The data on life expectancy of the prairie vole is very inconsistent because the measurements depend on predation at the specific region as well as climatic factors. The prairie vole mean life expectancy was established by Getz et al. [21] to be between 50 and 80 days depending on the season. Animals born in summer and autumn had higher life expectancy (61–80 days) than ones born in winter and spring (41–50 days). However, some voles born in the summer or autumn season live up to 450 days, while these born in the winter or spring live up to 120–150 days [21].

Voles are considered adults if they have reached 30 days of age or approximately 30 g weight.

An adult prairie vole produces in average 3–5 offsprings per litter and 3–4 litters per year. The gestation period is 21 days and is approximately equal to the generation time (period between pregnancies) [22,23].

The daily food intake depends on the quality of the food but for green (undried) grass it can be estimated as follows. Sawicka-Kapusta et al. [24] estimates for the caloric value of biomass of various fodders, for grass being 0.85–0.98 kcal/g biomass. Bradley [25] calculates the daily food intake of M. ochrogaster to be approximately 600 cal/g body weight. For 30 g body weight then the daily caloric requirement is 18 kcal. Using the data from Sawicka-Kapusta et al. [24] we obtain that the dailly grass intake for an adult vole will be about 20 g.

Breeding shows seasonal dependence and is defined as between February and November. Rose and Gaines [23] report no breeding in December and January and even through April in one of the years, while Krebs et al. [26] report very low percentage of breeding females in November–January and peaks of breeding in August, September and October. Keeping in mind the bioenergetics of these small mammals, which spend most of the produced metabolic energy for maintaining their body temperature, these reports are reasonable.

Species that exhibit site fidelity have a mean amount of territory that they move around daily and defend, the home range (HR). HR size has been demonstrated to be related to the energetic needs of the mammal [27]. Thus, it is gender and season dependent. The HR size has been reported to be somewhere between 10 and 50 m [28–30].

It is known from observations [21,26] that highest vole population densities occur in the late autumn (October, November), that densities decrease drastically in January–March, the lowest densities usually occurring in March and then start steadily increasing until late autumn.

The population density of the prairie vole depends on the quality and quantity of vegetation as well as on the presence of sympatric herbivore species and/or predators. Food quality is very important for the population dynamics of the prairie vole. For example, alfalfa has a higher nutrition value than tall grass [31]. The maximum vole density in tallgrass prairie as recorded in [32] was 90 ha⁻¹ while the mean density was 7 ha⁻¹ showing high variability during a 25 year study. Cole and Batzli [31] report maximum density 38 ha⁻¹ and 10 ha⁻¹ respectively in a next year for the above mentioned types of vegetation.

Experiments on enclosed grids show that dispersal plays an important role in population density regulation. Density can become quite high in enclosed areas. Krebs et al. [26] compared fenced and unfenced grids and observed a rapid increase in population density in the fenced case, which led to overgrazing. A very interesting experiment of introducing 18 prairie voles in an empty fenced plot of 0.8 ha led to an extreme density of 494 vole/ha (395 voles in the plot) after 7 months. Consequently the vegetation was overgrazed and the population fell to 11 voles in early April when it started to increase steadily.

Absence or presence of a predator also plays an important role on population density regulation. Desy and Batzli [33] conducted experiments on enclosed plots with mixed vegetation including blue grass and observed maximum density of 550 ha⁻¹ on enclosures with no predators and added food, while in enclosures with no supplemented food and with predators, the density was stable at about 90 ha⁻¹.

