MATHEMATICAL THEORY OF THE STRUGGLE FOR EXISTENCE1

Vito Volterra, Rome.

§ 1. There exist many applications of mathematics to biology. In the first place one may point to investigations of physiological questions connected with the sense organs, with blood circulation, and with the movement of animals; these investigations may be regarded as chapters of optics, acoustics, hydrodynamics, and the mechanics of a rigid body, and therefore these works could not have led to the appearance of new methods lying outside the circle of the classical methods of mathematics and physics. Conversely, biometrics, with its distinctive methods, applied the theory of probability and created a whole series of new and original methods of investigation2. The most recent geometrical investigations concerning the form and growth of organized beings also have an original character. Geometry was applied here to describe the forms themselves and their development, just as it had already long before the present time been applied in astronomy to describe the orbits and motions of heavenly bodies3. It is to be hoped that the methods, con—

data connected with the analysis of heredity may also be applicable to problems of biology1. Leaving aside other applications of mathematics, in the present article I shall speak of those applications which I regard as a typical problem of a special kind and which remain to be studied and developed further2.

These applications of mathematics can clarify various interesting data that are important at the present time for biologists.

§ 2. Biological associations (biocenoses) are formed by several species of animals living in one and the same environment. Usually the various individuals of these associations struggle for possession of one and the same food, and sometimes certain species of animals live at the expense of others, on which they feed. Nor is the possibility excluded that living beings may help one another. All this falls under the concept of the general phenomenon called the struggle for existence.

The quantitative character of this phenomenon is reflected in changes in the number of individuals composing the various species. Under certain conditions these changes consist in fluctuations of the number of individuals about certain mean values,

…in other cases they cause the disappearance or continuous increase in the number of individuals of the species. The study of these variations and of these diverse changes is important theoretically, but in many cases this study also has enormous practical importance, as we have in the case of different species of fish living in the same seas, the change in whose numbers is of interest to industry1. In exactly the same way, the change in the number of plant parasites is of interest to agronomy in the case when these parasites wage a struggle for existence with their own parasites. Infectious diseases (malaria, etc.) also show changes which depend, in all probability, on similar causes.

The question appears very complicated. Indeed, in the surrounding world there exist periodic causes, such as, for example, causes depending on the succession of the seasons and producing forced oscillations or oscillations of an external character in the number of individuals of various species.

These external periodic actions have been specially studied from the standpoint of statistics, but the question arises—are there not causes of an internal character, having their own periods—causes which would exist even if

external periodic causes would cease to act, and which are superimposed upon these external causes.

Observation gives grounds for thinking that this is so, and mathematical calculations confirm it, as we shall see in the present article. However, at first glance it may seem that, owing to the extreme complexity of the question, it is impossible to treat it mathematically, and that mathematical methods, because of their subtlety, can reveal only separate parts and details of the phenomena and may leave unexplained the very thing that is essential in the present question. To avoid this danger, it is necessary to proceed from hypotheses, perhaps crude but simple, and to schematize the phenomenon.

We shall begin our study with an investigation of a case that may be called a pure phenomenon, depending only on internal causes—a phenomenon determined only by the capacity for reproduction, on the one hand, and by the voracity of species, on the other. We shall regard these causes as the only ones acting. Then we shall study the simultaneous existence of these phenomena with an external periodic action, depending on the actions of the environment in which the animals live.

§ 3. What mathematical methods should be applied here? One may apply methods based on the theory of probabilities, which involuntarily come to mind first of all. I shall say from the very beginning that these are not the methods we shall use to solve the present question. I shall take the liberty of indicating how this question may be considered. Let us try to express in words the phenomena that characterize, in rough outline, the process under study, and then translate these words into the language of mathematics. This will allow us to establish the differential equations of the phenomenon. If, subsequently, we apply the methods of mathematical analysis, we shall go much further than ordinary spoken language and ordinary reasoning allow us to do, and we can formulate exact mathematical laws that do not contradict observations; at least the most important law remains in full agreement with the statistical resul-

...by predators¹). Thus, the path that we shall follow is indicated quite clearly by the preceding general considerations. We shall see below how the difficulties encountered can be overcome.

