Impact of Red Fox Predation on the Sex Ratio of Prairie Mallards
The Predictions
In Table 21, the line labeled "standard" gives summary statistics for 100 replications of the predictive model with parameters as just defined, which describe a situation believed to reflect that occurring in 1963-73 in our reference area. Fox populations averaged 10,195 families (SD = 3,998). Survival rates averaged 68% for adult male mallards and about 61% for adult females. The average sex ratio was 119.2:100 in Case I and 117.0:100 in Case II, values somewhat lower than those given by the model in Part One, but reflective of the situation throughout the 11-year period.
In addition to the standard situation we explored the effect on the sex ratio in the 11th year resulting from various changes in three variables: fox densities, mallard densities, and hunting rates. The model was replicated 100 times at each setting described below; the same sequence of pseudorandom variates was used in order to reduce random variation. The average male and female survival rates, as well as the sex ratio, are reported. To assess the influence of the three variables on the sex ratio, we performed a sensitivity analysis similar to that described in Part One.
Alterations in the Fox Population
Although the observed spring populations of red fox families during 1963-73 averaged about 9,900, the variation was rather dramatic; a low of 4,700 in 1969 and a high of 16,700 in 1964 were reached. Even more extreme populations are possible, of course. We examined the ramifications of two changes in the fox population submodel; one resulted in a sustained low population, the other in a sustained high population. For the sustained low population, we generated W from a normal distribution with mean given by Equation 5 evaluated at a = 1.50 and b = -0.00010; the standard deviation was 30% of the mean. The resultant fox populations averaged 4,686 (SD = 1,804), a 54% decrease from the standard. The high population was generated similarly, but with a = 1.50 and b=-0.000033, and averaged 14,071 families (SD = 5,466), a 38% increase.
Table 21 displays the results of the predictive model simulations under Case I and Case II. The reduced fox population resulted in an approximately balanced mallard sex ratio, 104:100 in Case I and 99:100 in Case II. The magnitude and sex disparity of the fox-related mortality rates nearly balanced the disparity in hunting rates. Holding fox populations at a high level caused more imbalanced sex ratios in the two cases, about 131:100. A sensitivity analysis ("response" values in Table 21)indicated that the sex ratio was responsive to changes in fox populations, particularly in Case II. The average response value was about 28%.
Alterations in the Mallard Population
The spring population levels of male mallards (PM) in the predictive situations were obtained more directly than were fox populations. For a low population, we took the mean to be 50% of the standard (200,000) and made a corresponding reduction in the standard deviation. Thus PM was randomly generated with mean 100,000 and standard deviation 25,000. The high population levels were obtained by doubling the standard, so PM was taken with mean 400,000 and standard deviation 100,000. Under the low population level, 95% of the spring populations of males ranged between 50,000 and 150,000 while the corresponding range under a high level was 200,000 to 600,000.
The consequences of these two situations are shown in Table 21. A reduced mallard population increased the distortion to 128:100 in both cases. A doubling of the mallard population, on the other hand, yielded more balanced sex ratios in both cases, 109:100 and 104:100.
The average response value corresponding to changes in the mallard population was 13% which indicates that this factor was not proportionally as influential in determining the sex ratio as was the fox population. If, however it is easier to manipulate the mallard population, or if there are other positive value associated with its change, then it might be better vehicle for changing the sex ratio the response value would suggest.
Alterations in the Hunting Rates
From Part One we know that hunting resulted in the annual loss of about 16% of the adult male and 10% of the adult female mallards, based upon the recovery rate data gathered in 1963-67, the reporting rate estimates for 1972-73, and the rates of unretrieved birds for 1965-69. More recently, hunting regulations have changed to give more protection to females and/or less protection to males. In this section we treat the effect of sex disparity in hunting loss rates on the spring sex ratio. This was done by altering values of r5 and r6 and observing the resulting change in the sex ratio. While this procedure is useful, it is not a complete solution to a prediction problem, the missing element being the relationship between hunting regulations and the hunting mortality rates. If this were also known, we could predict the sex ratio resulting from various sets of hunting regulations. As it is, we can see how the sex ratio might change as hunting rates change, regardless of the cause of change. We considered three sets of hunting loss rates. The first reflects a complete cessation of hunting: r5 = r6 = 0. The second, losses of 8% for males and 5% for females, represents uniformly reduced pressure. The third set results from increased pressure on males and unchanged pressure on females, and has r5 = 20% and r6 = 10%.
The simulated removal of hunting pressure resulted in a rise in the sex ratio to about 157:100 (Table 21), caused primarily by the increased survival of male mallards, 81% as opposed to 66% in the standard situation. Halving the hunting rates caused sex ratios averaging 134:100. When the male hunting rate was increased to 20% and the female rate left at 10%, a more even sex ratio resulted, about 107:100. The sensitivity analysis, based on the response in sex ratio as a function of the change in the difference between male and female hunting rates, showed that hunting exerted considerable influence on the sex ratio.
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