Vito Volterra and theoretical ecology

THEORETICAL

POPULATION

Vito

BIOLOGY

2, l-23

Volterra

(1971)

and Theoretical

FRANCESCO M. Departments

of Genetics and Mathematics, Received TO

LUISA

SCUDO+

Stanford March

VOLTERRA

Ecology*

University,

Stanford,

Calif.

94305

10, 1970

D'ANCONA

At times one has the impression that Volterra’s contributions to ecology are insufficiently known or improperly understood. With rare exceptions, such as Rescigno and Richardson (unpublished manuscript), direct references to his work are seldom found in the recent literature; the most commonly cited is Volterra (1928), an elementary exposition of some of his early work, written for . . apphcatlon to fisheries. In the following I shall try to give a concise and nontechnical description of Volterra’s major contributions to ecology1 and to indicate how they relate to the work of earlier authors. I shall also mention a few later developments in the field. I should like first, however, to comment briefly on the career of this extraordinary man2 and on the origins of his interest in ecology.

BIOGRAPHICAL

SKETCH

Vito Volterra was born in Ancona, May 3, 1860, and barely escaped death at age three months during a siege of the city. In his early youth he undertook the study of differential calculus and geometry, on his own initiative. Knowing nothing of integral calculus he was led to rediscover part of it in dealing with gravitational problems (he had started working at twelve on a restricted threebody problem). * Research supported in part under grant NIH 10452 at Stanford University, Stanford, California. + Fellow of the International Laboratory of Genetics and Biophysics, Pavia Section. Pavia, Italy. Present address: Department of Zoology, University of Massachusetts, Amherts, Massachusetts 01002. 1 Volterra’s major biological works are listed in a separate bibliography. Three monographs on his work are, to my knowledge, still available: Volterra (1930) in Italian; Volterra and d’Ancona (1935b) in French and d’Ancona (1954) in English. s An account of Volterra’s life and work is given in the obituary by Sir Edmund Whittaker (1941), which has been reprinted in the 1959 edition of Volterra 1930b.

1 0

1971 by Academic

Press, Inc.

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His father had died when he was two, leaving the family poorly provided. It was therefore only with difficulty that he pursued a higher education, first in the natural sciences at the University of Florence, later in physics at the Scuola Normale di Pisa. Once started, however, he rapidly established a worldwide reputation in various fields of pure mathematics, gaining special renown for having laid the foundations of the theory of functionals and integral equations. At the same time, Volterra was interested in a variety of mathematical applications to physics, the natural and social sciences, and economics. He also pursued the gravitational problems that had fascinated him from his early youth, and made significant contributions to ecology, fluid dynamics, elasticity, electrodynamics, and geophysics. Volterra did not limit himself to academic research. As an exceptionally young Senator of the Italian Kingdom (appointed 1905) he was also very active in politics. In this capacity he played a prominent role in bringing Italy into an alliance with the Western Powers, at the outset of World War 1. He then enlisted in the Air Force, where, among other pursuits, he made a number of contributions to the technology of aerial warfare, ballistics, and location of gun batteries by sound. His research and teaching were resumed only after the war. Although he had for some time been concerned with biological problems [see, Volterra (1901-1902), (1906a and b)] Volterra became interested in mathematical ecology only late in 1925. His interest in the field was stimulated by conversations with the young zoologist Umberto D’Ancona, then engaged to marry his daughter Luisa. D’Ancona, studying the records of the fish markets in the upper Adriatic, had noticed a curious phenomenon. He observed that during and after the war, when fishing was severely limited, the proportion of predators among the total catch had increased correspondingly, an effect predicted by Volterra’s models. D’Ancona was thus reinforced in his belief that the two facts were causally correlated. This beginning led Volterra to attack more general problems in ecology. The field soon became his major research interest, and it remained so for the rest of his life. I might conclude these remarks by mentioning Volterra’s outspoken opposition to the Fascist regime, which compelled him to resign his chair at the University of Rome as well as all of his memberships in Italian scientific societies. In the years that followed he spent most of his time working at home or abroad, until his death in Rome on October 11, 1940.

PRIOR STUDIES IN MATHEMATICAL

ECOLOGY

Attempts to explain the “balance of nature” through mathematics began to appear around the turn of the century. In 1911 Ross proposed a simple set of differential equations to describe malaria epidemics. Shortly later Martini

VOLTERRA AND ECOLOGY

3

[see d’Ancona (1942)] improved these equations by allowing for the immunity of individuals who had recovered from infection. A further refinement, the incubation lag, was introduced by Lotka and Sharp in 1923. In 1925 Lotka published his Elements of Physical Biology. A later edition of this work is still widely in use, especially in English speaking countries. In this work the interaction between two species is accounted for by a system of quadratic differential equations

where the E’Sare the “coefficients of self-increase,” the y’s account for the interactions, and the N’s are population sizes. Depending on the signs of the constants this can represent a species preying on or parasitizing another, or other types of interactions. Lotka worked out some of the mathematical properties of such systems. He showed that in the case of predation or parasitism the solution is a family of closed periodic curves, depending on the initial sizes of the two populations. He also recognized that small oscillations in the neighborhood of a unique fixed point are well approximated by ellipses, and that the period of the oscillations is almost independent of their amplitude. A few extensions of this simple case were also examined. Lotka noticed, for example, that the cyclic behavior ceasesif a cubic term is introduced into the above equations. The populations of the two species move in a spiral tending toward a stable equilibrium point. He also considered a special case of three species: two different preys of a common predator. One might suppose that the two preys together would survive better than each alone. In fact, the “weaker” prey is eventually eliminated. Lotka considered also more complex “food webs,” in which small animals are eaten by larger ones, which in turn are eaten by still larger ones. For such cases, however, he worked out only linear models. Volterra was not at first aware of Lotka’s findings. Though his point of departure was much the same, he probed far more deeply in certain directions. In describing his principal investigations and results I shall depart from the traditional order of presentation, beginning with various casesof a single isolated species. DYNAMICS OF A SINGLE ISOLATED SPECIES

In characterizing the population dynamics of an isolated species, the simplest case is that of an asexually reproducing organism for which age is irrelevant and for which behavior does not change with time nor with the number of the organisms. The number, N, must be sufficiently large that the process can be well

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approximated by a deterministic treatment and by real, rather than integer, numbers. A latitude of definition must be allowed for N: It may mean number of individuals, in which case it is an integer, but it may also mean total weight, total weight of certain parts, total metabolism, or some other measure of quantity of life. Under these assumptions the change in N is given by the Malthusian equation

dN EN, -= dt

(G > 0).

