Distribution of species diversity values: A link between classical and quantum mechanics in ecology

Despite the well-known thermodynamic traits of ecosystem functioning, their description by means of conventional physics should be regarded as incomplete, even if we take into account the most recent advancements in this field. The analytical difficulties in this field have been especially complex to get a reliable modeling of species diversity per plot (Hp) by endowing this indicator with a fully clear theoretical meaning. This article contributes to resolve such difficulties starting from (a) the previous proposal of an ecological state equation, and (b) the preceding empirical finding of an ecological equivalent of Planck's constant at the evolutionary scale. So, in the first instance, this article proposes an equation for density distributions of Hp values (EDH) based on a simple transformation of the Maxwell–Boltzmann distribution for molecular velocity values (M–BDv). Our results indicate that the above-mentioned equation allows an appropriate fit between expected and observed distributions. Besides, the transformation from M–BDv to EDH establishes connections between species diversity and other indicators that are consistent with well-known ecological principles. This article, in the second instance, uses EDHs from a wide spectrum of surveys as an analytical framework to explore the nature and meaning of stationary trophic information waves (STIWs) whose stationary nature depends on the biomass-dispersal trade-off in function of Hp values (B-DTO-H) that characterizes the most of the explored surveys. B-DTO-H makes these surveys behave as ecological cavity resonators (ECR) by trapping functional oscillations that bounce back and forth between the two opposite edges of the ECR: from r-strategy (at low biomass and diversity, and high dispersal) to K-strategy, and vice versa. STIWs were obtained by using the spline-adjusted values from the arithmetical difference between standardized values of species richness (S) and evenness (J′) in function of Hp values (i.e., a 2D scalar space Hp, S–J′). Twice the distance on the abscissas (2ΔHp) between successive extreme values on the ordinates (whatever a maximum or a minimum) along the above-mentioned spline adjustment was taken as the value of ecological wavelength (λe). λe was assessed in order to obtain the value of the ecological equivalent of Planck's constant (heec) at the intra-survey scale that was calculated as: heec=λe×me×Ie; where me: individual biomass, and Ie: an ad-hoc indicator of dispersal activity. Our main result is that the observed value of heec's mantissa is statistically equivalent to the mantissa of the physical Planck's constant (h=6.62606957E−34Js) in all of the discontinuous (i.e., with interspersed categories in which n=0) statistical density distributions of Hp values per survey. This means that heec=6.62606957EφJenat/individual, where φ=−xi, …, −3, −2, −1, 0,+1, +2, +3, …,+xi depending on the type of taxocenosis explored. That is to say, heec indicates the minimum amount of energy exchange allowed between two individuals. The exploration of the analytical meaning of this result in the final sections of the article explains why quantum mechanics (QM) is a useful tool in order to explain several key questions in evolutionary biology and ecology, as for example: the physical limit of adaptive radiation; the balance between competitive exclusion and functional redundancy to promote species coexistence by avoiding the negative effects of competitive exclusion; the apparent holes in the fossil record; the progression of body size along a wide spectrum of taxa as a general evolutionary trend; the non-continuous nature of net energy flow at the ecosystem level; the way in which the energy level is stabilized under stationary ecological conditions; the reasons of the higher sensitivity of high diversity ecosystems under environmental impact despite their higher stability under natural conditions; the tangible expression of complex concept as ecological inertia and elasticity; as well as the increased risk from pushing the biosphere until a rupture limit because of the potential discrete behavior of ecological resilience in the large scale due to the quantum nature of ecosystem functioning.