# Roots of Diversity Relations

Authors: Peter Würtz; Arto Annila
Journal: Journal of Biophysics
 (2008)

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## Abstract

The species-area relationship is one of the central generalizations in ecology; however, its origin has remained a puzzle. Since ecosystems are understood as energy transduction systems, the regularities in species richness are considered to result from ubiquitous imperatives in energy transduction. From a thermodynamic point of view, organisms are transduction mechanisms that distribute an influx of energy down along the steepest gradients to the ecosystem's diverse repositories of chemical energy, that is, populations of species. Transduction machineries, that is, ecosystems assembled from numerous species, may emerge and evolve toward high efficiency on large areas that hold more matter than small ones. This results in the well-known logistic-like relationship between the area and the number of species. The species-area relationship is understood, in terms of thermodynamics, to be the skewed cumulative curve of chemical energy distribution that is commonly known as the species-abundance relationship.


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## Body

## 1. Introduction

Species-area
relationships are frequently used to quantify, characterize, and estimate diversity
of biota [1–5]. Typically, the number of species (s) in a taxon is shown versus the size
of sampling area (A). For example,
the number of bird species increases mostly monotonically with a decreasing
slope on islands otherwise similar but increasingly larger in area [6].
The relationship is recognized as one of the few generalizations in ecology,
but its basis has remained obscure and hence, also its functional form has been
the subject of a long-standing debate [7, 8].Species
richness data from many ecosystems over a wide range of areas follow the power laws=cAz, where the slope z and intercept c are determined
empirically from a log-log plot [9, 10].
Nevertheless, this curve without an asymptote has been criticized as unphysical,
for example, because the globe
is finite [11, 12]. Logistic models and sigmoidal
curves are found to comply with observed species richness in large and bordered
communities [13–15]. Moreover, the small island effect,
that is, at the extreme of small sampling areas, the exponential form (s∝logA)
[16],
seems to account best for data [3, 17–19].Despite
the nonconformity among the three species-area models, it has been pointed out
that they could be approximations of a common but unknown functional form
[20].
Such an anticipated universal relationship would indicate similarity in overall
structural and functional organization of ecosystems rather than implying some common
parameters for all ecosystems. In any case, the species richness depends on
many other factors besides the area most notably insolation, temperature, and
rain fall. Species-energy theory [21]
aims at taking these factors also into account.Furthermore,
it has been realized that the species-area relationship is linked to species-abundance and
distribution-abundance relations [22–24].
Abundant species make large fractions of the total number of individuals in an
ecosystem, but curiously the probability density is skewed toward rarity in a log-normal-like
manner [5, 25–27].The
species-area relationship could hardly be rationalized without making a
connection to theory of evolution. Indeed, speciation as the source of
diversity and its relation to the size of area became recognized already early
on [28, 29].
Evolutionary effects have continued to interest and call for understanding how
nonequilibrium conditions affect the relationship [30]
by contributing to an imbalance between extinction and colonization [31–34].Thus,
the puzzle about the origin of species-area relationship appears particularly intricate because
many factors affect the species richness although all of them seem to associate
ultimately with energy, space, and time. Thus, we face the profound question,
where do the roots of diversity-area
relationships stem from.In
this study, the diversity relations are examined from the fundamental principle
of increasing entropy that was recently formulated as an equation of motion [35].
The statistical physics formulation places the theory of evolution by natural
selection [36]
on the 2nd law of thermodynamics. According to the 2nd law, flows of energy naturally
select the steepest gradients. These are equivalent to the shortest paths by
the principle of least action [37].
The thermodynamic formulation has been used to describe why natural
distributions are skewed [38]
and why standards such as chirality develop [39] as well as why genomes house
diversity of nonexpressed entities in addition to genes [40]. Also, the homeostatic nature of
the global system, including its abiotic and biotic mechanisms, has been
considered on the basis of imperatives in energy transduction [41].
These results are in agreement with earlier work based on the maximum entropy
principle [42–47].It
is no new idea to consider the species-area relationship to stem from a general
principle. The relationship has been understood by ecologists as a fundamental
pattern of nature that extends far beyond and below the length scales of
ecosystem organization [48, 49].
The objective here is to clarify the fundamental reason why the number of species
versus area is described by the aforementioned functional forms, not to suggest
a new species-area model. The description of an ecosystem as an energy-transduction
system is novel neither, but only until recently the thermodynamic formalism has
been available to derive the regularities of ecosystem organization from the
first principles.


