of the lactation period and the time until maturity. Parameter values are drawn from the respective probability distributions (Table 1) for each individual.
Update reproductive state
The start and end of the breeding season is drawn for each individual separately from the respective probability distributions (Table 1). Males are always fertile throughout the breeding season. Mature females are considered to be always fertile when they are not pregnant, because interbirth intervals in voles approximates usually the gestation length (Niethammer and Krapp 1982) and because ovulation is induced by mating behavior (Flowerdew 1987). The update of the reproductive state includes also giving birth after a given gestation length (Figure 1) after mating (see below). Gestation length and litter size are determined during mating.
Update of home ranges
Home ranges refer to a number of cells of the landscape used by an individual. The food resources available to an individual are calculated by the sum of the food resources of all home range cells. Only cells with a minimum vegetation cover can be part of a home range (parameter 14 in Table 1). The size of home ranges is determined by the amount of food available in the cells of the landscape. Each animal requires a certain amount of food. This amount is determined by a “consumption” parameter (parameter 16) that depends on the season of the year. The purpose of this parameter is to ensure realistic home range sizes. This parameter was calculated based on the food resources in the landscape cells and home range size of males and females during winter and summer in grassland from Jacob (2000) (average over the year: males, 193.9 m2; females, 141.6 m2) and data about the seasonal change of home range sizes from Reichstein (1960) (protected forest plantation and abandoned countryside: males, 880 m2 in summer and 360 m2 in winter; females, 360 m2 in summer and 270 m2 in winter). If the food resources available in a home range are below the needed value, an individual will try to increase its home range. If the food resources surpass the value needed, then the individual will release cells from its home range until the amount of food needed is met. The acquisition and release of cells to or from a home range follows an optimization procedure, i.e., an individual tries to accumulate the best cells available. For adding new cells to a home range, all cells neighboring the home range are ranked following certain quality criteria (Table 2) and then the best cells are selected. Similarly, before releasing cells from a home range, all home range cells are ranked and the worst ones are finally released. Cell quality is evaluated by food resources and social criteria, the ranking of both depending on the sex of the individual, the developmental stage, and the season (Table 2). Each day a maximum number of 6 cells can be added. This value was estimated by calibration to allow home ranges to increase sufficiently rapidly at the beginning of the breeding season, when marked home range increases are observed.
Table 2. Criteria for the evaluation of landscape cells in a descending ranking order
1. Absence of adult males
1. Absence of adult females
1. Absence of any individuals
2. Presence of adult females
2. Food resources
2. Food resources
3. Food resources
3. Absence of subadults
4. Absence of subadults
1. Absence of any individuals
1. Absence of any individuals
1. Absence of any individuals
2. Food resources
2. Food resources
2. Food resources
When an individual tries to add a cell to its home range that is also part of the home range of another individual, then the decision as to which individual obtains the cell depends on the developmental stage of both individuals: Adult females obtain the cell with a probability of 0.5. Adult females do not expel subadults or adult males from their home range and adult males or subadults do not expel any individuals.
To compare home range sizes to those of field studies, the widely used measure minimum convex polygon (MCP) was calculated for each home range. The area covered by the MCP is generally much larger than the area covered by the cells of the home range. For further details and an illustration of the home range submodel see Wang and Grimm (2007).
Dispersers are individuals that do not have a home range. Individuals may release their home ranges and become dispersers under 3 conditions. The first condition is when they are expelled from a home range by another individual. The second condition is when the food resources of a home range fall below a threshold of 38.4% (parameter 18) representing the absolute minimum amount needed for maintaining a home range. This parameter was estimated based on Briner et al. (2005), who reported individual home range sizes in wildflower strips. Because the smallest observed home range measured only 38.4% of the median home range size, it was assumed that a home range is not attractive if it provides less than 38.4% of the needed food resources. Dispersal is prevented, however, when a home range is still increasing compared to the previous day, because otherwise dispersers that had just acquired the first cells for their home range would disperse again. In the third condition, adult males disperse during the breeding season when their home range does not overlap with the home range of a female. Dispersers move in a random direction each day until a maximum distance is reached (parameter 17). During dispersal the “quality” of the cells crossed by the individual is evaluated, following the quality ranking of Table 2. If an appropriate cell is found it is used as the first cell of a new home range. It is considered that dispersing females do not reproduce, because settlement seems to be a prerequisite for reproduction in both sexes and animals that do not establish a home range in new populations disperse again (Hahne et al. 2010).
