Figure 10.6: Growth of each age class of sticklebacks. Model (10.16) is iterated for a stickleback population. The 1-year old class is depicted by the solid line, and 2-, 3-, and 4-year old classes are depicted by lines with increasingly long dashes, (a) Population size of each age class, (b) Proportion of the population in each age class (the 4-year-old class is too rare to appear in the figure). Parameter values: mq = 2, m2 = 3, m3 = 4, mA =4, pi = 0.6, p2 = 0.3, and p3 = 0.1.
where n is the number of age classes, = p, p2... P,-1 is the probability that an individual survives until age class i, and = 1 because we defined the fecundities m, as the number of offspring that survive to age 1. Regardless of the number of age classes, the n eigenvalues A of a Leslie matrix are the n roots of equation (10.17).
Equation (10.17) is sometimes referred to as the Euler-Lotka equation. It features prominently in life-history theory, which is devoted to understanding the evolution of age-specific patterns of fecundity and survival (Roff 1992; Stearns 1992; Charlesworth 1994). The Euler-Lotka equation also provides an alternative way of calculating the long-term growth of a population, by solving (10.17) for the largest root X1. If the population were not growing in size = 1), then the expected lifetime reproductive success of a newborn individual, would be one—that is, each individual would exactly replace itself. The Euler-Lotka equation shows us how to generalize this statement for populations that are growing or shrinking in size. Now, the expected lifetime reproductive success of a newborn individual, discounted by the amount the population has grown from when it was born until it reaches age i, S"=1/.m./A'1; will equal one.
For our model of sticklebacks, equation (10.17) can be written out as
Box 10.3: The Characteristic Polynomial of Leslie Matrices
Suppose '.\e have a I eslic matrix with n age classes:
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