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Bio demographic aspects of population entropy in quantifying population heterogeneity and its consequences for population fitness and species adaptation
1  Department of Environmental Conservation and Management, Faculty of Pure and Applied Sciences, Open University of Cyprus

Abstract: Entropy is a well-established measure of population variability and already used in contingency in life table analysis. Such an entropy, denoted H, is a measure of heterogeneity of the distribution of deaths in a cohort and consist of a touchstone to compare life strategies in different populations or species. Since environmental change affects directly life history traits of populations, entropy as crude demographic parameter may be used to quantitative such trends. Particularly for poikilotherms, entropy may serve as highly quantitative predictor of species net reproductive success under optimum and/or non favorable biotic conditions for growth and development. Nevertheless, entropy has been also used as a more general dynamic measure of species fitness and adaptation to variable ecological conditions. In principle, such demographic-population entropy is an analogue of the Gibbs–Boltzmann entropy in statistical mechanics, H=-Σpilnpi. Here, pi represents the probability density function of the age of reproducing individuals and therefore maximization of entropy is equivalent to maximization of the uncertainty of age reproduction. In such a context, population entropy consist of a dynamic measure and maximization of H under various constrains yield to different distributions of reproduction and survivorship. Moreover, considering that demographic dynamics are formally equivalent to the dynamics of a Markov chain, demographic entropy can be further used under an expression of the classical Leslie model by means of stationary Markov chains to estimate the convergence rate of population transitions to stable age distributions and demographic equilibriums. Populations may differ by terms of robustness captured by evolutionary entropy: the rate at which populations return to demographic equilibrium after a certain perturbation. Hence, under the hypothesis that resource abundance is unlimited and that the only factor affecting population dynamics are inherent properties of species, population entropy may be used to predict the selective advantage among different populations according to the entropic principle. From an applied population-ecological standpoint, considering that that entropy may differ among genera, species and populations, the capacity of each organism to adapt to new environments may be quantified. From an environmental management standpoint, demographic properties of a population do constrain to the rate of which species adapt to human disturbed environments. The adaptive value of a population can be interpreted as distance measure between the variability of the mortality distribution, where no environmental forces interfere and the conditional entropy, estimated given the known unperturbed reference mortality. Repeated use of pesticides for instance, can cause undesirable changes in the gene pool leading of a species due to artificial selection. Through this operation, populations with the favored demographic properties gradually develop resistance to the pesticide showing fitness advantages in the presence of the artificial selection factor. Thus, reproductive trends captured by demographic entropy may reflect macroevolutionary changes such as adaptation and extinction under variable conditions.
Keywords: Gibbs–Boltzmann entropy, Shannon entropy, artificial selection, evolution, pests
Comments on this paper
Petros Damos
coment by Prof Sherwin
16 Nov 2015:
Dear Dr Damos (cc:; I read your paper with interest. I cannot activate the "comment" function on the website, so I am sending this direct. You may mount my comments on the website if you wish. I have two points:

(1) I struck a few blockages:

- You write about the numerator of equation 6, but equation 6 is not a quotient

- Maybe I am misreading it, but equation 13 seems to imply that the system goes from i to j , r times, without every reverting to i. How can that be?

- P 8 Para 2 ".equilibrium of the upper population." Do you mean the treated population?

(2) Also, given the connections you make to selection, you might be interested in Barton, N.H.; De Vladar, H.P. 2009. Statistical mechanics and the evolution of polygenic quantitative traits. Genetics 2009, 181, 997-1011. If you delve further into genetics and entropy, my interest is in using entropy to express diversity of genes, eg (Sherwin WB. 2105. Molecular Ecology Resources 15:1259-1261. DOI: 10.1111/1755-0998.12458) AND (Chao A, et al 2015 PLoS ONE DOI:10.1371/journal.pone.0125471).

Regards, Bill Sherwin

Professor WB Sherwin
Evolution & Ecology Research Centre
Deputy Head, School of Biological Earth and Environmental Science, UNSW AUSTRALIA, Sydney NSW 2052 AUSTRALIA
PH:61-2-9385-2119 FX: 61-2-9385-1558
Petros Damos
November 19: Response by Petros DAMOS:

Dear Prof. Sherwin,
Thank you for your coments. I will tty to post them in the site portal along with the following coments:

-Yes, I agree, the quotient on the first version refers to equation 5. Thank you.
- If I have understand right your query: Indeed, the equation (13) estimates the evolution of the probabilities in one direction. The process is ergotic and entropy is estimated in the time domain for each time step. Therefore entropy convergence to equilibrium and don't reverts back as in the instance of a periodic chain. Nevertheless, the system per se (life cycle graph) can create any kind of transitions as defined by its transition matrix. This information is not captured in equation (13).

- Yes upper population coresponde to treated population
(2). Once the article will be prepared for official publication, I intend to connect the current approach to selection as well. I have noticed also your contribution in this interesting subject. Thank you for the article citations.
Kind Regards,
Petros Damos
19 Nov 15: Recomend by Prof Sherwin:
Dear Petros,
Thanks for that. If you are going to get further into selection, you might find it helpful to read section 3.3. "Selection" in the attached review (Sherwin 2010 Entropy 12:1765 doi:10.3390/e12071765). As well as an outline of the three basic types of selection, this paper includes a new equation for an entropic treatment of another basic type "balancing selection" (Eq 19), which has a component for the overall strength of selection, and a component relating to the fitness-differential between the two homozygotes.

NB of the three types of selection mentioned in that review, directional and balancing are probably the most prevalent (Barton & deVladar dealt with directional). The other type "disruptive selection" is expected to be very transitory (but important at the time it is occurring).

NB each selection type is divided into many sub-types, but you may not want to know about them.

Again, If you are able to post these comments on the conference website, please do so.
Petros Damos
Dear Bill
Thanks once more for your valuable references.
Actually, I found very interesting and informative your review article:
Entropy and Information Approaches to Genetic Diversity and its Expression: Genomic Geography
Alhough, I have worked on macroscopic scale with Shanon and other biodiversity measures, it will take me some time to deep on the genetic approach of entropy you work along (i.e. entropy as measure of genetic variability in sequences).
Still, it would be quite interesting deeping on allelic entropy that you define and if I have understand well, it also serves as a type of distance measure on survival functions to define selection strength between genetic diverse populations.
Thus, if I finally manage to get along so I will inquire for some remarks.