The Contribution of Models and Modelling: Some Examples

Abstract

In this chapter are several illustrations and/or results that were made possible thanks to models. The first part of this chapter, devoted to genetics, shows a typical probabilistic model, which we know and have demonstrated to be effective, but which says nothing about the mechanisms bringing about the random phenomena observed. With this in mind and as a basis for reflection, we can consider the transition chaos-randomness as we sketch it out in the second part of this chapter. Lastly, the major trends in the evolution of biodiversity can be modelled through simple mathematical expressions; for example, the logistic model. We can see that despite its simplicity, it can teach us something about the possible global mechanisms explaining these dynamics. In the last section, we propose a general plan for modelling living systems that includes the average “deterministic” trends and the random and chaotic components.

For the calculus of al-Jabr and al-Muqãbala, I have composed a short work which succinctly captures the subtle glories of that science; in this, I owe much thanks to Ma’mûn, the Guardian of the Faithful. He is a prince who brings men together, helps and protects them and encourages them to make the darkness light, the complex, simple.

Al-Kwârizmi, IXe century, in «Hisãb al-Jabr wa’l-Muqãbala»¹

¹

In this chapter are several illustrations and/or results that were made possible thanks to models. The first part of this chapter, devoted to genetics, shows a typical probabilistic model, which we know and have demonstrated to be effective, but which says nothing about the mechanisms bringing about the random phenomena observed. With this in mind and as a basis for reflection, we can consider the transition chaos-randomness as we sketch it out in the second part of this chapter. Lastly, the major trends in the evolution of biodiversity can be modelled through simple mathematical expressions; for example, the logistic model. We can see that despite its simplicity, it can teach us something about the possible global mechanisms explaining these dynamics. In the last section, we propose a general plan for modelling living systems that includes the average “deterministic” trends and the random and chaotic components.

In a way, in this chapter we see the clarifying and simplifying role of the model, a tool that is increasingly a “must”, particularly for analysing data and as a thinking aid.²

4.1 Genetics and Calculating Probability: Elementary Laws and Evolution During the Genetic Constitution of a Population

It is marvellous to note that, for discoveries made in the area of genetics, the calculation of probabilities provides effective mathematical models and statistics provide a strict framework for analysing experimental results. We can note that Mendel, the father of genetics, was a professor of natural science and that he also taught elementary statistics in a secondary school. It is not, then, by complete “chance” that he noted the strange and nearly reproducible proportions in the results of crossbreeding peas and other plants.

His discoveries, which can be considered among the greatest in the history of humanity, were not appreciated for their true value by his contemporaries.

We are going to illustrate these genetic bases and the relevance of probabilistic models in two simple examples.

4.1.1 The Mendelian Model

Let us consider a diploid, sexed population. In this population, individuals carry a gene, not linked to gender, that can present itself in the form of two alleles, A and a. Let’s suppose that we look at the descendents of the cross-breeding between two different heterozygous individuals, Aa. The result of this cross-breeding can be foreseen by constructing a double-entry table:

		Individual 1 (male)
		A	a
Individual 2	A	AA	Aa
(female)	a	aA	aa

The result of the Mendelian theory of gene transmission is that:

• (1) all of these possible results form a complete system of events (the results are mutually exclusive and the sum of their probability is 1); and

• (2) the different results are equally probable: P(AA) = P(Aa) = P(aA) = P(aa) = 1/4.

For all practical purposes, we see the phenotypic expression of the genome. First, the phenotypes Aa and aA are indiscernible, so that the probability of observing, in the descendents, a heterozygous individual is P(Aa)+P(aA) = 1/2.

If the allele A is dominant, then the probability of observing a descendent of phenotype A is: P(AA)+P(Aa)+P(aA) = 3/4, and for phenotype a it is 1/4.

For all practical purposes, when studying the descendents of heterozygous individuals, we do not see these exact proportions any more than we would see the exact proportions 1/2 and 1/2 in a game of “heads or tails”. But how do we decide that what we see can reasonably be interpreted as an experimental outcome of this theoretical schema? Statistics provide us with some appropriate tests (in this particular case the famous Chi ² test).

