Since the first organism appeared on Earth, life has diversified massively. Despite the different levels of fluctuations on different timescales, global diversity has increased exponentially on the macro-level (e.g., Hewzulla et al. 1999). However, this process is driven by extinction as well as by origination. Here we discuss the diversification patterns of individual groups to compare to our recent synoptic analysis of all terrestrial and marine data (Hewzulla et al. 1999). The logistic model is a widely accepted way to incorporate both exponential growth and the factor that limits the maximum growth allowed by the system. However, the logistic model still fails to explain how a particular group of organisms diversifies and eventually declines. Clearly the simple exponential model is no longer applicable at this level, otherwise no group would ever be driven to extinction.
The fact that old groups become extinct and new ones originate yields a more unstable sense of change. May and Lawton (1995) have suggested that the average life-span of a single species can be calculated using origination and extinction dates by assuming that a species has a higher risk of becoming extinct when its life span exceeds an average value. We have extended this idea to higher levels of classification, leading to the hypothesis that most organismal groups have a limited life-span at the macroevolutionary level. Under this model even the most successful groups may be destined to become extinct.
Our theoretical model is based on the observation that many families diversify according to such a distribution, diversifying relatively slowly at first and then radiating more quickly, reaching a maximum level over varying lengths of time, and becoming extinct slowly over longer periods of time (unless some environmental crisis disrupts the pattern). Nee and May (1997) have given a mathematical explanation of how an ecological community tends to preserve its tree structure over the loss of its member species that increases the chances of recovery after the extinction event. That can explain the slow decrease in the diversity of a group of organism during its final extinction. This supports our hypothesis that evolutionary changes in taxonomic groups resemble bell-shaped curves, with protracted origins prior to radiation and a long demise. Hence, we use a modified version of the logistic model to represent the bell-shaped diversification curves and test the hypothesis using the data from The Fossil Record 2:
Where: N(t) = is the estimated diversity at time.
N0 is diversity at t=0, determined by the
initial state of the model. Nf is called niche
capacity that corresponds to the equilibrium diversity when 
= 0.0.
is the extinction factor that determines how and when the
extinction takes place. The higher the
value, the stronger the suppression imposed onto the
increase of diversity at the earlier stage, and the quicker
the decrease of the diversity at the later stage. The 
is the origination factor that tends to increase
diversity. The model incorporates both the exponential and
the logistical interpretation of evolutionary change. When
= 0 it becomes a logistic model. The lower part of the
logistic model approaches the exponential curve when Nf
is very large.
According to our model, and according to common sense, a group reaches its peak and inevitably declines to extinction. However, apart from this deterministic component, an actual diversification process comprises indeterministic fluctuations that are caused both by internal dynamics and external perturbations (e.g., the effects of asteroids, continental movements, Hewzulla et al. 1999). The actual diversification pattern fluctuates around the trajectory determined by the analytic model. Therefore, the attainment of a local peak does not necessarily mean that the model will decline afterwards to extinction. When a local peak is reached, there is no way of telling whether further changes will lead to extinction or rise again to another local peak. In most cases, we will not know whether these fluctuations are caused by internal dynamics of the system or external perturbations. However, when there is a sudden large deviation from the model it is more likely caused by external (= environmental) factors.
Below we apply our mathematical model to the data and calculate the best-fit parameters in order to try to reveal a global trend that filters out the local fluctuations. When the global trajectory shows a clear downward trend to the present, we calculate the future extinction date of the group of organisms. However, in some cases the pattern does not show a clear global downward trend and, in these cases, the data are not sufficient to calculate the parameter that determines the group's extinction.
