# Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

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## Abstract

**:**

## 1. Purpose of This Paper

## 2. During the Last 3.5 Billion Years, Life Forms Increased as in a (Lognormal) Stochastic Process

^{9}) years ago, i.e., the accepted time of the origin of life on Earth. If the origin of life started earlier than that, for example 3.8 billion years ago, the following equations would remain the same and their numerical values would only be slightly changed. On the vertical axis is the number of species living on Earth at time $t$, denoted $L\left(t\right)$ and standing for “life at time $t$”. We do not know this “function of the time” in detail, and so it must be regarded as a random function, or stochastic process $L\left(t\right)$. This paper adopts the convention that capital letters represent random variables, i.e., stochastic processes if they depend on the time, while lower-case letters signify ordinary variables or functions.

## 3. Mean Value of the Lognormal Process L(t)

## 4. L(t) Initial Conditions at ts

## 5. L(t) Final Conditions at te > ts

## 6. Important Special Cases of m(t)

- (1)
- The particular case of Equation (1) when the mean value $m\left(t\right)$ is given by the generic exponential:$${m}_{\mathrm{GBM}}\left(t\right)={N}_{0}\text{\hspace{0.17em}}{e}^{{\mu}_{GBM}\text{\hspace{0.05em}}t}=\mathrm{or},\text{}\mathrm{alternatively},=A\text{\hspace{0.17em}}{e}^{B\text{\hspace{0.17em}}t}$$
**grew exponentially**(Malthusian growth). Upon equating the two right-hand-sides of Equations (4) and (22) (with t replaced by (t-ts)), we find:$${e}^{{M}_{\mathrm{GBM}}\left(t\right)}\text{\hspace{0.17em}}{e}^{\frac{{\sigma}_{GBM}^{2}}{2}\text{\hspace{0.17em}}\left(t-ts\right)}={N}_{0}\text{\hspace{0.17em}}{e}^{{\mu}_{\mathrm{GBM}}\text{\hspace{0.05em}}\left(t-ts\right)}.$$Solving this equation for ${M}_{\mathrm{GBM}}\left(t\right)$ yields:$${M}_{\mathrm{GBM}}\left(t\right)=\mathrm{ln}{N}_{0}+\left({\mu}_{\mathrm{GBM}}-\frac{{\sigma}_{\mathrm{GBM}}^{2}}{2}\right)\text{\hspace{0.17em}}\left(t-ts\right)\text{\hspace{0.17em}}.$$This is (with $ts=0$) the mean value at the exponent of the well-known GBM pdf, i.e.,:$$\mathrm{GBM}\left(t\right)\_pdf\left(n;{N}_{0},\mu ,\sigma ,t\right)=\frac{{e}^{-\frac{{\left[\mathrm{ln}\left(n\right)-\left(\mathrm{ln}{N}_{0}+\left(\mu -\frac{{\sigma}^{2}}{2}\right)\text{\hspace{0.17em}}t\right)\right]}^{2}}{2\text{\hspace{0.17em}}{\sigma}^{2}\text{\hspace{0.17em}}t}}}{\sqrt{2\pi}\text{\hspace{0.17em}}\sigma \sqrt{t}\text{\hspace{0.17em}}n},\text{}\left(n\ge 0\right).$$This short description of the GBM is concluded as the exponential sub-case of the general lognormal process Equation (2), by warning that GBM is a misleading name, since GBM is a lognormal process and not a Gaussian one, as the Brownian Motion is. - (2)
- As has been mentioned already, another interesting case of the mean value function $m\left(t\right)$ in Equation (1) is when it equals a generic
**polynomial in t starting at ts**, namely (with ${c}_{k}$ being the coefficient of the k-th power of the time t-ts in the polynomial)$${m}_{\mathrm{polynomial}}\left(t\right)={\displaystyle \sum _{k=0}^{\mathrm{polynomial}\_\mathrm{degree}}{c}_{k}\text{\hspace{0.17em}}{\left(t-ts\right)}^{k}.}$$The case where Equation (26) is a second-degree polynomial (i.e., a parabola in $t-ts$) may be used to describe the Mass Extinctions on Earth over the last 3.5 billion years (see [13]). - (3)
- Having so said, the notion of a b-lognormal must also be introduced, for t > b = birth, representing the lifetime of living entities, as single cells, plants, animals, humans, civilizations of humans, or even extra-terrestrial (ET) civilizations (see [12], in particular pages 227–245)$$\mathrm{b}-\mathrm{lognormal}\_\mathrm{pdf}(t;\mu ,\sigma ,b)=\frac{{e}^{-\text{\hspace{0.17em}}\frac{{\left[\mathrm{ln}\left(t-b\right)-\mu \right]}^{2}}{2\text{\hspace{0.17em}}{\sigma}^{2}}}}{\sqrt{2\pi}\text{\hspace{0.17em}}\sigma \text{\hspace{0.17em}}\left(t-b\right)}.$$