Integration yields a geometrical law of increase (or decrease, if E < 0) N = No exp{d)

It was Volterra’s practice to discuss at length the restrictive assumptions under which his formulas were derived. He then proceeded, in the best Baconian fashion, to remove one or two of the assumptions at a time. I shall describe, in some detail, only the more simple of his many generalizations. The above assumptions imply, for example, that unlimited environmental resources are available to the species. One can easily allow for a finite environmental capacity by taking for E a decreasing function of N. By assuming that c decreases linearly with N, one obtains the Verhlust-Pearl equation

$ = (c-

AN)N,

(A, 6 > 0).

The integral, often called the “logistic curve,” is widely used even outside ecology. Taking into account specific mechanisms affecting reproduction or mortality leads to much more complex functional relationships. Take, for example, a population living in a completely closed environment, such as some microorganisms confined to a test tube. The amount of nutrients available decreases with time in proportion to the total amount of “metabolism” that takes place in the tube from the beginning of the experiment. Total metabolism also determines the concentration of toxic waste in the medium. For simplicity, assume that the metabolic activity of the population is directly proportional to the number of individuals and that its total amount affects linearly the coefficient of self-increase. The system can be represented by the integro-differential equation -= dt

1~- /W(t) - P j; N(T) dj N(t),

(~94 P > 0)

which can be solved by successive approximations. Suppose now that the population is initially increasing. In the long run the effect of the integral term will be

VOLTERRA

AND

ECOLOGY

5

decisive, the population will reach a peak and then slowly decreaseto extinction. The assumption that individuals do not change with age is also untenable. The simplest improvement in this regard considers two life stages for the organism, the young do not reproduce and a constant fraction of them become adult in each unit time. Volterra considered in detail two special cases of this sort, young and adults competing for the same resources and adults cannibalizing the young, as in the flour beetle. In animals with separate sexes and dependence on cross-breeding, one has also to consider that the probability of mating depends on the size of a population scattered over a given area (or volume), Volterra (1938a). The simplest assumption in this respect is that the probability of any given female’s encountering a male per unit time is directly proportional to the density of males. Let 01be the constant proportion of males in the population, /3 = 1 - 01that of females. The chance of a female’s encountering a male per unit time is thus proportional to arN. If the chance of an encounter resulting in copulation remains constant, the number of matings per unit time is proportional to c@N2. If each mating produces, on the average, the same number of offspring, and if the death rate, E, is proportional to the number of individuals, one obtains finally dlV -z-=

--EN + hN2,

(c, h > 0).

This is, again, the Verhlust-Pearl equation, except that the constants are of opposite sign. The fate of the population is determined by its initial size. If E - XN,, > 0, the population will increase without bounds; if the reverse inequality holds, it will tend to extinction. Thus for nongregarious animals with compulsory cross breeding there is a critical density below which the number of matings is too small to maintain the population, no matter how large it is. As population density increases, however, this treatment breaks down. Fewer encounters will result in matings and/or each of them will produce fewer progeny. The addition of a negative cubic term is the simplest way to accommodate this effect in the above differential equation. This will generally produce two nonzero equilibrium points, the higher point, like extinction, is stable; the intermediate point is unstable. If the initial number of individuals is below the unstable point the population will tend to extinction. It is perhaps worth remarking that the “encounter” method of deriving the number of matings is a standard tool of classical statistical mechanics. Consider a gas or solution containing two kinds of particles, the number of collisions per unit time and volume between particles of different kind is proportional to the product of their densities. The constant of proportionality takes different

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names in different contexts. In chemistry, for instance, this product rule is known as the “mass action law.” In borrowing this method from classical mechanics, Volterra was deeply aware of its limitations when applied to living organisms.

Two

SPECIES: THE PREDATOR-PREY

CASE

I have discussed at some length Volterra’s treatment of an isolated species to illustrate the rigour of his thinking and his attention to biological detail. These qualities he carried over into his treatment of different forms of interaction among various numbers of species. Lotka’s equations for the predator-prey case afford a simple and convenient example of Volterra’s approach to more complicated cases. It is convenient to rewrite Lotka’s equation as

-

dlv, = (El- YlN2)Nl dt

and (predator)

-$$

= (-~a + r.JVJNs

,

where all the constants are positive. The assumptions implied by such formulas should be clear from what was said in the above section. Volterra integrated such equations showing, among other things, that the fluctuations are periodic, and that the period does not depend on the y’s. He also showed that, in one cycle, the averages of the numbers of the two species are the same as the equilibrium values,

jq = E2 Y2

jiJ2=‘. Yl

This result offers a simple and very useful interpretation. Suppose that constant proportions of the two species, S1 and 6, , are continuously removed from the system. Something of this sort could occur as a result of fishing or other forms of food gathering

by man. We have, then,

m

=

E2 i- 62

1 Y2

iiT2 = ~El

-

6,

Yl

provided that l l > 6, , the removal of constant proportions of the two species increases the numbers of the prey and decreases those of the predators. If, however, er < 6, , the system tends rapidly to extinction. As we have seen, it was d’Ancona’s observation of this effect that stimulated Volterra’s research in the field. The property has obvious implications for

