Northern Prairie Wildlife Research Center

In our model of Part One, the input parameters were fox and male mallard populations, which were fixed as described in Table 4 and Table 5, and predation rates per fox family, survival rates, and hunting mortality rates all of which varied randomly according to the probability distributions shown in Table 11 and Table 13. Rates of "other" mortality during summer and winter and the spring sex ratio were output variables of the system. In the revised model we examined the effects of various levels of fox populations, mallard populations, and hunting mortality rates, by varying these quantities according to prescribed distributions and including "other" mortality rates comparable to those calculated in Part One. The annual survival rates were determined within the model, as was the output variable of primary interest, the spring sex ratio.

We needed to resolve one further question: What proportion of the ducks found at dens represented fox predation and what proportion represented accumulation of scavenged birds? This question was irrelevant to the earlier model; whether fox-killed or scavenged, a duck brought to a fox den represented mortality in spring, and thus affected survival during the year. Now it is germane, however, because birds available for scavenging would be dead whether foxes were present or not, whereas predation depends upon the presence and numbers of foxes.

Objective evidence on the incidence of predation versus scavenging was lacking. We could only conjecture, basing our opinions on personal field experience. To start, we assumed that 50% of male mallards found at fox dens were killed by foxes and the remainder were scavenged. More of the females were probably killed by foxes; we assumed this to be 75%. Table 19, Case I, illustrates this situation (Case 0 is the standard used in the earlier model, in which all birds were ascribed to fox predation). Fifty percent predation on males implies that 0.48 male mallards (half of 0.959 in Table 11) were killed per fox family and the remainder died of other causes, and must be included in "other" spring mortality. The rate of "other" spring mortality is 50% of the average value of r7, the proportion of male mallards taken by foxes, calculated from the earlier model. This average was 5.04%, so "other" spring mortality was 2.52%, as shown in the penultimate column of Table 19. Correspondingly, since 75% of female mallards in food remains found at den' represented fox predation, the rate of "other" mortality in spring was 25% of 18.18%, or 4.54%. Under these assumptions the scavenging rate of foxes on females was nearly double the rate on males, which is consistent with high female susceptibility to road kill, agricultural operations, and stress.

A reasonable argument can also be made for the situation in which the predation rate on males remains 50% but the female rate is determined so that foxes scavenge the same numbers of males and females, i.e., 0.48 per fox family (Case II in Table 19). This situation is reasonable if we assume that foxes are equally likely to encounter a male as a female mallard for scavenging. Following this assumption, we obtained rates of "other" spring mortality in Case II as 2.52% for males and 2.35% for females. We explored the implications of these two cases. We simply added the rates of "other" mortality in spring to the corresponding summer rates.

We developed a simple stochastic submodel to generate annual values of F, the spring population of red fox families. The population in one year was determined from the previous year's population by

where W, the population change, incorporated the survival of adults plus the recruitment of young into the population. We randomly generated W from a normal distribution such that W was density-dependent. The following equation for the expected value of W, given F, was found after a trial-and-error procedure to generate realistic-appearing series of fox populations:

with a = 1.48 and b = -0.000044. (5)

The standard deviation was taken to be 30% of the expected value. In Fig. 5, E(W) is plotted against F(t). Notice that at F= 1,000, a low fox population, an increase averaging 43.6% is expected the following year. At F = 10,000, an average level, the population could decline or increase with nearly equal probability, and E(W)= 1.04, indicating an average change of +/- 4%. At high densities, F = 20,000, a decline averaging 40% [E(W) = 0.60] is anticipated, but further increase is possible. We imposed two additional constraints:

The first condition stipulated that the population could not decline more than 80% in 1 year. The second condition precluded complete extirpation of the fox in our reference area. The process was begun with F(1) = 10,000. Fig. 6 shows a typical 50-year series generated by the process just described. These values appear realistic, and the average of 10,195 fox families and standard deviation of 3,998 were very close to 9,905 and 4,019, respectively, the observed values in our 11-year period (Table 4).

As before, the female mallard population was determined within the system. After the surviving male and female mallards were calculated for 1 year, the recruitment of males was determined thus:

If this recruitment was positive, as was usually the situation, we simply set the female recruitment equal to it:

If recruitment was negative, indicating a net emigration, we assumed that the sex composition of emigrants was the same as that in the surviving population that year:

Because RM(t) was known, this implied

Although our original model did not incorporate any density dependence in the predation rates, perhaps because the mallard densities in the years of our data varied too little to detect any dependency, the predation rate clearly varies with the mallard density. In this predictive model we embodied such a relationship.

We assumed that E(r_{1}) and E(r_{2}), the expected numbers
of male and female mallards taken per fox family, vary with the respective
densities according to

and an analogous formula for females. The form of this curve is given in Fig. 8. Note that an absence of mallards implies a zero predation rate. The rate (number of mallards taken per fox family) increases with increasing mallard density, but as mallards become extremely abundant the rates begin to level off.

Ours is what Holling (1965) termed a Type 2 "functional response" of predators to prey density. Holling summarized several studies which found Type 2 responses characteristic of several invertebrate predators, while investigations of vertebrate predators suggested an S-shaped (Type 3) functional response. We chose the Type 2 curve, nevertheless, because fox predation on mallards does not seem to meet one of the assumptions of Holling ( 1965) leading to a Type 3 response: that learned associations decay if not reinforced. The net effect of this assumption is to insure that predators rarely take particular prey if the prey are uncommon because low prey densities cause the predator to unlearn the association between prey stimulus and edibility. We believe this situation is unlikely for the fox-mallard relationship because, even though mallards may be uncommon, other upland-nesting ducks and birds of similar size and habit may be common, so foxes will maintain the associations involved with preying upon nesting mallards. Thus, the assumption is violated, and a Type 2 functional response appears more appropriate. Both types of responses lead to similar conclusions except if prey become very scarce, in which instance the Type 2 response would indicate higher predation than would the Type 3 response.

We found that predation rates similar to those obtained from field studies could be generated by Equation 6 with the following values of a and b:

Case I |
Case II |
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Sex |
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^{-6} |
^{-6} |
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^{-6} |
^{-6} |

We included random variation in the predation rates by sampling r_{1}
and r_{2} from normal distributions with means as just specified and
standard deviations equal to 15% of the means.

Hunting mortality rates and rates of "other" mortality in summer and in
winter were generated randomly from bivariate normal distributions with means,
standard deviations, and correlations as specified in Table
20. Parameter values correspond to Case I; the only difference in Case
II was that the average of r_{10} (Table 3)
became 0.150. Means were obtained from data described in Part One and from
results of the model employed there. Standard deviations were taken to be
smaller than in the actual data to preclude the generation of negative rates
and to reduce the variation in output from simulations run under identical
conditions.

Previous Section -- Part Two: Using the Model to Predict Sex Ratios

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