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Unformatted text preview: 7.5. LIFE TABLES 953 7.5 Life tables In Section 6.1 we introduced the simplest differential equation model of population growth i.e. dN dt = r N This model, as well as the later models we considered, implicitly assume that all individuals whether young or old have the same mortality and fecundity rates. While this assumption is a useful first approximation, mortality and fecundity are often age-dependent. For instance, most animals have a life history that culminates in reproductively capable (or sexually mature) individuals only after they have reached a particular age. Additionally, for many organisms, the risk of mortality risk is higher at younger and older compared with intermediate ages. In this section we consider models that account for age-specific mortality and reproduction. Figure 7.23: Albertosaurus in action! Age-specific mortality In a recent Science article, Biology professor Gregory Erickson and colleagues studied fossils of four North American tyrannosaurs Albertosaurus , Tyrannosaurus , Gorgosaurus , and Daspletosaurus . Using the femur bones of these * G. M. Erikson et al. 2006. Tyrannosaur Life Tables: An example of nonavian dinosaur population biology. Science 313:213216 2010 Schreiber, Smith & Getz 954 7.5. LIFE TABLES Table 7.5: Life Table for Albertosaurus sacrophagus . Age t in years l ( t ) 2 1.0 4 0.96 6 0.91 8 0.86 10 0.77 12 0.73 14 0.64 16 0.45 18 0.32 20 0.18 22 0.11 24 0.08 26 0.06 28 0.04 correspond to interpolated values. fossils, the scientists estimated the life spans of the dinosaurs. The estimated life spans ranged from 2 years to 28 years. Using these estimates, the scientists created a life table for each of the dinosaurs. These life tables keep track of what fraction l ( t ) of individuals survived to age t . Although we may talk about fractions, these fractions are more properly interpreted as probabilities that individuals surviving from the starting age in the life table to age t . For example, the life table for Albertosaurus sarcophagus (see Figure 7.23) is reported in Table 7.5. This table asserts that 18% of these dinosaurs survived at least 20 years. The function l ( t ) in this table is an example of a survivorship function. Survivorship function A function l : [0 , ) [0 , 1] is a survivorship function if l (0) = 1 i.e. all individuals survive to age 0. l ( t ) is non-increasing i.e . if an individual survived to age t , then it survived to all earlier ages. lim t l ( t ) = 0 i.e. all individuals eventually die. Example 1. Aging dinosaurs Use Table 7.5 to do the following: a. Determine what fraction of dinosaurs die between ages 4 and 6. b. Determine what fraction of dinosaurs die between ages 10 and 14....
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