Analytical Note on Certain Rhythmic Relations in Organic Systems

Periodic phenomena play an important rôle in nature, both organic and inorganic.

In chemical reactions rhythmic effects have been observed experimentally, and have also been shown, by the writer¹ and others,² to follow, under certain conditions, from the laws of chemical dynamics.

However, in the cases hitherto considered on the basis of chemical dynamics, the oscillations were found to be of the damped kind, and therefore, only transitory (unlike certain experimentally observed periodic reactions). Furthermore, in a much more general investigation by the writer, covering the kinetics not only of chemical but also of biological systems, it appeared, from the nature of the solution obtained, improbable³ that undamped, permanent oscillations would arise in the absence of geometrical, structural causes, in the very comprehensive class of systems considered. For it seemed that the occurrence of such permanent oscillations, the occurrence of purely imaginary exponents in the exponential series solution presented, would demand peculiar and very specific relations between the characteristic constants of the systems undergoing transformation; whereas in nature these constants would, presumably, stand in random relation.

It was, therefore, with considerable surprise that the writer, on applying his method to certain special cases, found these to lead to undamped, and hence indefinitely continued, oscillations.

As the matter presents several features of interest, and illustrates certain methods and principles, it appears worth while to set forth the argument and conclusions here.

Starting out first from a broad basis, we may consider a system in the process of evolution, such a system comprising a variety of species of matter S₁, S₂….S_n of mass X₁, X₂….X_n. The species of matter S may be defined in any suitable way. Some of them may, for example, be biological species of organism, others may be components of the “inorganic environment.” Or, the species of matter S may be several components of an inorganic system in the course of chemical transformation.

We may think of the state of the system at an instant of time as being defined by statement of the values of X₁, X₂….X_n; of certain parameters Q defining the character of each species (in general, variable with time); and of certain other parameters P. The parameters P will, in general, define the geometrical constraints of the system, both at the boundaries (volume, area, extension in space), and also in its interior (structure, topography, geography); they will further define such factors as temperature and climatic conditions.

For a very broad class of cases, including those commonly treated in chemical dynamics, but extending far beyond the scope of that branch of science, the course of events in such a system will be represented by a system of differential equations of the form<mml:math display="block"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>…</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>;</mml:mi><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>…</mml:mo><mml:mo>.</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>dXidt=Fi(X1,X2….Xn;P,Q)(i=1,2….n)}(1)

If we restrict ourselves to the consideration of evolution at constant P's and Q's we may write briefly<mml:math display="block"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>…</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math>dXidt=Fi(X1,X2….Xn).(2)

The writer has elsewhere⁴ given a somewhat detailed discussion of the general case, in which no special assumption is made regarding the form of the functions F, that is to say, regarding the mode of physical interdependence of the several species and their environment.

We now proceed to consider a simple special case, as follows:

The system comprises

• 1. A species of organism S₁, a plant species, say, deriving its nourishment from a source presented in such large excess that the mass of the source may be considered constant during the period of time with which we are concerned.

• 2. A species S₂, for example a herbivorous animal species, feeding on S₁.

  In this case we have the following obvious relations

  Or, in analytical symbols,<mml:math display="block"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msubsup><mml:mi>A</mml:mi><mml:mn>1</mml:mn><mml:mo>′</mml:mo></mml:msubsup><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msubsup><mml:mi>A</mml:mi><mml:mn>1</mml:mn><mml:mo>″</mml:mo></mml:msubsup><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math>dX1dt=A1′X1−B1X1X2−A1″X1(5)<mml:math display="block"><mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>A</mml:mi><mml:mn>1</mml:mn><mml:mo>′</mml:mo></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mi>A</mml:mi><mml:mn>1</mml:mn><mml:mo>″</mml:mo></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>=(A1′−A1″)X1−B1X1X2(6)<mml:math display="block"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>=A1X1−B1X1X2(7)<mml:math display="block"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>=X1(A1−B1X2)(8)<mml:math display="block"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>dX2dt=A2X1X2−B2X2(9)<mml:math display="block"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>=X2(A2X1−B2)(10)The coefficients A₁, A₂, B₁, B₂ are in general functions of X₁ and X₂.

