In the mission statement for this Forum Alan Berryman asks"…whether there is or can be a fundamental theory of population dynamics." I propose the Opposition Principle as a candidate for that fundamental theory.

By Opposition Principle I mean this: The effect on the environment of evolutionary success is to alter that environment in a way that opposes the success. In what follows I will cast this principle in mathematical form so as to make it amenable to quantitative comparison with data from nature. A differential equation for population dynamics will be deduced. One of its solutions is compared to data.

This posting is in response to Mike Fowlers encouraging challenge: “This is an interesting statement, and I’d be even more interested to see how you’d frame it mathematically.”

The Principle concerns a society of living organisms that share an environment. The key feature of that society is that it consists of a number, n, of units which have the capacity and the drive to multiply by reproduction. Say a bacterial colony. A population of birds. Their number varies with time: n=n(t). We restrict ourselves to n=large»1 so that its integer nature is not pertinent to computations.

Because I don’t know whether population itself or some monotonic increasing function of n is the relevant parameter, let us define a population strength, s(n). With respect to its environment the population exhibits a certain strength in influencing it. This population strength, s(n), expresses the potency of the population in affecting the environment. Perhaps this strength, s, is just the number n, itself. The greater n is, the more the environmental impact. But it would take a lot of fleas to have the same environmental impact as one lion. So I would expect that the population strength is some function of n that depends upon the society under consideration.

Two things about s are clear. First, s(n) must be a monotonically increasing function of n; ds/dn > 0. This insures that if the population increases then its impact also increases. Albeit perhaps not linearly. Second, when n=0 so, too, is s=0. If the population is zero then certainly its impact is zero. One candidate for s(n) might be n raised to some positive power, p. If p=1 then s and n are the same thing. Another candidate is the logarithm of (n+1).

We need not specify the precise relationship, s(n), in what follows. Nature will tell us via experiment. The only way that s depends upon time is parametrically through its dependence on n. In what follows we shall mean by s(t) the dependence s(n(t)).

It’s convenient to define the population strength growth rate, g = g(t)

The colon indicates that the symbol on the left is defined to mean the expression on the right. Like s, g too acquires its time dependence parametrically through n(t). g = (ds/dn)(dn/dt)

Now we introduce the notion of environmental favorability. We’ll designate it by the symbol, f. Environmental favorability is what drives the growth rate to increase.

The variable, f may not be among those commonly recognized as a directly measurable quantity. Its usefulness is what justifies its introduction. At the moment f is understood simply by virtue of formula (2) above.

A blossoming population strength indicates evolutionary success. Mere growth, g, alone, is inadequate as a measure of success. The reason is that a falling growth implies that the population is not thriving even though the growth, itself, may be large. We take the rate of growth of the population strength – ‘the growth of growth’ – to define evolutionary success. The larger the one the greater the other. If dg/dt is large then the population is having great evolutionary success. In this formulation, of course, evolutionary success varies continuously in time.

Equation (2) says that evolutionary success is generated entirely by the favorability of the environment. In this view, then, f measures not only environmental favorability but also evolutionary success.

Based upon these notions we can caste the opposition principle as a mathematical statement. It 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. Here is the direct mathematical rendering of these two ideas:

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 redundant.)

Now we can integrate these partial differential equations to deduce that

The ‘constant’ (with respect to s and g) of integration, φ(t), has an evident interpretation. It is the gratuitous favorability provided by nature; the gift of nature. Equation (5) says that environmental favorability consists of two parts.

One part depends on the number and growth of the population being favored: the s(n) and its time derivative, g. This part has two terms both of which always act to decrease favorability. These terms express the opposition principle.

The other part is the gift of nature. It is independent of n. There must be something in the environment favorable to population growth but external to that population else the population would not exist in the 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 unexpected rains on a parched earth. All quite independent of the population under consideration.

Inserting equations (1) and (2) into (5) we arrive at the differential equation governing population dynamics under the opposition principle. It is this.

In the world of physical phenomena this equation is ubiquitous. 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 (6), yielding s(t) for any given environmental driving force φ(t), is known.

The easiest case is quite instructive. We suppose the gift favorability is simply constant over an extended period of time. Assume φ(t) = c independent of time. If α < 2w the solutions to (6) are periodic. Non-periodic solutions arise if α ≥ 2w. The periodic solution is this:

where the amplitude, A, and the phase, a, depend upon the conditions of the population at a designated time, say t=0. And the frequency, ω, is given by:

A graph of equation (7) for a particular hypothetical case is shown in the figure. We’ll assume 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 s. In the units chosen for s, the amplitude, A, is taken to be 0.6 and c is taken to be 0.6 yr^-2. The phase, a, is chosen so that there is a peak in the population in the year 1963, a=3.49 radians. The abscissa, t, is time for the years shown.

Because I am not very conversant with experimental results in population dynamics I rely for data on a graph in the paper mentioned by Alan Berryman in his posting, Real Data: The Larch Bud Moth

This graph is from that paper. It’s labeled “Population fluctuations of larch budmoth density”. Because the fluctuations are so large it is the 0.1 power of this LBM density that is plotted vs time.

I chose the parameters for the display in the first figure so as to make apparent its correspondence with the data in the second figure.

Since the authors plotted n^0.1 as their ordinate one might be led to conclude that the population strength, s(n), for the budmoth goes as the 0.1 power of n. But this conclusion would be in error. There is insufficient precision in the correspondence between the two graphs to warrant it.

The conclusions that may be warranted are these:

Considering that no information at all about the details of budmoth life have gone into the computation, the graphical correspondence is apparent. It suggests that those details of budmoth life are nature’s way of implementing an overriding principle. The graphical correspondence means that, under a constant external environmental favorabililty, a population should behave not unlike that of the budmoth.

Equation (7) admits of circumstances in which population extinction can occur. If A > c/w² then n can drop to zero. Societies with zero population are extinct ones. The governing differential equation, (6), doesn’t apply when n<0. On attaining zero n remains zero.

But the value of A derives from initial conditions; from s(t=0) and g(t=0). So depending upon the seed population and its initial growth rate the society may thrive or become extinct even in the presence of gift favorability, c.

This analysis reveals that periodic population oscillations can occur without a periodic driving force. Even a steady favorability produces population oscillations. The case explored demonstrates it.

Regardless of the details by which it is accomplished, there may be a general principle operative in determining population dynamics among living things. That is what is being proposed here. It remains for observational and experimental research to support it or refute it.

I am not expert in the field of population dynamics so I am curious to know whether this idea has a history. I need the learning of scholars in the field. Has such an idea been advanced before? Am I reinventing the wheel?

If not then I need a collaborator from among the experts. Someone who might gather the needed evidence to support or refute the proposal. Someone with whom to pursue implications. My email address is: chesterATphysics.ucla.edu