Bulletin of Mathematical Biology Vol. 45, No. 5, pp. 869-872, 1983. Printed in Great Britain

0092-8240/8353.00 + 0.00 Pergamon Press Ltd.

© 1983 Society for Mathematical Biology

T H E R M O D Y N A M I C A P P R O A C H TO B I O M A S S D I S T R I B U T I O N IN E C O L O G I C A L SYSTEMS

• DAVID LURII~, JOAQUIM VALLS and JORGE WAGENSBERG Departmento Termologia, Universidad de Barcelona, C[. Diagonal, 645, Barcelona (28), Spain

In the derivation of the biomass distribution function for an ecological populat ion critical use is made of an energetic constraint on the maximization of biomass diversity. The nature of this constraint is explored in detail using Kleiber's relation o ( m ) = c m 3" between animal metabolic rate o ( m ) and body weight m in conjunction with the Prigogine-Wiame thermodynamic paradigm for specific entropy production in biological stationary states. These two inputs fix the energetic constraint on the maximization of biomass diversity to be the constancy of the mean metabolic rate of the ecosystem. The resulting biomass distribution function is tested against observational data.

The B iomass Dis t r ibu t ion Model . In an earlier paper (Luri6 and Wagensberg, 1983) it was postulated that the biomass distribution function for an eco- system may be derived from the condition that the biomass diversity func- tional is maximal subject to an energetic constraint. The resulting biomass distribution,

1 p ( m ) = - ~m/m, (1)

r~

where rn is the mean biomass per individual, was tested against fishery data and found to be in good agreement.

In this note we wish to explore in more detail the nature of the energetic constraint to be used in the maximization of biomass diversity. If Kleiber's (1932) observed relation between body weight and metabolic rate is used in conjunction with the Prigogine-Wiame (Prigogine and Wiame, 1946) thermo- dynamic paradigm for local entropy production in biological stationary states, a new energetic constraint on biomass diversity is found to be the constancy of the mean metabolic rate of the ecosystem.

The results of Kleiber on the metabolic rates of various organisms in relation to their body weights indicate that this relation may be written in the form

o ( m ) = cm • , (2)

where o ( m ) is the metabolic rate and m is the biomass and c and 3' are

869

870 DAVID LURIE , J O A Q U I M V A L L S A N D J O R G E W A G E N S B E R G

constants. Figure 1, based on the data of Hemmingsen (1960), exhibits the universality of this relationship for the three large metabolic groups of animals covering a range of biomass values from 10 -~2 to 106 g. The slope of the straight lines is given by the constant 3' in (2). Numerically, 3" "~ 0.75 according to Kleiber.

i0 o

I0 -6

~ x _ % , . x o ~ . IO "~2 , ~x_ ",-'Sx'~

i ,~,° I I f

10-12 10-6 IO 0 10 6

Body weight (g)

Figure l. Metabolic rates of various organisms in relation to body weight [data from Hemmingsen (1960)].

Kleiber's experimental law connects the individual biomass whose prob- ability distribution function we are seeking to the energy cost of maintaining it, i.e. the metabolic rate. The latter is directly constrained by the possibilities of the environment. As noted by Prigogine and Wiame (1946), the mean metabolic rate of a stationary biological system can be identified with the local entropy production and the latter is determined by the external boundary conditions. In the case of a single organism the 'mean' metabolic rate must be understood as the metabolic rate per unit mass; for an ecosystem the 'mean' rate is the rate per individual. Since the latter must be constant in the stationary regime, the proper energetic constraint on the maximization of the biomass diversity functional

f? la = - - p ( m ) log2 p ( m ) d m , (3)

where p(m) is the biomass probability density, is simply

fo ' P ( m ) c m ' r d m = o , (4)

where a is the constant mean metabolic rate for the ecosystem.

THERMODYNAMIC APPROACH TO BIOMASS DISTRIBUTION 871

With this constraint the maximization of the biomass diversity functional yields the following expression for the biomass probability density in the stationary state:

where

k p(m) = ~ ~e m~p/a,

3"( a-Xl~) cll~ k -

r( l /3 ') and where 7(n) is the well-known gamma function defined as

> 0 F(n) = (a + 1)n f = tn-a e-(a+Ot dt n

2o a >- -1 .

(5)

(6)

(7)

The present result (5) differs slightly from that of our earlier paper. It reduces to the previous result when c = 3' = 1, which implies a reduction of the energetic constraint to a simple constraint on the biomass Fn.

Discussion. Observational tests of (5) require a discrete form for this result. We take o(m) as the continuum limit of Pi/Am for biomass values rn in the discrete interval i o f size Am. Then

r n / ~ = 3 ' ° ( - l n P t ) + 3'--~0 In A m + ln3" ~ (8) c c ( o / c ) l m "

We have tested this result against the same fishery data used to test (1). For the BENGUELA IV data (Luri6 and Wagensberg, 1983), for example, when m7 values are plotted on the y-axis the experimental points lie on a straight line, as predicted by (8), with a Pearsonian correlation coefficient of 0.986. We have also tested (8) against five other sets of data. In each case the conclusion is that the data cannot differentiate meaningfully between the models based on (2) and (1) respectively. The model of (1) provides a good approximation, even though it fails to take account of Kleiber's experi- mental relationship between biomass and metabolic rate. We do feel, how- ever, that more accurate data over a larger range of biomass values will enable one to differentiate between the two models, and based on the theoretical arguments presented above, we would tend to favor the model presented here.

872 DAVID LURIE, JOAQUIM VALLS AND JORGE WAGENSBERG

L I T E R A T U R E

Hemmingsen, A. M. 1960. "Energy Metabolism as Related to Body Size and Respiratory Surfaces, and its Evolution." Rep. Steno mere Hosp. 9, 1-110.

Kleiber, M. 1932. "Body Size and Metabolism." Hilgardia 6, 315-353. Lufi6, D. and J. Wagensberg. 1983. "On Biomass Diversity in Ecology." Bull. math.

BioL 45, 287-293. Prigogine, I. and J. M. Wiame. 1946. "Biologie et Thermodynamique des Ph6nom6nes

Irr6versibles." Experientia 2, 451--453.

R E C E I V E D 6-21-82

R E V I S E D 1 1-16-82