Ecology. Michael Begon

change in that element. Conveniently, too, with the matrix formulation, the elasticities sum to 1, so that the meaning of an elasticity of 0.5, say, is clear: the element concerned accounts for half the variation in R.

      

      APPLICATION 4.4 Elasticity analysis and population management

      elasticity analysis and the management of armadillo abundance

Elasticity analysis offers an especially direct route towards focused plans for the management of abundance. For example, an elasticity analysis has been applied to the population dynamics of nine‐banded armadillos, Dasypus novemcinctus, in Mississippi, USA. The armadillo is a reservoir of infection of the causative agent of leprosy, Mycobacterium leprae, and indeed the only known non‐human vertebrate host of the disease in the Americas (see Oli et al. (2017) who also carry out an elasticity analysis of leprosy dynamics within the armadillo populations). Globally, between 200 000 and 300 000 new human cases of leprosy are reported annually. Given the popular view of leprosy as a disease of a bygone era, a surprising number of these cases (around 200) are in southern USA, and these are increasingly being linked to infected armadillos. An understanding of the forces driving the population dynamics of the armadillos is important, therefore, because the risk of human infection increases with the abundance of infected armadillos, and hence with the abundance of the armadillos themselves. (The ecology of these ‘zoonotic’ infections, passed from wildlife to humans, is discussed in more detail in Section 12.3.2.) A life cycle graph and associated population projection matrix for armadillos is shown in Figure 4.19a. Three age classes are distinguished: juveniles (0–1 years old, prereproductive), yearlings (1–2 years) and adults (>2 years), though these adults may also transition into an infected (leprous) state that can also reproduce. Estimates for the various elements of the matrix, from field data, are shown in Figure 4.19b. The reproductive rates describe additions to the free‐living juvenile class, since these, rather than newborns, are the youngest animals that can be trapped. However, the survival rate from birth to becoming trappable is unknown. The matrix model was therefore run for low, medium and high values for this survival rate, γ (0.5, 0.8 and 1.0). The elasticities of the various elements of the matrix are shown in Figure 4.19c.

Figure 4.19 Elasticity analyses can guide the management of armadillo abundance. (a) Life cycle graph and population projection matrix for nine‐banded armadillos, Dasypus novemcinctus, in Mississippi, USA, comprising fecundities, F, and survival rates, S, for juveniles, J, yearlings, Y, non‐leprous adults, N, and leprous adults, L, and with ψ referring to the probability of non‐leprous adults becoming leprous. (b) Estimates, with standard errors, of these parameters from field data, except that α₁ (yearlings) and α₂ (adults) are probabilities of reproduction that are combined with litter sizes to generate the fecundities. (c) The elasticities of the population growth rate, R, to these parameters, to litter size (LS) and to the (unknown) probability of surviving to a trappable age, γ, for three values of γ (0.5, 0.8, 1.0).

      Source: After Oli et al. (2017).

      Of those elasticities, it is encouraging, first, that the elasticity for the unknown survival rate, γ, is low, indicating that our conclusions are not strongly dependent on our assumptions about γ. Next, it is apparent from Figure 4.19b that infected adults had a reduced survival rate (down 14.5%), and it is for this reason that the elasticity values for the probability of transition of adults into the infected state were negative (Figure 4.19c). However, these elasticities were especially low, indicating that R for the armadillo population would not be greatly affected by the infection rate. Rather, the parameter with an elasticity value indicating the greatest influence on R (approaching 0.5) was the survival rate of adults.

      The distribution of nine‐banded armadillos is expanding northwards in the USA, and the incidence of leprosy in these populations is increasing drastically. The elasticity analysis suggests that leprosy itself will do little to halt the spread of armadillos. If their abundance is going to be controlled, adult survival is likely to be the most effective, as well as perhaps the most practical target.

      elasticity analysis and thistle control

Elasticity analysis has been applied, too, to populations of the nodding thistle (Carduus nutans), a noxious weed that is prickly and unpalatable to most livestock, and that has expanded from its Eurasian origins to invade many parts of the world, including Australia and New Zealand. The question at issue in this case is why control measures for the thistle that are effective in one part of the world are not always effective elsewhere. The life cycle graph for the thistle has the same structure in the two countries (Figure 4.19), comprising four stages: a seed bank, and small, medium and large plants. In fact, ‘size’ is defined not literally but on the basis of their probability of flowering: <20%, 20–80%, and >80%, respectively, since the size–flowering relationship itself varies between the countries. Field data from sites in each country, summarised in the projection matrices in Figure 4.20, indicate that the detailed demography was also rather different in the two cases. In Australia, fecundity was relatively low compared with New Zealand, as indicated by transitions in the matrix from small, medium and large plants either into the seed bank, or directly into small plants, following germination (highlighted in bold in the matrices). Germination from the seed bank to small plants in Australia was also relatively low. On the other hand, the probabilities of surviving within a size‐class and of surviving and growing into the next size‐class were noticeably higher in Australia. This translates into values of R = 1.2 for the high survival, low fecundity Australian population, and R = 2.2 for the high fecundity, low survival New Zealand population, although despite this difference, the species is a highly invasive weed in both countries.

Figure 4.20 Elasticity analysis can guide the management of thistle abundance. (a) Life cycle graph and population projection matrix for the nodding thistle, Carduus nutans, in Australia, comprising a seed bank and small, medium and large plants. (b) The equivalent for a population in New Zealand. The arrows in the life cycle graphs are the transitions from year to year (survival, fecundity, growth and (for the seed bank) dormancy) and the numbers associated with them are the elasticities of R to these