Fertile females try to mate each day until they become pregnant or until the breeding season ends. In the model, mating is considered successful if a female's home range overlaps with the home range of a male. When females mate, the length of the gestation period and the litter size are determined (Table 1).
MODEL TESTING AND CALIBRATION
The code was verified by a thorough revision of the code and a manual inspection of variables during test runs. Subroutines and submodels were tested separately. Additionally visual debugging (Grimm 2002) was applied to test subroutines and submodels, for example a visual inspection of home ranges or landscape parameters.
Home range sizes were calibrated according to values reported by Jacob (2000) (average over the year: males, 193.9 m2; females, 141.6 m2) and data about the seasonal change of home range sizes from Reichstein (1960) (protected forest plantation and abandoned countryside: males, 880 m2 in summer and 360 m2 in winter; females, 360 m2 in summer and 270 m2 in winter). Calibration was necessary because in these references sizes of minimum convex polygon home ranges were given, i.e., the sizes of the smallest area enclosing all points where an individual was observed. Calibration was conducted by varying the “home range adjustment factor,” which corresponds to the amount of food being required per home range (parameter 16; Table 1).
SENSITIVITY ANALYSIS AND VALIDATION
A local sensitivity analysis of the model was carried out by varying all input parameters within a range of ±30% in steps of 1% and carrying out a 1-year simulation per parameterization, i.e., 61 simulations per parameter. Regression coefficients were calculated for each parameter using a linear regression.
Validation was conducted focusing on all parameters and model behaviors that are relevant for the purpose of the model, i.e., the simulation of population dynamics and spatial behavior of voles to predict potential impacts of pesticides and recovery. Consequently, reproduction, survival, spatial behavior, and population dynamics were considered, because any of these processes may influence the population development or the magnitude of effects by pesticides (e.g., spatial distribution). Because data availability from field studies is often limited, validation experiments were usually specifically designed to allow a comparison with the available field data from the literature. For example, to compare age structure in the model with field data from Adamczewska-Andrzejewska and Nabalgo (1977), the same age classes used by these authors had to be implemented in the model to make a direct comparison possible. This approach to design or adapt simulation experiments according to field observations has also been described as the “virtual ecologist approach” (Zurell et al. 2010). Validation was conducted on all parameters for which data from the literature were found. Because in many cases information from the literature is restricted to mean values or ranges, the validation of the model is based on a quantitative comparison of the model output and field observations.
For model validation, both 50-year simulations and 50 1-year simulations were used, recording the range, mean, median, and standard deviation for each parameter. Simulations in grassland were carried out using an initial population density sampled from a lognormal distribution (µ = 3.454 and σ = 1.003) calculated based on winter densities (November–February) from the habitats pasture, set-aside, grassland, and meadow from Schön (1995), Jacob (2000), Adamczewska-Andrezejewska (1981), Mackin-Rogalska (1981), and Horvath and Pinter (2000). This distribution corresponds to a mean of 60.5 N/ha. For simulations including the habitat pasture the initial density in pasture was also sampled from a lognormal distribution (µ = 3.6284, σ = 0.9246) calculated from winter densities (November–February) reported by Adamczewska-Andrezejewska (1981), Jacob (2000), Haitlinger (1981), and Mackin-Rogalska (1981). In simulation with arable fields it was assumed that no voles are present when the simulation starts (January, bare soil). This corresponds to the crops regime in Jacob (2000), from where vegetation height was obtained as a measure of food availability.
The results of the local sensitivity analysis are shown in Table 3. Population density, the production of offspring and population growth were mostly affected by the length of the breeding season, although also other parameters, such as gestation length and mortality had a marked influence. As expected, home range sizes were most sensitive to the amount of food available in cells.