4.1.2 Genetic Evolution of an Autogamous Population

Let’s continue to consider the same schema, but with an autogamous population; that is to say, individuals that are self-fertilised. This situation can be found in numerous plant species where the same individual bears both the male and the female gametes. Let’s suppose that we start with one heterozygous population, Aa. We will try to anticipate the genetic evolution of this type of population.

Lastly, we will study the evolution of the genetic structure for a bi-allelic gene that we will write as A (and) a. The different possible genetic structures are then:

• AA and aa for the homozygous individuals, and

• Aa for the heterozygous individuals.

If, in addition, we hypothesize that the generations are distinct (discrete time), these different structures can constitute the states of what is known as a Markov process with the trials being the passage from one generation to another. In fact, the make-up of a population at a certain generation depends only on the make-up of this same population at the previous generation (based on the hypotheses):

• all homozygous individuals will have homozygous descendents in the same category; and

• all heterozygous individuals will have

  • 1/4 probability of having descendents AA,

  • 1/2 probability of having descendents Aa, and

  • 1/4 probability of having descendents aa.

We can then construct Table 4.1 with transition probabilities from generation G _k to G _k+1:

Table 4.1 Genotype probabilities for descendents, in relation to the alleles from gene A, according to the Mendelian theory of disjunction and independent recombination of alleles

So the matrix for the passage from generation k to generation k+1 is:

$$P = \left( {\begin{array}{*{20}c} 1 & {1/4} & 0 \\ 0 & {1/2} & 0 \\ 0 & {1/4} & 1 \\\end{array}} \right)$$

Let’s suppose, for example, that we start with a population made up only of heterozygous individuals at generation 0. This hypothesis can be shown in the form of the following single column matrix:

$$V_0 = \left( {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\\end{array}} \right)$$

The probabilities for the first generation will then be:

$$V_1 = P\,\,V_0 = \left( {\begin{array}{*{20}c} 1 & {1/4} & 0 \\ 0 & {1/2} & 0 \\ 0 & {1/4} & 1 \\\end{array}} \right)\left( {\begin{array}{*{20}c} 0 \\ 1 \\ 0 \\\end{array}} \right) = \left( {\begin{array}{*{20}c} {1/4} \\ {1/2} \\ {1/4} \\\end{array}} \right)$$

We can easily see how evolution occurs over the course of time. For that, it is only necessary to calculate P ⁿ. By using a diagonal matrix in an appropriate coordinate system, that of eigenvectors of P, we get:

$$P^n = \left( {\begin{array}{*{20}c} 1 & {\frac{{2^n - 1}}{{2^{n + 1} }}} & 0 \\ 0 & {\frac{1}{{2^n }}} & 0 \\ 0 & {\frac{{2^n - 1}}{{2^{n + 1} }}} & 1 \\\end{array}} \right)$$

We see that:

$${\textrm{when}}\,n \to \infty \,{\textrm{then}}\frac{{2^n - 1}}{{2^{n + 1} }} \to \frac{1}{2}\,{\textrm{and}}\,{\textrm{that }}\frac{1}{{2^n }} \to 0,$$

So

$$\mathop {\lim }\limits_{n \to \infty } P^n = \left( {\begin{array}{*{20}c} 1 & {1/2} & 0 \\ 0 & 0 & 0 \\ 0 & {1/2} & 1 \\\end{array}} \right)$$

We can then study the proportions of homozygous individuals AA and aa after many generations based on the initial composition:

p 0	for	AA
q 0	for	Aa	with p 0 +q 0 +r 0 = 1
r 0	for	aa

After many generations, the frequencies of the different genetic compositions will tend towards:

$$\mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} {p_n } \\ {q_n } \\ {r_n } \\\end{array}} \right) = \left( {\begin{array}{*{20}c} 1 & {1/2} & 0 \\ 0 & 0 & 0 \\ 0 & {1/2} & 1 \\\end{array}} \right)\left( {\begin{array}{*{20}c} {p_0 } \\ {q_0 } \\ {r_0 } \\\end{array}} \right) = \left( {\begin{array}{*{20}c} {p_0 + \frac{1}{2}q_0 } \\ 0 \\ {r_0 + \frac{1}{2}q_0 } \\\end{array}} \right)$$

Note that if p ₀ = r ₀, we obtain a population that always includes equal proportions of individuals AA and aa.