VOLTEFtRA

AND

ECOLOGY

7

many problems of exploitation and control of natural environments. The idea was not a new one; it was foreshadowed, for instance, by Darwin (1882, pp. 53-54). This was the first time, however, in which an ecological model matched reliable empirical data (see discussion). On the more theoretical side, Volterra discovered that systems like the one above possess a biologically interesting mathematical property. First we rewrite the system as

Then, summing the two equations gives

or, in an integral form,

which is analogous to many conservation laws of physics. Let us denote the first two terms by

which Volterra calls “actual demographic energy” and the last two terms by

which he calls “potential demographic energy.” The changes in numbers of the two species will leave the “total demographic energy” H=T+V unchanged. In a still more theoretical vein, consider the quantity

A = j-”0 &N log NI + Bzrv,log W dt which, by analogy to classical mechanics, Volterra calls “vital action.” It is easily verified that Lotka’s equations describe the motion for which vital action is at a minimum. To systems of this type the “canonical” formulation of mechanics applies.

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8

SIMPLE EXTENSIONS TO MORE THAN Two SPECIES

The most general analogue of Lotka’s equations in the case of n species is a system of the form

dNr dt

(G + $

$ ~2s)

N,

(r = 1, 2 )...) n),

r.71

where all yrr = 0 and Y,.~= -yST for all r f s. The /3’s are positive constants that can be interpreted in terms of weight or nutritional value of the individuals. Such systems can be derived from the variational principle described in above section. They are called “conservative” since they satisfy the conservation law of total demographic energy. Such a system is said to have a nontrivial equilibrium if all the roots of the linear system of equations

are positive. This can occur only if the number of species is even and there are both positive and negative E’S.When the number of species is odd, the matrix of the system, being antisymmetric, will have a null determinant. Volterra showed that the existence of such an equilibrium is enough to guarantee that the numbers of each species will always remain bounded above and away from zero (unless initially zero). If the initial numbers of a conservative association differ from the equilibrium ones, a state of undamped oscillations will persist indefinitely. Since such oscillations will be, in general, aperiodic, one must compute asymptotic means for the numbers of individuals. As in the twospecies case these means turn out to be the same as the equilibrium numbers. Although conservative systems have very appealing mathematical properties, Volterra did not think them good representations of nature. He found particularly disturbing that (a)

A stationary state could exist only for an even number of species;

(b) When no such state exists the numbers of some (possibly all) species will, sometime, tend to infinity. Many later authors confine their treatment to the even more restrictive class in which all /3’s are 1. They seem not to be particularly disturbed by the peculiar properties of the systems nor aware of most of their limitations. Such drawbacks are most easily disposed of by assuming that the coefficients of increase, E, , of the species are linear decreasing functions of their numbers (equivalently that all the yr7 are negative). Actually it is often sufficient that just one of the yr,. be negative to obtain a biologically more meaningful behavior.

9

VOLTERRA AND ECOLOGY

To illustrate the point we may consider the simple food web of a carnivorous animal that hunts an herbivorous one, which in turn grazes on a single plant species. One must assume self-limitation for the plant, since, even without grazing, its numbers would be limited by the space available. Without loss of generality, a quadratic system of equations for such a web can be given as (carnivore)

& -

(herbivore)

(plant)

= (-AZ

+ 4W,

& 7

= (-f&m

- 4

P3 $h

= W

dt

- %

> + WW,

- bN,)N,

where all the constants are positive. The equilibrium

,

,

point

a m

= 3

will be admissible-that

aP3k - b&l ah

’

is, the three species will survive-if ab/?,k - b2,8,1-

aA/3,m > 0.

The values of the parameters determine whether the equilibrium monotonically or through damped oscillations.

point is attained

THE GENERAL CASE OF n SPECIES The most general quadratic model for an association by the system of differential equations

+

of n species is given

= (G+ f,l~d’L)N, (r = 1, Z-v

where no restriction is imposed equilibrium conditions are ~1.+ t ~-3s s=l

n),

on the E’S and the y’s. The corresponding

= 0

(7 = 1, 2 )..., n).

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We say that a nontrivial equilibrium state exists if all the roots of this system of linear equations, fil , fi2 ,..., fin , are positive. What follows can be more easily understood by considering from the outset the quantity V = i

a,N, ,

r=l

where all the 01’sare arbitrary positive constants. We shall call this the “value” of the association. It is apparent from the original system that the change in V with time can be broken down into the two components - dV, dt

= 1n ~3, r=1

and

Depending on the values of the y’s, we might be able to choose special values for the CL’S such that the second of the above expressions is zero for any positive N,. , N, . Thus for each species, i, there could be a certain LY~such that the values of V would not be affected by the interactions among the species. Volterra proved that this occurs, when all yrS (r # S) are nonzero, if and only if all 3 x 3 combinations of the y’s satisfy

It is easily verified that the subclass defined by this condition is that of the conservative systems examined in above section. The approach here, however, has the advantage of exposing the restrictive and somehow artifical nature of these systems. It asserts that a scale can be found for counting or weighing each species such that the effects of interactions cancel out in determining the total weight or numbers of the association. There is a far less restrictive class of general systems that Volterra felt to have more practical value. In these systems, which he called dissipative, the quadratic function dV,/dt enjoys the mathematical property of being positive-definite.3 When an equilibrium state exists, a dissipative association tends toward it, 3 The function dV,/dt can be rewritten as & aifNsNj , where aij = aji , and it is said to be positive-definite if it takes positive values whenever the N’s are not all zero. When this is the case all the eigenvalues of the a il matrix are positive. The necessary and sufficient condition for this to occur is that the “principal” minors of all orders have a positive determinant.

