Bulletin o[ Mathematical Biology, Vol. 45, No. 2, pp. 287-293, 1983. Printed in Great Britain.

0092-8240/83/0202874)7503.00/0 Pergamon Press Ltd.

�9 1983 Society for Mathematical Biology

N O T E

ON BI OMASS D I V E R S I T Y IN E C O L O G Y

DAVID LURII~ Dep. Fisica Te6rica, Universidad de Barcelona

JORGE WAGENSBERG Dep. Termologia, Universidad de Barcelona, c /Diagonal , 645, Barcelona (28), Spain

We postulate that the biomass distribution function for an ecological population may be derived from the condition that the biomass diversity functional is maximal subject to an energetic constraint on the total biomass. This leads to a biomass distribution of the form p(m) = ff~-t exp ( - m/Va), where r~ is the mean biomass per individual. The same con- dition yields a unique value for the biomass diversity functional. These predictions are tested against fishery data and found to be in good agreement. It is argued that the existence of a unique value for biomass diversity may provide a preliminary theoretical foundation for the observed upper limit to species diversity.

A Model .for Biomass Diversity. The use of the Shannon (Shannon and Weaver, 1963) formula as a measure of ecological diversity was originally introduced by MacArthur (1955) and Margalef (1958) and has proved to be a useful tool in analysing the structure of ecosystems (Margalef, 1972). If the number of individuals of species 1, 2 . . . . . s is N1, N2 , . . . ,N~ and

N = ~: Ni, then the diversity per individual can be expressed in either of the i = 1

two alternate forms:

N~ D = l~ N , ! N ~ ] . . N~! (la)

s

D = - ~ P i log2 Pi, (lb) i = l

where Pi = Ni/N. Both forms can be shown to be equivalent for large populations using Stirling's formula. In this note we wish to call attention to certain interesting consequences which follow from shifting the focus of attention from the above species diversity index to a quantity which we shall call the biomass diversity per individual. To define this quantity

287

288 D. LURIt~ AND J. WAGENSBERG

we divide the range of biomass values available to an ecosystem into r discrete intervals of size Am. Then the biomass diversity per individual is defined as

1 N ! = ~ l o g 2 nl !n2! . . . nr! ' (2a)

where n~ is the number of individuals in the ecosystem whose biomass falls in the interval i independently of species. Needless to say, this definition of tz depends both on the size of the interval Am as well as on the range r A m of biomass values available to the ecosystem, and we shall concern ourselves with this dependence when we consider concrete examples further on. For large populations an equivalent form for /z is as in (lb)

/'/i 1 /'/i /x = - ~- lOg2 ~-. i = l

(2b)

For the sake of simplicity we assign one biomass m~ to each individual in class i. The quantity ~ is more closely related than the species diversity index to the physical constraints which operate on the ecosys- tem; for example, a natural constraint would appear to be the finiteness of the energy input (solar radiation, nutrients, etc.) and the resulting limitation on the total biomass M which the system can support. Let us now assume as a working hypothesis that biomass diversity is maximized subject to the above constraint on the total biomass, i.e.

M = ~ n,m,, (3) i = l

where rn~ is a representative biomass for individuals in the interval i. In assuming that diversity is to be maximized, we are guided by analogies with entropy-like quantities in other fields (Demetrius, 1978a, b). Indeed, the maximization of t~ under the constraint (3) and the obvious restric-

tion that ~ n~ = N is completely analogous to the derivation of the i ~ l

canonical ensemble in statistical mechanics. The standard variational technique leads to the result

- - ~ r n i /li e

N ~ e_t3,~,' i = l

(4)

ON BIOMASS DIVERSITY IN ECOLOGY 289

where/3 is a constant Lagrange multiplier. Defining a second constant

a = ~ , (5) i = 1

we can write (4) in the form

1 1 m, = - -a- In P, - w In a,

P P (6)

where we have defined P~ = ndN ; P~ is just the occupation probability for the interval i. Our hypothesis therefore leads to the prediction that there should exist a linear relation between mi and In P~.

Comparison with Observations. We have checked the validity of this prediction against two sets of observational data: the report of the ATLOR VI expedition (Manriquez and Rucabado, 1976) on the weight distribution of fishes off the northwest coast of Africa and the report of the BENGUELA II expedition (MacPherson, 1981) on the weight dis- tribution of fishes off the coast of Namibia. As we have noted earlier, the definition (2a) of the biomass diversity tz depends on both the size of the mass interval Am and on the range r a m of biomass values available to the ecosystem. A corresponding dependence on the choice of these two parameters is also evident in (6) since both P, and a by their very definition depend on the manner in which the biomass continuum is discretized. The choice of ram, i.e. the total range of biomass, is relatively straightforward--it is suggested by the analysis of the fishery data itself. The ATLOR data, for example, consists of 50 hauls in a zone of upwelling containing a total of 398,000 fishes. This sample contains at least 20 different species, having a minimum representation of at least 1% of the total number of fish. The lower limit of the biomass range is established by the size of the fishing mesh and the upper limit is determined by the fact that biomass values greater than 0.085 kg are represented by occupation numbers n, which are less than 0.1% of the total population sample, and are therefore sufficiently improbable that they may be neglected.