## 2. Thermodynamic Description of an Ecosystem

Many spontaneous processes in nature, commonly
referred to asnatural processes [50], evolve toward more probable states
by leveling differences in energy. The universal phenomenon of energy dispersal
is also known by the principle of increasing entropy and by the 2nd law of
thermodynamics. In accordance with classical texts [51–54], an ecosystem is regarded by thermodynamics
as an open energy transduction network. Populations are diverse repositories of
chemical energy and individual organisms are energy transformers that tap into available
potentials to drain them. Flows of energy direct down along gradients when chemical
reactions transform species from one repository to another. At the level of cells
and organisms, the energy equalizing process is customarily referred to as
metabolism. At the level of an ecosystem, the energy transforming structure is
known as the food web.The
description of energy transduction by statistical physics remains at a formal
level. All entities of an energy transduction system are described as energy
densities [55]. In this way, they can be compared with
one and another to deduce which way energy will flow. In nature, potential
energy differences among the entities, for
example, populations of species are diminished by numerous processes
that take place at molecular level, for example, by photosynthesis, or at
macroscopic level, for example, by grazing.An
energy-transduction network is thermodynamically self-similar in its structure
at all levels of hierarchy. For example, atoms are the base constituents that
make molecules. Likewise at a higher level of hierarchy, cells are the base
constituents that make organisms that make populations. Owing to the scale-independent formalism,
one may, at each and every level of hierarchical organization, transform the
formal description to a model where entities are assigned with parameters and
functions to account for their properties and mutual interactions.The
amount of chemical potential energy associated with a population ofNj individuals is given by the
chemical potential [56] μj=RTln[Njexp(Gj/RT)], where the Gibbs-free energy Gj is relative to the average energy RT. The concept of chemical potential is
not restricted to molecules, but applies to
all entities such as plants and animals that result from chemical reactions. A
population of plant or animal species is associated with a chemical potential
just as a population of molecular species. The chemical potential denotes essentially
the trophic level height. In other words, the species at the top of food chain
are thermodynamically “expensive” to maintain by the long dissipative chain of
energy transduction. The chemical potential is a valuable concept to deduce the
structure of an ecosystem because the flows of energy equalize potentials. The stationary-state
condition for chemical reactant populations [56] determines also plant and animal populations as
results of numerous reactions.In
an ecosystem, many reactions convert quantaΔQjk of high-energy
radiation from the Sun to chemical energy. Subsequently, many additional reactions
redistribute the resulting base potential among diverse repositories of
chemical energy (Figure 1). The overall energy transduction from the base
production potential μ1 toward
all other potentials μj takes
the direction of increasing entropy S [35]: (1)S≈1T∑j=1Nj(∑k=1μk−μj+ΔQjk+RT)=R∑j=1Nj(AjRT+1). The
chemical potential difference, that
is, the free energy, experienced by species j, is, in this context, usually referred to as affinity Aj=∑μk+ΔQjk−μj
or free
energy relative to the average energy RT.
The concept of RT means that the
system is sufficiently statistic [57],
that is, a change in the energy influx is rapidly distributed within the
entities of the system. Thus, no major potential differences will amount
between the populations of species that interact with each other more
frequently than the total energy content of evolving ecosystem changes. Nevertheless,
a large variation in the energy influx due to the annual rhythm may drive huge
population fluctuations. Also abrupt changes in conditions or mechanistic
failures, for example, due to a disease, may bring about a
large imbalance.Figure 1
Schematic distribution
of chemical energy in a simple model ecosystem is described by an energy level
diagram. The governing thermodynamic principle is exemplified by considering
only one type of base constituents (atoms), but the result has been generalized
for diverse base constituents [35]. The number of individuals at trophic level j makes a population Nj.
The corresponding density-in-energy Njexp(Gj/RT) amounts from the number of base constituents nj=jNj that are needed to assemble the population and from
the invested energy Gj. For
a species at a level j in the food
web many atoms and much energy are needed to propel its growth and to maintain it
in the mature state. Species are equipped with mechanisms to generate these vital
flows of energy by numerous reactions (arrows) that absorb high-energy or emit low-energy
quanta (wavy arrows). Systems on larger areas, hence having access to more base
constituents N=∑nj,
will evolve to larger and more effective energy transduction machineries
comprising more species. Coloring emphasizes that species differ from each
other by their energy transduction properties, that is, phenotypes.According
to (1), the population Nj may
proliferate by acquiring ingredients Nk and external energy ΔQjk from
the surroundings, as long as Aj>0. Likewise,
when Aj<0, the population Nj is in for downsizing. When Aj=0, the potential μj associated with Nj of
species j matches the sum of
potentials ∑μk of
species k and external energy ΔQjk that
are vital for maintaining the population Nj.
Finally, when all Aj=0, the ecosystem has reached via
numerous chemical reactions the maximum entropy state  S=R∑Nj,
the stationary state of chemical nonequilibrium powered by solar flux. The
species-area relationship, as will be shown below, is a consequence of the
stationary-state structure of the ecosystem.