Microtine species, one of which is the prairie vole species, exhibit seasonal and multi-annual population size fluctuations. The physiology of the vole, a very small mammal, which needs an enormous amount of energy (relative to its body mass) to maintain constant body temperature, is clearly seasonally dependent. At low temperatures, the vole has to diminish or abandon certain high-energy-consuming activities in order to survive. Breeding (as seen above), deaths due to predation and life expectancy are seasonally dependent phenomena. Several authors have reported multiannual fluctuations in population density with occurring peaks with frequency 2–4 years. Rose and Gaines [23] note that microtine cycles vary from 2 to 5 years in duration, with many having 3–4 year intervals between peaks. Getz and Hofmann [32] report high-peak densities appearing with periods of 4.3 years and secondary peaks at 2.1–2.4 years on three different habitats and that these fluctuations are synchronized for the three different habitats. The periodicity of the fluctuations does not depend on the external manipulations of the habitat (burning of tall grass, mowing of blue grass and alfalfa) but their amplitude does depend on the quality of food (alfalfa has the highest caloric quality, tall grass the lowest) and are positively correlated with it [34]. Cole and Batzli [31] also note that food availability may influence the amplitude of fluctuations but not the periodicity of the cycle. The multi-annual population size oscillations of the microtine species is a continuing puzzle to ecologists.

There are various hypotheses and studies related to them concerning vole population fluctuations. Some authors study the connection of population density and juvenile survival (Krebs et al. [26], Getz et al. [21]), while others put forward as a possible cause changes of food quality [31].

Getz and McGuire [22] show that the majority of single males are wanderers while a considerably smaller proportion of single females wander. Wanderers frequently visit the nests of pairs but are kept away by the male [35]. Unrelated adults join communal groups only if they include already grown offspring male [35]. New pairs were formed after the winter period from surviving individuals mainly from different social groups [22]. Most of the new pairs formed in summer-autumn were made from two wandering individuals.

Prairie voles are monogamous animals. Paired males and females cohabit a nest and a home range [22]. McGuire and Getz [35] find that paired males exclude nonresident males from the areas around their nests. This was also confirmed in [22].

There is little overlap in use of space by social groups even at high population densities [35]. Spatial behavior limits interaction between members of adjacent social groups. Jacquot [36] found that vacant territories were fast occupied. Density of population was positively correlated with the time necessary to emigrate. Female prairie voles immigrated in response to vacant territories and not to potential mates. Males also immigrated in response to vacant territories. It is generally accepted that males show preference to territories with present females.

At each time moment n the total amount of present animals is K₁(n). The animal objects

Thus, at any time moment n the animal objects with their data represent a K₁(n)×9 matrix, the matrix of animal objects.

K₁(n) can take very big values depending on the amount of spatial cells (area size) we consider. In a typical simulation K₁ reaches several hundred thousands up to a million. The lower bound is naturally 0, but in general, even if K₁ does not become so small, it is highly variable, falling to several thousands in the winter months.

All animal objects share constant data characteristic of the species: average daily food intake DFI (in grams per day), life expectancy LE, maturation age MA, generation time GT (in days), litter size LS. The values of these quantities are taken to be in the ranges outlined in the previous section. Additionally, the following controlling parameters relevant for the energetic budget enbudget are introduced: MEB, maximum energetic budget, MeanEB, mean energetic budget, JEB, juvenile energetic budget. We explain the meaning of the controlling parameters in a following subsection. The values of the parameters are reported in Section 5.1.

Space is represented as a collection of a constant number K₂ of square cells, C₁,…,C_K2, each one with an area roughly equal to the home range of the herbivore. The location of each cell is represented by two coordinates (row and column). The cells are individual objects possessing the following 5 variables: quantity of vegetation veg in the cell, total number pop of animals in the cell (juveniles, male and female), number of present male herbivores mpop, female herbivores fpop, and a variable p indicating whether the spatial cell has been polluted and therefore should be removed (i.e. it has been developed, polluted or otherwise rendered uninhabitable). Thus, at any time moment n the cells with their data represent a K₂×5 matrix, the matrix of cell objects.

Note that we address the spatial cells in two different ways: we order the cells sequentially as C₁,…,C_K2, and we also address them by using their coordinates when defining an animal object. We need to have a way to link the coordinates with the cell.

Thus, given row and column coordinates, nrow, ncol, let #(nrow,ncol) be a function whose value is the sequential number of the cell with coordinates nrow, ncol. Otherwise said, C_#(nrow,ncol) has coordinates (nrow,ncol). An animal object A_j will then be located in the cell C_{#(nrowj,ncolj)}.

