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Reference Area and Time Period

The time period used is 1963-73, an 11-year period for which estimates of all necessary parameters could be determined. Included are years when fox populations were high and years when they were low; mallard densities also varied widely.

Although we could estimate the necessary parameters, we did not know their values with certainty. To assess the potential influence of errors, we perform in a later section a sensitivity analysis which determines how markedly the results differ if an input parameter is misspecified by a certain amount.

Components of the Model and their Relationships

The components of the model (Table 1) assume a value for each year, 1963-73. The number of red fox families (F) and the spring population of male mallards (PM) each year were exogenous to the model and held fixed in all simulations. All the other components were determined within the model, according to the relationships described in Table 2. The rates involved are defined in Table 3.

We assumed that the mortality causes followed the sequence: fox predation in spring, "other" mortality in summer, hunting during fall, and "other" mortality in winter. This assumption is somewhat unrealistic, of course, as certain mortality forces operate throughout the year at varying levels of intensity. We made this simplifying assumption largely because (as we will later demonstrate) fox predation is the major cause of mortality in spring and hunting the major cause in fall.

Within each year of the simulation, the first quantities determined were the losses of male and female mallards to fox predation, FM and FF (Table 2). These were taken as the products of the spring red fox population F and the corresponding predation rates per fox family, r₁ and r₂.

The mallards surviving fox predation were then subjected to summer mortality from other causes, at rates r₉ and r₁₀. This determined "other" mortality in summer for males and females (OM and OF), and the survivors were then available at the beginning of the hunting season.

Hunting mortality was then applied, at rates r₅ and r₆, to obtain the hunting losses HM and HF. Remaining birds were then subjected to winter mortality at rates r₁₁ and r₁₂; thus, winter losses WM and WF were determined.

After all losses were subtracted from the original spring populations, the survivors, SM and SF, remained. The proportion of the initial spring population surviving is denoted r₃ for males and r₄ for females. By adding recruitment, RM and RF, to the surviving populations, we obtained the next spring's populations, thus completing one annual cycle.