§ 4. Let us take one species of animals and suppose that it increases and decreases continuously, i.e. let us suppose that the number \(N\) of individuals of this kind is not an integer, but is represented by a positive number varying continuously. Generally speaking, births occur at definite moments, in definite intervals of time separated from one another. We shall neglect these circumstances, assuming that births occur continuously at every moment and that, other things being equal, the number of births is proportional to the number of individuals of the given species existing at that moment. The same we may say with regard to mortality, and depending on whether the birth rate is greater than the death rate or conversely, we shall be dealing either with an increase or with a decrease in the number of individuals of the species. In addition, we shall assume the homogeneity of the individuals of each kind, neglecting variations of age and growth. If we are dealing with only one species, or if another animal species does not affect the development of the species under study, so that the circumstances connected with birth rate and mortality are constant, we obtain for the rate of increase of the kind, i.e. for the number of individuals by which the kind increases per unit time, the expression:

\[ V = nN - mN = (n - m)N, \]

where \(n\) is the coefficient of birth rate, \(m\) the coefficient of mortality, and both these quantities are constant.

¹ As we shall see in detail in § 7, D’Ancona finds, from a statistical study of the markets of Trieste, Venice, and Fiume, that during the war in the northern part of the Adriatic Sea there was a relative percentage increase in selachians, which should be regarded as the most predatory kind. These results agree with the law of perturbation of mean values, which we shall set forth below.

If we assume that \(n-m=\varepsilon\), we shall have the expression:

\[ V=\varepsilon N, \]

from which one can obtain the well-known law of exponential change in the number of individuals of a species, so that, if time increases in an arithmetic progression, the number of individuals increases in a geometric progression1. The quantity \(\varepsilon\) is called the coefficient of increase of the species. If this quantity is positive, the geometric progression will be increasing. If this quantity is negative, then the geometric progression will be decreasing, so that in the first case the species increases numerically, while in the second case it dies out.

Fig. 1.

We may give a geometric interpretation of the indicated relations. Suppose the first species finds in the surrounding environment sufficient food, so that the coefficient of increase will be constant and positive. If \(N_1\) represents the number of individuals, the exponential curve of Fig. 1, having time \(t\) as abscissa and \(N_1\) as ordinate, gives us a representation of the change in the number of individuals of the given species in the case when these individuals remain alone in the environment. Thus the curve of Fig. 1 may be represented by the equation:

\[ V_1=\varepsilon_1 N_1, \]

where \(V_1\) is the rate of increase in the number of individuals of the species.

\[ \frac{dN}{dt}=\varepsilon N, \]

whence, separating the variables, one obtains

\[ N=N_0 e^{\varepsilon t}. \]

If \(t\) changes in an arithmetic progression, then \(N\) changes in a geometric one. (P. L.).

MATHEMATICAL THEORY OF THE STRUGGLE FOR EXISTENCE

We can easily determine the time \(t_1\) necessary for the number of individuals to increase from \(N_1\) to \(2N_1\). The construction for this case is shown in Fig. 1. The time \(t_1\) is independent of the initial value \(N_1\) and depends only on the coefficient \(\varepsilon_1\). In fact, the time \(t_1\) is determined by the quantity:

\[ \frac{\lg 2^{1)}}{\varepsilon_1}. \]

Consider a second genus, which does not find food for itself in the environment, so that if it were living alone in the environment, its coefficient of growth, \(-\varepsilon_2\), would be constant and negative (\(\varepsilon_2\) may be called the coefficient of extinction). If \(N_2\) represents the number of individuals, then the exponential curve of Fig. 2 will represent the changes over time in the number of individuals of the species when it lives alone in the environment. The course of the curve shows in an obvious way the continuous extinction of the genus. This curve geometrically represents the equation \(V_2 = -\varepsilon_2 N_2\), where \(V_2\) is the negative rate of increase of the species.

Fig. 2.

