Northern Prairie Wildlife Research Center

We have thus far elaborated a hypothesis for testing, established our reference area and time period, defined the components and rates of our model, and estimated certain components and rates from real data. In this section we explain how the components and rates were related within the model and describe how random variates were generated for the simulations.

Of the components listed in Table 1, we had annual estimates for 1963-73 of F, the spring population of red fox families (Table 4), and PM, the spring population of male mallards (Table 5). These two components were taken to be exogenous and fixed; values varied year-to-year as shown in the tables, but not from one simulation run to another. All remaining components were determined within the model and varied with each simulation.

Of the rates defined in Table 3, r_{1}- r_{6} were generated
randomly, r_{7} - r_{l2} were determined within the model,
and r_{13} was set by the experimenter. The predation rates per fox
family of male and female mallards (r_{1} and r_{2}) were
generated from a bivariate normal distribution with means, standard deviations,
and correlation as given in Table 11. The survival
and recovery rates r_{3} - r_{6} were taken from a four-variate
normal distribution with means, standard deviations, and correlation matrix
as given in Table 13. We used normally distributed
rates even though all rates are non-negative (and rates r_{3}-r_{6}
cannot exceed unity), while normal variates can fall outside this range. The
smallness of the standard deviations, however, insured that unreasonable values
occur only rarely in the simulations. Further, we lacked sufficient data to
warrant use of another distribution. The normal distribution was employed
because of its simplicity- two parameters completely characterize it- and
the ease with which random variates from it can be generated (see below).

The proportions of male and female mallards taken by foxes (r_{7}
and r_{8}) were determined within the model; e.g.,

These rates were not directly employed by the model but were calculated to monitor the simulation.

In addition to fox-related and hunting mortality, death from other causes occurs throughout the year. For our modeling purposes we restricted this "other" mortality to two periods, summer- after the fox denning season and before the waterfowl hunting season, and winter- after the waterfowl hunting season and before the fox denning season of the following spring. We suggest that the mortality rate was higher in the summer period than in the winter period because of botulism, predation, haymowing, etc. Although we do not know precisely how much higher the summer rate was, we set it at three times the winter rate. A later analysis demonstrates that this assumption was not critical to our results. This multiple was treated as a constant, but in reality can vary from year to year.

For each year in the simulation, we calculated the rates of "other" mortality
after we determined the rate of loss to foxes, the hunting loss rate, and
the annual survival rate. For males, the total annual survival rate is r_{3},.
But a male surviving the entire year must survive fox-inflicted mortality
(at rate 1-r ), summer mortality (at rate 1-r_{9}), hunting (at rate
1-r_{5}), and winter mortality (at rate 1-r_{11}). Thus, total
survival was the product of the separate survival rates, i.e.,

With this equation and our subjectively set relation r_{9} = 3r_{11},
we calculated r_{11}, as a root of the quadratic equation,

An analogous formulation determined "other" mortality rates for females,
r_{10} and r_{12}.

Random deviates were generated in the following way (e.g., Naylor et al.
1966:92, 98). For each normal deviate desired, 12 independent deviates from
a uniform pseudorandom generator defined on the unit interval were obtained
and summed. Adding a constant and scaling suitably formed an approximately
normal deviate with proper mean and standard deviation. Multivariate random
deviates were obtained by multiplying a vector of independent normal deviates
with zero mean and unit variance by the matrix P, where P is such that P^{T}P
equals the desired variance-covariance matrix. The required mean vector was
then added.

In all settings of input parameters, we used the same value to initialize the first call to the random variable generator to insure that differences between simulation runs made at different settings of input parameters reflected the settings, rather than simply random variation.

Each year in a simulated 11-year cycle began with populations of male mallards and fox families specified. In the first year of a cycle, the number of female mallards was set equal to the number of males. Subsequent female populations were determined within the model.

The randomly generated rates of predation per fox family, r_{1}
and r_{2}, were multiplied by the number of fox families F to determine
the numbers of male and female mallards taken by foxes (FM and FF). Expressed
as proportions of spring mallard populations, the losses to foxes occurred
at rates r_{7} for males and r_{8} for females.

The mallards surviving fox predation were then subjected to summer mortality
from other causes, at rates r_{9} for males and r_{10} for
females. These rates were calculated as described above, after setting mortality
rates from foxes and from hunting, and survival rates. Summer losses from
"other" mortality are denoted OM and OF.

Mallards not succumbing to fox predation or summer "other" mortality were
then available at the beginning of the hunting season, and suffered hunting
mortality at rates r_{5} for males and r_{6} for females,
these rates having been randomly generated. Hunting losses are denoted HM
for males and HF for females.

Remaining birds were then subjected to "other" mortality in winter, at rates
r_{11} for males and r_{12} for females. The losses in winter
are denoted WM and WF.

Subtracting all losses from the original spring populations left SM males and SF females surviving. The population next spring was the sum of the number recruited and the surviving population. For males we knew the next spring's population, so the recruitment was simply the number necessary to add to the surviving population to bring it to the level observed the next year. Thus, in year t,

The recruitment of female mallards was assumed to be in proportion to male
recruitment: RF = r_{13} x RM, where the recruitment sex ratio r_{13}
can be specified. We later explore the effect of various values of r_{13}
on the simulation results.

Note that we have defined a generalized recruitment rate, which incorporated not only the addition of young to the population but also the net effect of emigration and immigration. Thus, recruitment can be positive or negative, the latter event likely in a year of good wetland habitat conditions followed by a year of poor conditions, when large numbers of mallards desert the area.

After performing these calculations for the tth year, we could determine the population of female mallards the next spring by adding to the survivors the number recruited:

The initial condition PF(1) = PM(1) was used.

The primary output of the model was the spring sex ratio PM:PF in the final year of the 11-year cycle. We sought to determine how the sex ratio varied under the prescribed conditions. We also examined various other outputs of the model to judge their reasonableness and to monitor the behavior of the model.

So far we have not considered any density dependence between the predation rate and the size of the mallard population. Predation rates obviously vary with prey densities but the data we examined were inadequate to indicate the relationship. For the interim, we placed a maximum on the predation rates: in any given year the mortality rate due to foxes could not exceed 25%. In Part Two we develop a model with density dependence.

Previous Section -- The Data

Return to Contents

Next Section -- Results of the Simulations