VOLTERRA AND ECOLOGY

11

either through damped oscillations or monotonically. The value of the quadratic form &‘a/& gives a measure of the rate at which equilibrium is attained. Such a quadratic plays much the same role as friction in mechanical systems. We may look upon a conservative association then, as an ideal, limit case of a dissipative one. In this light it is not surprising that conservative associations can be at a stationary state only under very stringent conditions. A minute change in these conditions could easily bring on what, in the real world, would be called a catastrophe. In dissipative systems, on the other hand, it is enough that one of the E’S be positive to guarantee the coexistenceof all the species. No further restriction needs to be imposed, since the determinant of the yrs matrix is always positive. Another welcome property of dissipative systems is that no species can increase without bound. The “worst” that can happen is that all species tend to zero when all the E’S are negative, but this is hardly surprising.

OTHER EXTENSIONS OF THE PREDATOR-PREY MODEL Before examining in some detail a radically new model, it is convenient to mention a few other cases of dissipative and conservative associations treated by Volterra. He worked out, for example, the behavior of a dissipative association when a stable state does not exist for all of the species. This situation can occur where the successful introduction of one or more new species upsets the stability of a previously existing association. He also extended the treatment of conservative associations to account for periodic fluctuations in the environment, and to more general conservative interactions in which the y’s are functions of the numbers of the species. The precarious nature of the oscillations in the conservative case can be further emphasized by considering a simple migration model. Take a number of partially isolated populations for each of which, if alone, a Lotka model would apply. In each of these populations, emigration, which depends on the numbers of the species, determines new E’S and y’s. Assume, on the other hand, that immigration does not depend on the state of the “host” population. Unless all subpopulations have the same parameters and are in phase, immigration into each population would be well approximated by a constant. By adding such constants to Lotka’s system one always obtains a unique, stable equilibrium state. Volterra emphasized consistently that differential equations are, at best, only rough approximations of actual ecological systems. They would apply only to animals without age or memory, which eat all the food they encounter and immediately convert it into offspring. Anything more realistic would yield integrodifferential rather than differential equations. The appropriate modifications

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emerge quite naturally when we examine critically [see Volterra (1927b)] the derivation of Lotka’s equations. Consider first the prey, neglecting complications due to age. If isolated, the change in their number, N,(t), in a small time interval dt is eJVi(t) dt, where e1 measures the balance between births and all causes of death except being eaten by the predators, which number N,(t). The number of encounters between predator and prey is proportional to the product of their numbers. Combining the two causes of variation we obtain dN,(t) = clNl(t) dt - y&(t)

N,(t) dt

(9 ) Yl > 0).

Consider now the predators. Alone, they would die of starvation, c&s(t) dt of them in a time interval dt. On the other hand, the survival, growth, and reproduction of each individual predator up to the “present” time depends on how he fed in all his previous life. As a very rough first approximation suppose that his voracity (tendency to eat the prey when encountered) does not depend on age, 7, nor on the state of the association. Assume also that the age distribution of the predators, h(~-,i), can be considered as independent of time, h(7). The individuals of age not younger than r will be in the proportion

so that of the N,(t) predators active at time t a number proportional to f(t - T) N,(t) will have been active at time t - 7. Their feeding rate, which is proportional to f(t - r) N,(t) Ni(r), must be multiplied by a positive function, p)(t - T), measuring the effect of feeding through all previous time on the chances of survival and the rate of reproduction at a subsequent time. Setting yp(t - ~)f(t - T) N,(t) N,(T) dr = F(t - T) N,(t) N,(T) dT and integrating over all previous time we obtain, as a positive term for the predators’ equation,

N,(t)1’ F(t --m

7)N,(7)

d7.

Similar considerations can justify a negative integral term for the prey’s equation. It is convenient to write the resulting system of integro-differential equations as follows

and

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VOLTERRA AND ECOLOGY

where both y’s are generally positive (yz might be zero), and bothF’s are generally positive for t - 7 < T (Fi can be zero) and zero when t-7 >, T. Here T is the maximum age the individuals of either species can attain. Volterra worked out quite a few properties of this system. He showed that it has a unique fixed point

iq =

E2 f12= Yz+ r* ’

El r1+

l-1

where

r,=s=F,(5) 8

and

0

J’s = s =F2(5) df 0

which is always unstable. Thus Ni end N, do not tend to a limit but, rather, oscillate forever about the equilibrium point. He also calculated the maxima and minima of the “stationary” oscillation and demonstrated that the above equilibrium values coincide with the asymptotic means. It is again easily verified that removing, up to a certain limit, constant proportions of the individuals of the two species will increase the numbers of the prey and decrease those of the predator. The above treatment in Volterra (1927a), was the first to describe a stable oscillatory behavior. In later works Volterra applied the method of successive approximations to the case of two species [Volterra (1931a)l and outlined the extension to n species [Volterra (1939)]. In the following we shall see that this behavior can be found even in differential equation models of greater complexity that the quadratic. In such cases it will be possible to show that the oscillations are asymptotically periodic-that is, that they are limit cycles of some sort.