## 7. Boundary Conditions when m(t) is a First, Second, or Third Degree Polynomial in the Time (t-ts)

- (1)
**The mean value is a straight line.**This straight line is the line through the two points, $\left(ts,Ns\right)$ and $\left(te,Ne\right)$, that, after a few rearrangements, becomes:$${m}_{\mathrm{straight}\_\mathrm{line}}\left(t\right)=\left(Ne-Ns\right)\frac{t-ts}{te-ts}+Ns.$$- (2)
**The mean value is a parabola**, i.e., a quadratic polynomial in $t-ts$. Then, the equation of such a parabola reads:$${m}_{\mathrm{parabola}}\left(t\right)=\left(Ne-Ns\right)\frac{t-ts}{te-ts}\left[2-\frac{t-ts}{te-ts}\right]+Ns.$$Equation (30) was actually firstly derived by this author in [13] (pp. 299–301), in relation to Mass Extinctions, i.e., it is a decreasing function of time.- (3)
**The mean value is a cubic**. In [13] (pp. 304–307), this author proved, in relation to the Markov-Korotayev model of Evolution, that the**cubic**mean value of the $L\left(t\right)$ lognormal stochastic process is given by the cubic equation in $t-ts$:$${m}_{\mathrm{cubic}}\left(t\right)=\left(Ne-Ns\right)\cdot \frac{\left(t-ts\right)\left[2{\left(t-ts\right)}^{2}-3\left({t}_{\mathrm{Max}}+{t}_{\mathrm{min}}-2\text{\hspace{0.17em}}ts\right)\left(t-ts\right)+6\left({t}_{\mathrm{Max}}-ts\right)\left({t}_{\mathrm{min}}-\text{\hspace{0.17em}}ts\right)\right]}{\left(te-ts\right)\left[2{\left(te-ts\right)}^{2}-3\left({t}_{\mathrm{Max}}+{t}_{\mathrm{min}}-2\text{\hspace{0.17em}}ts\right)\left(te-ts\right)+6\left({t}_{\mathrm{Max}}-ts\right)\left({t}_{\mathrm{min}}-\text{\hspace{0.17em}}ts\right)\right]}+Ns.\text{\hspace{0.17em}}$$

## 8. Peak-Locus Theorem

**upon**the mean value curve (1), is given by the following three equations, specifying the three parameters $\mu \left(p\right)$, $\sigma \left(p\right)$, and $b\left(p\right)$ appearing in Equation (27) as three functions of the peak abscissa, i.e., the independent variable $p$. In other words, we were actually pleased to find out that these three equations may be written directly in terms of ${m}_{L}\left(p\right)$ as follows:

## 9. EvoEntropy(p) as a Measure of Evolution

- (a)
- The constant term$$\frac{1}{4\pi N{s}^{2}}$$$$\frac{1}{4\pi}=0.079577471545948$$
- (b)
- The denominator square term in Equation (44) rapidly approaches zero as ${m}_{L}\left(p\right)$ increases to infinity. In other words, this inverse-square term$$-\frac{1}{4\pi {\left[{m}_{L}\left(p\right)\right]}^{2}}$$
- (c)
- Finally, the dominant, natural logarithmic, term, i.e., that which is the major term in this EvoEntropy Equation (45) for large values of the time $p$.$$\mathrm{ln}\left(\frac{{m}_{L}\left(p\right)}{Ns}\right).$$

## 10. Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential (GBM): This Is Just the Molecular Clock

**is exactly a linear function of the time**$p$, since the first two terms inside the braces in Equation (44) cancel each other out, as we now prove.