The dimensions of the cell objects matrix do not change over time. The number of columns of the animal objects matrix may change at each time step because of births and deaths. The newborn objects are added as last columns of the matrix, while the dead objects (with remove=1) are deleted and the list of animals is updated. Thus, at time n an animal may have index j in the list, while at time n+1 this index may be different.

The time unit used in the model is 1 day. We simplify the simulations by assuming that a month has 30 days and an year has 360 days.

The mapping functions describe the transition from matrices $𝒜\left(\mathrm{n}\right)$ and $𝒞\left(\mathrm{n}\right)$ to $𝒜(\mathrm{n}+1)$ and $𝒞(\mathrm{n}+1)$. The order in which the maps are executed is important and we are keeping to it in the following description.

In this model the only random map is the one that controls the animals' movements between cells.

(0) If at time n animal A_j has index j and remove_j=0, then we define img(j) as the function which calculates the new index, at time n+1, of animal j. Keeping in mind the way we do the rearrangement of the list, we find that img(j)=j− the number of animals that died and had index less than j=img(j)=j−∑_k<jremove_k.

(1) Cell population numbers

(2) Aging

This is the simplest mapping from time n to time n+1.

(3) The energetic budgets

To model the effect of food and starvation on survival, we use the variables enbudget_j(n), j=1,…,K₁(n).

enbudget_img(j)(n+1)=enbudget_j(n)−1, if veg_{#(nrowj(n),ncolj(n))}<DFI×pop_{#(nrowj(n),ncolj(n))}

Otherwise,

That is, if the animal feeds, it increases its energy store (unless it is at the maximum) and if it cannot feed because of lack of enough resource, its energy store drops. Additionally, there is a limit to the energy an animal can store, which is smaller for the juveniles.

(4) Survival or removal

remove_j(n+1)=0 if age_j(n+1)<LE and if enbudget_j(n+1)>0. Otherwise, remove_j(n+1)=1. That is, an animal dies if its energy budget became 0 or if its age reached the life expectancy.

(5) Status change

The variable stat_j changes its value as follows.

stat_img(j)(n+1)=resident, if age_img(j)(n+1)<MA.

stat_img(j)(n+1)=wanderer, if age_img(j)(n+1)=MA.

If age_img(j)(n+1)>MA and if stat_j(n)=wanderer, then

Otherwise, stat_img(j)(n+1)=wanderer.

If age_img(j)(n+1)>MA and if stat_j(n)=resident, then

stat_img(j)(n+1)=wanderer if veg(#(nrow_img(j)(n),ncol_img(j)(n))=0.

Otherwise stat_img(j)(n+1)=resident.

In plain words, all juveniles are residents until they reach adult age, when they become wanderers. Adults who were wanderers can become residents if they find themselves in a cell with no other individuals of the same gender, otherwise they continue to wander. Thus, each cell can contain only one pair of adult residents (plus their immediate offspring) and many wanderers. Adult residents become wanderers if the cell gets overgrazed and has no vegetation left.

Wanderers do not contribute to the population growth but use the resource, thus controlling the population in certain bounds.

(6) Births (4.3)

In other words, a female vole gives birth only in the months between February and October and if it is an adult resident, with energetic budget not less than MeanEB and if a male animal is present in the cell.

(7) The time between pregnancies τ (4.4)

(8) Location change

Wanderers change their location at each time moment until (if) they become residents.

If (nrow_j(n),ncol_j(n)) are the coordinates of the location of animal A_j at time n, then the 8 surrounding cells have pairs of coordinates (nrow_j−1,ncol_j−1),…,(nrow_j+1,ncol_j+1).

Let C_{#(nrowj−1,ncolj−1)},…,C_{#(nrowj+1,ncolj+1)} be the 8 neighboring cells. To ease notation, let us denote them by C_j(1),…,C_j(8).

If stat_j(n)= resident, then

(9) Vegetation growth and depletion

The variables veg_i(n), i=1,…,K₂, are transformed into veg_i(n+1), i=1,…,K₂, as follows

(10) Genders

Naturally,

(10) Finally, p_j(n) is an externally defined function.

