Northern Prairie Wildlife Research Center
Before discussing the validity of our results, we consider their reliability. To wit, is the model sensitive to small changes in the parameter values, or dependent upon the length of the 11 year period used? To resolve questions on the adequacy of the model, we performed a sensitivity analysis to determine the effects of each input parameter on the output sex ratio and on the rates of "other" mortality. We then explored the stability of the model, both with respect to errors in parameter values and temporally, i.e., would the sex ratio continue to change if the 11 year period were lengthened? We also employed a mathematical result concerning the limiting behavior of the sex ratio, which confirmed our sensitivity and stability analyses as well as provided a useful framework for analysis of other species.
Sensitivity analysis is an important aspect of modeling research in which input parameters are altered one at a time and resultant changes in the output variable are recorded. An input parameter which, when varied, effects only small changes in the output variable is deemed relatively unimportant, and further research to determine its value more precisely will be wasteful. If, however, the output variable is sensitive to small changes in the input parameter, we need to have a precise estimate of the true value of that parameter. Thus, sensitivity analysis serves to isolate the more critical parameters of a system. This knowledge (Table 15) is useful for determining most profitable lines of investigation and detecting weaknesses in the model.
As formulated, our model was highly dependent upon male and female survival rates, r3 and r4. Fox predation and hunting mortality were fitted into total mortality, and the remainder by subtraction was grouped as "other" mortality. For this reason, none of the input variables except r3 and r4 (and the recruitment sex ratio) actually affected the sex ratio; they did, however, affect rates of "other" mortality for males and females. In our sensitivity analysis, therefore, it was important to determine how OM and OF were affected by changes in the input variables. The effect on winter mortality rates was similar because WM = 0M/3 and WF = 0F/3.
We generally varied the input parameters by -10% and +10% and compared to it the percentage change in the sex ratio and "other" mortality rates. The "standard" was achieved by setting all input parameters to the values as presented earlier. Means are each based on 50 replications. To reduce the random variation between averages for various settings, the same sequence of pseudorandom numbers was used for each setting.
As a measure of sensitivity, we used:
This quantity, expressed as a percentage and labeled "response" in Table 15, gives the ratio of the relative change in the output variable to the relative change in the input parameter. Large values indicate that the output is highly dependent on that input parameter; small values indicate an insensitivity to the parameter. For example, a response of 100% means that variation in the input parameter causes the same amount of variation in the output variable; a response of zero means that none of the variation is conveyed to the output variable. Three output variables were considered: the sex ratio, "other" summer mortality of males, and "other" summer mortality of females.
As our model is formulated, the size of the fox population did not affect the simulated sex ratio, because annual mallard mortality was fixed. Rates of "other" mortality were only slightly affected, response values indicating that only 31% of the variation in F was transmitted to r9, and about 60% to r10. The density of male mallards also did not affect the sex ratio; its effect on the rates of "other" mortality was of about the same magnitude as F. although in the opposite direction.
The various factors comprising the predation rates turned out to have no impact on the sex ratio and, with the exception of the sex composition of birds found at dens, none seriously affected rates of "other" mortality. For the incidence of mallards found on the surface at dens, percentage detectable at the surface, percentage brought to dens, and number of dens, the response values ranged from about 28% to 35% for r9 and from 56% to 65% for r10. The sex composition of birds at dens (expressed as percentage female) had a marked effect on r9, with response values averaging about 116% and a lesser effect on r10, response values averaging about 61%.
The survival rates of mallards did affect the spring sex ratio and rates of "other" mortality. Varying male and female survival rates up or down together did not markedly change the sex ratio, but setting the female rate further below the male rate caused the sex disparity to increase. Conversely, bringing the rates close together caused the sex disparity virtually to vanish. The response values (Table 15) for survival rates are expressed as a function of the difference in male and female rates. The response values pertaining to the sex ratio are meaningful, but those pertaining to rates of "other" mortality are not. The latter can be evaluated by direct examination of the average values. As would be expected, changes in the survival rates produced marked changes in rates of "other" mortality.
As the model is formulated, the expansion factors for hunting rates were not particularly influential. The recovery rate did not affect the sex ratio and affected the "other" mortality rates only moderately. The rate of crippling loss was negligibly important. The reporting rate effect turned out to be comparable to that of the recovery rate.
In addition to assessing the sensitivity of the parameters for which we had estimates, we examined the importance of the following assumptions made in developing our model: (1) setting an upper bound (BOUND = 0.25) on the proportion of female mallards that could have been taken by foxes in any year, (2) setting the rate of "other" mortality in summer at three times the winter rate, and (3) allowing the sex ratio of recruited birds to be balanced. The first assumption was an artifact to include some density dependence in the predation rates; the other two we made subjectively, lacking quantitative data on the true situation. All assumptions were examined to determine if our conclusions strongly depended upon these somewhat arbitrary decisions.
Varying the value of BOUND did not affect the output variables appreciably. A ± 10% change did not affect the sex ratio or the male "other" mortality rate, and altered the female "other" mortality rate moderately. BOUND did not appear to be a critical determinant of our model.
To examine the import of our assumption that summer "other" mortality was thrice that of winter, we contrasted the results based on a 3 to 1 ratio (Alpha = 3.0) and the results based on a 1 to 1 ratio (Alpha = 1.0 in Table 15). The sex ratio was unchanged, but the summer "other" rates were greatly reduced. Winter rates, of course, increased accordingly. Thus, this assumption is irrelevant to the sex ratio, but does influence any conclusions we reach about "other" mortality rates.