KOLMOGOROFF'S TREATMENT OF THE PREDATOR-PREY INTERACTION

In a little-known paper published in Italian, Kolmogoroff (1936) discusses a simple and intriguing dissipative extension of a conservative model of Volterra mentioned in section above. In the equations

- dt

= N,K,(N, , N,)

*

= NaK,(N, , Na)

and (predator)

KI and K2 , besides being continuous and having continuous first derivatives

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(for nonnegative Nr , NJ must satisfy a number constraints. Those for Kr are that

of biologically

reasonable

(Al) For any given size (number, weight, etc.), the rate of increase of the prey species is a decreasing function of the number of the predator species. (A2) For any given ratio between the two species, the rate of increase of the prey is a decreasing function of size. (A3) When the sizes of the two species are both very small, the rate of increase of the prey is positive (K,(O, 0) > 0). (A4) There is a critical size A for the predator species that blocks further increase in the prey, even when the prey is very rare (K,(O, A) = 0). (A5) There is a critical size B at which the prey species can no longer increase, even in the absence of predators (K,(B, 0) = 0). Three constraints (Bl) ber, etc.).

are imposed on K, :

The rate of increase of the predators decreases with their size (num-

(B2) For any given ratio between the two species, the rate of increase of the predators is an increasing function of size (the reverse of A2). (B3) There is a critical size, C, for the prey species that blocks further increase in predators, even when the predators are very rare (K,(C, 0) = 0, the analogue of A4). We must also assume that C < B, for, otherwise, the predators are bound to disappear. Under such conditions only three equilibrium points are possible, the origin, the critical number B, and the unique intersection of Kl = 0 with K, = 0, denoted by 2. From the behavior of the integral line originating at B (line L, Fig. I), Kolmogoroff claims that one of the following possibilities will occur, according to the functional form of the K’s : initial

(a) Point 2 is a stable fixed point that is appoached, value, through damped oscillations (a focus, Fig. la).

(b) Fig. lb).

Point 2 is again stable and is approached

from any nonzero

asymptotically

(a node,

(c) There is a closed integral curve, F, enclosing 2, that will be approached from the outside. Various behaviors are possible inside, and further restrictions are necessary to discriminate among them. The dashed line in Fig. Ic describes the most simple and practically interesting case of a unique, bilaterally stable limit cycle. Finally,

Fig.

Id represents

a center, the solution

of Lotka’s

conservative

VOLTERRA

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ECOLOGY

system. This is a special limit case of both (a) and (c), but not of (b). Solutions (a) and (b), by contrast, characterize Volterra’s dissipative systems. Kolmogoroff does not state analytical conditions for the existence and stability of the closed integral F and resulting subcases. His paper is thoroughly reported

-

N,

A

0

c

!3

Ni

0

N,

FIG. 1. The four types of solutions of Kolmogoroff’s predator-prey model: (a) a stable node; (b) a stable focus; (c) a stable limit cycle (for graphical convenience the quite arbitrary lines Kl = 0 and KS = 0 have been drawn equal); (d) a center, the solution of Lotka’s equations, which is a limit case of both (a) and (c).

in Rescigno and Richardson (1967). The same equations, under somehow different conditions, again give periodic solutions [Waltman (1964)]. The stable oscillatory behavior of (c) is very similar to-when not identical to-that of Volterra’s integro-differential model. Limit cycles, or their n-dimensional equivalents, arise also in models having a simple algebraic structure [Kostitzin, (193611. They can be made to occur, for example, by imposing quite simple restrictions on quadratic models, as shown in a discrete time by S. Hubbell (personal communication). Summing up, it seems that a stable oscillatory behavior is likely to be found in models more complex than quadratic ones. It is difficult to characterize this

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behavior analytically, and almost impossible for more than two dimensions. This may explain in part why stable oscillations have not found much favor among practical ecologists.

THE

COMPETITIVE

EXCLUSION

PRINCIPLE

AND THE NICHE

CONCEPT

I have chosen to close with an early result by Volterra (1927a) because it seems to have had, quite indirectly a great impact on ecology. Consider first the case of two species, in numbers Nr and Na , competing for the same environmental resources (food, space, etc.). The two species exploit the resources at different rates and utilize them with different efficiencies. Assume, for simplicity, that the depletion of resources increases linearly with the numbers of the competing species. Depletion, in turn, decreases linearly the coefficients of increase of both species. We have

and

where all constants are positive. Suppose also that clys # esyr . It is easily proved that the numbers of the species having the larger c/y, say the i-th, will tend to E&&J while those of the other tend to zero. The same occurs in the case of any number of species and where a much more general function, F(N, , N, ,..., Nd, measures depletion in place of a linear one. It is enough that F increases without bound as any of the Nr does so, while F(0, O,..., 0) = 0. Only the species with the largest c/y will survive. This simple model, in conjunction with some modest experiments, led Gause (1934) to formulate his famous “exclusion principle”: If two speciescoexist they must occupy dz@rent “niches.” This apparently innocuous statement lead to an explosion of literature on the nature and properties of the niche or, more rarely, of the “species.” It would be impossible here to discuss, or even to cite, such literature in detail. One gets the feeling that Volterra’s result was taken to imply much more than what it actually meant. Thus, for instance, age does not enter the model. If it would (even in a simple way like two life stages)drastically different results would be obtained. Also the depletion of the environment is measured by a single function. This means that all such species utilize the different components of the environment (space, food of all kinds, etc.) in exactly the same proportions, or, equivalently, that they are all competing for exactly the same limiting factor.