Table 3. Sensitivity of population size, growth, the number of offspring, and home range sizes
The regression coefficient βi is a measure of the (positive or negative) sensitivity of the model to changes of a parameter. R2 is the coefficient of determination for the regression of a parameter and the model output.
Breeding start of females
Breeding end of females
Breeding end of males
Mortality of adult males
Number of offspring
Breeding start of females
Breeding end of females
Breeding end of males
Breeding start of males
Mean monthly growth rate
Breeding end of females
Breeding start of females
Mortality of adult females
Breeding end of males
Female home range size
Mean food in cells of the landscape
Male home range size
Mean food in cells of the landscape
For model validation, model outputs were compared with observations from field studies to evaluate and discuss whether and to which extent the behavior of the individuals and the population matches field observations and empirical knowledge (Table 4).
Table 4. Key variables for model validation
Variable for model validation
Values from literature
Estimate, no exact measurement.
Number of litters per breeding female and lifetime
Litter size was imposed in the model (data from Spitz 1974) and therefore directly reflects field data. The number of litters per breeding female and lifetime, however, indirectly emerged from the model as a result of the behavior of the individuals, survival, and other parameters. The number of litters per breeding female and lifetime approximates the values observed in free ranging common voles (Table 4). Values of 2 litters seem to be most common, although larger values have been observed. The average value emerging from the model was 1.8 litters. The percentage of pregnant females, which has only rarely been reported from field studies, emerges from the model. Corbet and Harris (1991) report a range of 60% to 100% in August and 30% to 50% in early September. Model simulations resulted on average in 75% in August and 57% in September. Reproduction, together with survival or population dynamics, can also be evaluated indirectly by observing age structure in the course of the year, because age structure is a result of the all reproductive parameters, such as the start and end of the breeding season, litter size, the fraction of pregnant females, and other parameters. For the comparison of the age structure, data from Adamczewska-Andrzejewska and Nabalgo (1977) were applied, who measure the minimum age of animals over 2 years based on recaptures. Adamczewska-Andrzejewska and Nabalgo (1977) subdivided all animals into the 4 groups (recruits, 1 week to 1 month, 1 month to 3 months, older than 3 months). Recruits referred to all individuals captured for the first time (juveniles and adults). The age of animals was estimated based on how long they were captured. Therefore, the ages reported by Adamczewska-Andrzejewska and Nabalgo (1977) are minimum ages. For the comparison of age structure in the model all individuals were classified using the age classes 0 days to 1 week, 1 week to 1 month, 1 month to 3 months, and greater than 3 months. The age structure emerging from the model revealed a general correspondence with the field observation for the breeding season (Figure 2). Differences were observed during the beginning of the year (January–March), when recruits and animals less than 3 months were observed, whereas the model simulations showed almost only animals of over 3 months (animals produced during the breeding season in the preceding year). However, taking into account that recruits in Adamczewska-Andrzejewska and Nabalgo (1977) included adults captured the first time and that the reported ages represent minimum ages, it can be concluded that age structure in the simulations generally corresponds to the observation in the field.
Survival rates were imposed in the model. The applied values can probably be considered realistic, because survival rates were estimated using capture-mark-recapture (CMR) models (Adamczewska-Andrezejewska and Nabaglo 1977). In contrast to survival rates based on minimum number alive (MNA), which are usually underestimated, survival rates estimated by CMR models can be considered fairly realistic, because these methods take account of dispersal and the probability of capturing an animal (see Pollock et al.  for review).
Minimum convex polygon home range sizes were on average 197.8 m2 in males and 140.6 m2 in females in simulations with grassland. For this habitat Jacob (2000) reports home range sizes of 193.9 m2 in males and 141.6 m2 in females. The seasonal changes of home range sizes were imposed in the model, resulting in larger home ranges in summer. For grassland seasonal changes of home range size are not reported. However, Reichstein (1960) observed markedly increased home ranges in males in other habitats. From other small mammals it is known that home ranges increase during summer, especially in males (Niethammer and Krapp 1982; Wolton 1985). These increases are interpreted as a mechanism to ensure sufficient food resources for reproduction in females and to guarantee access to females by males (Ostfeld 1990).
