The behavior of the maximum body size (body length) in an evolving clade is exemplified by the evolutionary histories of Bivalvia, Cetacea, and Camerata (Crinoidea). Changes of the maximum size with time track closely diversification history: when a clade diversifies exponentially, the maximum size also increases exponentially, and when the number of species changes irregularly (at varying rates), the maximum size also changes in that manner. However, within any given clade, the maximum body size changes at lower rates than diversity does. The observed shifts in maximum body size approximate the rate of diversification per unit of time to the power of about 0.5.

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Abstract. The behavior of the maximum body size (body length) in an evolving clade is exemplified by the evolutionary histories of Bivalvia, Cetacea, and Camerata (Crinoidea). Changes of the maximum size with time track closely diversification history: when a clade diversifies exponentially, the maximum size also increases exponentially, and when the number of species changes irregularly (at varying rates), the maximum size also changes in that manner. However, within any given clade, the maximum body size changes at lower rates than diversity does. The observed shifts in maximum body size approximate the rate of diversification per unit of time to the power of about 0.5.Key words. Change in maxima, Cope's rule, diversification, evolutionary size change, morphological rates, taxonomic rates.Received April 20, 2004. Accepted February 8, 2005 In an earlier paper (Trammer 2002), I presented the maximum body size (body length) in radiating clades as a function of diversity according to the empirically obtained equation:(1) where S is the maximum size, S 0 is the initial size, N is the clade diversity or the number of species, N 0 is the initial diversity, and a and b are parameters of the relationship between the maximum size and diversity.As parameter a approximates one (Trammer 2002), it may be omitted for simplicity and thus equation (1) reduces to: The aim of the present study is the further analysis of changes of the maximum body size in an evolving cladein this case the maximum size is described as a function of time. The present paper may be seen as a quantitative continuation of the pioneering work of Stanley (1973), who first documented clearly the cladogenetic nature of long-term trends of body-size increase as an increase in the maximum size. CHANGES OF MAXIMUM SIZE UNDER CONDITIONS OF EXPONENTIAL DIVERSIFICATION The ModelAs stated by Stanley (1979; for a discussion, see also Walker 1985), exponential growth may be a good approximation for the diversification history of some clades. Consequently, Stanley (1979) (6) where N is the number of species, N 0 is the initial number of species, r is the Malthusian parameter or per species rate of increase per unit of time, t is time, and e is the base of the natural logarithms. Assuming monophyly or paraphyly of a given clade, N 0 will equal one and equation (6) reduces to:To model the pattern of increase of the maximum body size with time in the context of an exponentially diversifying clade, N in equation (4) may be modified by substituting N with the right side of equation (7):Equation (8) may also be deduced from equation (2) without assuming that the initial number of species equals one, with N substituted by the right side of equation (6). Thus, when a clade diversifies exponentially (eq. 7) at the rate r (e r times per unit of time), then its maximum body size also increases exponentially (eq. 8) but at the rate rb (e rb times per unit time). Given that parameter b is approximately 0.5, the approximate version of equation (8) Thus, the rate of body size...

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INTRODUCTIONAn important question of evolutionary palaeobiology is whether originations and extinctions of species were concentrated during short periods within the time intervals (pulsed) or if they were distributed broadly throughout the intervals (continuous). To solve the problem, investigators have used the origination and extinction data of all known genera (e.g. Raup and Sepkoski 1984;Raup 1986;Foote 1994Foote , 2005Gradstein and Ogg 1996) as a proxy for species' speciation and extinction.Essentially, the expectation of a continuous turnover of species within the time intervals is that longer intervals should have higher origination and extinction rates than shorter ones. This is, however, not the case ( Raup and Sepkoski 1984;Raup 1986;Foote 1994;Gradstein and Ogg 1996;Alroy 2008Alroy , 2010, which is consistent with the idea of pulsed origination and extinction. Likewise, the scrupulous investigation of Foote (2005; see also Alroy 2008Alroy , 2010 clearly supports this model. The idea and the aim of the present contribution is to use stratigraphic biozones to verify whether the turnover of biostratigraphically significant species is pulsed or continuous. METHODSThere are many types of biozones (see e.g. MacLeod 2005) but almost every biozone records an origination and an extinction in its total range. For example, the lower boundary of a range zone denotes the origination of the index taxon and the upper limit of this zone identifies the extinction of the same taxon. Concurrent range zones and partial range zones record the origination of one taxon and the extinction of another taxon. As a result, the count of the number of biozones is also a rough proxy for the number of originations and extinctions of the index taxa on which the biozones are based.Genus-level ranges define some biozones, for example the standard Devonian, Carboniferous and PerContinuous versus pulsed origination and extinction and biozones JERZY TRAMMERInstitute of Geology, University of Warsaw, Al. Żwirki i Wigury 93; ABSTRACT:Trammer, J. 2015. Continuous versus pulsed origination and extinction and biozones. Acta Geologica Polonica, 64 (4), 361-366. Warszawa.The count of biozones as an approximate measure of the origination and extinction rates of index taxa (mostly species) is proposed. The mean number of extinctions of these taxa per m.y. is 1.136 (Palaeozoic mean 0.902, Mesozoic mean 1.680, Cenozoic mean 0.789). In contrast with the majority of taxa, the originations and extinctions of index taxa, as well as of other taxa belonging to biostratigraphically important systematic groups, seems to be continuous.