We can also represent the evolution of such a population graphically (cf. Fig. 4.1).

Fig. 4.1

Example of the evolution of a diploid, autogamous population. We start with a population of heterozygous individuals; these die out asymptotically and only the homozygous individuals remain

This model then permits us to anticipate the evolution in the genetic composition of an autogamous population. It can be tested through an experiment and the data resulting from that experiment. It also allows us to anticipate the consequences of this type of reproduction. In particular, the tendency towards homozygosity will generally decrease the resistance and “performance” of these populations.

This is what permits us to explain that even though diploid, autogamous species do exist, immunological mechanisms limit and even hinder self-fertilisation.³ This is a means of avoiding consanguinity knowing that, with the progressive accumulation of regressive alleles, it leads rather quickly to weak individuals. Evolution “invented” autogamy, which could also prove to be a solution for the preservation of the species: in the case of a catastrophe, a single individual is theoretically enough to reconstitute it. It also invented the means of limiting autogamy so as to preserve the genetic mix. On the whole, where randomness was restricted, the means of restoring it were selected.

4.2 From Chaos to Randomness: Biological Roulettes – An Example from the Discrete-Time Logistic Model

The question of the existence of chaotic dynamics in biological and ecological systems was judiciously posed by Carl Zimmer (1999) after two decades of limited success in finding them. In fact, it would have been more practical to envisage where and why chaos could provide an advantage to these systems before searching for it, than to examine observed data without, a priori, a hypothesis. In fact, some may argue that Robert Costantino’s experiment (1997) on the particular dynamics of flour beetles did just that: the erratic population dynamics is the consequence of adults cannibalising larvae and pupae, a “natural” mechanism limiting the size of the population. So, the experiment, based on a non-linear model, was developed to show the different dynamics (e.g. equilibrium, periodic and chaotic oscillations) by adjusting the mortality rate of adults and the density of pupae, and then the rate of the cannibalism. Nevertheless, in this example, chaos is a consequence and not an evident evolutionary advantage.

Nevertheless, we can always assume the presence of chaotic systems – or more generally dynamical systems that enhance unpredictability (e.g., the game of “heads or tails”) – because they are possible solutions to non-linear dynamical systems which are very representative of large classes of biochemical, biological and ecological phenomena, but mainly because they are a way to generate a kind of randomness that benefits living things.

Here, by using a well known and simple mathematical model from mathematical biology, we show the relationships and analogies between chaotic and stochastic behaviours.

4.2.1 Discrete-Time Logistic Model

The discrete-time logistic model was developed for population dynamics. Let’s recall that it was proposed by May in 1976 and that its principal purpose was to question ecologists and specialists in population dynamics on the interpretation of erratic dynamics. This article met with great and, moreover, deserved success because the introduction of deterministic chaos into population dynamics corresponded to a veritable epistemological split. We are going to present some details about this model and use it to examine the transition chaos-randomness.

Let’s write as x(t) the size of a population at time t, and consider a time interval (t, t+1). If the increase in this population x(t+1) – x(t) is proportional to its size x(t), the model is linear (i.e., x(t+1) − x(t) = a x(t), where a is a constant). In the case where proportionality is not proven, the model is non-linear. This is the case for the discrete-time logistic model which is expressed as: x(t+1) − x(t) = a x(t)(K − x(t)). We can standardise the model, and by writing x(t+1) in terms of x(t) and changing the scale for x where K = 1, the model is then written: x(t+1) = r x(t)(1 − x(t)) where r is a positive constant whose value regulates the model’s behaviour. Depending on the values of the parameter r, we can observe different behaviours in this model. They are summarized in Fig. 4.2.