Dispersal was limited to, at most, 537 m according to Boyce and Boyce (1988b). However, in model simulations, dispersal distances were smaller, because dispersal stopped once an adequate cell was found to start a new home range. The average dispersal distance amounted to 143.9 m in the model. In comparison, Spitz (1977) reported that dispersal distances range between 100 m and 200 m. Hahne et al. (2010) observed dispersal when population density was highest, i.e., from July to November. In the model, dispersal was also observed mainly during that period, but dispersal occasionally occurred during other times of the year too.
To analyze habitat preference in the model, simulations were run in a 25 ha landscape with grassland, pasture, and arable fields (Figure 3). Because food abundance in the arable fields, measured as vegetation height from Jacob (2000) (cereals), was very low in winter and spring, the home ranges of most animals were concentrated in grassland or pasture. However, with increasing crop height in late spring and summer and larger population density, a large number of animals also settled in the arable fields. In summer, home ranges in the arable fields were smaller than in grassland or pasture, because the fields provided more food than the surrounding habitats. The population densities for these habitats obtained from 50 1-year simulations are shown in Figure 4. These results from the simulation are in agreement with observations from Zejda and Nesvadbova (2000), Jacob and Hempel (2003), and Jacob (2003), showing that voles mainly enter in fields when crop provides sufficient food and cover and that densities rapidly decline after harvest.
Population regulation and cycles
For the analysis of population dynamics, the timing of the annual population peak and the amount of fluctuations from 1 year to the next were observed. Based on a breeding season starting in April 1 and ending in October 1, peak densities were reached between July and September. Field studies have shown that peak densities may be observed after September (Table 4), however, this is evidently not possible if breeding ends after September. Therefore, a population peak after September is probably due to an extended breeding season.
The magnitude of population fluctuations depends on the area in which population abundance is measured (Wang and Grimm 2007) due to stochastic effects that are stronger the less animals are assessed. In field studies with a small trapping, grid fluctuations seem to be larger than in studies with large trapping grids (Schlesser et al. 2002). Therefore, for the analysis of natural fluctuations of population density, an artificial trapping area of 0.5 ha was considered in the simulations, which is comparable to trapping grids used in field studies. This trapping area was located in the center of the landscape. For the assessment of population fluctuations, only animals being located in this trapping grid were considered. Population fluctuations were assessed based on the change of the maximum density from 1 year to the next based on a 50-year simulation. The change of population abundance ranged between −44.7% and +102.4%. This indicates that populations may double from one year to the next or that they may halve. These fluctuations are comparable to natural fluctuations in similar habitats (meadow, set-aside, grassland), which ranged between −78.5% to 89.8% (calculated from Giraudoux et al. 1994; Schön 1995; Horvath and Pinter 2000; Jacob 2000; Hellstedt et al. 2002).
After parameterization of the model multiannual population cycles emerged, when running simulations over several years (Figure 5A). Such cycles are frequently observed in natural population of several vole species. The length of the population cycles, i.e., the time from 1 peak to the next was mostly 3 years, exceptionally 4 years. Niethammer and Krapp (1982) and Tkadlek and Stenseth (2001) report population cycles with a period length of usually 3 years for free-ranging common voles in central Europe. Mackin-Rogalska and Nabaglo (1990) observed cycles of up to 4.9 years. Tkadlek and Stenseth (2001) observed a geographical gradient of population cycles in M. arvalis in central Europe, with population cycles increasing toward the South. Tkadlek and Stenseth (2001) explain the stronger population cycles in the South with a longer breeding season, whereas in the North a shorter breeding season may destabilizes cycles. To test this hypothesis, simulations with the model were run with different breeding season lengths, i.e., with the breeding season length of the original model (167 days) and with breeding season lengths increased by 1 month (197 days) or decreased by 1 month (137 days). These breeding season lengths correspond to the values for the Polish, Czech, and Slovakian populations in Tkadlek and Stenseth (2001) (populations 1, 13, and 24 in the original reference). A comparison of the model simulations with the population fluctuations reported by Tkadlek and Stenseth (2001) shows that population cycles in the model followed the general pattern observed by Tkadlek and Stenseth (2001) (Figure 6). The population cycles were stronger for increasing breeding season length.
