Vol.1, No.3, 77-84 (2011)
Open Journal of Ecology
opyright © 2011 SciRes. OPEN ACCESS
A law of nature?
Marvin Chester
Physics Department, University of California Los Angeles, Los Angeles, USA; chester@physics.ucla.edu
Received 20 September 2011; revised 21 October 2011; accepted 30 October 2011.
Is there an overriding principle of nature, hith-
erto overlooked, that governs all population be-
havior? A single principle that drives all the re-
gimes observed in nature—exponential-like gr-
owth, saturated growth, population decline, po-
pulation extinction, oscillatory behavior? In cur-
rent orthodox population theory, this diverse
range of population behaviors is described by
many different equations—each with it s own sp-
ecific justification. The signature of an overrid-
ing principle would be a differential equation
which, in a single statement, embraces all the
p anoply of regimes. A candidate such governing
equation is proposed. The principle from which
the equation is derived is this: The effect on the
environment of a populations success is to
alter that environment in a way that o ppose s the
Keyw ords: Population Dynamics; Ecology; Population
Evolution; Exponential Growth; Eco-Evolutionary
Dynamics; Biology
What are the conceptual foundations in ecological stu-
dies? Are there laws of nature governing ecological sys-
tems? What are they?
Over the years there has grown a community of sch-
olars who have grappled with these profound questions.
[1-7] Some have concluded that concern for such laws is
not the business of ecological research. [8,9] Others ha-
ve concluded that Malthusian exponential growth con-
stitutes an essential law [10]. That exponential growth
and the logistics equation are of great conceptual utility
but not laws of nature is argued cogently by still others
Focussing on the lofty notion, Law of Nature, may be
a distraction from a more elemental pursuit: to unders-
tand nature. That is surely the goal of science. Progress
in understanding is marked by conceptual coalescence:
the quest to embrace an ever larger body of findings with
ever fewer statements of principle. Paraphrasing Mark
Twain, the task of science is to describe a plethora of
phenomena with a paucity of theory.
Newton showed that the motion of things on earth are
governed by the same rules as the motions of heavenly
bodies. Formerly these two had appeared to be unrelated
domains. Newton showed that a single principle gover-
ned them both. This synthesis was magnificently fruitful.
It underlies our understanding of anything mechanical or
structural. Much of our material well being depends on
Darwin’s principle of natural selection explained a
wealth of biological phenomena by a single idea. Throu-
gh his synthesis the concept of evolution became part of
our intellectual heritage.
Wegener showed us that continental drift—plate tecto-
nics—is the underlying reason for such diverse phenom-
ena as the distribution of fossils in the world, the shape
of continents, earthquake belts and volcanic activity. That
insight has proved remarkably beneficial.
Mendeleev gave structure to the chaos of chemistry
with his table of the elements. He consolidated a profu-
sion of chemical data into an all encompassing tabular
statement of principle. This undertaking led to the under-
standing that matter was made of atoms. (Mendeleev,
himself, never believed this!)
James Maxwell brought electricity and magnetism
together by an overriding formalism that covered them
both. The undertaking gave rise to an understanding of
the nature of light.
Laws of Nature are not immutable. They may lose
their status. This process of conceptual coalescence is an
ever evolving one. Newton’s laws on mechanical motion
and Maxwell’s on electromagnetism are incompatible. In
1905 Einstein produced a theory that embraced both of
these vaste domains. In it Newton’s principles become a
limit behavior of a more all inclusive theory—relativity.
So a law of nature can be dethroned—albeit still cher-
ished and useful. It can be subsumed under a principle
which embraces a larger domain of phenomena. The
broader the scope of applicability the more valuable is
M. Chester / Open Journal of Ecology 1 (2011) 77-84
Copyright © 2011 SciRes. OPEN ACCESS
the theory. Einstein’s laws of nature absorbed Newton’s.
Relativity spawned nuclear energy, a deeper understand-
ing of stellar processes and much more.
All of these examples have in common that a wide
breadth of empirical observation is accounted for by a
single idea. We see in them that conceptual coalescence
is a foundation stone of scientific understanding. In that
spirit, offered here is a candidate synthesis: a single equ-
ation that brings together the disparate domains of popu-
lation behavior. We suggest that the panoply of popula-
tion behaviors all issue from a single principle.
In current orthodox population theory, the diverse ran-
ge of population behaviors is described by many differ-
ent equations—each with its own specific justification.
