Conclusions sensitivity analysis

We have learned that age- or stage-structured growth is common in most plant and animal populations, but the details of both survivorship and fertility differ greatly across species. For example, natural populations display many variations in survivorship, although a type III survivorship schedule is most common. Most of the "charismatic megafauna" of interest to conservation biologists and the general public shows age structured growth. Thus the techniques outlined in this chapter are necessary before we can make predictions about the potential for growth and recovery of an endangered population. We must understand the effects that age-specific survivorship and fertility have on the behavior of a population before we are in a position to implement a management plan that would actually be effective in promoting its long-term survival.

For example, examine Table 4.12. These data were gathered by Schmutz et at. (1997) on a population of the emperor goose (Chen canagica) (analysis from Morris and Doak 2002). Given this information, is this population growing or declining? What aspects of its life history are most important to its long-term growth rate? To answer these questions, we will use the matrix format from Section 4.10. It turns out that the long-term l for this population is 0.989.

We now ask, "How sensitive is population growth (or extinction risk) to particular demographic changes?" Specifically, will a particular change in survivorship or fertility have a large impact on the growth rate, l? Using the above matrix, we have substituted survivorship values from 0.136 to 1.00 for both hatchlings (S0) and older birds (Sal), and fertility rates of 0.136-2.000 for two-year-old (F2) and three-year-old and older birds (F>3). Figure 4.13 summarizes this analysis. Obviously survival, especially that of the older birds, has the greatest effect on the growth rate (l). Increases in fertility have virtually no effect. A related point is that, since adult survival is so critical to population growth, errors in our estimates would have a very large impact on our conclusions about this population. As it stands now, from this matrix our estimate of l is 0.989 < 1.000, and we expect this population to slowly decrease. A small change in our estimation of survivorship, however, would lead us to believe that this population is stable or growing. For example, a

Table 4.12 Survivorship and fertility of an emperor goose population (Schmutz et al. 1997).

Survival of

Survival of

Fertility of

Fertility of three-

hatchlings = S0

one-year-old

two-year-old

year-old and

and birds = Ss1

birds = F2

older birds = FS3

0.136

0.893

0.639

0.894

Vital rates

Figure 4.13 The effect of changing vital rates on the value of lambda in an emperor goose (Chen canagica) population. An increase in survivorship of adults (one-year-old and older birds) has the greatest effect. Increases in fertility have a negligible effect on l.

change in adult survival from 0.893 to 0.905 changes lambda from 0.989 to 1.001. A change in juvenile survivorship from 0.136 to 0.155 changes the expected lambda to 1.000.

An examination of this sort is known as a perturbation or "sensitivity analysis." Although Fig. 4.13 is convincing, conservation biologists have sought to summarize the kind of analysis we have done above into a single number that would summarize the sensitivity of lambda to particular vital rates. The most common basic measure of sensitivity is the slope of the tangent taken on the curve of lambda as a function of each vital rate. Problems with this approach include the possibility of nonlinearity in the relationship between lambda and a particular vital (survivorship or fertility) rate. A second issue is the scaling of sensitivity values. Obviously survivorship scales on a strict 0.0-1.0 scale, while reproduction can scale to very large numbers (number of acorns produced by an oak tree). These comparisons can be made more meaningful by examining the proportional change in lambda as a proportion of change in the vital rates. These calculations result in a measure known as elasticity. Elasticity, then, is a standardized sensitivity that measures the effects of proportional changes in vital rates. That is, elasticities tell us the effect of perturbations in vital rates that are all of the same relative magnitude. Elasticities are standardized to sum to 100%.

For the emperor goose population discussed above the vast majority (92% of elasticity) of sensitivity was in the survival of the one-year-old and older birds (Morris and Doak 2002). An analysis of the Amboseli baboon population by Alberts and Altmann (2003) led to similar conclusions. That is, fertility represented just 9% of the total elasticity for both males and females. Survival of the pre-reproductive age classes accounted for 37% of the total elasticity for females and 62% for males. Details on the calculation of both sensitivity and elasticity values can be found in Morris and Doak (2002) or Alberts and Altmann (2003).

What should be done to promote the long-term survival of these two populations, or of other populations described in the first paragraph of this chapter? What evolutionary forces have led to a particular life history in the first place? These are just two of many questions for which there are no easy answers. Still, the analyses outlined in this chapter should have given you the tools necessary to at least begin to address these issues.

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