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17

ECOLOGY

The one thing that is allowed to vary in Volterra’s model is efficiency in the use of the limiting factor. It is not surprising, then, that the most efficient species will prevail. Volterra’s treatment asserts only that a balance among more than one species cannot be due simply to a unique limiting factor. If the above is not true, i.e., if different species deplete the environment in different ways, they might as well coexist. This ovbious logical alternative had been explicitly considered, in a simple model by Lotka, as early as 1932. The next general step, by Rescigno and Richardson is a very recent one (1965). These authors consider an ecological “space” as made up of several operationally distinguishable components, each of which they term a “niche”. Assume that we can distinguish among m niches and consider n different species competing for them. The utilization of niche j is measured by a nonnegative function N,), as in Volterra’s model, except that not all species need to Fj(% 3 N, ,‘a’, utilize all niches. The depletion of the niches decreases linearly the coefficients of increase of the species. We have then the system

where all the E’S and the y’s are positive. Under such hypotheses Rescigno and Richardson prove that if the species outnumber the niches, at most as many species as niches can survive. How many of them actually do so (possibly only one) depends on the form of the F’s, the values of the constants, and the initial numbers. The conditions are worked out in detail for the special case of three species competing for two niches. The model by Rescigno and Richardson states that if a number of species are limited by as many qualitatively different limiting factors, they might all survive if such factors affect their numbers in ways that are sufficiently different. These writers go further, showing that a qualitative distinction among resources is not necessary to explain the coexistence of more species. This result, achieved in Rescigno and Richardson (1967) by considering an appropriate specialization of Kolmogoroff’s equations, yields the following simple qualitative conclusions: (a) If the curves K1 = 0 and K, = 0 do not intersect, species will survive.

only one of the

(b) If the curves intersect at one point, two results are possible: either one species will survive, depending on their initial numbers, or both species will survive indefinitely. (c) If the curves intersect at several points, depending again on the initial numbers. 653/2/l-2

any outcome

can result,

18

SCUD0 SYMBIOSIS AND PARASITISM

In the foregoing I have omitted discussion of most theoretical treatments of symbiosis, a form of interaction for which Volterra showed little interest. “Perfect symbiosis, ” i.e., an association of two species that is to the advantage of both, was first considered in detail by Kostitzin (1934) together with a few other extensions of Volterra’s models. Kolmogoroff’s approach has also been applied to symbiosis, in Rescigno and Richardson’s 196’7 paper. Haldane and Jayakar (1963) undertake a partial treatment of the problem, limiting their considerations to the proportions between the two species in discrete time. It is interesting to note that in their approach a stable oscillatory behavior can be obtained. As remarked by Ilan Eshel (personal communication), this does not occur in the continuous time analogue. This and other simple cases, as discussed for example in Mainard Smith (1968), lead one to suppose that such discrepancies occur frequently, and to ponder their biological implications. The host-parasite caseis a form of interaction that has attracted more attention than any other. Epidemiology has progressed steadily from the modest beginnings which I have mentioned earlier and remains an active field of research. Here the relationship between empirical observation and model building is much closer than in the field considered by Volterra. A stochastic approach to the problem has lately proved to be the most useful. The field is thoroughly covered by a number of good monographs, such as Bartlett (1960).

CONCLUDING REMARKS

Very few of the approaches tried or suggested by Volterra have inspired further researches, besides those mentioned. A large proportion of later works concentrate on simple cases of dissipative or conservative associations. Stochastic analogues are usually confined to the latter class. I shall not attempt here a systematic survey of this class of literature; most of it can be traced back from the few references I have given. A remark by Whittaker (1941) on Volterra’s ecological work could well serve as a starting point for some more general considerations. He observed, that “It would be rash to say whether the analogies with physical sciences which he unearthed will remain what they appear to be at first, and certainly are, at least, a clever and remarkable tour deforce-or whether they will eventually be seen as the germs of a profound biodynamics, essential to the theoretical and economical biology of the future... .” If Whittaker were alive today he would be astonished to fmd how pressing the need for such an approach has already become. Localized ecological disasters are premonitory signs of impending

0.16-

-016

-6

FIG. 2. Yearly data on the bottom dwellers sold on the Trieste market from 1904 to 1961. The upper part gives the fraction of selachians, a group at the top of the food web or very close ot it. The lower part gives the total catch of selachians, bony fishes, mollusks, and crustaceans in 1Oaton units. The four point “window” Y, = (Y&/6 + (Y,-i)/3 + Y,/3 + (Y,,+i)/6, applied both on selachians and the total catch, makes the data more readable. Beginning with 1904 the weight of the seafood sold on the Trieste market has been, and is, reported monthly by the Department of Statistics of the City Council. Most data on adults are by species. Only the data for 1918 and part of 1939 are missing. The initial and final points and those involving the year 1918 are three-point rather four-point weighted averages. The Italianized vernacular names for the different types of catch are also given in the statistics. In some cases they are indicative of the form in which the animal is sold. This may be just part of the animal, called “tail”, as for prawns (-0.33 of the animal’s total weight) and for angler fishes (-0.45 of the weight). Similarly there are two different words for crabs, to distinguish between those molting and those not. No correction for such partial weights has been made. Imported and frozen seafood has been disregarded. All genuses listed as such in D’Ancona (1926, see also 1964) have been considered bottom dwellers. Very little information is available on their feeding habits. The yearly data, selachians and totals, in tons, are as follows: (505; 5.249), (524; 4.935), (470; 4.823), (470; 5.469), (800; 9.834), (457; 6.472), (369; 6.663), (624; 7.224) (749; 7.925), (1.003; 6.357), (760; 5.162), (69; 914), (138; 846), (116; 744), ( missing), (484; 2.499), (896; 5.747), (853; 6.449), (867; 8.216), (770; 7.593), (704; 7.987), (901; 7.547), (1.000; 7.699), (913; 8.052), (990; 8.049), (965; 6.073), (723; 8.400), (716; 10.018), (790; 7.631), (519; 7.530), (582; 8.642), (699; 7.073), (809; 9.413), (764; 9.940), (913; 10.078), (1.172; 9.684, partial), (715; 7.843), (196; 5.331), (326; 4.418), (140; 2.312), (49; 1.361), (149; 1.414), (1.062; 7.846), (988; 9.979), (1.159; 11.050), (934; 8.468), (863; 10.477), (995; 9.279) (866; 8.648), (631; 8.195), (580; 8.195), (754; 9.452), (754; 7.280), (833; 7.369), (826; 8.487), (806; 7.899), (853; 7.509) and (844; 7.513). The homogeneity of the series can be questioned on different grounds. Thus fishing techniques changed steadily, but not notably for the “community” under consideration. The change in national borders after World War II may be a major source of discontinuity. How seriously this change affected the real (rather than official) relationships between the fishing fields and the market is a difficult and politically sensitive question.