To test which parameter was responsible for the emergence of population cycles in the model, several parameters that may result in delayed or undelayed density dependent population regulation were deactivated. Only 2 parameters were shown to affect population cycles: dispersal behavior (minimum amount of resources needed) and social behavior (probability of females to acquire a cells that is occupied by another female). Although the removal of the minimum amount of resources needed, which triggers dispersal once the home range resources are below that value, destabilized population cycles (Figure 5B), the removal of the probability of females to acquire cells caused an unrealistic model behavior, resulting in extreme population density (data not shown). These results demonstrate that dispersal and home range behavior alone are sufficient to cause population cycles. Mechanistically, cycles emerging in the model can be explained by increased dispersal and consequently reduced reproduction at high density during and after peak phases.
To test the resilience of the model, simulations were run applying an additional mortality on all animals on April 1 to simulate a hypothetical impact of a pesticide application. Ten pairwise simulations were run in grassland: each simulation pair including a control simulation without applying an additional mortality, and a simulation with applying an additional mortality. The simulations for each pair started with the same initial parameterization, such as initial density. It was assumed that simulated populations had recovered when no statistical significant population density was observed between control simulations and the simulations with the additional mortality using a Wilcoxon matched-pairs test. The simulations showed that populations recovered from an additional mortality of up to 50% within the same breeding season (Figure 7). These results demonstrate that regulatory mechanisms are included in the model.
Finally, the effect of reduced litter size on the population development was analyzed using the same simulation scheme as above (10 simulation pairs, Wilcoxon matched-pairs test). Litter size was reduced for either 1 or 2 weeks starting on June 1, a time when many animals are reproducing. A reduction of litter size of up to 70% applied for 1 week caused only a temporary change of population density, which disappeared within 1 month. However, when litter size was reduced over 2 weeks, effects persisted until the next year, except for a reduction of only 10% (Figure 8).
The need to validate population models has already been discussed over a decade ago by Bart (1995). Bart (1995) stressed the danger of overestimating the reliability of the models whereas in reality it is only poorly understood. Although validation is often considered an essential requirement for applying population models in risk assessment (e.g., Kramer et al. 2010), a thorough validation of such models is still rare (Schmolke et al. 2010), and only few published models were systematically validated, including estimation of the realism of all or at least most submodels and emerging parameters. Here, we exemplarily showed how a population model can be validated systematically by addressing all areas that are relevant for the purpose of the model. Because the primary purpose of this model was to realistically reproduce the population dynamics of the common vole, validation included survival, reproduction, spatial behavior, habitat preference, and finally, population dynamics. In general, this validation showed that simulation results approximated data from field studies. This included parameters that emerged from the model, such as habitat preference, age structure, or population dynamics.
Because for certain parameters no or only biased data are available from field studies, validation may be difficult for certain parameters of a model. For example, in small mammals, analyses of age structure in field studies are usually based on body weight (e.g., Boye 2000), because it is not possible (or only with much more effort) to measure the real age of individuals. For the validation of the present model data from Adamczewska-Andrzejewska and Nabalgo (1977) were available, which were based on an extensive set of capture-recapture data. Therefore, a rather unbiased analysis of age structure was possible after creating age classes in the model that corresponded to the classes used by Adamczewska-Andrzejewska and Nabalgo (1977). However, in some cases only biased data may be available if any. Indirect approaches can help in such cases to conduct a validation. For example, if no field data are available for determining if gestation length is realistic, then observations of interbirth intervals or the number of offspring produced per year can help to indirectly determine gestation length. This demonstrates that the parameterization and the methods chosen for validation depend to a large extent on data availability from field studies. However, field methodology must also be taken into account when validating a population model. Wang and Grimm (2007) have shown that population fluctuations in a model of the common shrew (Sorex araneus) depended to a large extent on the size of the area in which fluctuations are measured. For field studies, the size of the trapping grid has been shown to have a marked influence on the resulting observations (Schlesser et al. 2002). In the present study, we used an artificial trapping grid, i.e., a smaller area within the model landscape that corresponded to typical trapping grid sizes, to be able to validate population fluctuations.