Quite analogously to the preceding case, one may obtain the time during which the species decreases by half. The construction in Fig. 2 gives the time \(t_2\), during which the number of individuals of the species from \(N_2\) becomes \(\dfrac{N_2}{2}\). The time \(t_2\) is likewise independent of the initial value \(N_2\) and is equal to:

\[ \frac{\lg 2}{\varepsilon_2}. \]

\(^{1)}\) From the equation \(\dfrac{dN_1}{dt}=\varepsilon_1 N_1\) we have: \(\lg N_1=\varepsilon_1 T_1 + A\) and \(\lg 2N_1=\varepsilon_1 T_2 + A\); whence \(\lg 2=\varepsilon_1(T_2-T_1)=\varepsilon_1 t_1\); \(\lg\) denotes the sign of the Napierian logarithm. (N. K.)

§ 5. Let us suppose that the species live together and that individuals of the second species feed at the expense of individuals of the first species. What will happen in the biological association? Let us try to express in words what may be expected in this phenomenon: it is evident that the coefficient of increase \(\varepsilon_1\) of the first species will be changed, and moreover it will no longer be constant and will be the smaller, the more numerous are the individuals of the second kind, which devour individuals of the first kind: the coefficient \(\varepsilon_1\) may even become negative. The number \(-\varepsilon_2\) also will no longer be constant; but it will increase (it may even change its sign), and will be the greater, the more numerous are the individuals of the first species, since with an increase in the latter the amount of food for individuals of the second species increases. We can now translate what has been said into mathematical language by replacing the constant quantity \(\varepsilon_1\) by a quantity which decreases when \(N_2\) grows, and in a first approximation we may replace \(\varepsilon_1\) by \(\varepsilon_1-\gamma_1 N_2\), and \(-\varepsilon_2\) by \(-\varepsilon_2+\gamma_2 N_1\), where \(\gamma_1\) and \(\gamma_2\) are positive coefficients. If \(V_1\) and \(V_2\) represent the rates of increase in the number of individuals of both species, i.e. represent the number of individuals of the two species by which these species increase per unit time, then the two preceding equations must be replaced by the following:

\[ V_1=(\varepsilon_1-\gamma_1 N_2)N_1 . \tag{1} \]

and

\[ V_2=(-\varepsilon_2+\gamma_2 N_1)N_2 . \tag{2} \]

These two equations must be considered simultaneously. Thus one obtains what are called two simultaneous differential equations. Starting from these equations, analysis makes it possible easily to obtain a definite relation between \(N_2\) and \(N_1\).

We can form a clear idea of the relation between the quantities \(N_2\) and \(N_1\) by means of a geometrical interpretation of the phenomena. Indeed, from equations (1) and (2) it is easy to find:

\[ \varepsilon_2 V_1\left(\frac{\gamma_2}{\varepsilon_2}-\frac{1}{N_1}\right) +\varepsilon_1 V_2\left(\frac{\gamma_1}{\varepsilon_1}-\frac{1}{N_2}\right)=0 . \]

If we divide this equation by \(\varepsilon_1\varepsilon_2\) and put \(\dfrac{\varepsilon_2}{\gamma_2}=k_1\) and \(\dfrac{\varepsilon_1}{\gamma_1}=k_2\), then the preceding equation will be transformed into the following:

\[ \frac{1}{\varepsilon_1} V_1\left(\frac{1}{k_1}-\frac{1}{N_1}\right) +\frac{1}{\varepsilon_2} V_2\left(\frac{1}{k_2}-\frac{1}{N_2}\right)=0 . \tag{3} \]

Let \(x\) and \(y\) be two quantities which vary with velocities \(V_x\) and \(V_y\) such that

\[ V_x=V_1\left(\frac{1}{k_1}-\frac{1}{N_1}\right) \quad \text{and} \quad V_y=V_2\left(\frac{1}{k_2}-\frac{1}{N_2}\right). \]

The relations between \(x\) and \(N_1\), and \(y\) and \(N_2\), are represented by curves I and II (Fig. 3)1.

The minima of their ordinates correspond to the points \(M_1\) and \(M_2\), where \(N_1=K_1\) and \(N_2=K_2\).

They become infinite when the abscissas are equal to zero or infinity.