20

SCUD0

catastrophes on much larger scales. Yet a profound biodynamics is far from emerging from Volterra’s “germs.” It also seems that, at times, the powers of quantitative prediction of some of Volterra’s models have been overestimated. Somethinkthat theirlimitations make themodels of doubtlessworth as compared, for instance, to those in celestial mechanics or atomic theory. Not many people seem to understand the value of models dealing mainly with qualitative features, which are successfully applied, for example, in many problems of astrophysics. But we need not pursue analogies so far afield. Let us, rather, return to a concrete qualitative problem in ecology: that of persistent, stable oscillations. We have seen that the “neutral” oscillations of ideal, conservative systems cannot be of any practical interest. On the other hand stable oscillations often arise in models taking into account simple biological realities-like, say, the time lag in the effect of feeding on reproduction. Only long after their prediction have stable oscillations been observed in well-documented studies of natural and artificial populations. Nicholson’s experiments (1957) left little doubt of their reality under controlled laboratory conditions. Yet no rigorous statistical analysis [like Moran (1953)] of the only well-documented natural case, that of the Canadian lynx, seems enough to convince many ecologists of their occurrence in nature [see, e.g., Andrewartha and Birch (1954)]. And if we are to accept the oscillation of the lynx, are we to suppose it an oddity or a common property of many predators? Certainly the widespread occurrence of such oscillations would greatly affect many current approaches to the observation, prediction, and control of ecological systems. To answer the question purely on sound, empirical grounds would require lengthy field work at a prohibitive cost. In this respect it might be interesting to reexamine the (now greatly extended) series of data on Trieste’s fish market that inspired Volterra’s research. In Fig. 2a the proportion of large predators in a loosely defined community is reported year by year. This compares with the total catch for the same community in the lower part (Fig. 2, b). The relative increase of predators following the slump in fishing during World War I repeats itself, to a lesser extent, during World War II. The data also suggest a persistent oscillation, with a period of about ten years. The data, of course, are not vey clean, and even if accepted as clean they are unlikely to have high statistical significance in either respect. It would be remarkable, however, if the similarities of the Trieste data with the much more reliable data on the Canadian lynx and other mamals [Elton, 19421 were just a coincidence.

ACKNOWLEDGMENTS

Space prevents me from acknowledging individually the many contributors to this work. I shall simply list them in alphabetic order: Mrs. Luisa d’Ancona, Mr. W. W. Carver, Mr. E. Eshel, Professor G. Goodman, Professor S. Hubbell, Professor J. B.

VOLTERRA AND ECOLOGY

21

Keller, Professor S. Karlin, Professor N. Keyfitz, Miss Gail Lemmond, Professor J. McGregor, Professor G. Mozzi, and Professor I. W. Richardson. To all of them, my warmest thanks. An even greater debt, however, is due the late Professor U. D’Ancona, who introduced me to this fascinating field while at the Department of Zoology, Padua University.

REFERENCESTO VOLTERRA'S WORKP 1901-1902. Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali, Annuario R. Universita, Roma, 3-28, and Giorn. Economisti 3, 463-458. 1906a. Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali, Arch. Fisiol. 3, 175-191. 1906b. Les mathematiques dans les sciences biologiques et sociales, Revue du mois. 1, l-20. 1926a. Variazioni e fluttuazioni de1 numero d’individui in specie animali conviventi, Mem. Accad. Lincei 2, 31-113. 192613. Fluctuations in the abundance of a species considered mathematically, Nature 118, 558-560. 1927a. “Variazioni e fluttuazioni de1 numero d’individui in specie animali conviventi,” R. Comit. Talass. Italiano, Memoria 13 1, Venezia. 1927b. Sulle fluttuazioni biologiche, R. C. Accad. Lincei (6), 5, 3-10. 1927~. Leggi delle fluttuazioni biologiche. R. C. Accad. Lincei (6), 5, 61-67. 1927d. Sulla periodicita delle fluttuazioni biologiche, R. C. Acad. Lincei (6), 5, 463-470. 1927e. Essai mathematique sur les fluctuations biologiques, Bull. Sot. Ockanograde France ? ?. 1927f, 1928. Una teoria matematica sulla lotta per l’esistenza, Scientia 41, 85-102; French transl. 41, Suppl., 33-48; Russian Transl., Prog. Sci. Fis. 8, 13-34 (1928). 1927g. Lois de fluctuations de la population de plusieurs especes coexistant dans la mCme milieu, Ass. Franc. Avant. Sci. Paris ? ?, 96-98. 1928. Variations and fluctuations of the number of individuals in animal species living together, J. Conseil Int. Explor. Mer. 3, 1-51. 1930a. Sulle fluttuazioni biologiche, R. C. Semin. Mat. Fis. Milan0 3, 154-174. 1930b. “The Theory of Functionals and of Integral and Integro-Differential Equations,” 1959 ed., Dover, New York. 1931a. “Lecons stir la theorie mathematique de la lutte pour la vie,” Gauthiers-Villars, Paris (compiled by M. Brelot). 1931b. (with U. d’Ancona) La concorrenza vitale tra le specie nell’ ambiente marino, Proc. 7th Congr. Int. Aquiculture (Paris-Orleans), 1-14. 1931~. Ricerche matematiche sulle associazioni biologiche, Giorn. Ist. Ital. Attuari 2, 295-355. 1935. (with U. d’Ancona) “Les Associations Biologiques au Point de Vue Mathematique,” Hermann, Paris. 1936. La theorie mathematique de la lutte pour la vie et l’experience, Scientia 59, 169-174. 1936a. Les equations des fluctuations biologiques et la calcul des variations, C. R. Acad. Sci. Paris 202, 1953-1957. a These works are all by V. Volterra and are listed chronologically. Most of Volterra’s papers have been reprinted, with corrections, in a series of volumes by the Accad. Lincei, Rome. A complete, although not very precise, list of references is also given there. of the I have not been able to check directly some of them, hence the questions marks in some references.