Fig. 4.2

Discrete-time logistic model x _t+1 = r x _t (1 − x _t ). The appearance of the graph x(t) for t = 0, …, n, …, 20, depending on various values for the constant r, changes notably

When a mathematical object – in this case, differential equations or, like here, recurrent equations – changes behaviour (especially asymptotic behaviour) based on variations in one or more parameters of this equation, we speak of bifurcation; for example, the solution to this equation changes, as shown in Fig. 4.2: from a plateau, or a horizontal asymptote for r ≤ 3 to sustained oscillations for r > 3 with, first of all, a simple period, and then a period a little more complicated; oscillating signals that repeat themselves, but with a longer period, and, in each period, two and then four oscillations of different amplitudes. Lastly – and rapidly – we see a multiplication of the intermediary states corresponding to what we call chaos. For r > 4, we record an exponential “implosion”. To study the nature of the solutions, we can draw what is known as a bifurcation diagram (cf. Fig. 4.3). Analytically, it is not always easy to calculate the precise values of the parameters for which we observe a bifurcation; it is for this reason that we often use numerical calculations to obtain approximations.

Fig. 4.3

Bifurcation diagram of the discrete-time logistic model. This diagram can easily be obtained numerically through recurrence: r _t+1 = r _t + h; x _t+1 = r _t x _t (1 − x _t ), with 0 < x ₀ < 1 and r ₀ = 2, h being “small” (on the order of 10⁻⁶). It provides a precise idea of the points where changes occur in the regime of a dynamical system

This simple model has become a reference in the study of chaotic systems. Coming out of population biology, it has led biologists to question certain dynamics observed in nature that have strong oscillations and that were thought to result from a simple, monotonous function subject to random environmental factors. In fact, irregular oscillations can also come from chaotic regimes stemming from the dynamics of these populations. This model was published by May in 1976. We had to wait some 20 years to have the experimental confirmation for a similar model (Costantino et al., 1997). Since then, we have continued to search for other examples of such dynamics, but we have found only a few. As has already been mentioned, however, searching for them directly in the data is not the best way.

4.2.2 Analysis of the Simultaneous Dynamics of Two Populations

We can use this basic model as a thinking aid to approach more complex situations; for example, to analyse the simultaneous dynamics of two populations. Let’s consider the following system first (cf. Fig. 4.4):

$$x_{n + 1} = r\,x_n (1 - x_n )$$

$$y_{n + 1} = r\,y_n (1 - y_n )$$

Fig. 4.4

Comparison of two processes, the first chaotic (left) and the second random (right). The chaotic process comes from the discrete-time logistic model (with r = 3.98, x ₀ = 0.2 and y ₀ = 0.1). The random process is created by the procedure ALEA() using Excel software, which furnishes a reasonably uniform distribution, as shown in the bottom right figure (1000 numbers created for x and y); x and y are not correlated and the internal autocorrelation of the series of values for x and y are almost nil. The chaotic process provides a U distribution with an accumulation at the edges; x and y are not correlated. On the other hand, there is obviously a strong internal autocorrelation to the x and y series. A chaotic process this simple does not result in randomness, but something that begins to resemble it. A more complex dynamic system would probably better simulate it

We recognise the discrete-time logistic model for two simultaneous and independent populations; by using only the property of sensitivity to initial conditions for this type of equation, we create pairs of values (x _n , y _n ) that are widely distributed over the unitary square.

In the case of a single variable, the space where a structure appears is the plane (x _n+1, x _n ); we call this the “phase space”. In the case of two variables, the corresponding phase space is four dimensional: (x _n+1, x _n , y _n+1, y _n ). Thus, when we look at the successive points in the space (x _n , y _n ), it is in fact a projection of the phase space⁴ of this plane.

The simultaneous dynamics of the two independent populations appears messy, as we might expect. And now if we introduce an interaction, a pairing of two populations, what happens?

4.2.3 From the Erratic to the Regular: The Effect of Pairing

In the case of the discrete-time logistic model, we can see that, based on the value of parameter r, we can go from a uniform trajectory to an oscillating trajectory, and then to a chaotic trajectory. But what happens when two chaotic regimes are paired?