The reasons for selecting the form (5), (9) for the analytical formulation of (3), (4) require perhaps a little explanation. For small changes the rate of formation of new material of a given species of organism under given conditions is proportional to the existing mass of that species. In other words, the growth of living matter is a typically autocatakinetic⁵ process. This term has, therefore, been put in the form <mml:math><mml:msup><mml:mi>A</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math>A′ X₁ for the species S₁. Proportionality does not hold for large changes of X₁, X₂, and this is duly provided for in that <mml:math><mml:mrow><mml:msubsup><mml:mi>A</mml:mi><mml:mn>1</mml:mn><mml:mo>′</mml:mo></mml:msubsup></mml:mrow></mml:math>A1′ is a function of X₁, X₂.

Similarly the mass of matter rejected per unit of time from the species S₁ is proportional to X₁, and has been put in the form <mml:math><mml:mrow><mml:msubsup><mml:mi>A</mml:mi><mml:mn>1</mml:mn><mml:mo>″</mml:mo></mml:msubsup><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math>A1″X1, where A″ is in general a function of X₁, X₂.

Again, the mass of S₁ destroyed by S₂ feeding upon it will, for small changes, be proportional to X₂ and also to X₁. This term has, therefore, been set down in the form B₁X₁X₂. Here again the departures from proportionality are taken care of by the variations of B₁ with X₁ and X₂, of which variables B₁ is a function.

Similar remarks apply to the formulation (9) of (4). If there were no waste in the feeding process, and assuming that S₂ consumes no other substance than S₁, we would have B₁ = A₂; but in the more general case B₁ ± A₂. Approaching now the analytical treatment of equations (5), (9), or their equivalents (8), (10), we note first of all that there are two ways of satisfying the condition for equilibrium, namely:<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="italic">X</mml:mi><mml:mi mathvariant="italic">1</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>X1=X2=0(11)and<mml:math display="block"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mo>;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math>X1=B2A2;X2=A1B1(12)

We shall return later to the condition (11).

Condition (12) we will employ to define a new origin. Accordingly we introduce into (8), (10) the variables:<mml:math display="block"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math>x1=X1−B2A2(13)<mml:math display="block"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math>x2=X2−A1B1(14)and obtain<mml:math display="block"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>112</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>212</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math>dx1dt=a12x2+a112x1x2dx2dt=a21x1+a212x1x2(15)where<mml:math display="block"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>B</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math>a12=−B1B2A2(16)<mml:math display="block"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>112</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math>a112=−B1(17)<mml:math display="block"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math>a21=A1A2B1(18)<mml:math display="block"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>212</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>a212=A2(19)

Note the significant fact that in (15) the linear terms in the dexter diagonal are lacking. It is this circumstance which imparts an oscillatory character to the process.

For, since a₁₂ and a₂₁ are in general functions of x₁, x₂, let us expand them by Taylor's theorem and put<mml:math display="block"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>o</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mo>…</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math>a12=po+p1x1+p2x2+….(20)<mml:math display="block"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>o</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mo>…</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math>a21=qo+q1x1+q2x2+….(21)

A general solution of the system of differential equations (15) is then<mml:math display="block"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mi mathvariant="normal">1</mml:mi></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mo>…</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math>x1=P1eλ1t+P2eλ2t+P11e2λ1t+P22e2λ2t+….(23)<mml:math display="block"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mo>…</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math>x2=Q1eλ1t+Q2eλ2t+Q11e2λ1t+Q22e2λ2t+….(24)where <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">λ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>λ1,λ2, are the roots of the determinental equation for <mml:math><mml:mi mathvariant="normal">λ</mml:mi></mml:math>λ<mml:math display="block"><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant="normal">λ</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">o</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:msub><mml:mi mathvariant="italic">q</mml:mi><mml:mi mathvariant="normal">o</mml:mi></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant="normal">λ</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>|−λpoqo−λ|=0(25)that is to say,<mml:math display="block"><mml:mrow><mml:mi mathvariant="normal">λ</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="normal">±</mml:mi><mml:msqrt><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">o</mml:mi></mml:msub><mml:msub><mml:mi>q</mml:mi><mml:mi mathvariant="normal">o</mml:mi></mml:msub></mml:mrow></mml:msqrt></mml:mrow></mml:math>λ=±poqo(26)

Now, according to (20), (21) p_o, q_o are the equilibrium values of a₁₂, a₂₁. Hence, if we denote by Ā₁, B₂ the equilibrium values of A₁, B₂, i.e., those values which correspond to x₁ = x₂ = 0, then we have, by (16), (18)<mml:math display="block"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">o</mml:mi></mml:msub><mml:msub><mml:mi>q</mml:mi><mml:mi mathvariant="normal">o</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>A</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>B</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>poqo=−A¯1B¯2(27)and hence<mml:math display="block"><mml:mrow><mml:mi mathvariant="normal">λ</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:msqrt><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>A</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>B</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:msqrt></mml:mrow></mml:math>λ=±−A¯1B¯2(28)Now the coefficient B₂ is, in the nature of things, a positive number, as follows from its definition by (4), (9).

