Several models have been proposed to explain the diversification of life during geological periods.

Efan Halliday 2021-02-11 Answered

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 fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification.
(a) (Exponential Model) This model is described by dNdt=reN (8.86). Solve (8.86) with the initial condition N(0) at time 0, and show that re can be estimated from re=1t ln [N(t)N(0)] (8.87)
(b) (Logistic Growth) This model is described by dNdt=rlN (1  NK) (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=1t ln [K  N(0)N(0)] + 1t 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=10000.
(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 NK is small compared with 1.

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Expert Answer

Daphne Broadhurst
Answered 2021-02-12 Author has 109 answers

Step 1
Given: N(t) denotes the diversification function, which counts the number of taxa and r denotes the intrinsic rate of diversification.
Step 2
Calculation:
a) The exponential model is described as:
dNdt=reN
 dNN=redt
Integrate both sides
 N(0)N(t) dNN=re 0t dt
 ln |N|N(0)N(t)=ret0t
 ln [N(t)N(0)]=ret
 re=1t ln [N(t)N(0)]
Step 3
b) The logistic model is described as:
dNdt=rlN (1  NK)
 dNdt=rlK (N)(K  N)
 [KN(K  N)]dN=rldt
 dN [1N + 1K  N]=rldt
 dNN + dNK  N=rldt
Integrate both sides
N(0)N(t) dNN + N(0)N(t) dNk  N=rl 0t dt
ln|N|N(0)N(t)  ln |K  N|N(0)N(t)=rl0t dt

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