Let us now consider the system:

$$x_{n + 1} = r\,x_n (1 - x_n ) - \alpha x_n y_n$$

$$y_{n + 1} = r\,y_n (1 - y_n ) - \alpha x_n y_n .$$

This is an extension of the preceding model with two competing populations. This interaction is represented by the term: α x _n y _n . By playing with the values for r and α, we can create different figures in the plane (x _n , y _n ) and different chronicles; that is to say, graphs of x and y as a function of n; however, if we progressively increase the value α, more structured forms appear, up to the point of forming a straight line. The pairing introduced by α “destroys” the “almost random” structure.

We find ourselves faced with a situation a little different from that mentioned above. Indeed, the near random structure that we observe depends first of all on the choice of the projection plane. In other planes, in particular (x _n , x _n+1) and (y _n , y _n+1), we would have observed parabolic organisations characteristic of discrete-time logistic models. In fact, it is the combination of these two simple structures that results in this distribution, and not the fact that a structure exists in a larger space (even if it does exist). Finally, we note that by pairing these two chaotic dynamics, strange forms appear and we arrive at a linear relationship between x and y. Pairing seems to introduce order into the behaviour of the system and its non-linearity creates diversity (Fig. 4.5).

Fig. 4.5

Discrete-time competition model: the two images at the top, not paired, show an “erratic” distribution. Then, the pairing of two chaotic systems creates an apparent order (Pavé and Schmidt-Lainé, 2003). We only present a few figures here. In fact, the dynamics of the system are more richly varied. The non-linear can also create diversity, like randomness. Nevertheless, greater pairing, measured by the value of parameter α, rapidly synchronizes the two variables and the relationship in space (x _n , y _n ) becomes linear

Yet, this observation cannot be made for values of r that create a very erratic regime (e.g., for values close to four, like that used to create Fig. 4.4 where r = 3.98). There is a limit at which order does not appear through pairing. For all practical purposes, it is not necessary to have chaos that is too unstructured.

Using this formulation, it is possible to study situations representing other types of interactions; for example, a predator with its prey. Yet, we need to be wary of piling up pretty simulations that add only little to the biological or ecological reality.

4.2.4 From Chaos to Randomness

To analyse the properties of generators of chaotic distributions and compare them with those that generate random variables, we can study how the theorems established for random variables can be verified for chaotic variables. We can take the example of linear combinations and examine how these combinations tend or do not tend towards normal distributions (central limit theorem). Figure 4.6 illustrates a numerical experiment on the sums of chaotic or random variables. We see such a tendency in both cases, but it is a little less rapid in the chaotic case than for random variables.

Fig. 4.6

Comparison between distributions of linear combinations of chaotic and random variables. Chaotic dynamics are created by the formula: x _n+1 = r x _n (1 – x _n ), where r = 3.98 and with different initial conditions. The chosen values correspond to a chaotic domain over [0,1]. Like before, the random dynamics were obtained thanks to the Random Number generator software ALEA in Excel. The first line shows the distributions obtained for single variables: a chaotic variable is asymmetrical and U-shaped and nearly uniform for random variables. The other results correspond to weighted sums (to remain in the [0,1] domain), chaotic (left column) and random variables (right column). Convergence towards Gaussian law was expected in the random case, but not – at least not as quickly –in the chaotic case

Biologically, we can interpret this approach in the following manner: we analyze the dynamics of independent populations with chaotic regimes, and look at the sums of the densities according to time. We thus added two, then four and finally eight variables together. These sums were then weighted in a way that kept the values between 0 and 1.

We see from this numerical experiment that the difference between randomness and chaos holds. In the end, the common characteristic is a priori the unpredictability of a result. Yet, as we have seen, the word randomness also has many other meanings; in particular, an effect resulting from multiple causes that are little or not known. On the other hand, chaos has the advantage of being created by a mechanistic model that can be interpreted in physical, chemical or even social terms. This is also the case for a roulette wheel in a casino that conforms to mechanical laws.

