# Towards a Mathematical Description of Biodiversity Evolution

## Abstract

**:**

## 1. Introduction

## 2. Second Order Formalism in Population Dynamics and Its Extension to Biodiversity

#### 2.1. Motivating Basic Ecological Forces

#### 2.2. Simple Solutions

## 3. Comparison with Existing Fossil Records

**Figure 1.**Best fits to the long-term recovery after the Triassic-Jurassic (left red segment) and Cretaceous-Paleogene (right red segment) extinctions. The black curve is the data by Sepkoski [22], which suggests a more dramatic growth for the latter and overall for ${N}_{G}$. The model performs well within a quadratic evolution steaming from the solutions of $({d}^{2}{N}_{G}/d{t}^{2})=constant$.

^{−1}) in the late curve are present.

**Figure 2.**A comparison between the Green function response Equation (5) (blue curve) with the time-resolved data corresponding to the Permo-Triassic extinction event analyzed by Burgess, Bowring and Shen [23] (black curve). The origin of the time axis has been set to the end of the extinction interval given by these authors. See the text for details.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Thomas, I. Greek Mathematical Works: Thales to Euclid, 1st ed.; Harvard University Press: Cambridge, MA, USA, 1939; Volume I. [Google Scholar]
- Turchin, P. Does population ecology have general laws? Oikos
**2001**, 94, 17–26. [Google Scholar] - Lotka, A.J. Elements of Physical Biology, 1st ed.Williams & Wilkins Company: Baltimore, MD, USA, 1925; pp. 1–495. [Google Scholar]
- Volterra, V. Fluctuations in the abundance of a species considered mathematically. Nature
**1926**, 118, 558–600. [Google Scholar] - Kingsland, S.E. Modeling Nature: Episodes in the History of Population Ecology, 1st ed.; University of Chicago Press: Chicago, IL, USA, 1985. [Google Scholar]
- Clark, G.P. The second derivative in population modeling. Ecology
**1971**, 52, 606–613. [Google Scholar] - Ginzburg, L.R. The theory of population dynamics I: Back to first principles. J. Theor. Biol.
**1986**, 122, 385–399. [Google Scholar] - Yee, J. A non-linear second-order population-model. Theor. Pop. Biol.
**1980**, 18, 175–191. [Google Scholar] - Guinzburg, L.R.; Colyvan, M. Ecological Orbits: How Planets Move and Populations Grow, 1st ed.; Oxford University Press: Oxford, UK, 2004. [Google Scholar]
- Barret, M.; Clatterbuck, H.; Goldsby, M.; Helgson, C.; McLoone, B.; Pearce, T.; Sober, E.; Stern, R.; Weinberger, N. Puzzles for ZFEL, McShea and Brandon’s zero force evolutionary law. Biol. Phil.
**2012**, 27, 723–725. [Google Scholar] - Turchin, P. Complex. Population Dynamics: A Theoretical/Empirical Synthesis, 1st ed.; Princeton University Press: Princeton, NJ, USA, 2003. [Google Scholar]
- Mc Shea, D.W.; Turchin, P.N. Biology’s First Law: The Tendency for Diversity and Complexity to Increase in Evolutionary Systems, 1st ed.; University of Chicago Press: Chicago, IL, USA, 2010; p. 184. [Google Scholar]
- Goldstein, H.; Poole, C.P., Jr.; Safko, J.L. Classical Mechanics, 3rd ed.; Addison Wesley & Co.: Reading, NY, USA, 2001. [Google Scholar]
- Kleppner, D.; Kolenkow, R. An Introduction to Mechanics, 1st ed.; Mc Graw-Hill: Columbus, OH, USA, 1973. [Google Scholar]
- Okasha, S. Does diversity always grow? Nature
**2010**, 466. [Google Scholar] [CrossRef] - Benton, M.J. Diversification and extinction in the history of life. Science
**1995**, 268, 52–58. [Google Scholar] - Sepkoski, J.J., Jr. Patterns of phanerozoic extinction: A perspective from global data bases. In Global Events and Event Stratigraphy, 1st ed.; Walliser, O.H., Ed.; Springer: Berlin, Germany, 1996; pp. 35–52. [Google Scholar]
- Alroy, J.; Aberhan, M.; Bottjer, D.J.; Foote, M.; Fursich, F.T.; Harries, P.J.; Hendy, A.J.W.; Holland, S.M.; Ivany, L.C.; Kiessling, W.; et al. Phanerozoic trends in the global diversity of marine invertebrates. Science