Every regime has its own special theoretical rationale.
Exponential growth has a limited range of validity. The
Logistics Equation describes another restricted regime.
Oscillatory behavior demands that a new paradigm be
requisitioned; the Lotka-Volterra equations [13,14] or,
because their solutions are not structurally stable, their
later modifications [15,16]. And none of these describe
population decline, nor population extinction. Contem-
porary theory offers no overriding principle that governs
the entire gamut of population behaviors.
As long ago as 1972 [17], in a challenge to orthodox
convention, L. R. Ginzburg took the bold step of pro-
posing that population dynamics is better represented by
a second order differential equation. All accepted for-
mulations relied on first order differential equations as
they still do today. He developed his thesis over the
years [18-21] culminating in the pithy and persuasive
book, “Ecological Orbits” [4].
When the family of solutions to a differential equation
is found to fit empirical reality then that equation is ex-
pressing a truth about nature. It can give us insights and
enable us to make predictions. Producing a second order
equation whose solutions characterize a variety of popu-
lation behavior is equivalent to uncovering a principle of
nature governing those populations.
In the following we take a route different from Gin-
zburg’s and arrive at a substantially different equation -
albeit a second order one. We procede from a guess at
what may be the underlying principle and then derive the
second order differential equation that expresses that
principle. If empirical reality is well fit by the progeny
of that equation then we may conclude that the principle
is true. And we will have produced a conceptual coales-
cence: a tool for better understanding nature.
Call the number of members in the population, n. At
each moment of time, t, there exist n individuals in the
population. So we expect that n = n(t) is a continuous
function of time.
The rate of growth of the population is dn/dt; the in-
crease in the number of members per unit time. That this
is proportional to population number, n, is the substance
of Malthus’ idea of “increase by geometrical ratio”. Call
the constant of proportionality, R. Then the well known
differential equation that embodies the idea is:
dndt =Rn (1)
It is a first order differential equation and when R is
constant, its solution yields the archetypical equation of
exponential growth.
Now, common experience tells us that exponential gr-
owth cannot proceed indefinitely. “Most populations do
not, in fact, show exponential growth, and even when
they do it is for short periods of time in restricted spatial
domains,” writes R.D. Holt [12]. No population grows
without end.
The first efforts to expand the breadth of applicability
of theory to observation—to acheive some conceptual
coalescence—was to allow R to vary with time. The
motivation was to retain that appealing exponential-like
form and seek to explain events by variations in R. “The
problem of explaining and predicting the dynamics of
any particular population boils down to defining how R
deviates from the expectation of uniform growth” [10].
The concept is that exponential growth is always taking
place but at a rate that varies with time. The idea is
ubiquitous in textbooks [15,22-24].
An object example of this process is provided by the
celebrated Verhulst equation.
dn n
=nR n=nr 1
dt K
Here the constant, r, is the exponential growth factor
and K is the limiting value that n can have - “the carry-
ing capacity of the environment” [16]. The equation in-
sures that n never gets larger than nMAX = K. A popula-
tion history, n vs. t, resulting from this first order differ-
ential equation is the black one of Figure 1. The Ver-
hulst equation—often cited as the Logistics Equation—is
regularly embedded in research studies [25-30].
The textbook mathematical structure outlined in the
last section has acquired the weight of tradition. Keeping
the exponential-like form by allowing R to vary is cer-
tainly appealing. But it has this serious failing: the prac-
tice forbids description of several known regimes of po-
pulation behavior. It denies further conceptual coales-
cence. For example, unless R is taken as imaginary the
observation of population oscillations cannot be descri-
M. Chester / Open Journal of Ecology 1 (2011) 77-84
Copyright © 2011 SciRes. OPEN ACCESS
bed in this formalism.
Another proscribed regime is extinction. A phenome-
non well known to exist in nature is the extinction of a
species. “... over 99% of all species that ever existed are
extinct” [31]. But there exists no finite value of R—posi-
tive or negative—that yields extinction! It cannot be rep-
resented by R except for the value negative infinity; –.
So, in fact, there is good reason to avoid R as the key
parameter of population dynamics.
In the continuous n perspective the mathematical con-
ditions for extinction are these: n = 0 and dn/dt < 0. No
infinities enter computations founded on these statements.
Hence embracing n(t) itself as the key variable directly
allows one to explore the dynamics of extinction.