In both cases, statistics permit us to study the results of these processes. Typical probabilistic approaches do not model the mechanisms creating randomness. They only make hypotheses and construct models on the results, processing them elegantly. In the Mendelian model on hereditary transmission presented in Section 4.1, for example, we do not make a hypothesis based on the underlying biological and biochemical mechanisms, and we model them even less. On the other hand, we can construct a model of the results from simple probabilistic hypotheses: everything takes place as though we had drawn from the lot of gametes and recombinations that do not depend on the nature of the genes carried by the gametes.

Statistical analysis can provide us information on the transition between chaos and randomness. In this way, Figs. 4.6 and 4.7 show us how a chaotic system evolves towards displaying properties close to those of a stochastic system.

Fig. 4.7

Analysis of the correlations and autocorrelations between chaotic series and sums of chaotic series created by the discrete-time logistic model. The left column shows point clouds obtained between chaotic dynamics and sums of chaotic dynamics with different initial conditions. The point clouds are only slightly slanted. We see a structure appear that is close to what would result in a 2-dimensional Gaussian distribution. The correlation coefficients are all lower, in absolute values, than 0.15. The right column shows the autocorrelations between successive values for chaotic dynamics. The structure of the point cloud fades when the number of terms of the sum increases. Although the correlation is not linear, at least in the first three cases, it seems to become linear (the point cloud has a tendency towards an elliptical appearance)

Thus, in this example, we show that purely deterministic, non-linear dynamical systems functioning in chaotic regimes and associated with a simple linear combination can exhibit quasi-stochastic properties. It is possible to think that “biological roulettes” have analogous modes of functioning.

Figure 4.7 shows another aspect of the appearance of stochastic-type properties in the sums of chaotic series: the structuring of series created independently in a Gaussian, two-dimensional point cloud and the disappearance of the autocorrelation between successive values of the sums of chaotic variables.

4.3 The Continuous-Time Logistic Model and the Evolution of Biodiversity

Several authors have attempted to create models using paleontological data from the database created by J. John Sepkoski Jr.⁵ The first, by Michael Benton (1995), proposed making a change to the exponential model whose differential form is written dN/dt = α N (where N represents the number of families and a is a real positive constant). The implicit hypothesis, formulated in everyday language, is that the speed at which the number of families increases is proportional to this number. In 1996, Vincent Courtillot and Yves Gaudemer used the logistic model to represent data from over the last 500 million years (the beginning of the Ordovician). They focused on the ascending phases, knowing that the descending phases have been widely studied elsewhere. The model is written: dN/dt = αN (K–N) where K represents the asymptote; that is to say, the number of families after a sufficient amount of time. If N ₀ < K, which is the case here, we see the famous sigmoidal curve where K represents the maximum number of families. Then, these authors fit the model for the different periods: (1) from the beginning of the Ordovician to the start of the Permian, (2) the ascendant phase of the Triassic, (3) the Jurassic-Cretaceous, and (4) the Tertiary-Quaternary (cf. Fig. 4.8). We can also represent the first data with a logistic model (0).

Fig. 4.8

Data from http://Fig. 1.1 (but restricted to marine biodiversity), and the first models representing these data: the exponential model (Benton, 1995). The chained piecewise logistic model was proposed by Pavé et al., in 2002. Previously, Courtillot and Gaudemer (1996) used different logistic models representing only the ascending parts of the dynamics. Recently, Alroy et al. (2008) obtained new estimations of data at the genus level. We have represented the corresponding points and dynamics, after corrections to scale permitting them to be included in this graph. We have adjusted the linear and exponential models to these data. The second model, represented here, adjusted better than the linear model. In fact, we will have to have the validity of these new estimations confirmed before revising our previous conclusions based on Sepkoski’s data. In any case, biodiversity is, on average, always growing exponentially

We have adopted the same point of view in continuing their analysis, particularly by proposing several interpretations of the logistic model in this context. Thus, the parameterisation, known as “r, K”, typically adopted in ecology can be used: dN/dt= r N (1–N/K). Parameter r represents the intrinsic rate of variation in the biodiversity; parameter K represents the environmental possibilities in terms of ecological niches.