On the other hand, equation (3) may be rewritten as follows:

\[ \frac{V_x}{\varepsilon_1}+\frac{V_y}{\varepsilon_2}=0, \]

whence one obtains

\[ \frac{x}{\varepsilon_1}+\frac{y}{\varepsilon_2}=\text{const.} \]

The straight line \(A_x A_y\), which intersects the \(x\)- and \(y\)-axes at distances from the origin proportional to \(\varepsilon_1\) and \(\varepsilon_2\), represents this relation. Draw the tangents \(M_1P\) and \(M_2Q\) to the two curves (I) and (II) at the points \(M_1\) and \(M_2\), and through the points where they meet the straight line \(A_xA_y\), draw straight lines parallel to the \(y\)- and \(x\)-axes. These lines meet curves II and I at the points \(P'\), \(P''\) and \(Q'\), \(Q''\).

If through \(P'\) and \(P''\) we draw straight lines parallel to \(y\), and through \(Q'\) and \(Q''\)—straight lines parallel to \(x\), we obtain a rectangle. The curve \(\Lambda\), which represents the relation between \(N_1\) and \(N_2\) and which therefore has abscissas \(N_1\) and ordinates \(N_2\), is enclosed in this rectangle and is tangent to the sides of the rectangle at the points \(M_1'\), \(M_1''\), \(M_2'\), \(M_2''\), where the straight lines parallel to the axes and drawn through the points \(M_1\) and \(M_2\) meet the sides of the rectangle.

Fig. 3.

Hence it may be inferred that \(N_1\) and \(N_2\) oscillate between positive values, which represent the рас-

distances of the sides of the indicated rectangle from the axes parallel to them, through any point \(R\) belonging to the segment \(PQ\). Let us draw straight lines parallel to the axes, just as through the four points where they meet curves I and II. These latter straight lines form a rectangle whose vertices belong to the curve \(\Lambda\). By changing the position of the point \(R\), one can in this way find as many points of the curve \(\Lambda\) as desired.

This circumstance makes it possible easily to obtain the curve \(\Lambda\), and it turns out to be a closed cyclic curve. If we denote by \(T\) the time during which a complete cycle is traversed, then at the end of this period the quantities \(N_1\) and \(N_2\) again assume their initial, original values, so that the final conditions are the same as the initial ones. The phenomenon is thus periodic, and the period \(T\) can be found by using a formula of integral calculus expressing this period as a function of \(\varepsilon_1, \varepsilon_2\) and the initial conditions.

If small fluctuations are involved, then the period is given approximately by the formula

\[ T = 2 \frac{2\pi}{\sqrt{\varepsilon_1 \varepsilon_2}} = 9{,}06 \sqrt{t_1 t_2}, \]

from which it follows that the period of the fluctuation is proportional to the geometric mean of the two times during which, respectively, the first species doubles in number, and the second decreases by half.

One may represent, by means of a graph, the changes of \(N_1\) and \(N_2\) with time. This relation is given by Fig. 4, in which times are plotted along the abscissas, while \(N_1\) and \(N_2\) are ordinates. If we change the initial conditions, the cycle changes.

In order to construct different cycles, it is sufficient to move the straight line \(A_xA_y\) parallel to itself. In this way, moving \(A_xA_y\) into the position \(B_xB_y\), we obtain in Fig. 3 a new cycle \(\phi\), and in the same way one could obtain as many cycles as desired.

The point \(\Omega\), lying inside this infinite system of curves, has as its coordinates the ratios:

\[ K_1=\frac{\varepsilon_2}{\gamma_2}\quad \text{and}\quad K_2=\frac{\varepsilon_1}{\gamma_1}; \]

we shall soon see the significance of the numbers \(K_1\) and \(K_2\).

§ 6. We shall first consider the significance of the quantities \(\varepsilon_1, \varepsilon_2, \gamma_1\) and \(\gamma_2\). It is obvious that the quantity \(\varepsilon_1\) is the coefficient of increase in the number of individuals of the first species under the assumption that it exists alone in the environment, and \(\varepsilon_2\) is the coefficient of disappearance of individuals of the second species under the assumption that it too exists alone in the environment.