## 3. Distribution of Chemical Energy

During the course of evolution, free energy is consumed
and entropy increases at the rate [35] (2)dSdt=∑j=1dSdNjdNjdt=1T∑j=1vjAj, as the
ecosystem moves to increasingly more probable states via numerous chemical
reactions that adjust populations of species relative to one and other. During the
evolutionary processes toward the thermodynamic steady state also new species may
appear and old ones may disappear. New species will gain ground only when they
are equipped with mechanisms that allow to them contribute to S. The old species will perish if their
potentials are exhausted by others that have more efficient means of energy
transformation.To
satisfy the balance equation, the populationNj of species j changes at the rate [35], (3)vj=dNjdt=rjAjRT, proportional
to thermodynamic driving force Aj,
that is, a potential
difference, by a mechanistic coefficient rj>0. The rate equation differs from phenomenological differential equations
based on the law of mass-action that are used in population dynamics, for example, for modeling colonization
and extinction. The flow equation differs also from the logistic equation, where
a constant carrying capacity is taken
proportional to the sampling area [51, 58, 59].
However, in reality there is no fixed carrying capacity but thermodynamic
driving forces keep changing with changing populations that in turn affect the driving
forces. In other words, the flows down along gradients keep changing due to the
changing free energy landscape.The
interdependency among densities-in-energy means that when one species is consuming
in its processes common resources, for example,
base constituents the others have less. Even a small change in the initial
conditions will affect the outcome later, hence by the definition [60], evolution is chaotic. For these reasons,
it is in principle impossible to predict precisely trajectories of evolution
and ensuing detailed structure of an ecosystem. Accordingly, there is no analytical
form for the species-area relationship because it results from nonintegrable
and nondeterministic processes [35]. However, an effective approximation, in addition to
the logistic and power law forms, is available.