**2008**, 321, 97–100. [Google Scholar] - Raup, D.M. Taxonomic diversity during the phanerozoic. Science
**1972**, 177, 1065–1071. [Google Scholar] [CrossRef] - Sepkoski, J.J., Jr. A kinetic model of phanerozoic taxonomic diversity. III. Post-paleozoic families and mass extinctions. Paleobiology
**1984**, 10, 246–267. [Google Scholar] - Alroy, J.; Marshall, C.R.; Bambach, R.K.; Bezusko, K.; Foote, M.; Fürsich, F.T.; Hansen, T.A.; Holland, S.M.; Ivany, L.C.; Jablonski, D.; et al. Effects of sampling standardization on estimates of phanerozoic marine diversification. PNAS
**2001**, 98, 6261–6266. [Google Scholar] - Sepkoski, J.J. A compendium of fossil marine animal genera. Bull. Amer. Paleontol.
**2002**, 363, 1–560. [Google Scholar] - Burgess, S.D; Bowring, S.; Shen, S.-Z. High-precision timeline for earth’s most severe extinction. PNAS
**2014**, 111, 3316–3321. [Google Scholar] - Shen, S.-Z.; Crowley, J.L.; Wang, Y.; Bowring, S.A.; Erwin, D.H.; Sadler, P.M.; Cao, C.Q.; Rothman, D.H.; Henderson, C.M.; Ramezani, J.; et al. Calibrating the end-Permian mass extinction. Science
**2011**, 334, 1367–1372. [Google Scholar] - Wang, Y.; Sadler, P.M.; Shen, S.-Z.; Erwin, D.H.; Zhang, Y.; Wang, X.-D.; Wang, W.; Crowley, J.L.; Henderson, C.M. Quantifying the process and abruptness of the end-Permian mass extinction. Paleobiology
**2014**, 40, 113–129. [Google Scholar] - Chen, Z.Q.; Benton, M.J. The timing and pattern of biotic recovery following the end-Permian mass extinction. Nat. Geosci.
**2012**, 5, 375–383. [Google Scholar] - D’Hondt, S. Consequences of the Cretaceous/Paleogene mass extinction for marine ecosystems. Annu. Rev. Ecol. Evol. Syst.
**2005**, 36, 295–317. [Google Scholar] - Ruhl, M.; Deenen, M.; Abels, H.A.; Bonis, N.R. Astronomical constraints on the duration of the early Jurassic Hettangian stage and recovery rates following the end-Triassic mass extinction (St Audrie’s Bay/East Quantoxhead, UK). Earth Planet. Sci Lett.
**2005**, 295, 262–276. [Google Scholar] - Erwin, D.H. Temporal acuity and the rate and dynamics of mass extinctions. PNAS
**2014**, 111, 3203–3204. [Google Scholar] - Rohde, R.A.; Muller, R.A. Cycles in fossil diversity. Nature
**2005**, 434, 208–210. [Google Scholar] - Melott, A.L.; Bambach, R.K. A ubiquitous 62-Myr periodic fluctuation superimposed on general trends in fossil biodiversity: I, Documentation. Paleobiology
**2011**, 37, 92–112. [Google Scholar] - Kirchner, J.W.; Weil, A. Delayed biological recovery from extinctions throughout the fossil record. Nature
**2000**, 404, 177–180. [Google Scholar] - Alvarez, L.W.; Alvarez, W.; Asaro, F.; Michel, H.V. Extraterrestrial cause for the retaceous—Tertiary extinction. Science
**1980**, 208, 1095–1108. [Google Scholar] - Tollmeier, F.; Geisel, T.; Nagler, J. Possible origin of stagnation and variability of earth’ biodiversity. Phys. Rev. Lett.
**2014**, 112, 228101–228104. [Google Scholar]

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**MDPI and ACS Style**

Horvath, J.E.
Towards a Mathematical Description of Biodiversity Evolution. *Challenges* **2014**, *5*, 324-333.
https://doi.org/10.3390/challe5020324

**AMA Style**

Horvath JE.
Towards a Mathematical Description of Biodiversity Evolution. *Challenges*. 2014; 5(2):324-333.
https://doi.org/10.3390/challe5020324

**Chicago/Turabian Style**

Horvath, Jorge E.
2014. "Towards a Mathematical Description of Biodiversity Evolution" *Challenges* 5, no. 2: 324-333.
https://doi.org/10.3390/challe5020324