Next consider the eponymous Verhulst Eq.2, the Lo-
gistics Equation. As Verhulst himself pointed out [32] it
is motivated only by the observation that populations
never grow to infinity. They are bounded.
But there are other ways—not describable by Verhul-
st’s equation—in which population may be bounded. For
example, n(t) may exhibit periodicity. Or, as in Figure 1,
a curve essentially the same as Verhulst’s may arise from
an entirely different theory where no K = nmax limit ex-
ists. One of the possible population histories resulting
from the alternative theory offered below—which con-
tains no nmax—is shown in blue. Data fit by one curve
will be fit just as well by the other. The limited validity
of r/K Selection Theory has been noted by researchers
over the years [33,34].
Thus the accepted Malthusian Structure of population
dynamics has, and will always have, only a limited do-
main of validity. Many population histories cannot be fit
with this structure no matter how R is allowed to vary.
So, in current orthodox population theory, to describe
the entire range of population behaviors requires many
Figure 1. Two population histories: number vs. time. The black
curve is the Verhulst (Logistics) Equation. The blue curve is one
of the solutions to the Opposition Principle differential Equation
(8). Where one curve fits data so will the other.
different equations—each with its own specific justifica-
tion. Exponential growth has a limited range of validity,
as does the Logistics Equation, and any other equation of
first order. Oscillatory behavior demands that a new pa-
radigm be requisitioned; the Lotka-Volterra equations.
And none of these describe population decline, nor ex-
No structure exists that embraces—in one single sta-
tement—all possible behaviors. Contemporary theory of-
fers no overriding principle that governs the gamut of
population behaviors. To produce such a structure is the
aim of what follows.
We seek a mathematical structure to embrace all of the
great variety of population behaviors. The equation is
built on some foundational axioms. Empirical verifica-
tion of the equation they produce is what will measure
the validity of these axioms. The axioms are:
First: Variations in popu lation number, n, are due en-
tirely to environment.
Conceptionally we partition the universe into two: the
population under consideration and its environment. We
assume that the environment drives population dynamics;
that the environment is entirely responsible for time
variations in population number—whether within a sin-
gle lifetime or over many generations.
The survival and reproductive success of any individ-
ual is influenced by heredity as well as the environment
it encounters. This statement doesn’t contradict the axi-
om. The individual comes provisioned with heredity to
face the environment. Both the environment and the po-
pulation come to the present moment equipped with their
capacities to influence each other; capacities derived fr-
om their past histories.
That the environment molds the population within a li-
fetime is clear; think of a tornado, a disease outbreak, or a
meteor impact. That the environment governs population
dynamics over generations is precisely the substance of
“natural selection” in Darwinian evolution.
That principle may be summarized as follows: “... the
small selective advantage a trait confers on individuals
that have it...” [31] increases the population of those
individuals. But what does ‘selective advantage’ mean?
It means that the favored population is ‘selected’ by the
environment to thrive. Ultimately it is the environment
that governs a population’s history. Findings in epigene-
tics that the environment can produce changes transmit-
ted across generations [35,36] adds further support to
this notion.
Much productive research looks at traits in the pheno-
type that correlate with fitness or LRS (Lifetime Repro-
M. Chester / Open Journal of Ecology 1 (2011) 77-84
Copyright © 2011 SciRes. OPEN ACCESS
ductive Success) [37,38]. The focus is on how the orga-
nism fits into its environment. So something called “fit-
ness” is attributed to the organism; the property of an
organism that favors survival success. But environmen-
tal selection from among the available phenotypes is
what determines evolutionary success. The environment
is always changing so whatever genetic attributes were
favorable earlier may become unfavorable later. Hence
there is an alternative perspective: fitness, being a matter
of selection by the environment, is induced by it and
may, thus, be seen as a property of the environment.
Although, fitness, in some sense, is “carried” by the
genome, it is “decided” by the environment. Assigning a
fitness to an organism rests on the supposition of a static
environment; one into which an organism fits or does not
fit. A dynamic environment incessantly alters the “fitness”
of an organism.
This is the perspective underlying the axiom that va-
riations in population number, n, are due entirely to en-
In this view, although birth rates minus death rates
yield population growth they are not the cause of popu-
lation dynamics; rather birth and death rates register the
effect of the environment on the population. This view
parallels that of N. Owen-Smith who writes: [39] “...
population growth is not the result of a difference be-
tween births and deaths (despite the appearance of this
statement in most text-books), but rather of the differ-
ence between rates of uptake and conversion of reso-
urces into biomass, and losses of biomass to metabolism
and mortality.”