## 4. Species-Area Relationship

Under a steady external flux of energy, the ecosystem
will eventually reach a stationary state, the climax corresponding to the maximum
entropy. Then all thermodynamic driving forces have vanished and potentials across
reactions are equal:(4)dSdt=0⟺μj=∑kμk+ΔQjk⟺Nj=∏k=1Nkexp(−ΔEjkRT), where
ΔEjk=ΔGjk−ΔQjk. The condition
of chemical nonequilibrium stationary state expresses the familiar pyramid of
numbers by giving species in the order of increasing thermodynamic costs. The
climax state corresponds to the thermodynamically most optimal populations Nj at all trophic levels j. The nonequilibrium stationary state
is maintained by incessant energy transduction powered by an external source.
Such a system resides in the free energy minimum and will rapidly abolish any
emerging energy differences. High through-flux is powering the climax state in
agreement with the maximum power principle [61, 62].
However, the stationary state does not have to house the maximum number of
species that may have been encountered earlier during succession to the maturity.
The succession culminates to the system of fewer species that are highly effective
in energy transduction.All potentialsμj in the
ecosystem ultimately tap into the base potential μ1, that
is, couple to reactions that absorb solar energy (or extract from some other
high-energy external source). Since the form given by (4) is difficult to
analyze, we simplify the decreasing exponential partition (4) by an average thermodynamic
relation by expressing all interacting species Nj in terms of stable (i.e., G1=0) base constituents N1, that is, atoms and external energy that is incorporated in the assembly
processes. The average relation (4)  is merely a
simplification of the energy transduction network but
this analytical form allows us to depict species-area relationship
and compare it with the relationships that are known to
account for the data.The condition of thermodynamic stationary state(5)Nj=N1jexp[(j−1)ΔQ1RT]=exp[γ(j−1)] says how
many stable base constituents N1 and energy quanta  ΔQ1 are
required to maintain the population Nj of species j at the (trophic) level Gj, given concisely in units
of the average base potential γ=lnN1+ΔQ1/RT. The simplified stationary-state
condition (5) takes into account the larger number of base ingredients on
larger areas but not that mechanisms of energy transduction evolve on larger
areas more effective and efficient on larger areas than on small ones. Furthermore,
the approximation that all species would have the same stoichiometric
composition of base constituents N1 on the average is reasonable for many biotic systems but it is not without
exceptions. Therefore, parameters in the models of species-area relationships are not universal
as is apparent from many field studies.The
species-area relationship is essentially a consequence of conservation of
matter. For a given influx of energy, the populationsNj of all species j (5) each having the base constituents in numbers nj=jNj (Figure 1) are summed up to the total amount N=∑nj that is taken proportional to the area A:
(6)N=∑j=1sjNj=∑j=1sjexp[γ(j−1)]=αA. When (6)
is solved for the total number of species s and plotted against increasing area A,
the average thermodynamic relation gives understanding to the commonly used functional
forms species-area curves (Figure 2). However, it should be emphasized that (6) is
not a model; it is the instructive approximation of (4) to deduce the structure
of ecosystem’s energy transduction network. The proportionality constant α consumes implicitly many factors. For example,
the diverse base constituents originate mostly from the atmosphere above A, not from the ground that supplies
nutrients. Therefore, species-area relationships are customarily extracted from samplings,
ideally alike in constituents and energy input, differing only in their areas. Also
different abiotic constituents, for example, water and carbon dioxide that
couple to external energy, require different amounts of energy for activation. The
many ingredients, in a form of base constituents and energy, influence how far
the natural process may advance. They all are contained in (1)–(3), but
obviously it would be extremely challenging to model a large system in such a
great detail.Figure 2
Species (s) versus area (A) relationship (black) is a cumulative
curve of nonequilibrium stationary-state distribution of chemical energy in an
ecosystem. The total amount of base constituents N in the system is taken proportional to the area A. The cumulative curve follows mostly
the power law (green) but at large areas the logistic form (blue) accounts
better for the statistical series. The units on axes depend on the energetics
given by γ, units of measurements, and
proportionality constants.The
definition of species, implied by the indexj,
would mean that any two entities that can be distinguished from each other are
distinct. In nature, entities distinguish from each other in interactions. Thus,
the definition of species is subject to the resolution that is available in the
subjective detection process. The increment in index j is, therefore, not of primary interest when examining the
functional form of species-area relationship.The
obtained form for the species-area curve (Figure2) is consistent with the data
[5, 12] and theoretical considerations [15, 20]. At small areas, it rises nearly exponentially,
turns into the power-law form at larger areas, and finishes in the logistic
manner at the largest areas. The slope lns/lnA diminishes with the increasing number
of species. The correspondence to the power-law slope z is obtained from the derivative of s(A) and the relations to
the parameters of logistic or exponential model by best fit of a particular
data.The
debated question, does the species-area relationship have an asymptote, is not
particularly meaningful because the thermodynamic objective is not to maximize
the number of energy transformers of different kind but to arrive at the system
in a stationary state with respect to its surroundings whatever number of
species it takes. Thus, it is the surroundings that will ultimately dictate how
high the system may possibly rise with its ingredients to make energy
transformers. It is also emphasized that the sum over the species in (6) is
open to the energy influx from the surroundings that is an ingredient along
with the substances bound by earth’s gravitation.