Fig. 4.

As for the quantities \(\gamma_1\) and \(\gamma_2\), it may be seen directly that they increase together with the predacity and voracity of the second species and decrease when the means of protection in the first species become greater. In other words, they may be called the coefficients of predacity. With respect to \(K_1\) and \(K_2\), one can see that if \(N_1=K_1\) and \(N_2=K_2\), then \(V_1\) and \(V_2\) are equal to zero; \(K_1\) and \(K_2\) are the numbers of individuals corresponding to the stationary state, i.e. the state in which both species neither increase in number nor decrease. On the other hand, although the curves of Fig. 3 are not symmetrical with respect to the point \(\Omega\), it can be proved that \(K_1\) and \(K_2\) are the mean values of \(N_1\) and \(N_2\) over one period.

Thus one may conclude: 1) that the fluctuations in the number of individuals of the species are periodic, and 2) that the mean value of the number of individuals of the two species does not depend on the initial condi-

whether, and only if, the coefficient of increase and the coefficient of predation remain the same, i.e., if one starts from an initial state in which there is a greater or lesser number of individuals, then the mean value of the number of individuals, after the cycle corresponding to the period has been completed, will remain the same. On the other hand, the magnitudes of the periods will change when the initial conditions change.

Now let us suppose that we shall artificially destroy individuals of the two species. For example, if the matter concerns fish, then let us suppose that they are being caught; then $\varepsilon_1$—the coefficient of increase of the first kind—will decrease, while $\varepsilon_2$—the coefficient of decrease of the second kind—will increase. Thus, if the voracity of the first species and the defensive means of the second do not change, $K_1$ will increase, and $K_2$ will decrease, i.e., the mean number of individuals of the first kind (the species being eaten) will increase and the mean number of individuals of the second kind (the species that eats) will decrease. Hence follows the third law: 3) if one strives to destroy, at one and the same time, individuals of both species, then the mean number of individuals of the species being eaten increases, and the number of individuals of the species that eats decreases.

§ 7. As was indicated above, this third law is in complete agreement with the result obtained from the study of the state of fishing in the northern parts of the Adriatic Sea. D’Ancona investigated the statistics of the markets of Venice, Trieste, and Fiume, which collect a large part of the fish catch in the northern parts of the Adriatic Sea. The statistics concerned periods preceding and following the war, and he was able to observe that toward the end of the war predatory species had become predominant, chiefly selachians (Acanthias vulgaris, Scyllium sp., Mustelus sp., Squatina angelus, Tigon sp., Myliobatis sp., Raja sp.), and, conversely, a relatively smaller quantity of other, less predatory species of fish was observed. D’Ancona explains these facts by saying that the conditions of fishing during the period 1914–1918 temporarily shifted the equilibrium among the various biolo-

gical species in favor of those most predatory at the expense of those most defenseless among the fishes, which are economically the most important. Hence it follows that the biological equilibrium which can become established among the fish species of the Adriatic Sea has shifted, owing to fishing, toward the least protected: the cessation of fishing during the period of the war restored, on the contrary, the original conditions, i.e., produced an increase in predatory fishes.

§ 8. It is easy to understand that the indicated ratio is justified up to a certain limit, since beyond a known boundary the two species tend toward disappearance. The boundary above which the causes acting destructively upon both species are most favorable for the preyed-upon genus can easily be computed in advance, and it can be shown that, if this boundary is crossed, then the two species diminish in number; but if this boundary is only reached, then the predatory species disappears, while the preyed-upon tends toward a limit smaller than the average value previously attained. In other words, there exists an upper limit, which, however, is not a maximum.

§ 9. We have assumed that when two species live together—$\varepsilon_1$ and $\varepsilon_2$ must respectively decrease and increase when the values $N_1$ and $N_2$ respectively increase, which in fact does occur and which compelled us to replace $\varepsilon_1$ by the expression $\varepsilon_1-\gamma_1 N_2$ and $-\varepsilon_2$ by the expression $-\varepsilon_2+\gamma_2 N_1$. The circumstance that we have taken these quantities to be linear with respect to $N_1$ and $N_2$ gives us not only a first rough representation of the phenomenon, but is also confirmed by the fact that the rate of increase of the two genera must be measured proportionally to the probability of encounters between two species of individuals and, consequently, will be proportional to the product $N_1$ and $N_2$, which is precisely proportional to the number of encounters of the species. Equations (1) and (2), thus, are rigorously proved.

