Density Dependence

The lack of periodicity in turnover rates does not imply that these rates are random and therefore does not imply that the diversity curve evolves randomly. The reason is that at least three causal relationships might regulate diversity and turnover while not producing marked auto-

216 / John Alroy correlation. First, origination rates may correlate negatively with preceding diversity levels (Rosenzweig, 1975; Sepkoski, 1978). An equilibrium will result, because high diversity will lead to an origination deficit, low diversity will lead to a burst of origination, and intermediate diversity will lead to a balance of origination and extinction. Second, diversity and subsequent extinction rates may correlate positively, producing an equilibrium for similar reasons (Rosenzweig, 1975; Sepkoski, 1978). Both relationships have been found in Sepkoski's family- and genus-level data (Foote, 1994b, 2000b). Third, origination and extinction may be positively correlated (Webb, 1969; Flessa and Levinton, 1975; Mark and Flessa, 1977; Alroy, 1996, 1998). Such a correlation will greatly slow the net movement of the diversity curve and has been observed in Sepkoski's data (Kirchner and Weil, 2000b; Foote, 2003).

These hypotheses can be tested by computing simple correlations among the turnover rates (k and |J.) and diversity (Ns). There are no strong correlations in the raw data between either extinction or origination and diversity in the immediately preceding, current, or succeeding bins. However, the picture changes after detrending the turnover rates (Fig. 11.2) and diversity curve; as it should, detrending markedly reduces the correlation between neighboring diversity values, with p = 0.769, P < 0.001 instead of p = 0.464, P = 0.002.

Now there is a correlation between past origination and current diversity (p = 0.327, P = 0.035). However, this weak relationship may be influenced by analytical biases, and there is no correlation between past extinction and diversity (p = -0.214, n.s.). Of more interest is a stronger match (Fig. 11.3A) between current diversity and immediately following extinction (p = 0.439, P = 0.004) but not origination (p = -0.039, n.s.). The diversity/extinction relationship is unlikely to result from a commonly encountered bias called regression to the mean (Freckleton et al., 2006). Concurrent diversity and extinction also correlate with the predicted positive sign, if not significantly (p = 0.243, n.s.), and likewise concurrent origination is not predictable (p = 0.223, n.s.). Finally, there is a match (Fig. 11.3B) between past extinction and future origination (p = 0.337, P = 0.029) but not the other way around (p = 0.016, n.s.), with independence of concurrent rates (p = 0.111, n.s.).

The two strong relationships are crucial, because they are responsible for the suggested equilibrium: high diversity will be brought down by high extinction rates, and large extinctions will be compensated by high origination rates. After splitting the data at the Permo-Triassic boundary, sample sizes are too small to establish significance, but consistent patterns in both relationships are still seen (Fig. 11.3). Furthermore, the predicted relationships hold up after removing the outlying end-Permian extinction rate (diversity and future extinction: p = 0.396, one-tailed P = 0.006; extinction and future origination: p = 0.291, one-tailed P = 0.033).

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