## 5. Species-Abundance Relationship

To
relate the species-area relationship with the species-abundance relationship, the
sum over all speciesj in (6) is
approximated by a convenient continuous function: (7)∑j=1sjexp[γ(j−1)]≈∫1sP(j)dj. The
density function P(j) is the distribution of chemical
energy. The skewed function peaks at the fractions that contribute most to
entropy, that is, to energy dispersal and tails toward rare species’ fractions
(Figure 3). The populations are in relation to their potentials. Those species
that have mechanisms to tap into rich potentials on large areas are abundant,
and they are also likely to find some resources on smaller areas to support a correspondingly
smaller population. The thermodynamically expensive species consume large
potentials hence they are rare even on large areas and unlikely to be found on
smaller areas with insufficient potentials.Figure 3
Distribution of chemical energy among diverse chemical repositoriesj, that is, species according to (6). The probability density P(j)
of species-area curve is characteristically skewed toward rarity at high-energy
trophic levels j. The integral of P(j)
sums up all matter that is distributed among populations of all species s in an ecosystem. When the total matter
is taken proportional to the area A, the
species-area relationship is obtained as the cumulative curve.According
to the self-similar formulation of thermodynamics, also distributions of
individuals are skewed, approximately log-normal, functions [38] in agreement with observations [5]. The most abundant bins of a distribution correspond
to those individuals, that is,
mechanisms that contribute most to energy transduction. Likewise, within a
taxon, the density function P(j) versus j displays a characteristic
peak at the species richness that is identified to the intermediate size
species [5]. It is these intermediate fractions that contribute
the most to energy transduction. The variation of densities-in-energy among
individuals in the same species is small in comparison with the total dispersal
of energy in the entire ecosystem. This is to say that the individuals of the
same species have approximately similar mechanisms of energy transduction
whereas individuals of different species have distinctively different
mechanisms. The skewed distributions have also been found in genomes [63]
and rationalized using the 2nd law [40]. The ubiquitous characteristics imply that the
species-area and species-abundance relationships are not only ecological relationships, but
also account for hierarchical organization of matter to dissipative systems in
general.