Moreover, to keep the number of parameters to be estimated to a minimum, we have constructed two chained⁶ models, an exponential model and a logistic model, taking into account the descending phases. The exactness of the parameters is also evaluated to permit proper comparisons.

It seems that the values for parameter r can be considered identical for periods (1), (2) and (4), but one of them seems significantly smaller for period (3). Parameter K, representing the plateau, is significantly higher for periods (3) and (4) than for the previous periods. The plateau is not reached during the third period; the Cretaceous-Tertiary (K-T) crisis interrupted this process, but it rapidly started up again during the Tertiary-Quaternary Period with the same value for r as for periods (1) and (2).

Another way of formulating this model permits the number of ecological niches to be explicitly entered as a state variable. We show, then, how it is possible to find the previous formulations again easily. The advantage of this formulation (proposed in Pavé, 1993, 1994) – all in all, rather commonplace – is that it permits other things to be developed, in particular new models and a better interpretation of the underlying mechanisms:

$$\begin{array}{l} \dfrac{{dN}}{{dt}} = \alpha \,N\,S \\ \dfrac{{dS}}{{dt}} = - \,\alpha \,N\,S \\ \end{array}$$

where S represents the number of ecological niches free at time t. This is really another formulation of the logistic model.⁷ Indeed, we have, dS/dt = – dN/dt so, S–S ₀ = – (N – N ₀) and S = S ₀ + N ₀ – N, so that we can write:

$$\frac{{dN}}{{dt}} = \alpha \,N\,(K - N)$$

with K = S ₀ + N ₀.

S ₀ represents the number of niches initially free. N ₀ represents the initial number of families that occupy the same number N ₀ of niches (the units are the same for the number of niches and the number of families). The total number of possible niches, and of families, is K.

We can even introduce a term describing the “spontaneous” dying out of families; we’re back to the logistic model if we assume that when a certain number of families die out, an equal number of ecological niches are freed up. The model is then written:

$$\begin{array}{l} \dfrac{{dN}}{{dt}} = \alpha \,N\,S - \beta \,N \\ \dfrac{{dS}}{{dt}} = - \alpha \,N\,S + \beta \,N \\ \end{array}$$

We still have dS/dt = – dN/dt and S = S ₀ + N ₀ – N then dN/dt = α N (S ₀ + N ₀ – N) – β N or even dN/dt = α N (K – N) now with K = [α(S ₀ + N ₀) – β]/α; this model is more general and especially permits us to represent, according to the values of the parameters, the ascending and descending phases, but the simplified expression remains the same.

Thus the increase in biodiversity can be interpreted as a consequence of the:

• creation of ecological niches after environmental disturbances or even by living things themselves;

• appearance of new genetic mechanisms; and

• emergence of new ecological relationships.

On the one hand, we can reasonably suppose that environmental disturbances destroy as many, if not more, ecological niches as they create, and that, even in imagining a process of restoration, these disturbances are produced in a “regular” fashion and are not enough to explain explosions in biodiversity. On the other hand, genetic mechanisms (at the molecular level) can affect the speed of diversification – for all practical purposes, the rate of diversification r. On an ecological level, we can suggest that relationships evolve, at least in part, from competition to cooperation by way of co-existence. New relationships are progressively and successively established at all levels in the organisation of living things right up to the ecosystem. We might think that this would lead several species, genera and families to occupy the same ecological niche, and thus to an apparent multiplication of these niches. We suggest then that mechanisms of coexistence and cooperation, from an ecological standpoint, became established during the “recent” Tertiary-Quaternary Period (r “normal” and K elevated), perhaps, and to a lesser extent as early as the Jurassic-Cretaceous (r weak and K elevated), but were only slightly present in the preceding periods (r “normal” and K weak) (Fig. 4.9).

