GEOS 26400/36400/BIOS 23255/EVOL 32300: Principles of Paleontology

Evolutionary Rates, II
I. Introduction to taxonomic rates
      -Initial study (Simpson 1944) as proxy for morphological rates
            -implicitly implies species=chronospecies, extinctions=pseudoextinctions
      -Later interest in taxonomic rates themselves
            -E.g., differences in rate may reflect differences in population structure,
		ecological specialization etc.
      *Some milestones
            -Lyell's faunal percentages (see below)
            -Simpson's survivorship curves
            -Van Valen's survivorship curves on semi-log axes: near linearity suggests
            possible rate constancy.
      *"species as particles" (study of aggregate properties of large numbers of species) 

II. Measuring taxonomic rates: General

      A. Number of events should depend on number of lineages and elapsed time
            -Therefore, define relative rate (per-capita rate)
                  * p = originations per lineage per time unit (usually, m.y.)
                  * q = extinctions per lineage per time unit (usually, m.y.)
      B. Counting method
            -count number of lineage-million-years (LMY) (analogue: person-hours)
            -count number of origination or extinction events
            -rate = (#E or O)/(#LMY)
            -Disadvantage: Requires good enough record that precise time of O and E can be identified
            -Advantage: direct method, doesn't require model of E or O process

III. Measuring taxonomic rates: Long-term average rates for a group

      A. Survivorship methods: average taxonomic rate for a group of taxa over time
            -Forward survivorship (depends on extinction rate)
                  -Historical development in context of Van Valen's Law.
                  -Of all taxa originating in a given interval, how many survive beyond a certain time? 
                        -P(s,t)=exp(-qt)
                        -analogy to radioactive decay
                        -Mean duration = 1/q
                        -Median duration (half-life) = ln(2)/q
                  -Problems
                        -left side of curve limited by stratigraphic resolution.
                        -right side of curve highly affected by sampling error
                  -Advantage: by assuming model, we need not place events precisely within intervals
                  *Even if constant extinction does not hold, cohort approach allows you
                        to see when there are large changes in extinction rate.
		  **Raw versus cumulative survivorship (e.g. Cz mammals)
		  **Dynamic versus cohort survivorship (e.g. Cz mammals)
        B."Backward survivorship" (depends on origination rate)
                  -Just like forward survivorship analysis, except that taxa are followed
                  backward in time and we solve for origination rate.
	C. Relationship between species-level and higher-level survivorship
		-Even if species rates constant, higher-level survivorship
		expected to be age-dependent ("higher"=genus, family etc.)
		-If age-dependent model assumed, it's possible to infer
		species level rates from higher-level survivorship
	    -Example of camerate vs. non-pinnulate crinoids (dynamic)
		-non-pinnulates have longer average duration
		-densely pinnulate crinoids require stronger current to 
		feed, therefore probably ecologically more specialized

            **It is a good idea to omit taxa confined to a single interval of time, since
            these are highly sensitive to variation in the quality of preservation
            (extreme example: Lagerstaetten).
         D. Special case of forward survivorship: Lyellian percentages
                  *Of species alive at a given time in the past, what proportion are still alive today?
                  -Advantage: fauna today nearly completely known, so whether taxon is
                  extinct is known nearly with certainty (no effect of incompleteness).
                  -Percentage reflects cumulative extinction over elapsed time.
                  -Lyell (1833) characterized Tertiary epochs by their percentages.

IV. Measuring taxonomic rates: Interval-by-interval methods

	A. Four fundamental classes of taxa:
                  --"b" refers to crossing bottom boundary
                  --"t" refers to crossing top boundary
                  --"F" refers to making first appearance within interval
                  --"L" refers to making last appearance within interval
            1. Those crossing lower boundary into interval and last appearing within
            interval. Let the number of these be denoted N(bL).
            2. Those first appearing within interval and crossing upper boundary
            out of interval.  Let the number of these be denoted N(Ft).
            3. Those crossing both interval boundaries.  There are N(bt) of these.
            4. Those confined to the interval. There are N(FL) of these.
           
   	B. Combinations:
                  -Total crossing into interval = Nb = N(bL)+N(bt).
                   -Proportion of these that survive through interval = N(bt)/Nb.
                  -Total crossing out of interval = Nt = N(Ft)+N(bt).
                   -Proportion that "survive backward" through interval = N(bt)/Nt.
                  -Total extinctions (not very useful) = N(bL) + N(FL).
                  -Total originations (not very useful) = N(Ft) + N(FL).
            
	C. Three requirements of origination-extinction theory:
                  (p=orig. rate, q=extinct. rate, d=interval duration)
            1. Nt = Nb x exp[(p-q)d]
            2. N(bt) = Nb x exp(-qd)
            3. N(bt) = Nt x exp(-pd)

            **Requirements 2 and 3 imply a simple way of estimating p and q:
                  p = -ln(N[bt]/Nt)/d
                  q = -ln(N[bt]/Nb)/d

	D. Choice of rate measure involves tacit assumption about distribution of event within time interval
		-Dividing by interval length makes sense if turnover rates more or less constant through interval.
		-This does not make sense if turnover concentrated at interval boundaries.
		-If normalization by interval length appropriate, then normalized per-capita rates 
		should be uncorrelated with interval duration.