## 6. Species-Area Relationship in Evolution

At
this point, it is insightful to describe effects of migration, speciation, and
extinction on the species-area relationship using thermodynamics. Customarily, the
species-area relationships are considered when there is a balance between
immigration and in situ
speciation and extinction. Obviously ecosystems evolve in space and time.
Nonequilibrium conditions are expected to show in species-area relations.According
to the basic thermodynamic rationale, evolution as a whole is an energy-transduction
process. For any flow of energy, there is only one reason—an energy difference. Diverse differences in
energy drive diverse flows that manifest, for example, as migration, speciation, and extinction.To
begin with, the question, why there are so many species, calls for the answer. Functionalities
of entities, for example,
organisms, appear in mutual interactions when they tap into various potentials
by their phenotypic mechanisms. However, no single entity due to its finite
composition may exhibit all possible functionalities to drain all conceivable sources
of energy. This limits utilization of resources and promotes segregation of
species for specialized and efficient functional roles to acquire chemical
energy from specific sources. The populations of species themselves are repositories
of energy for others to be consumed. Hence, diversity builds on diversity. In
the quest to reduce all possible energy gradients, species evolve to thrive in
ecological niches that are, thermodynamically speaking, basins in the free
energy landscape. The diversification may also proceed within a species and manifest,
for example, as behavioral specialization, that
is, “division of labor”.The
characteristic mechanisms of energy transduction are referred to as phenotypes
that distinguish a species from another in the same system. According to the
Lyapunov stability criterion that is given in terms of entropy [50, 60], for any two species having nearly similar
mechanisms, one will inevitably be excluded because such a system is unstable.
The competitive exclusion principle is not limited to animals and plants but
has been shown to account for the ubiquitous handedness of amino acids and
nucleic acids as well [39].The
fitness criterion for natural selection, equivalent to the rate of entropy
increase (2), gives rise to increasingly economical and effective dissipative
systems to consume various sources of free energy. Nevertheless, it may appear
odd that species tend to evolve by retaining their ancestral ecological characteristics.
From the thermodynamic viewpoint, an organism must sense an energy gradient for
it to evolve. If there is not even a rudimentary or indirect mechanism
available for a species to tap into a potential, the specific source of free
energy provides no gradient for the species to direct its evolution. Hence, the
particular species continues to diversify more readily along those gradients that
are sensed by the mechanisms resulting from the ancestral development.The
phylogenetic conservatism may lead to an unusual species-area relationship. When
a species that is equipped with superior migratory mechanisms, such a bird
species, happens to colonize a rich remote location, such as a large isolated
island, phylogenetic conservatism may confine the ensuing diversification so
that numerous mechanisms, that is,
species will emerge, however, all with avian characteristics and none with
truly optimal mechanisms for full terrestrial activity. Under those
circumstances, the number of species may become larger than expected on the
basis of the islands area. Therefore, the ecosystem appears to be in a
nonequilibrium state. To be more precise in wording, the ecosystem is stable, that is, not subject to driving
forces, but it is vulnerable to an eventual later colonization by more potent
species from other ancestral lines that are more suited for terrestrial life. A
single nonnative species with superior mechanisms may rapidly drive numerous
native species to extinction by consuming previously ineffectively and inefficiently
used potentials. Obviously, a pioneering immigrant species that has specialized
far away from its ancestral habitant and thus has given up its valuable virtues
may fall as an easy prey for newly emerged predators.It
is also conceivable that a small remote location holds a lower number of
species than expected on the basis of its area. Nowadays, it is less likely
that such an isolated and intact location could be found but certainly a newly
surfaced volcanic island displays initially anomalously low species-area
relationship. When the
area is small, all potentials are small and limited as well. Flows between the
potentials are few and their rates are low. Also the rate of speciation is low
and owing to the remote location, immigration rates are very low as well. It
may then happen that the island lacks, for
example, an entire genus. Then the ecosystem appears to be in a
nonequilibrium state having too few species. More specifically, the state is
stable until members of the “missing” genus appear and expose the ecosystem to
novel energy gradients. Then, the diversification begins and brings up with
time the number of species to the expected level.The
interdependent thermodynamic description takes into account effects that a new
species introduces on all other species in an ecosystem. The new transduction
mechanism puts the system in motion toward a new stationary state (2). The
species-area relationships
essentially states that for a new species (s+1) to appear on increasingly larger areas, it will become increasingly more
demanding, in thermodynamic terms, to meet the differentiation condition dS/dNj+1>∑dS/dNj. For the new species to gain
ground it must be able to increase entropy, that is, to disperse energy by its characteristic mechanisms
more than could be achieved by increasing the populations of existing species.A
continent has more ingredients and more energy to fuel diverse flows that may
combine so that a new species will emerge in comparison with a small island
that is more likely to acquire new species by migration. An island next to the
main land or a mountain top above a plain may acquire frequently new species. The
small area may support some immigrants even below the aforementioned differentiation
condition, but only for a limited time period. When the immigrants have overdepleted
their vital potentials at the small location, they must leave to tap into
potentials elsewhere or they will perish. Therefore, an adjacent island, just
as a mountain top, that enjoys from a continuous influx of species may hold a
larger number of species than would be expected only on the basis of its area.
Such a state is usually referred to as a nonequilibrium state but when the
influx is steady, the state is also steady.