As regards the coefficient A₁, we have two possible alternatives.

If A₁ is negative for all values of X₁, X₂, then <mml:math><mml:mi mathvariant="normal">λ</mml:mi></mml:math>λ, as defined by (28), would be real; but this inference is nugatory. For B₁, like B₂ is, by definition (3), (5), an essentially positive quantity, and hence the equilibrium defined by (12) would in this case occur at a negative value of X₂. But this is physically impossible, since X₂ is a mass.

By referring to (5), (7) it will be seen that this case, in which A₁ is negative for all values of X₁, X₂, and in which an equilibrium of the type defined by (12) is physically impossible, corresponds to a species S₁ incapable of maintaining itself even in the absence of the tax placed upon it by the species S₂ feeding upon it. This is a case of minor interest.

If, on the contrary (12) can be satisfied by a positive value of A₁, so that an equilibrium of the type (12) is physically possible, then, evidently, by (28), λ is a pure imaginary. The solution (23), (24) then takes the form of Fourier's series; the process is an undamped oscillation continuing indefinitely.

In this connection, it is interesting to recall a passage in Spencer's “First Principles,” chapter 22, paragraph 173:

“Every species of plant and animal is perpetually undergoing a rhythmical variation in number—now from abundance of food and absence of enemies rising above its average, and then by a consequent scarcity of food and abundance of enemies being depressed below its average…………amid these oscillations produced by their conflict, lies that average number of the species at which its expansive tendency is in equilibrium with surrounding repressive tendencies. Nor can it be questioned that this balancing of the preservative and destructive forces which we see going on in every race must necessarily go on. Since increase of numbers cannot but continue until increase of mortality stops it, and decrease of numbers cannot but continue until it is either arrested by fertility or extinguishes the race entirely.”

A question now arises. Do the curves representing the solution (23), (24) dip below the zero axes of X₁, X₂? This would mean that one or the other, or both, of the species S₁, S₂ would become extinct through the violence of the oscillations.

To answer this question we consider the relation:<mml:math display="block"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math>dX2dX1=X2(A2X1−B2)X1(A1−B1X2)(29)which is obtained from (8) and (10) by division. From the periodicity of x₁, x₂ (and, therefore, X₁, X₂) it follows that the curve defined in rectangular coördinates X₁, X₂ by (29) is a closed curve. Furthermore, this curve can never cross the X₁ axis, for at all points of this axis the first, and all the higher derivatives of X₂ with regard to X₁ vanish, as can be seen from (29) directly and by successive differentiations.

Similarly it can be seen that the curve defined by (29) can never cross the X₂ axis.

Hence, if any point on any integral curve of (29) lies within the positive quadrant, the whole of that curve lies in that quadrant. Thus the oscillations can never exceed the limits of positive values X₁, X₂.

We conclude, therefore, that under the conditions of the problem as here set forth, neither the species S₁ nor the species S₂ can become extinct through severity of the oscillations alone. In practice the eventuality might arise, however, that in the course of these oscillations one or the other species might be so thinned out as to succumb to any extraneous influence that might arise such as has not been taken into account in our present considerations.

We return now briefly to the consideration of the equilibrium defined by the equation<mml:math display="block"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>X1=X2=0(11)Applying here the criterion set forth by the author elsewhere,⁶ it is seen that when A₁ is positive the determinental equation for <mml:math><mml:mi mathvariant="normal">λ</mml:mi></mml:math>λ has at this point two real roots of opposite sign, which is characteristic of unstable equilibrium. If, on the other hand, A₁ is negative in the neighborhood of the origin of X₁, X₂, then the equilibrium here is found to be stable, the two roots for <mml:math><mml:mi mathvariant="normal">λ</mml:mi></mml:math>λ being both negative.

In conclusion it may be remarked that a system of equations identical in form with (8), (10) is obtained in the discussion of certain consecutive autocatalytic chemical reactions. Here, however, the coefficients A, B are constants and the integration can be reduced to a quadrature. Aside from a certain number of periodic reactions which have been observed more or less as laboratory curiosities, a certain interest is also attached to this matter from the fact that rhythmical reactions (e.g., heartbeat, which may continue after excision), play an important rôle in physiology. We cannot, of course, say whether in such case geometrical (structural) features are the dominating factors.