Second: An increasing growth rate is what measures a
populations success.
The “success” of a population is an assertion about a
population’s time development; it concerns the size and
growth of the population. A reasonable notion of success
is that the population is flourishing. We want to give
quantitative voice to the notion that flourishing growth
reveals a population’s success.
Neither population number, n, nor population growth,
dn/dt, are adequate to represent “flourishing”. Population
number may be large but it may be falling. Such a popu-
lation cannot be said to be flourishing. So we can’t use
population number as the measure of success. Growth
seems a better candidate. But, again, suppose growth is
large but falling. Only a rising growth rate would indi-
cate “flourishing”. This is exactly the quantity we propo-
se to take as a measure of success; the growth in the gr-
owth rate. By flourishing is meant growing faster each
year. That the rate of change of growth is a fundamental
consideration in population dynamics has been advoca-
ted in the past [4].
A corollary of these two foundational hypotheses is
that change is perpetual. Equilibrium is a temporary
condition. What we call equilibrium is a stretch of time
during which dn/dt = 0. Hence “returning to equilibrium”
is not a feature of analysis in this model. “Biological
persistence (is) more a matter of coping with variability
than balancing around some equilibrium state.” [40]
Another corollary is this: The environment of one
population is other populations. It’s through this mecha-
nism that interactions among populations occur: via reci-
procity—if A is in the environment of B, then B is in the
environment of A. So the structure offers a natural set-
ting for “feedback” [41]. It provides a framework for the
analysis of competition, of co-evolution and of preda-
tor-prey relations among populations. These obey the sa-
me equation but differ only in the signs of coefficients
relating any pair of populations.
Based on the understandings outlined above we pro-
pose that an overriding principle governs the population
dynamics of living things. It is this: The effect on the
environment of a populations success is to alter that
environment in a way that opposes the success. In order
to refer to it, I call it the Opposition Principle. It is a
functional principle [42] operating irrespective of the
mechanisms by which it is accomplished. In the way that
increasing entropy governs processes irrespective of the
way in which that is accomplished.
The Principle applies to a society of living organisms
that share an environment. The key feature of that soci-
ety is that it consists of a number, n, of members which
have an inherent drive to survive and to produce off-
spring with genetic variation. Their number varies with
time: n = n(t).
Because we don’t know whether n, itself, or some
monotonically increasing function of n is the relevant
parameter, we define a population strength, N(n). Any
population exhibits a certain strength in influencing its
environment. This population strength, N(n), expresses
the potency of the population in affecting the environ-
ment—its environmental impact.
N. Owen-Smith [39] urges us to “assess abundance”
in terms of biomass rather than population. What is here
named “population strength” is related to that idea.
Population number, itself, may not be a measure of en-
vironmental potency.
Perhaps this strength, N, is just the number n, itself.
The greater n is, the more the environmental impact. But
it takes a lot of fleas to have the same environmental
impact as one elephant. So we would expect that the
population strength is some function of n that depends
upon the population under consideration. Biomass is one
such function.
M. Chester / Open Journal of Ecology 1 (2011) 77-84
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Two things about the population potency, N, are clear.
First, N(n) must be a monotonically increasing function
of n; dN/dn > 0. This is because when the population
increases then its impact also increases. Albeit, perhaps
not linearly. Second, when n = 0 so, too, is N = 0. If the
population is zero then certainly its impact is zero. One
candidate for N(n) might be n raised to some positive
power, p. If p = 1 then N and n are the same thing. An-
other candidate is the logarithm of (n + 1).
We need not specify the precise relationship, N(n), in
what follows. Via experiment it can be coaxed from na-
ture. The only way that N depends upon time is para-
metrically through its dependence on n. In what follows
we shall mean by N(t) the dependence N(n(t)). We may
think of N as a surrogate for the number of members in
the population.
The population strength growth rate, g = g(t), is de-
fined by
g:= dt (3)
Like N, g too acquires its time dependence paramet-
riccally through n(t). g = (dN/dn)(dn/dt).
To quantify how the environment affects the popula-
tion we introduce the notion of “environmental favorabi-
lity”. We’ll designate it by the symbol, f. It represents
the effect of the environment on the population.
A population flourishes when the environment is favo-
rable. Environmental favorability is what drives a popu-
lation’s success. We may be sure that food abundance is
an element of environmental favorability so f increases
monotonically with nutrient amount. It decreases with
predator presence and f decreases with any malignancy
in the environment—pollution, toxicity.