It is possible to investigate the same equations by making all possible hypotheses about the values of the coefficients $\varepsilon_1$, $\varepsilon_2$, $\gamma_1$ and $\gamma_2$, and then one has to consider various cases,

when encounters between different individuals of two genera are favorable or unfavorable for them.

If we imagine how many cases of interest for medicine may be brought under a phenomenon that depends on encounters and mutual action between different species of living beings (the human species and a pathogenic germ, the species on which the parasite lives, and the parasitizing species), then it is easy to understand in what relation the variations of epidemics may stand to the theories that we have set forth here.

§ 10. If we consider only two species living together, then we restrict the problem extremely. It is possible to treat mathematically the general case in which an arbitrary number of species, acting upon one another, live together. Let their number be equal to \(n\), and suppose that the encounter of two individuals of different species produces either a favorable result for the species to which one of the individuals belongs and an unfavorable one for the species to which the other individual belongs, or else produces a zero result for both species. Let us take two of these species, for example the first and the second, and determine the ratio between the number of individuals by which one species, for example the first, will increase, and the number of individuals by which the other species, for example the second, will decrease, as a consequence of their mutual encounters, during which the first species will devour individuals of the second kind, which will produce a decrease of some species and an increase of others proportionally to the food that they thereby obtain. Suppose that the indicated ratio is always expressed by the quantity \(\frac{\gamma_1}{\gamma_2}\), in which \(\gamma_1, \gamma_2 \ldots \gamma_n\) represent \(n\) positive numbers corresponding to the first, second, ... and \(n\)-th species. Suppose also that, by finding the ratio between these same numbers, one can obtain analogous ratios relating to encounters of individuals of the corresponding species. Under this hypothesis the numbers \(\gamma_1\) and \(\gamma_2 \ldots \gamma_n\) form equivalents of individuals of the different species. Indeed, the supposition,

that individuals of the first species, owing to their voracity, can destroy \(\gamma_2\) individuals of the second kind, increasing at the same time by the number \(\gamma_1\), is equivalent to saying that \(\gamma_1\) of the first kind is equivalent to \(\gamma_2\) of the second kind.

The same thing may be expressed in another way, by calling the quantities \(\beta_1=\dfrac{1}{\gamma_1},\ \beta_2=\dfrac{1}{\gamma_2},\ldots,\ \beta_n=\dfrac{1}{\gamma_n}\) the common value of isolated individuals of the 1st, 2nd ... kinds, and, if \(N_1\) and \(N_2\) are the numbers corresponding to the number of individuals of the various species, by calling the expression \(W=\beta_1N_1+\beta_2N_2+\cdots+\beta_nN_n\) the magnitude of the biological association.

In this case the preceding hypothesis leads to the supposition that encounters of two species do not change the magnitude of the biological association.

We may say that an association of this kind is conservative. The change in the number of individuals of the various species is in this case determined by a system of simultaneous differential quadratic equations. Their properties depend on the properties of a certain determinant, and one must distinguish the case in which we are dealing with an even number of species from that with an odd number. In both these cases integrals of the differential equations can be found.

§ 11. If the number of species is even, then three laws are easily obtained, which are generalizations of the laws indicated above. The first law says: in the case of a conservative biological association of order \(n\), for which there exists a stationary state, the variations in the number of individuals of the various species are enclosed between positive numbers, and there always exist fluctuations of these numbers, which do not die out.

In this generalization, the properties of periodicity are lost, and there remains only fluctuation in the number of individuals.

When the fluctuations are small, they can be obtained by approximate calculation, by superposing \(\dfrac{n}{2}\) undamped fluctuations, each of which has its own period, independent of the initial conditions.

