Diversification of Life Several models have been proposed to explain the diversification

Chapter 8, Problem 3

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Diversification of Life Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models for diversification. They are analogous to models for population growth; however, the quantities involved have a different interpretation. We denote by N(t) the diversification function, which counts the number of taxa as a function of time, and by r the intrinsic rate of diversification. (a) This model is described by dN dt = reN (8.86) Solve (8.86) with the initial condition N(0) at time 0, and show that re can be estimated from re = 1 t ln _ N(t) N(0) _ (8.87) [: To find (8.87), solve for r in the solution of (8.86).] (b) This model is described by dN dt = rl N _ 1 N K _ (8.88) where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that rl can be estimated from rl = 1 t ln _ K N(0) N(0) _ + 1 t ln _ N(t) K N(t) _ (8.89) for N(t) < K. (c) Assume that N(0) = 1 and N(10) = 1000. Estimate re and rl for both K = 1001 and K = 10, 000. (d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of [r ] to represent underestimates of exponential rates, because some radiation will have followed distinctly sigmoid paths during the interval evaluated. (e) Explain why the exponential model is a good approximation to the logistic model when N/K is small compared with 1.