22

SCUD0

1936b. Les equations canoniques des fluctuations biologiques, C. R. Acad. Sci. Paris 202, 2023-2026. 1936c. Sur l’integration des equations des fluctuations biologiques, C. R. Acad. Sci. Paris 202,2113-2116. 1936d. Le principle de la moindre action en biologie, C. R. Acad. Sci. Paris 203, 417-421. 1937a. Sur la moindre action vitale, C. R. Acad. Sci. Paris 203,480-481. 193713. I miei studi piu recenti di Biologia Matematica, Gazzetta del Pop010 della Sera ? 2. 1937c. “Preface a la Biologie Mathematique de V. A. Kostitzin,” A. Colin, Paris. 1937d. Principes de biologie mathkmatique, Acta Biotheor. 3, l-36. 1937e. Applications des mathematiques a la biologie, Emeign. Math. 36, 297-330. 1937f. Leggi delle fluttuazione e principii di reciprocita in biologia, Riv. Biol. 22, 265-380. 1938a. Population growth, equilibria and extinction under specified breeding conditions: A development and extension of the theory of the logistic curve, Human BioZogy 10, l-l 1. 1938b. Lois des fluctuations hiologiques et leur consequences. 1939. The general equations of biological strife in the case of historical actions, Proc. Edinburgh Math. Sot. 6, 4-10. 1938c. Remarques sur l’action toxique du milieu a propos de la Nate de M. Regnier et Mlle Lambin, C. R. Acad. Sci. Paris 207, 1146-l 148. 1938d. Calculus of Variations and the Logistic Curve, Human Biology 11, 173-178. 1938e. Fluctuations dans la lutte pour la vie, leurs lois fondamentales et de reciprocite. Conf. Rtun. Int. Mathem., Paris? ?.

OTHER

REFERENCES

H. G. AND BIRCH, L. C. 1954. “The Distribution and Abundance of University of Chicago Press, Chicago. BARTLETT, M. S. 1960. “Stochastic Population Models,” Methuen, London. CHAPMAN, R. 1931. “Animal Ecology,” McGraw-Hill, New York. D’ANCONA, U. 1926. “Dell’ in.fIuenza dell stasi peschereccia de1 period0 1914-18 sul R. Comit. Talass. Italian0 Italian0 Mem. 126, patrimonio ittico dell’Alto Adriatico,” Venezia. D’ANCONA, U. 1942. “La Lotta per l’Esistenza,” Einaudi, Torino. D’ANCONA, U. 1954. “The Struggle for Existence,” Brill, Leiden. D’ANCONA, U. 1964. Dagli Equilibri Biologici Alla Teoria Della, Pesca, Ric. Sci. 34, 251268. DARWIN, C. 1882. “The Origin of the Species by Means of Natural Selection,” 6th ed., J. Murray, London. ELTON, C. 1942. “Moles, mice and lemmings. Problems in population dynamics,” Clarendon Press, Oxford. GAUSE, G. F. 1934. “The Struggle for Existence,” Williams and Wilkins, Baltimore, Md. HALDANE, J. B. S. AND JAYAKAR, S. D. 1963. Polymorphism due to selection depending on the composition of a population, Genet. 58, 318-323. KOLMOGOROFF, A. 1936. Sulla Teoria di Volterra della Lotta per l’Esisttenza, Giorn. 1st. Ital. Attuari 7, 74-80. KOSTITZIN, V. A. 1934. “Symbiose, Parasitisme, et Evolution,” Herman, Paris. KOSTITZIN, V. A. 1936. Sur les solutions asymptotiques d’equations differentielles biologiques, C. R. Acad. Sci. Paris 203, 112-l 127. LOTKA, A. J. AND SHARPE, F. R. 1923. Contributions to the analysis of malaria epidemiology. IV: Incubation lag, Amer. J. Hyg. 3, 96. ANDREWARTHA,

Animals,”

VOLTERRA

LOTW, A. J. 1925. “Elements

AND

ECOLOGY

23

of Physical Biology,” Williams and Wilkins, Baltimore, Md., reprinted 1956, Dover, New York. tiTKA, A. J. 1932. The Growth of mixed populations: Two species competing for a common supply, J. Wash. Acad. Sci. 22, 461-469. MAINARD SMITH, J. 1968. “Mathematical Ideas in Biology,” Cambridge University Press, London. MORAN, P. A. P. 1953. The statistical analysis of the Canadian lynx cycle, Amt. j.2001. 1, 163-173, 291-298. NICHOLSON, A. J. 1957. The self-adjustment of populations to change, Cold. Sp. Harb. Symp. 22, 153-173. RFSCIGNO, A. AND RICHARDSON, I. W. 1965. On the competitive exclusion principle, Bull. Math. Bioph. 27, 85-89. RFSCIGNO, A. AND RICHARDSON, I. W. 1967. The struggle for life: I, Two species, Bull Math. Bioph. 29, 377-388. RICHARDSON, I. W. ANLI RESCIGNO, A. The deterministic theory of population dynamics (unpublished manuscript). Ross, R. 1911. The Prevention of Malaria, 2nd ed., London. WALTMAN, P. E. 1964. The equations of growth, Bull. Math. Biophys. 25, 75-93. WHITTAKER, R. 1941. Biography of Vito Volterra, Obit. Not. Fellows Roy. Sot.