**Proof.**

**This is the same linear behaviour in time as the molecular clock**, which is the technique in molecular evolution that uses fossil constraints and rates of molecular change to deduce the time in geological history when two species or other taxa diverged. The molecular data used for such calculations are usually nucleotide sequences for DNA or amino acid sequences for proteins (see [16,17,18]).

## 11. Markov-Korotayev Alternative to Exponential: A Cubic Growth

## 12. EvoEntropy of the Markov-Korotayev Cubic Growth

## 13. Comparing the EvoEntropy of the Markov-Korotayev Cubic Growth, to the Hypothetical (1) Linear and (2) Parabolic Growth

- (1)
- The LINEAR (= straight line) growth, given by the mean value of Equation (29)
- (2)
- The PARABOLIC (= quadratic) growth, given by the mean value of Equation (30).

- (1)
- LINEAR EvoEntropy:$$\mathrm{STRAIGHT}\_\mathrm{LINE}\_\mathrm{EvoEntropy}\left(t\right)=\frac{1}{\mathrm{ln}\left(2\right)}\left\{\frac{1}{4\pi N{s}^{2}}-\frac{1}{4\pi {\left[{m}_{\mathrm{straight}\_\mathrm{line}}\left(t\right)\right]}^{2}}+\mathrm{ln}\left(\frac{{m}_{\mathrm{straight}\_\mathrm{line}}\left(t\right)}{Ns}\right)\right\}.$$
- (2)
- PARABOLIC (quadratic) EvoEntropy:$$\mathrm{PARABOLA}\_\mathrm{EvoEntropy}\left(t\right)=\frac{1}{\mathrm{ln}\left(2\right)}\left\{\frac{1}{4\pi N{s}^{2}}-\frac{1}{4\pi {\left[{m}_{\mathrm{parabola}}\left(t\right)\right]}^{2}}+\mathrm{ln}\left(\frac{{m}_{\mathrm{parabola}}\left(t\right)}{Ns}\right)\right\}.$$
- (3)
- CUBIC (MARKOV-KOROTAYEV) EVOENTROPY, i.e., Equation (56).

## 14. Conclusions

## Supplementary Materials

## Conflicts of Interest

## References

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**Figure 1.**

**Increasing DARWINIAN EVOLUTION as the increasing**

**number of living species on Earth between 3.5 billion years ago and now.**The red solid curve is the mean value of the GBM stochastic process ${L}_{\mathrm{GBM}}\left(t\right)$ given by Equation (22) (with t replaced by (t-ts)), while the blue dot-dot curves above and below the mean value are the two standard deviation upper and lower curves, given by Equations (11) and (12), respectively, with ${m}_{\mathrm{GBM}}\left(t\right)$ given by Equation (22). The “Cambrian Explosion” of life, that on Earth started around 542 million years ago, is evident in the above plot just before the value of ′0.5 billion years in time, where all three curves “start leaving the time axis and climbing up”. Notice also that the starting value of living species 3.5 billion years ago is

**one**by definition, but it “looks like” zero in this plot since the vertical scale (which is the true scale here, not a log scale) does not show it. Notice finally that nowadays (i.e., at time $t=0$ ) the two standard deviation curves have exactly the same distance from the middle mean value curve, i.e., 30 million living species more or less the mean value of 50 million species. These are assumed values that we used just to exemplify the GBM mathematics: biologists might assume other numeric values.

