In a finite limited environment, the real number of new families is only a part of the potential exponential increase in the number of families: only those that can find a free well suited niche can survive. For a newborn family, the probability to find an occupied niche is, in a first approximation, N/N_f, then the probability to find a free one is 1 – N/N_f. So, the number of really new families becomes:

This other form of logistic model can be formally deduced from (1) if we assume α = r/N_f (cf. Fig. 3A for a graphic interpretation).

If we consider the graph in Fig. 3B, the projections on the r axis (ordinate) are embedded. So, we have tested the following hypothesis: is it possible to consider r as globally constant during all the considered geological interval of time? The model does not fit well (the sum of square is 75 150, to compare to 20 295 – cf. Table 1). However, we tried other possibilities by continuing to apply the principle of parsimony to select models that fit well, with a minimum number of parameters. Among them, the following ones, where the extinction process is also represented, yield good results (cf. Table 1).

These results suggest, to a first approximation, that r (the inner biological mechanism of differentiation) is globally constant, except during the third period: six times lower than during other periods (the approximate confidence intervals are: 0.0639–0.0735 (model (3)) or 0.0630–0.0738 (model (4)) for periods 1, 2 and 4, and 0.0057–0.0153 (model (3)) or 0.0047–0.0155 (model (4), for period 3). In the next section, we will examine possible interpretations by using the following alternative expressions of the logistic model.

Parameter b represents the apparent time constant of the decreasing processes from data (b = 0.0158 Myr^–1). Half of the initial number of families is reached after time T_0.5 = (1/b) ln 2, that is to say T = 43.9 Myr. The Cretaceous–Tertiary extinction corresponds to the disappearance of about 9% of families, 91% remains untouched, then the duration of extinction is estimated to be T_0.91 = –1/b ln (0.91) ≈ 6 Myr. These values are greater than estimates from chrono-stratigraphic data (about or less than 1 Myr); for example, Olsen et al. evaluate at 0.04 Myr the duration of the extinction period at the end of the Triassic 〚8〛. The discordance between estimations from global model and field chrono-stratigraphic data can be explained: data are too rough to enable precise estimations, so the evaluation of parameter b is common to the three major extinction periods chosen by Courtillot and Gaudemer, whose durations are different; for instance, the Permian–Triassic transition is longer than other ones; it covers, in fact, five successive events, as noted by Hallam and Wignall 〚9〛: three extinctions (whose time is less than 1 My) and two radiations. Data on more recent periods also show a great variability, both in time and amplitude of extinction phenomena (see 〚10〛 on the Palaeogene mammalian record). So b can be interpreted more as a statistical parameter, which permits to link periods and then to obtain a chained model, than a characteristic quantity of extinction processes. In fact, the resolution of data does not allow evaluating such an estimate.

But other forms of the logistical model can also be explored. Among them, if we introduce an additional variable, named for example S, and if N still represents the observed variable, for instance the number of families, the following differential system is another form of this model 〚6〛:

In this context, S can be interpreted as the number of free ecological family niches (a family ecological niche is the set of niches of ecological species which belong to a same family) with the initial conditions N = N₀ and S = S₀.

The concept of ecological niches is a corner stone in theoretical ecology (a recent presentation can be found in Begon et al. 〚11〛). It was proposed by Hutchinson 〚12〛, and completed after (Vandermeer presented an historical compilation 〚13〛). It can be seen like “an n-dimensional domain of environmental parameters and variables whithin which organisms of a species can maintain a viable population” (adapted from Begon et al., op. cit.). At the beginning, these parameters and variables were physicochemical, then necessary resources were introduced and also other species which can interact with this species are progressively considered as elements of the niche (e.g., competitors, predators). Progressively the concept of niche leads to describe environmental conditions that permit the survival and development of organisms and populations of a species. This concept was first introduced to describe the actual world, from an ecological point of view. But it can be introduced in long term analysis, particularly for the evolution of living systems by introducing environmental changes during the past and responses of living systems to these changes. In such a context, changes of genetic structures, which lead progressively to new species and then to be more or less adapted to environmental changes (i.e. to be able or not to find new niches), has to be considered as dynamics processes. Here we propose to extend the concept of niche, defined for species, to other taxonomic levels, particularly families one. Finally, this concept is global, its concrete realisations are habitats, referenced in space and time. The set of habitats constitute the niche of a taxon.