The sex ratio of recruited birds was unknown, but arguments can be made for either a male or female preponderance. Recruitment was applied to the spring population and included both immigrants (minus emigrants) and returning juveniles. It is reasonable to suppose that the sex composition of immigrants and emigrants is roughly the same as that in the population; hence, a disparity favoring males is indicated. Sex ratios of young birds before the hunting season, on the other hand, are only slightly unbalanced in favor of males (Dzubin 1970). Because juvenile males are hunted more heavily than juvenile females (Anderson 1975), more yearling females might enter the next breeding season. It was thus impossible to reach a firm conclusion as to the sex ratio of birds recruited into the spring population. Accordingly, we tested our model with two fairly extreme values of the sex ratio in recruitment: 90:100 and 110:100. These cases generated output sex ratios of 113:100 and 133:100, respectively. Any sex disparity in the recruitment heightened the sex disparity occurring with even recruitment. The response of "other" mortality rates was zero for males and minor for females. We conclude that the sex ratio of recruited birds did exert an important influence on the output sex ratio, but we cannot make a strong case for much departure from parity.
In summary, it appears that our model, and particularly the spring sex ratio, are relatively insensitive to small changes in the value of input parameters. The sex ratio did respond to changes in survival rates, but a disparity persisted as long as male and female rates differed. Any disparity in the sex composition of recruits essentially heightened the disparity under standard conditions. No other variables affected the spring sex ratio, because of the way we constructed our model.
It is reasonable to question the stability of the model in two regards. First, what would happen to the sex ratio if we allowed the model to run on for longer periods of time? By the 11th year, the sex ratio changed from 100:100 to 126:100; would this increase continue indefinitely? Second, how stable is the model with respect to changes in the parameters? Suppose one of our parameter estimates was in error by a moderate amount. Would such an error cause the model to yield misleading answers?
We assessed the long-term stability of the model by running 50 11-year cycles sequentially. The sex ratio at the beginning of each cycle was not taken as even, but rather as the sex ratio resulting from the previous cycle. The average sex ratio of the 50 11th years was 126.2:100, only slightly higher than the average in which initial sex ratios were even. This indicates that the model was indeed stable with respect to time, and that the sex ratio would not "explode" if the conditions specified in the model continued indefinitely. This result can be verified directly by using mathematical techniques, as will be shown shortly.
The stability of the model with respect to changes in the values of the parameters was confirmed by the sensitivity analysis. The model did not disintegrate when input parameters were altered ± 10%. In fact, the only appreciable changes in any of the three output variables resulted from altering the survival rates so that male and female rates were either more or less disparate than before. No major change resulted from changes of similar magnitude in the male and female rates. That female survival rates are generally lower than those of males was convincingly demonstrated by Anderson (1975), who found that condition to be nearly universal among mallard populations he examined.
Thus, we conclude that the model was reasonably stable, with respect both to time and to changes in the input parameters.
The main result of the sensitivity analysis (that the simulated sex ratio depends only upon survival rates) and the result of the stability analysis with respect to time (that the sex ratio would not "explode") can be derived directly from mathematical theory, without reference to a simulation model. This section describes how the asymptotic, or limiting, sex ratio can be calculated as a function of male and female survival rates and recruitment. The method is analogous to that proposed by Wight et al. (1965), who also demonstrated that under general conditions the sex ratio of adults in a population tends to stabilize. We need the following additional notation. Let
Sm = adult male survival rate,
Sf = adult female survival rate,
Dm = ratio of juvenile male survival rate to adult male survival rate,
Df = ratio of juvenile female survival rate to adult female survival rate,
Rm = number of young males recruited per adult female,
Rf = number of young females recruited per adult female,
Thus, DmSm is the survival rate of juvenile males (Johnson 1974). We assume either that these values do not change annually or that they represent long-term averages. Then the number of males or females in the population one year is the surviving adults from the previous year plus those juveniles produced the previous year that survive. In mathematical notation,
Then the sex ratio in year t+1 is given by
This equation indicates how the sex ratio changes from one year to the next. By assuming that p, Dm, Rm, Df and Rf are essentially constant, we can take the limit of the sex ratio as t gets large. The limit exists (Wight et al. 1965) if
Applying the averages in Table 13, we see that the limit exists for our data if DfRf > 0.10. Denoting the limit by without a time index, and taking the limit in Equation 3, we have
which has the solution
The asymptotic sex ratio thus depends upon p, the ratio of male to female survival rates, and upon DmRm and DfRf, which are essentially measures of recruitment to the spring population.
We can view this relationship graphically by setting DmRm = DfRf = DR. Fig. 4 displays Φ as a function of p, for various plausible levels of DR. Although DR is generally not very well known, inspection of Fig.4 shows that DR has little effect on the sex ratio over much of the range of p. For example, Table 13 indicates that for our data p (rho)= 0.678/0.617 = 1.1. For values of p (rho) near 1.1, the limiting sex ratio is insensitive to reasonable changes in DR. This equation predicts a sex ratio of approximately 120:100, based on the observed discrepancy in survival rates.
Although it is unrealistic to assume constancy of certain rates, this mathematical exercise serves to substantiate the results of our sensitivity analysis (that the sex ratio depends only upon differences in male and female survival rates, for a fixed level of recruitment), and the temporal stability of the model (the sex ratio converges).