The second law remains unchanged in the case when, as the mean value of the number of individuals of different species, one takes the limits of the mean values for an interval of time equal to infinity (asymptotic mean values). As for the third law, it assumes the following form: if, in a conservative association of even order, in which there are predator species and prey species, all the species tend toward destruction, then among these latter there always exist species whose mean asymptotic number of individuals increases. At the same time, among the predator species there always exist certain species whose mean asymptotic value decreases1.

When the number of species of the conservative system is odd, it is impossible for the number of individuals of each species to be enclosed between two positive numbers; therefore the association is unstable2.

§ 12. The case of a conservative system may be regarded as a limiting case approached by natural associations; but still closer to actually existing systems, apparently, are dissipative associations, the value of \(\beta_1, \beta_2 \ldots \beta_n\) in which decreases at such an encounter of individuals of two species, in which the devouring of one species by the other follows. Then one can see that fluctuations die out around the stationary state and that the system tends by itself toward the stationary state.

§ 13. In a similar way one may study the internal causes of fluctuations, which are sufficient to explain various observed phenomena and to

...to predict new ones which, very probably, can be subjected to experimental or observational control. We have already spoken earlier of external periodically acting causes. They can be computed by assuming that the coefficient of increase of the species is periodic, instead of remaining constant; and then, in the case of small fluctuations, the principle of superposition of the proper variations and the forced variations holds, so that small fluctuations can be obtained by superposing upon the proper variations the forced variations, whose period is that of the coefficients of increase of the species, in the case when this latter does not coincide with any of the periods of the proper fluctuations.

§ 14. A case which can be studied mathematically to the end and in the greatest detail is that in which three species of animals live in a bounded environment, such as, for example, an island, the first species feeding on the second and the second on the third. As an example we may take carnivorous animals that feed on a certain species of herbivorous animals, which in turn feed on a certain species of plants, if we allow that for the plants we apply the very same method of investigation as has been applied to the animals. The same device may be applied to insects parasitic on plants and to the parasites of these insects.

We shall next present various cases and special forms of these cases which may occur in reality and which are characterized by the values of the coefficients occurring in the above equations, i.e. the coefficients of increase and of predation and the numbers of individuals of the various genera.

First case: if we assume that plants cannot increase in number indefinitely, then the food supplied to carnivorous animals through the mediation of herbivorous animals is insufficient to maintain the carnivorous species, and the latter disappear, while the numbers of herbivorous animals and of plants tend to give periodic undamped fluctuations.

Second case: if the coefficient of increase of the plant species is constant, then the number of individuals of this species unboundedly

increases; one may also suppose that the indicated coefficient decreases proportionally to the number of individuals. The second case, subdivision a): the food left to the plants is insufficient to support the existence of herbivorous animals; in this case the species of herbivorous animals and the species of carnivores disappear, while the plant species tends toward a constant value. The second case, subdivision b): plants occur in sufficient quantity to support the existence of herbivorous animals, but there is not enough food for carnivores at the expense of the herbivorous animals; in this case the species of carnivorous animals dies out, while the herbivores and plants tend to form a damped fluctuation tending toward a stationary state. The second case, subdivision c): there is enough food for all species, and these latter, owing to asymptotic variation or damped fluctuations, all tend toward a stationary state.^1)

§ 15. It should be noted, from the analytical point of view, that the study of fluctuations and changes in the number of individuals of species living together falls outside the domain of the ordinary study of oscillations,^2) since the general equations are not linear, whereas the classical schemes of the theory of oscillations lead to linear equations. Indeed, the oscillations under study are not, in the general case, small oscillations. Only in the case when we make the hypothesis of the existence of small fluctuations can we neglect quantities of the second order and use linear equations.

§ 16. Before concluding this article, we should like to warn the reader against objections that may arise and that will present in a false light the results of the preceding investigations, these results perhaps appearing inaccurate or even devoid of meaning. We should like to forestall and guard against this.

For example, in the above-mentioned case of two species, of which one devours the other, we find that it is always estab—

^1) We shall not write out here the algebraic inequalities which must be satisfied by the coefficients characterizing the various cases and subdivisions; they may be found in the cited memoir from the Royal Academy, p. 6.