We have dS/dt = – dN/dt, then S – S₀ = – (N – N₀) and S = S₀ + N₀ – N. S and N have the same units; N₀ is a measure of occupied niches, S₀ is a measure of free niches at t = 0. S(t) represents the amount of free niches at time t (t > 0). Finally, it leads to the logistic model:

In a first approximation, the rate of emergence of new families is proportional to the total number of existing families and to the amount of free niches (S). Newborn families occupy free niches, and then the amount of these niches decreases as they become occupied (formally: S(t → +∞) = 0 and N(t → +∞) = N_f; cf. Fig. 1: increasing period).

But this model does not include a possible spontaneous disappearance of families. Introducing such a mechanism is consistent with actual experimental data on extinction dynamics (see, for example 〚14〛). Now we propose to introduce such a mechanism:

As above, we have dS/dt = –dN/dt, S = S₀ + N₀ – N or S = k – N, and

This is always the same equation as 〚2〛, with r = (α k – β) and N_f = k – β/α.

If r > 0 and N_f > N₀, we observe a growing phase. It is the case for the periods (1), (2), (3), and (4).

If we suppose now that during some periods, called extinction ones, there is in addition a ‘mortality’, consequence of an environmental perturbation as we introduced in model 〚4〛, $\frac{\mathrm{d}N}{\mathrm{d}t}=\mathrm{r}N\left(1-\frac{N}{N_{\mathrm{f}}}\right)-\mathrm{b}\mathrm{N},$ it also leads to an analogue expression:

Finally, the classical formula, can be proposed to describe all phases:

This formula can describe both the growth parts and decreasing parts of the change of number of families. If r > 0, this is the classical logistic model, K represents the asymptotic equilibrium point, which can be denoted N_f (the final number of families which could occupy the same number of ecological niches). If K = N_f > N₀ we observe a growing process from N₀ to N_f, and if K = N_f < N₀, N decreases from N₀ to N_f (cf. Fig. 1: decreasing period). But r can be less than 0 (i.e. r < b) and for the same reason K < 0. In this case, the asymptotic number of families is 0 (the other equilibrium point is – K; obviously, it has no significance). This is an exponentially decreasing process. In practice, and as observed from data, environmental variations and speciation processes (i.e. biological diversification) have avoided reaching such a fatal fate, up to now...

Finally, another way to obtain the logistic model, in this framework, is to consider a constant variation of the number of ecological niches S during an interval of time:

We still have dS/dt = –dN/dt and S = S₀ + N₀ – N. The parameter u can be positive or negative. Simple calculus leads to the expression: dN/dt = α N (K – N), where K = S₀ + N₀ + u, if u < 0 then K < N₀ + S₀ and if u > 0 then K > N₀ + S₀. So, the logistic model can also take into account constant, positive or negative, additional variations of ecological niches during an interval of time and not only its initial value S₀.

Then this model can represent a succession of positive and negative environmental stresses. Practically, one can see the variations of the number of families as a succession of increasing and decreasing phases described by the logistic model, following the values of K and r. These values depends on the ‘mortality’ of families, internal or induced by an environmental perturbations, and of the amount of free niches and its variations, which can also be consequences of such perturbations (Fig. 1: succession of increasing and decreasing periods). No hypothesis is formulated on the size of the interval of time t₁ – t₂. It can be smaller if the precision of data is sufficient. In fact, as mentioned above, we use data that just enable to evaluate an average process of biodiversity decreasing and increasing on very long intervals of time (e.g., the last case in Fig. 1). It is important to underline that it is a choice to use global data in the goal of analysing a global process, even if there are more precise data. For example, to analyse the variation of the vegetation at the planetary level, we do not use individual data concerning each plant or tree; data are aggregated to represent large biome levels. Here the approach is similar.

As already observed by Courtillot and Gaudemer, the logistic model seems to be well adapted to describe global biodiversity changes all along the geologic ages, both on a quantitative basis (it fits well the data) and on a qualitative one (it describes well increasing and decreasing phases). As shown in this article, it can be interpreted in terms of ecological mechanisms, with an explicit introduction of ecological niches. Moreover, the last expressions allow further developments. For example, by introducing in equation (6) a difference in family disappearance and in corresponding ecological niches releasing, we obtain the Kostitzin’s model. System (6) becomes:

However, if this is theoretically satisfying, data do not permit such subtlety (even if we retain equation (6), we can just estimate α K – β and α – K/β).

There is a little difference between the two chained models (3) and (4) whose fitting results are summarised in Table 1. The advantage of the second one is to have a simple representation and to be more satisfying, because there is no evidence that during a global decreasing phase, there is no appearance of new families. The advantage of the first one is to fit better, particularly for the growth phases (3) and (4). However, as shown in Table 2, the differences are globally small and related interpretations are the same.