But in the last section we arrived at a quantitative
measure of success. The rate of growth of the population
strength—“the growth of growth” or dg/dt—measures
success. Hence, that a population’s success is generated
entirely by the environment can be expressed mathema-
tically as:
dg =f
dt (4)
By omitting any proportionality constant we are de-
claring that f may be measured in units of (time)–2. Since
Eq.4 says that success equals the favorability of the en-
vironment, it follows that f measures not only environ-
mental favorability but also population success. One can
gauge the favorability of the environment—the value of
f—by measuring population success.
We’re now prepared to caste the Opposition Principle
as a mathematical statement. The Principle has two parts:
1) Any increase in population strength decreases favora-
bility; the more the population’s presence is felt the less
favorable becomes the environment. 2) Any increase in
the growth of that strength also decreases favorability.
Put formally: That part of the change in f due to an
increase in N is negative. Likewise the change in f due to
an increase in g is negative. Here is the direct mathema-
tical rendering of these two statements:
0 and 0
We can implement these statements by introducing
two parameters. Both w and
are non-negative real
numbers and they have the dimensions of reciprocal time.
(Negative w values are permitted but are redundant.)
These partial differential equations can be integrated.
The result is:
  (7)
The “constant” (with respect to N and g) of integra-
tion, F(t), has an evident interpretation. It is the gratui-
tous favorability provided by nature; the gift of nature.
Eq.7 says that environmental favorability consists of two
One part depends on the number and growth of the
population being favored: the N and its time derivative,
g. This part has two terms both of which always act to
decrease favorability. These terms express the Opposi-
tion Principle.
The other part—F(t)—is the gift of nature. There must
be something in the environment that is favorable to
population success but external to that population else
the population would not exist in th e first place. This gift
of nature may depend cyclically on time. For example,
seasonal variations are cyclical changes in favorability.
Or it may remain relatively constant like the presence of
air to breathe. It may also exhibit random and sometimes
violent fluctuations like a volcanic eruption or unex-
pected rains on a parched earth. So it has a stochastic
component. All of these are independent of the popula-
tion under consideration. In fact, however, dF/dt may
depend on population number since this is the rate of
consumption of a limited food supply.
Inserting Eq.3,4,7 we arrive at the promised differen-
tial equation governing population dymics under the
Opposition Principle. It is this.
dN dN
dt wFt
In the world of physical phenomena this equation is
ubiquitous. Depending upon the meaning assigned to N
it describes electrical circuits, mechanical systems, the
production of sound in musical instruments and a host of
M. Chester / Open Journal of Ecology 1 (2011) 77-84
Copyright © 2011 SciRes. OPEN ACCESS
other phenomena. So it is very well studied. The exact
analytical solution to (8), yielding N(t) for any given F(t),
is known.
To explore some of the consequences of this differen-
tial equation we consider the easiest case; that the gift
favorability is simply constant over an extended period
of time. Assume F(t) = c independent of time. Non-peri-
odic solutions arise if
2w. As displayed in Figure 1
these produce results similar to the Verhulst equation—
the Logistics equation. And like that equation they exhi-
bit an exponential-like growth over a limited range.
Empirical data on such an exponential-like growth is
exhibited in Figure 2 showing the population of musk
ox (Ovibos moschatus) isolated on Nunivak Island in Al-
aska [43]. The data, gathered every year from 1947 to
1968, is in Table 1 of Spencer and Lensink’s paper. It is,
indeed, well fit by an exponential curve showing the
formidable growth rate of 13.5% per year.
The population cannot possibly fit such a curve in-
definitely. Nothing grows without end. The exponential
curve fits the population data only in the domain shown.
But, in this domain, the data is also well fit by an Oppo-
sition Principle curve where
= 0.02 per year, w =
0.0076 per year and n = N.
< 2w the solutions to (8) are periodic and are
given by:
 
 a
Principle curve may be fit to the data.
gnated time, say
where the amplitude, A, and the phase, a, depend upon
Figure 2. Data points, gathered every year from 1947 to 1968,
reporting the number of musk ox on Nunivak Island, Alaska.