## 7. Discussion

The
thermodynamic description of an ecosystem as an energy transduction network and
the view of species as energy transformers are not new ideas [51–54]. The new insight to biotic systems is provided by
the 2nd law of thermodynamics given as the equation of motion [35, 37]. It reveals that the principle of increasing entropy
and the theory of evolution by natural selection are in fact stating one and
the same imperative; not describing opposing forces as it is often mistaken.It
is important to realize that the 2nd law only states that differences in energy
tend to diminish. Often it is one-sidedly thought that the 2nd law would
describe only the evolutionary course leading to diminishing
densities-in-energy. This is the scenario at the cosmic scale. Here on earth
next to Sun, the imperative is the same but it is perceived differently. The flow
of energy is also downward when the high-energy solar flux couples via chemical
reactions to the low-energy matter on earth. Consequently, chemical potential
of matter is bound to increase when mechanisms that couple to the influx happen
to emerge.The
quest to diminish the energy difference with respect to the insolation directs
evolution. Over the eons, the machinery for the base production has emerged.
The base production, in turn, provides the high potential for other mechanisms
to be consumed. In this way, energy is distributed by diverse mechanisms
downward to other repositories within the ecosystem and eventually dumped in as
low-energy radiation in space. The imperative to level gradients increasingly
more effectively and efficiently results in the characteristic regularities and
relationships of nature. Intriguingly, such skewed distributions, for example, of plants and animal
populations, and sigmoid dispersion relations, for example, species-area relationship, are not only encountered in ecology, but also
found in many other contexts [38, 64, 65]. The thermodynamic formulation for the intricate and
complex network of energy transduction of an ecosystem resembles the
power-series derived from the concept of self-similarity [22] in accordance with the simplifying form of (6).Despite
the holistic view provided by thermodynamics, the self-consistent scale-independent
description of energy transduction systems may appear abstract, especially as
it seems to take no account on biological mechanisms, structures, and
functions. However, the entropy formula (1) is deceptive in its conciseness. It
describes energy densities in an entire ecosystem by every unit of matter Nk and Nj and by every quantum of energy Gk and Gj,
as well as by indexing all interactions by j and k. Obviously it would require a
detailed knowledge of all reactions, for example, the full atomic description
of energy transduction, to establish the precise relationship between s and A for a particular ecosystem. Such a network of nested summations
over all entities in (1) would be enormous and impractical, but the abridged
form of (6) reveals the sigmoid diversity-area relationship. It is, in terms of physics, a
dispersion relation, that is,
the energy response function.Properties
of atoms, characteristics of molecules, functions of organisms, phenotypes of
animals, and so on obtain their definitions in interactions. Also our
observations are dissipative interactions [66]
that classify individuals in diverse species. Increasingly, powerful
experimental methods allow us to distinguish finer and finer details. Consequently,
the species is only a practical definition that refers to a particular class of
densities-in-energy by emphasizing reactions of reproduction. Certainly,
hereditary mechanisms are powerful, however, irrespective of reproduction mechanisms
the overall structure of any energy transduction is governed by the universal
imperative to disperse energy down along gradients most rapidly.Thermodynamic reasoning is
simple. Systems, at all scales, evolve toward stationary states in their respective
surroundings. Evolution is a natural process, a sequence of successive steps
that makes no difference between inanimate and animate when devouring free
energy. A small system will rapidly acquire mechanisms in succession, whereas
for the global ecosystem it has taken eons to emerge via random variation with de novo mechanisms in the quest for a
stationary state. For all systems, it is the superior surrounding energy
densities that command evolution. However, it takes mechanisms for energy to
flow between the system and its surroundings. Intrinsic emergence of mechanisms
or acquisition of them from the surroundings, unleash flows in the quest for
the stationary state. However, the equation of evolution, that is, the 2nd law
as the equation of motion cannot be solved because the flows affect the driving
forces that in turn redirect the flows. Therefore, the courses of evolution are
intricate and difficult to predict in detail.For a long time, there has been
a search for the common ground to establish the many laws of ecology. The thermodynamics
of open systems meets the early expectations of ecology as pronounced a century
ago by Oscar Drude, an eminent plant ecologist. “Ecology has arisen from the
need to unite originally separate branches of science in a new and natural
doctrine; it is characterized by the breadth of its aims, and its peculiar
power and strength in its ability to unite knowledge of the organic life with
knowledge of its home, our earth. It assumes the solution of that most
difficult as well as most fascinating problem which occupies the minds of
philosophers and theologians alike, namely, the life history of the plants and
animal worlds under the influences of space and time” [48].



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