The surprising fact consists in the importance of diversification, despite drastic environmental perturbations. This diversification is evaluated both by parameters r and K (or N_f).

Since the beginning of the Cretaceous, N_f has increased strongly: from 459 during the first phase and 365 during the growth part of the Triassic, to more than 1000 for the last increasing periods 3 and 4. This can be interpreted in the following different ways: (i) an increase of ecological niches following environmental perturbations (e.g. parcelling out of the environment), and/or (ii) the appearance of new mechanisms of diversifications.

Another explanation (iii) consists in a limitation of growth during period (1), as a consequence of drastic successive perturbations; then, the observed plateau is lower than the one which would have been observed without them, and might be of the same magnitude than for periods 3 and 4.

In the same way, is it possible to propose an explanation for the low value of r during period 3, associated to a very long growth? Does it correspond to an internal biological mechanism that makes lower the rate of diversification (e.g., genetic diversification), or, as suggested by hypothesis (iii) above, to successive perturbations, not sufficient to stop diversification but which curb the increase in the number of families?

In fact, the low value or r (0.0101) compared to the value for other phases (0.0684), can be related to the well known regular ‘minor’ extinctions during this period 〚9, 15, 16〛, which was not sufficient to stop the growth of biodiversity, but which could make it slower.

From data used in this article, a possible evaluation of perturbations can be obtained from the residual sums of squares. They result from sampling errors, from second order variations (e.g., the continuous background of minor extinctions) and from bias introduced by the non-linearity of the model. First, if we examine these variations, the larger ones can be associated to ‘minor’ extinctions, whereas the non-linearity bias can be corrected by calculating the centred sum of squares. Finally, these centred sums of squares (CSQ) correspond to different time intervals, and then it has been weighted by dividing by the durations of periods (RCSQ/Δt (My)). So from models (3) and (4), and only on the growth parts corresponding to phases 1, 3 and 4, excluding the Triassic period (too poor in data), we obtain the results consigned in Table 3.

We can note that the bias introduced by the non-linearity of models and measured by residual averages is very low. This confirms the adequacy of these models to describe data. The ‘sample errors’ are reported on Benton’s graphs 〚4〛, but to ensure the legibility of the graphs they have not been reported in Fig. 2. We just note that after the Devonian, they have more or less the same amplitude than the size of the points. So the oscillations around the curve can mainly be attributed to perturbations.

The values of the residuals by Myr have the same magnitudes for the first and fourth periods, but this value is lower for the third period. In some way, it confirms the hypothesis of a slowing down of the growth during this period, but which is not sufficient to stop the biodiversity increase.

But, even if there are only few data for the fourth period, why is diversification speed so large as compared to the first one, despite the observed perturbations? And why does the apparent number of ecological niches become so large? In fact, if newborn families strictly occupy niches released after extinction, the new plateau would have the same magnitude as the first one. For example, after the Triassic, the biodiversity overshoot this plateau. So, its recovery and overshoot cannot be explained only by the colonisation of released niches, but by the appearance of new niches and/or by ecological mechanisms. After the Triassic, the parcelling out of the super-continent increases the area of coastal zones and consequently the number of marine niches. For instance, the North Atlantic Ocean was formed during the third period and led to coastal zones expansion 〚17〛. But, on the one hand, we have no evidence to consider that the total number of niches is larger than the one in the first period before Pangea, formed in the mid-Palaeozoic. And, on the other hand, we have to explain also the acceleration and the overshoot of the last phase.

Now, let us consider biological and ecological points of view. Among many explanations, we can think of two complementary hypotheses:

• • the first one concerns genetic aspects (the appearance of new genomic mechanisms that favour the emergence of new variants, new species, new genera and new families);
• • the second one is related to the emergence of new ecological relations that could lead to an apparent increase in ecological niches.

If we consider these hypotheses, we can refer to Michod’s work on Darwinian dynamics 〚18〛. Michod proposes that, during geological times, the appearance of cooperative mechanisms between living entities is a major mechanism of evolution. He demonstrates that this hypothesis is consistent with Darwinian theory. So, coexistence and cooperation between macromolecules led to cells, and coexistence and cooperation between cells led to multicellular organisms. He assumes that cooperation appeared progressively and probably sequentially for each level of organisation and therefore allowed the emergence of these levels. Then we may think that what permitted the emergence of cells and organisms could act in the same way at higher levels of organisation, particularly at ecological levels. Even the general tendency is to privilege competition and predation as major mechanisms in ecosystem functioning, it is well known that exist also few mechanisms of coexistence (parasitism) or cooperation (e.g., symbiosis, mutualism). And perhaps we have to re-estimate the relative importance of these mechanisms to explain the functioning of ecosystems and their evolution.