The next step is the selection of the biomass interval Am. For the ATLOR data, a choice of Am = 0.01 kg yields the set of data points in Figure 1. Inspection of this figure shows that the points do indeed fall in a straight line, as predicted by the model. The best linear regression fit to the data has a slope of 0.014 and a y-intercept of -0 .002 with a Pearsonian correlation coefficient of r = 0.99, indicating a near optimum

290 D. LURII~ AND J. WAGENSBERG

m 5

. J . J

i t L I I I t I 2 5 4 5 6

- i n Pi

Figure I. The biomass probability distribution of the fishery data (ATLOR VI) for Am = 0.01 kg. The mass per individual m~ is plotted in Am-units.

fit. Before discussing how the analysis of the ATLOR data would be affected by a different choice of Am, we note that the parameters of the straight line (its slope and y-intercept) can be computed theoretically on the basis of the model. To do this we consider the mean biomass r~ = M / N and use (4) to write it in the form:

• m i e -t3"~ = i= l (7)

~ e-/3mi i = l

Multiplying the numerator and denominator by Am and approximating the summations by their continuum limits, we obtain

1 ,n = w . (8 ) /a

A similar manipulation for a using (5) yields

ffl c~ = A m " (9 )

Hence, once tfi is known and Am is selected, the slope and y-intercept in (6) are calculable. For the ATLOR data, fit = M[N is found to be 0.015 kg. The theoretical values for the slope and y-intercept are there- fore r~ = 0.015 kg and r~ In Am[fit = - 0.006 kg, which compare satis- factorily with the observed values cited above. This provides a con-

O N B I O M A S S D I V E R S I T Y IN E C O L O G Y 291

sistency check on the analysis of the data. A further consistency check can be performed by varying the mass window Am and observing whether or not the data continue to fall on a straight line having the same slope fit and a displaced y-intercept in accordance with (6). We have verified that this is indeed the case for mass windows of 0.02 kg (four data points) and 0.005 kg (sixteen data points).

The B E N G U E L A II data consists of 69 hauls with a total of 510,000 fishes and is analysed in the same way as the ATLOR VI data. As is seen in Figure 2, the data points again fall in a straight line, with the best linear regression fit having slope rfi = 0.035 kg and y-intercept -0.006 kg. The Pearsonian correlation coefficient is r = 0.98, which again indicates a good fit. The theoretical values for the slope and y-intercept in this case are rh = 0.038 kg and ~ In Am/ffz = - 0.012 kg.

Discussion. The model appears to perform quite well for the two sets of data which we have analysed. The hypothesis that ecosystems tend to maximize biomass diversity subject to an energetic constraint has a further interesting consequence. If instead of the Am-dependent biomass diversity (2a) or (2b) we define a new 'normalized' biomass diversity

rp, -i /X = - ~:,~ P, log2 {_ Am J' (10)

m i 4

#

/

I I I I I 0 3 4 5 6

- L n P i

Figure 2. T h e b i o m a s s probabi l i ty d is t r ibut ion of the f i shery da ta ( B E N G U E L A II) for Am = 0.03 kg. T he m a s s per individual mi is plot ted in Am-un i t s .

/ /

I I I 2

292 D. LURII~ AND J. WAGENSBERG

we can easily show that (a) the above quantity is independent in the continuum limit of the choice of Am and that (b) in nature it must have a fixed value of

k - log2 e = I/In 2 = 1.443 bits per individual.

To see this, note that (4) for Pi = nJN can be rewritten using (8) and (9) a s

-mi /~

P~ - e------Am. (11)

Substituting (11) into (10) we obtain

/ 2 = - k ~ P , - =k , (12) i = l

whose value is just 1.443 bits per individual. We have computed 12 for the two sets of fishery data analysed above. The values found in each case are respectively 1.101 and 1.221 bits per individual, both in reasonably good agreement with the theoretical value predicted by (12) for the continuum limit.

In this connection it is worth pointing out that Margalef (1972) has previously called attention to the existence of a rough band of values for the species diversity index (1) having an approximate upper bound of 5 bits per individual. Although the species diversity index is not directly related to the normalized biomass diversity and we are therefore not in a position to claim that the existence of a universal value of k for the latter is necessarily connected to the existence of an observed band of values for the former, we do not wish to dismiss this possibility out of hand. This point definitely requires further study. One possible line of attack would be to relate species and biomass indices to each other under the admittedly unrealistic assumption that all members of a given species have the same biomass.

Leaving this question aside, we do wish to argue that the remarkably good fit to the two sets of fishery data reported here provides a good measure of support for the model which we have presented and that, despite the scepticism often expressed by biologists regarding maximiza- tion principles in general, the biomass diversity maximization principle might prove to be a useful tool for treating ecological and more generally biological systems. To test the validity of the principle for other ecosys- tems much more data are, of course, needed.

ON BIOMASS DIVERSITY IN ECOLOGY 293

We are indebted to Professor Margalef for several stimulating dis- cussions on the subject of this paper. We are also indebted to Mr. J. Valls for assistance in analysing the BENGUELA II data.

LITERATURE

Demetrius, L. 1978a. "Entropy and Life Table." Naturwissenschaflen 65,435--436. 213_~llA978b.,..,_ "Adaptive Value, Entropy and Survivors Curves." Nature, Lond. 275,

MacArthur, R. 1955. "Fluctuations of Animal Populations and Measure of Community Stability." Ecology 36, 533-536.

MacPherson, E. 1981. "Informe de Benguela II." Direcci6n General de Pesca, Barcelona. Manr~quez, M. and J. Rucabado. 1976. "Datos Informativos No. 1 del Afloramiento N. O. de

Africa." Instituto de Investigaciones Pesqueras, Barcelona. Margalef, R. 1958. "Information Theory in Ecology." Gen. Syst. 3, 36-71.

. 1972. "Homage to Evelyn Hutchinson, or Why There is an Upper Limit to Diversity." Trans. Conn. Acad. Arts Sci. 44, 211-235.

Shannon, C. E. and W. Weaver. 1963. The Mathematical Theory of Communication. University of Illinois Press. Urbana.

RECEIVED 2-3-82