The curves show that both an exponential and an Opposition
the conditions of the population at a desi
t = 0. And the oscillation frequency,
is given by:
In Figure 3, Eq.9 is compared to empirical data. The
fluctuations are so large the authors plot-
ed are these:
s of
ces in which popu- lation
ure shows the population fluctuations of larch bud-
moth density [44] assembled from records gathered over
a period of 40 yr. The data points and lines connecting
them are shown in black. The smooth blue curve is a
graph of Eq.9 for particular values of the parameters.
We assumed
is negligibly small so it can be
ual to zero. The frequency,
is taken to be 2/(9 yr) =
0.7 per year. The vertical axis represents N. In the units
chosen for N, the amplitude, A, is taken to be 0.6 and c
is taken to be 0.6 per year2. The phase, a, is chosen so as
to insure a peak in the population in the year 1963; a =
3.49 radians.
Because the
d n0.1 as the ordinate for their data presentation. The
ordinate for the smooth blue theoretical curve is N.
Looking at the fit in Figure 3, suggests how population
potency may be deduced from empirical data. One is led
to conclude that the population strength, N(n), for the
budmoth varies as the 0.1 power of n. But the precision
of fit may not warrant this conclusion.
The conclusions that may be warrant
Considering that no information about the detail
dmoth life have gone into the computation the gra-
phical correspondence is noteworthy. It suggests that
those details of budmoth life are nature’s way of imple-
menting an overriding principle. The graphical corres-
pondence means that, under a constant external environ-
mental favorability, a population should behave not un-
like that of the budmoth.
Eq.9 admits of circumstan
tinction can occur. If A > c/w2 then N can drop to zero.
Societies with zero population are extinct ones. (On at-
Figure 3. Observational data on the population fluctuations of
larch budmoth density is shown as black circles and squares.
The smooth blue curve is a solution of Eq.9.
M. Chester / Open Journal of Ecology 1 (2011) 77-84
Copyright © 2011 SciRes. OPEN ACCESS
ing differential
initial conditions; fr-
se explored reveals that periodic population os-
he plethora of solutions to the governing dif-
monds) of Figure 5 record the
taining zero, N remains zero. The govern
Eq.8, doesn’t apply when N < 0.)
But the value of A derives from
N(t = 0) and g(t = 0). So depending upon the seed
population and its initial growth rate the population may
thrive or become extinct even in the presence of gift fa-
vorability, c. This result offers an explanation for the ex-
istence of the phenomenon of “extinction debt” [45] and
a way to compute the relaxation time for delayed exti-
The ca
llations can occur without a periodic driving force. Ev-
en a steady favorability can produce population oscilla-
Among t
rential Eq.8, is this one: Upon a step increase in envi-
ronmental favorability—say, in nutrient abundance—the
population may overshoot what the new environment
can accommodate and then settle down after a few cy-
cles. Figure 4 illustrates this behavior. That there are
such solutions amounts to a prediction that population
histories like that of Figure 4 will be found in nature. In
fact it has been found.
The data points (dia
pulation of Escherichia coli (using optical density, OD,
to measure it) maintained over 30,000 generations on a
nutrient containing citrate which it could not exploit [46].
Around generation 33,100 a mutation arose allowing a
Figure 4. A theoretical population history that can result f om
Figure 5. Fit of Opposition Principle curve to the data on a str-
occurs in the favorability of its environment.
sparate regimes of population
(2000) Universal laws and predictive
nd evolution. Oikos, 89, 403-408.
the Opposition Principle differential equation.
ain of E coli for which, because of a mutation, a step increase
strain of the species to exploit this nutrient component.
For those with this mutation the environment became
suddenly more favorable and they flourished. Shown in
the same figure is the theoretical curve arising from the
Opposition Principle Eq.8 with parameters
= 0.028
per generation and w = 0.023 per generation and n =
7. CON
We noted at least five di
history—each with it’s own individual and disjoint desc-
riptive equation: exponential-like growth, saturated growth,
population decline, population extinction, and oscillatory
behavior. Another regime, without a theory, is population
overshoot. It’s argued here that these regimes can be
brought under the embrace of a single differential equa-
tion describing them all.
That equation is the mathematical expression of gen-
eral concepts about how nature governs population be-
havior. Being quantitative it offers us a framework with
which to validate or refute these concepts. They are
itemized as axioms and principles (Section 4). Some of
these run counter to accepted convention thus making
empirical refutation a substantive matter. In short: a re-
futable proposition about the nature of populations is
offered for assessment by the scientific community.
Verification of the proposed equation would establish a
basic understanding about the nature of living organ-
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