So a similar ecological niche could be shared by different species belonging to different families (no exclusion, but coexistence and/or cooperation). Then biological diversification will proceed not only from multiplication of ecological niches, but also from a kind of ‘niche-sharing’ mechanism resulting from coexistence and cooperation. It may be a way to understand why diversified ecosystems exist today and how they evolved, like present-day intertropical forests, although some ecological mechanisms, such as competition, should lead to simplification (e.g., competitive exclusion). In fact, we can think that scale factors are badly taken into account, particularly spatial ones. Competition is certainly determining at the local scale of the individual tree: it occupies progressively a piece of space during an interval of time (during its life), and therefore limits the possibility to another tree to grow in its immediate neighbouring. Except specific edaphic or climatic conditions, competition is probably less important at higher scales. We can also denote that cooperative processes between species at the root level have been detected (e.g. by the way of mycorrhiza 〚20–22〛).

Moreover, diversification and interactions between elements lead to complex systems, which seem to make them more resilient. It is a hypothesis widely shared by ecologists to promote the interest of diversified systems. Although it is not completely demonstrated at this level of organisation, it has been shown recently that a network of interrelated (cooperative) elements allows large variations of parameters, without significant consequences on the systems (for example, cf. von Dassow et al.’s work on a gene network of Drosophila m. 〚19〛). Then cooperative mechanisms, and resilience of ecosystems, as a consequence of interrelations between organisms, are in favour of diversification, at least in the long term.

In this framework, it is reasonable to assume that cooperative and evolutive mechanisms at the molecular and cellular levels have been stabilised for a rather long time (e.g. for the cellular level, since the first multicellular radiation), and that ecological ones are probably more recent and would explain large diversification, as a consequence of appearance of ecological niche-sharing during the last period (and perhaps already during the third one). Perhaps they are always in an evolution phase, not yet stabilised.

So, in a first approximation, we can propose the following scheme of biodiversity increase: molecular processes are more or less established for a long time, then the rate of diversification, estimated by parameter r, is more or less constant (the low value during the third period could be the consequence of successive minor extinctions). But to survive and to be observed in natural archives, a new taxon needs to be ecologically stabilised during a sufficient time. Then the observed increase of biodiversity should result from new ecological relationships, such as coexistence and cooperation, which favour niche-sharing and then ecological stabilisation and the increase of the apparent number of niches (parameter K).

However, we cannot exclude the emergence of new processes at the genomic level, such as transposon moving or virus integration, and we must not forget, in the end, that differences between taxa are expressed at the genomic level, whichever the involved mechanisms. Then diversification could be seen as a progressive adaptation between genomic dynamics and the emergence of new ecological relations (e.g., co-evolution). This enhances the interest of coupling researches between these two ‘distant’ levels: the molecular one and the ecological one.

At the molecular level, it would be of great benefit to find at which time genomic mechanisms of diversification appeared. Are they old, as suggested by Michod, or more recent? Searching pieces of sequences that look like moving parts of genomes, dating them, and testing if the incorporation process is homogeneous during the envisaged period (from Cambrian to actual) could help, through the study of known DNA sequences, to test this assumed consequence of Michod’s hypothesis.

At the ecological level, an exciting topic of research should be to make an effort in palaeoecology. The concept of niche is related to spatial distribution. So, one of the main problems is to reconstruct spatial distributions of organisms and populations in palaeoecosystems and the evolution of these structures, in order to verify the progressive appearance of niche-sharing, and then to assume the correlated appearance of coexistence and cooperative mechanisms. Conversely, if such relationships are not still stabilised, which is very plausible, we may have the opportunity to observe an evolutionary mechanism in action. In fact, more and more studies on ecological interactions, called ‘long-term interactions’, such as host-parasite ones, already go in this direction (cf., for example, 〚23〛). This leads one to ask whether present actions of Man, which results in new kinds of perturbations on ecosystems, could disturb the existing relationships, or stop the natural emergence of new ones, and then lead to the ‘Third Mass Extinction’, following P. Ward 〚24〛, or the ‘sixth extinction’ feared by Leakey and Lewin 〚25〛.