**Figure 2.**

**The Darwinian Exponential is used as the geometric locus of the peaks of b-lognormals for the GBM case.**Each b-lognormal is a lognormal starting at a time b (birth time) and represents a different

**species**that originated at time b of the Darwinian Evolution. This is

**cladistics**, as seen from the perspective of the Evo-SETI model. It is evident that, when the generic “running b-lognormal” moves to the right, its peak becomes higher and narrower, since the area under the b-lognormal always equals one. Then, the (Shannon)

**entropy**of the running b-lognormal is the

**degree of evolution**reached by the corresponding

**species**(or living being, or a civilization, or an ET civilization) in the course of Evolution (see, for instance, [14,15,16,17,18,19]).

**Figure 3.**

**EvoEntropy (in bits per individual) of the latest species appeared on Earth during the last 3.5 billion years if the mean value is an increasing exponential, i.e., if our lognormal stochastic process is a GBM.**This straight line shows that a Man (nowadays) is 25.575 bits more evolved than the first form of life (RNA) 3.5 billion years ago.

**Figure 4.**

**According to Markov and Korotayev, during the Phanerozoic, the biodiversity shows a steady, but not monotonic, increase from near zero to several thousands of genera**.

**Figure 5.**

**The Cubic mean value curve (thick red solid curve) ± the two standard deviation curves (thin solid blue and green curve, respectively) provide more mathematical information than Figure 4.**One is now able to view the two standard deviation curves of the lognormal stochastic process, Equations (11) and (12), that are completely missing in the Markov-Korotayev theory and in their plot shown in Figure 4. This author claims that his Cubic mathematical theory of the Lognormal stochastic process $L\left(t\right)$ is a more profound mathematization than the Markov-Korotayev theory of Evolution, since it is stochastic, rather than simply deterministic.

**Figure 6.**

**The EvoEntropy Equation (44) of the Markov-Korotayev Cubic mean value Equation (31) of our lognormal stochastic process**$L\left(t\right)$

**applies to the growing number of Genera during the Phanerozoic.**Starting with the left part of the curve, one immediately notices that, in a few million years around the Cambrian Explosion of 542 million years ago, the EvoEntropy had an

**almost vertical growth**, from the initial value of zero, to the value of approximately 10 bits per individual. These were the few million years when the

**bilateral symmetry**became the dominant trait of all primitive creatures inhabiting the Earth during the Cambrian Explosion. Following this, for the next 300 million years, the EvoEntropy did not significantly change. This represents a period when bilaterally-symmetric living creatures, e.g., reptiles, birds, and very early mammals, etc., underwent little or no change in their body structure (roughly up to 310 million years ago). Subsequently, after the “mother” of all mass extinctions at the end of the Paleozoic (about 250 million years ago), the EvoEntropy started growing again in mammals. Today, according to the Markov-Korotayev model, the EvoEntropy is about 12.074 bits/individual for humans, i.e., much less than the 25.575 bits/individual predicted by the GBM exponential growth shown in Figure 3. Therefore, the question is: which model is correct?

**Figure 7.**

**Comparing the mean value**${m}_{L}\left(t\right)$

**(A) and the**$\mathrm{EvoEntropy}\left(t\right)$

**(B) in the event of growth with the CUBIC mean value of Equation (31) (blue solid curve), with the LINEAR Equation (29) (dash-dash orange curve), or with the PARABOLIC Equation (30) (dash-dot red curve).**It can be seen that, for all these three curves, starting with the left part of the curve, in a few million years around the Cambrian Explosion of 542 million years ago, the EvoEntropy had an almost vertical growth from the initial value of zero to the value of approximately 10 bits per individual. Again, as is seen in Figure 6, these were the few million years where the bilateral symmetry became the dominant trait of all primitive creatures inhabiting the Earth during the Cambrian Explosion.

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Maccone, C.
Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI. *Life* **2017**, *7*, 18.
https://doi.org/10.3390/life7020018

**AMA Style**

Maccone C.
Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI. *Life*. 2017; 7(2):18.
https://doi.org/10.3390/life7020018

**Chicago/Turabian Style**

Maccone, Claudio.
2017. "Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI" *Life* 7, no. 2: 18.
https://doi.org/10.3390/life7020018