Fig. 4.9

Chained logistic model, distance between data and the model with parameterisation (r, K), variations in the parameters r and K of the model

Thus, the evolutionary schema would be as follows: the emergence of new ecological relationships permitting new taxa to become stabilised over a period sufficiently long for them to be visible in the fossil record. The establishment of these new relationships (coexistence and cooperation) would explain the apparent increase in the number of ecological niches during the last two periods (K elevated). The low α value during the Jurassic-Cretaceous Period could then be interpreted as a result of a succession of environmental disturbances leading to “minor” extinctions blunting growth. It would seem then that the mechanisms of genetic diversification were all in place as early as the Cambrian, if we assume that α represents the constant for the speed of the diversification. In fact, a correlation exists between the two parameters, α and K, well known to those who make estimations of non-linear models that we must take into account to counterbalance this latter conclusion.

Finally, as we have already pointed out, by analyzing the most recent, most numerous, and especially the most precise data (e.g., the number of genera instead of the number of families; Rohde and Muller, 2005), the oscillations seem to be significant (Fig. 4.10). This analysis is founded on a classic technique: the general tendency is modelled through a 3rd degree polynomial, and then the remainders (differences compared to the model) are calculated, and the oscillating components are found through a Fourier analysis. First, we find a component with a period of some 62 million years, and then another of some 140 million years. Like others before us, we suspected the presence of such oscillations, but the data used did not permit us to show them. The starting point for our study was even the research and modelling of such oscillations. All that remains is to provide an explanation: Rohde and Muller lean towards an explanation coming from astronomy (e.g., meteorites periodically falling to the Earth); in their commentaries on the results, Kirchner and Weill encourage also looking for a biological and ecological explanation (Kirchner and Weill, 2000, 2005). This is also what we propose, as we indicated in Section 4.4 of Chapter 2 devoted to the dynamics of biodiversity. This historic survey is “to be continued” in the next episodes.

Fig. 4.10

Results of analysing the oscillating components in the dynamics of biodiversity on a geological scale. Adjustments to these components after the period 62 MY and 140 MY are shown in the lower part of the graph (based on Rohde and Muller, 2005)

4.4 Towards a General Schema for Modelling Living Systems and Their Diversities

It seems important, based on what we have just shown, to examine how the modeller can take part in furthering the understanding of processes. We can use a highly schematic diagram to help us (Fig. 4.11).

Fig. 4.11

Diagram of the principle for modelling living systems and the processes considered to be important in these systems. A living system is a cell, an organism, a population or an ecosystem. The model can be a general model (very rare), or concern certain functions

Approaches I and II are typical and efficacious. Approach I shows the classical representations: y = f(x) + e where f is a mono or multi-dimensional analytical function known explicitly or implicitly (i.e., as an ordinary differential equation or with partial derivatives); x is an independent variable, also mono- or multi-dimensional (often time and/or one or several dimensions in geometric space); and e is a random “error term”. Approach II shows the probability of occurrence of an event or a set of events based on one or more independent variables; for example, the exponential law, P(T < t) = 1–e ^−at, provides the probability of occurrence for an event during an interval of time [0, t] if the distribution is uniform and stationary. Still concerning this same process, Poisson’s Law is the law for the number of events for a given interval of time; however, the mechanism that creates the event is not represented. It is thus a purely phenomenological approach, like in the genetic examples provided in Section 4.1.

On the other hand, approach III is less frequent. It concerns representing, and obviously analysing, the processes that generate chaos or randomness. We find such examples in population dynamics, the most simple of which is the discrete-time logistic model that we have used as an illustration. The advantage of this approach is, obviously, understanding this category of processes, but, also, through the model, analysing the consequences of modifications, optimisations, etc., with a view towards practical applications; for example, increasing the speed of diversifications or, on the contrary, decreasing it. Lastly, a final comment: we can imagine representing a living system through a system of differential equations of major dimensions. We know that, for ordinary differential equations, we can record chaotic behaviours for dimensions greater or equal to three. Theoretically, such behaviours could be very frequent in an organism, but, as we have seen, this is not the case. They would probably be the source of functional problems. We can, then, imagine that processes of regulation, including retroactions, were selected to avoid erratic regimes harmful to the proper functioning of the “machine-organism”. As we have seen, this is not the case for certain processes (e.g., chromosome reshuffling, the immune systems of vertebrates) or at other levels of organisation where randomness plays a major role.