Sensitivity analysis showed that in the present model, population density was mainly influenced by the length of the breeding season. Because breeding season seems to vary between different locations within Europe (Tkadlec and Stenseth 2001) the selection of the breeding season is probably the most relevant parameter regarding the uncertainty of the model. The length of the breeding season did not only affect population density in general but also the amplitude of population cycles. As a result, breeding season length will have marked influence on a potential recovery of the simulated populations after impacts by pesticides or agricultural practices. In the present model, the breeding season corresponds to an average breeding season length for Central Europe. However, when applying this model for risk assessment in other regions, the length of the breeding season should be carefully adjusted to match the most realistic value for the region of concern.
The observation of population cycles in model simulations was a surprising result. In real populations, several factors causing population cycles have been discussed, including delayed density-dependent mechanisms affecting either reproduction or survival or the impact of predators and many more (Krebs and Myers 1974). However, the causes leading to population cycles are still discussed controversially. Some authors hypothesized that weak density-dependence or delayed density-dependent responses lagging behind density (such as the regeneration of food resources after heavy exploitation) may be responsible for population cycles in small mammals (Andrewartha and Birch 1954; Hassell 1986; Turchin 1990). However, Ostfeld et al. (1993) found no evidence for delayed-density dependence based on field observations of Microtus pennsylvanicus. Inchausti et al. (2009) observed reduced body weight, small litter sizes, and a reduced number of breeders in spring after population peaks. The authors hypothesized that reduced reproduction and also reduced survival could be causes of population cycles in M. arvalis. However, the observed life history traits showed no direct density-dependent variation. In Scandinavia population, cycles were mostly explained by the interaction of predators (e.g., Hanski et al. 2001). However, such cycles are usually caused by the relationship of one or few predators feeding predominantly on a specific prey species. When several generalist predators feed on a prey species, predation can even have a stabilizing effect on usually cyclic populations (Erlinge et al. 1991). In consequence, there is no agreement on which factors are the main ones driving cyclic population behavior in voles.
The population cycles produced in the present population model were mainly determined by home range behavior and dispersal. A systematic deactivation of parameters in the model showed that population cycles disappeared when dispersal at low food resources was disabled. Hence the density-dependent mechanism driving the population cycles was the dispersal of animals at high densities, which resulted in a smaller number of reproducing residents. Interestingly, Tkadlec and Stenseth (2001) observed that interannual population cycles in central Europe correlated with the length of the breeding season. In regions with a longer breeding season population cycles were pronounced, whereas in regions with a shorter breeding season population cycles were small or even disappeared. The authors hypothesized that shorter winters allow stronger demographic variations, which would destabilize cycles. To test this hypothesis, model simulations were run with different breeding season lengths. Remarkably, also the model population produced stronger cycles when breeding season was long and shorter cycles when breeding season was short, supporting the hypothesis of Tkadlec and Stenseth (2001). The length of cycles matched the cycle lengths observed in real populations. These results demonstrate that population models can be useful tools for analyzing and testing ecological hypothesis and for understanding possible mechanisms in population ecology.
Although we demonstrated how models can be validated and tested in relative detail to estimate population impacts or recovery times for the use in chemical risk assessment, further models and validation would be valuable for many other regulatory important species. Additionally, it is necessary to define relevant scenarios for population-level risk assessments that reflect the currently established protection goals (EFSA 2010). Models offer the unique advantage that different landscapes or other conditions can easily be tested, either to evaluate the risk of a specific environmental condition or landscape or to reveal relevant or worst-case scenarios. However, for the acceptance of population models in risk assessment validation may often be an important requirement.