In the simulations we use numeric values of the parameters close to the ones reported in the literature (compare with Section 3). The area of each cell is 900 square meters (30 m by 30 m cells), roughly corresponding to the prairie vole's HR size. The initial time is set as January 1, 1960. The life expectancy LE was varied across simulations between 90 and 140 days (which is close to the maximum observed life span of voles born in winter). Additionally, GT=21 days, MA=30 days as described by real data. The litter size LS is taken to be equal to 2, a twice lower value than the number of newborns, to account for a high mortality of offspring reported in many publications. DFI=10 is also taken at a lower value than the one known for adult voles, because it is an average value for both adults and juveniles. The controlling energetic budget values were taken as MEB=20; MeanEB=15; JEB=2. These are rather arbitrary because of lack of data. MEB=20 has the meaning that an adult vole would die after 20 days of starvation, while a juvenile would endure only 2 days of starvation (JEB=2).

Initially all cells were assumed to contain the same amount of vegetation (no spatial heterogeneity was assumed).

Two types of initial distributions of animals in the spatial cells were generated and used in the simulations.

The first type was a random (‘uniform’) distribution of animals on the whole region, created by using a random number generator. This was achievd by generating a random number between 0 and 3, 4 or 5 (in various experiments) of animals in each spatial cell.

The second type of initial distribution was ‘a single source’; i.e. generating a random number (as above) of animals only in a certain small number of aggregated cells usually in one corner of the region. This was considered as a simulation of an experiment where voles were let into an empty enclosed region to colonize it.

All initial distributions consisted of wandering adults.

5.2.1 General observations

2) The time series of the vole densities showed annual peaks (probably due to the seasonality of the vegetation data) and multi-annual fluctuations with peaks separated by periods ranging between 2 to 5 years, similar to what is described in Section 3. Fig. 2 shows a typical time series of a 120×120 cells simulation with an initial ‘single source’ distribution of 348 animals. Similar time series patterns are seen in Figs. 5 and 7 (the non-extinct case). Fig. 7 shows an interesting observation: the peaks in density appear in the same years for the three plots. That is, the amplitude of fluctuations depends on the size of the plot, but the periodicity does not. This is comparable to the observations mentioned in Section 3 according to which the periodicity of fluctuations does not depend on the quality of the plot. Our simulations show that the peaks in population density do not coincide with the peaks in the resource density. One can hypothesize that the multi-annual fluctuations are due to the combined effect of the oscillatory vegetation forcing and the intrinsic population density oscillations due to the periodicity of breeding.

4) Specific spatial patterns similar to wave motion formed in all simulations. Simulations starting with uniform random distribution over the whole plot showed the formation of wave patterns after the first year when separate individuals survived and started reproducing and dispersing randomly. The random dispersal in all directions led to the formation of oval populated regions with highest density in the center of the ovals. The subsequent overgrazing led to the removal of animals from the central part of the ovals thus forming rings of nonzero density, which we refer to as ‘waves’. The waves further expanded and their central parts were removed as results of overgrazing and deaths. The waves hit one another subsequently forming various patterns of irregular structure. Fig. 3 shows typical patterns illustrating this observation.

5.2.2 Optimal conditions for survival

5.2.3 The importance of the area size

In one simulation we compared three ‘fenced plots’ of 200×200, 300×300 and 400×400 cells respectively. We started the simulations with the same initial distribution of voles which was randomly generated in the smallest plot. The plots are contained into one another, with the bottom left corner being common for all plots. That is, the 300×300 plot was five ninths empty and the 400×400 plot was three quarters empty in the beginning. The parameters of the model were the same as described in the previous section with the life expectancy being 120 days. As an illustration, Fig. 4 shows the density distributions in the largest plot in December of the second, tenth and fifteenth year.

5.2.4 The importance of dispersal and predation

5.2.5 Fragmentation is not necessarily a bad thing

We compared the results of simulations on plots of various sizes with identical initial distributions but with various types of ‘obstacles’. The obstacles are parts of the landscape unaccessible (or avoided) by the animals. Here we discuss the comparison between three simulations, two without obstacles and one in the form of a ‘checkerboard landscape’, see Fig. 6 for a visual representation.