A Law of Nature?

Is there an overriding principle of nature, hitherto overlooked, that governs all population behavior? A single principle that drives all the regimes observed in nature - exponential-like growth, saturated growth, population decline, population extinction, oscillatory behavior? In current orthodox population theory, this diverse range of population behaviors is described by many different equations - each with its own specific justification. The signature of an overriding principle would be a differential equation which, in a single statement, embraces all the panoply 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 population's success is to alter that environment in a way that opposes the success.

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 profusion of chemical data into an all encompassing tabular statement of principle. This undertaking led to the understanding 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 cherished 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 the theory. Einstein's laws of nature absorbed Newton's. Relativity spawned nuclear energy, a deeper understanding 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 equation that OppositionPrinciple p. 4 January 1, 2012 brings together the disparate domains of population behavior. We suggest that the panoply of population behaviors all issue from a single principle.
In current orthodox population theory, the diverse range of population behaviors is described by many different 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 (Lotka 1956, Volterra 1926 or, because their solutions are not structurally stable, their later modifications (Murray 1989, Vainstein et al 2007. And none of these describe population decline, nor population extinction. Contemporary theory offers no overriding principle that governs the entire gamut of population behaviors. As long ago as 1972 (Ginzburg 1972), in a challenge to orthodox convention, L. R. Ginzburg took the bold step of proposing that population dynamics is better represented by a second order differential equation. All accepted formulations relied on first order differential equations as they still do today.
Suppose the family of solutions to a single second order equation should match population behavior just as well as the many accepted first order equations do. That family of solutions have in common their single progenitor. Embedded in them is the principle that generated them.
When the family of solutions to a differential equation is found to fit empirical reality then that equation is expressing a truth about nature. It can OppositionPrinciple p. 5 January 1, 2012 give us insights and enable us to make predictions. Producing a second order equation whose solutions characterize a variety of population behavior is equivalent to uncovering a principle of nature governing those populations.
In the following we take a route different from Ginzburg'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 coalescence: a tool for better understanding nature.

Traditional Perspective
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 increase 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: It is a first order differential equation and when R is constant, its solution yields the archetypical equation of exponential growth.
OppositionPrinciple p. 6 January 1, 2012 Now, common experience tells us that exponential growth 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 (Holt 2009). 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" (Berryman 2003). The concept is that exponential growth is always taking place but at a rate that varies with time. The idea is ubiquitous in textbooks. (Britton 2003, Murray 1989, Turchin 2003, Vandermeer 1981 An object example of this process is provided by the celebrated Verhulst equation. Here the constant, r, is the exponential growth factor and K is the limiting value that n can have -"the carrying capacity of the environment" (Vainstein et al. 2007). The equation insures that n never gets larger than n MAX = K. A population history, n vs. t, resulting from this first order differential equation is the black one of Figure 1.The Verhulst equation -often cited as the Logistics Equation -is regularly embedded in research studies. (Nowak 2006;Torres et al. 2009;Jones 1976;Ruokolainen et al. 2009;Okada et al. 2005; Ma 2010) p. 7 January 1, 2012

Shortcomings of the traditional perspective.
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 certainly appealing. But it has this serious failing: the practice forbids description of several known regimes of population behavior.
It denies further conceptual coalescence. For example, unless R is taken as imaginary the observation of population oscillations cannot be described in this formalism.
Another proscribed regime is extinction. A phenomenon well known to exist in nature is the extinction of a species. "... over 99% of all species that ever existed are extinct" (Carroll 2006). But there exists no finite value of Rpositive or negative -that yields extinction! It cannot be represented 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 conditions 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 Equation (the Logistics Equation). As
Verhulst himself pointed out (Verhulst 1838)

Conceptual foundations for an overriding structure
We seek a mathematical equation to embrace all of the great variety of population behaviors. The equation is built on some foundational axioms.
Empirical verification of the equation they produce is what will measure the validity of these axioms. The axioms are: First: Variations in population number, n, are due entirely 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 single lifetime or over many generations. propose to take as a measure of success; the growth in the growth 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 advocated in the past. (Ginzburg and Colyvan 2005) 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." (Owen-Smith

2002)
Another corollary is this: The environment of one population is other populations. It's through this mechanism that interactions among

The Opposition Principle: quantitative formulation
Based on the understandings outlined above we propose that an overriding 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 society is that it consists of a number, n, of members which have an inherent drive to survive and to produce offspring 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, 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 OppositionPrinciple p. 14 January 1, 2012 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. Another 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 nature. The only way that N depends upon time is parametrically 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 defined by Like N, g too acquires its time dependence parametrically through n(t). 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 favorability; 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 mathematical rendering of these two statements: We can implement these statements by introducing two parameters. Both w and α are non-negative real numbers and they have the dimensions of reciprocal time. (Negative w values are permitted but are redundant.) !f !N = "w 2 and !f !g = "! (6) These partial differential equations can be integrated. The result is: The 'constant' (with respect to N and g) of integration, F(t), has an evident interpretation. It is the gratuitous favorability provided by nature; the gift of nature. Equation (7) says that environmental favorability consists of two parts.
One part depends on the number and growth of the population being Inserting equations (3) and (4) into (7) we arrive at the promised differential equation governing population dynamics under the Opposition Principle. It is this.
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 other phenomena. So it is very well studied. The exact analytical solution to (8), yielding N(t) for any given F(t), is known.

Fits to empirical data.
To explore some of the consequences of this differential equation we consider the easiest case; that the gift favorability is simply constant over an 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 Alaska (Spencer and Lensink 1970). The data, gathered every year from 1947 to 1968, is in Table 1   If α < 2w the solutions to (8) are periodic and are given by: where the amplitude, A, and the phase, a, depend upon the conditions of the population at a designated time, say t=0. And the oscillation frequency, ω, is given by: In Figure 3, equation (9) is compared to empirical data. The figure shows the population fluctuations of larch budmoth density  assembled from records gathered over a period of 40 yrs. The data points and lines connecting them are shown in black. The smooth blue curve is a graph of equation (9) for particular values of the parameters.
We assumed α is negligibly small so it can be set equal to zero. The frequency, ω, is taken to be 2π/(9yrs) = 0.7per 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 year 2 . The phase, a, is chosen so as to insure a peak in the population in the year 1963; a=3.49 radians. Because the fluctuations are so large the authors plotted n 0.1 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 warranted are these: But the value of A derives from initial conditions; 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 favorability, c. This result offers an explanation for the existence of the phenomenon of 'extinction debt' (Kuussaari et al. 2009) and a way to compute the relaxation time for delayed extinction.
The case explored reveals that periodic population oscillations can occur without a periodic driving force. Even a steady favorability can produce population oscillations.

Conclusion
We noted at least five disparate regimes of population history -each with it's own individual and disjoint descriptive 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 equation describing them all.
That equation is the mathematical expression of general concepts about how nature governs population behavior. 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.