^2) For example, mechanical, acoustic. (P. I.).

a periodic cycle is poured into it, thanks to which the number of individuals of the two species oscillates about a certain mean value. One may object that it is easy to imagine a predatory species so numerous and so predatory that in a short time it will devour, one after another, the individuals of the other species and, in this way, will make the above-mentioned periodic oscillations impossible.

We must here recall that the law of the cycle follows from the assumption that one of the species, if it lacks food, can disappear only in the course of an infinite time, and this may seem still further removed from reality than the law of the cycle itself. This circumstance arises from the fact that among the hypotheses laid at the foundation of our whole exposition there is the hypothesis that the number of individuals is a positive number varying continuously, whereas in reality this number can only be an integer and cannot be less than one. Thus we must imagine that if the number of individuals of a species has become sufficiently small, it must be regarded as zero, and the adoption of values smaller than one is an assumption of a purely theoretical character, devoid of any real meaning. Let us return to the example set forth in § 5. If the coefficient of predation \(\gamma_1\) is very large and the initial value \(N_2\) is also extraordinarily large, then \(N_1\) may quickly become less than one; and this circumstance in practice is equivalent to the extermination of one species and, consequently, the cycle which theoretically would have continued will in fact not be closed and will end at a point.

What we are saying here is characteristic not only of those applications of mathematics to biology which we have set forth here. Analogous phenomena occur in all cases when, instead of a discontinuous quantity, a continuous quantity is taken. In many cases this substitution is necessary, since without it it does not seem possible to use the most powerful instrument possessed by mathematics—the infinitesimal calculus; on the other hand, in all classical cases, the consequences,

MATHEMATICAL THEORY OF THE STRUGGLE FOR EXISTENCE

which can be obtained from the theory are practically fulfilled.

Questions of this kind arise not only in the case when one makes a substitution similar to the one indicated above, but also in every application of mathematics to the natural sciences. This depends on the fact that the application of mathematics to any question is possible in the case when the application is connected with hypotheses about the properties of objects to which qualities are ascribed similar to those which these objects possess in reality. Thus, for example, rigid bodies in mechanics must possess such properties that, when subjected to the action of some force, they do not change their shape at all; this property, obviously, is not fulfilled for any material body. How, then, is one to proceed at the present time with a theory that has long been applied, when it is necessary to overcome the difficulties encountered by it?

Here it is necessary to distinguish two phases. During the first phase the problem is solved by analysis, considering the adopted hypotheses as absolutely exact. When the solution has been obtained, it is necessary during the second phase to investigate this solution more closely, and if it then turns out that certain limits of the assumptions have been passed, so that the hypotheses made are too far removed from the truth, then it is necessary to abandon the solution of the problem or to alter the solution itself.

In this way, for example, we may calculate the forces that hold the parts of an ideal beam together, assuming that this beam is infinitely resistant and infinitely rigid. If the solution has been obtained, then during the second phase it is necessary to see whether some of these forces exceed known limits, since in that case equilibrium is impossible and the beam will break; and this circumstance is extremely important to foresee. In exactly the same way in the case of fluctuations of species, which are regarded as ideal species formed by some positive number of individuals. If, after the calculation has already been made, we find that the number of individuals of a certain kind falls below a value less than unity, we may say-

as we saw above, that the variations of the species cease, since the species can no longer exist. The solution found is thus not an incorrect solution, but here there arises a limiting condition that is of extremely great practical importance and significance. The first phase, of which we spoke above, is what we may call the theoretical phase; the second phase is the applied phase. And we in fact have, on the one hand, theoretical mechanics, and on the other hand—applied mechanics. The investigations in mathematical biology carried out by me and briefly set forth in the present article, according to this classification, belong to the theoretical phase.

1. As we have already said in § 8, this law is valid up to a certain limit; if one goes further in destruction, it is possible to make all species perish. ↩↩↩↩↩↩

2. This result is explained by the absolute character of conservative systems. ↩↩↩

3. D’Arcy Thompson Wentworth. On Growth and Form, Cambridge, 1917. ↩