Estimation of marine production from size spectrum

Ecological Modelling~ 42 (1988) 33-44

33

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

ESTIMATION OF MARINE PRODUCTION FROM SIZE SPECTRUM

J.E. PALOHEIMO

Department of Zoology, University of Toronto, Toronto, Ont. M5S 1AI (Canada) (Accepted 4 November 1987)

ABSTRACT Paloheimo, J.E., 1988. Estimation of marine production from size spectrum. Ecol. Modelling, 42: 33-44. Size composition or size spectrum is routinely collected in marine ecology. It is considered a major characteristic of the community structure as whole. The present paper provides a basis for a method to estimate total production from the size spectrum when the size-specific growth rate is known. The method generalizes many previously used numerical procedures and extends their domain of applicability. In particular, the previous methods based on cohort size composition are modified so as to be applicable for estimating current production based on the current size composition. The formulae are applied to estimated total production in those parts of the oceans for which size spectra are available.

INTRODUCTION

Understanding energy and material flows in an ecosystem is essential for any dynamic a n d / o r predictive model of the system. Biological production by species, population or assemblage of species provides a measure of material or energy that is actually available for circulation. It is an integrated measure of the total output from a part of the ecosystem. While it still needs to be partitiooned among its recipient pools the total production nevertheless provides us a first basis for further studies. Diverse techniques have been developed to measure production. These depend on the organism involved and on its trophic level. Primary pelagic production is typically measured in situ by rates of a4C incorporation (Steemann-Nielsen, 1963). Secondary production is often inferred from data The research was supported by a grant from the National Science and Engineering Research Council (Canada, Grant No. A4900). 0304-3800/88/$03.50

© 1988 Elsevier Science Publishers B.V.

34 on growth and mortalities or from known or estimated feeding rates and gross conversion efficiencies (e.g. Nees and Dugdale, 1959; Waters, 1977; Borgmann, 1982). All the techniques for measurements of secondary production are laborious. They are primarily applicable to estimate production only by a cohort that has been censused throughout its lifespan. In view of the relative ease with which data on size of organisms can be collected (Sheldon et al., 1972) it is important to develop techniques to measure both primary and secondary production based on current size and abundance compositions of organisms. A mathematical basis for such a technique is proposed here. When the production estimates are based on current rather than cohort size composition, size-specific growth rate is required. In an extensive review of secondary production, Waters (1977) classifies the various techniques based on population statistics into: removal-summation method; increment-summation method; instantaneous growth or Ricker's method (Ricker, 1946), Allen curve method (Allen, 1950, 1951) and Hynes' method. The Allen curve method is represented as a graphical extension of Ricker's method while Hynes' method is likened to the removal-summation method with an 'innovative approach'. Apart from the instantaneous growth method due to Ricker (1946) and the Allen curve method other remaining methods are presented as numerical procedures for calculating the production by a cohort. This fact has very much obscured both the generality and basic similarity of these apparently diverse procedures, as already pointed out by Mann (1969). To anticipate our more detailed treatment and to illustrate the points just made, consider a cohort the initial size of which is N(0) and the size at age a is N(a); also let w(a) be the mean weight at age a. By integration by parts we get:

-- foN(am)w(a)dN(a)=w(O) N(O)-w(am)

N(am)+

fOw(am)N(a)d w ( a )

where a m is the maximum age. Note that W(am) N(am) is the remaining biomass at age a m. If the census record covers the complete lifespan, the remaining biomass is zero and can be ignored. The first term on the right is the production due to recruitment and the second the production due to growth. The left-hand side gives the total production. This is shown graphically in Fig. 1. We note the figure that the production due to growth is the area bounded by the size spectrum curve and the weights axis while the total production is the area bounded by the size spectrum and the numbers axis. Although not always apparent from the literature (however, see Mann, 1969), it can be shown that the previously proposed numerical procedures are equivalent to evaluating one or the other

35 11

9

Z

. . . .

:~"

. . . . . . . . . . . . . . . . . . . . . . .

2

4

6

8

10

w(a) Fig. ]. Production f r o m a cohort. The hatched area on the fight estimates the to growth and the one on the left, the production due to recruitment.

production due

above integrals. The integral on the left-hand side represents the removalsummation method and Hynes' method (Waters, 1977) while the integral on the right represents the increment-summation method and Allen curve method (Allen, 1950, 1951; Waters, 1977). When the rate of growth is constant the integral is also equivalent to Kicker's method. The difference between the two integrals, N(0) w(0), is the production due to recruitment by the cohort's parent generation. In what follows we extend the above formula for a cohort to evaluate the total production and its components for equilibrium and non-equilibrium populations sampled either at a point in time or over the life span of organisms in case of univoltine species, i.e. species with one generation per year. The formulae generalize many of the previous numerical procedures. They also remove many of the restrictive assumptions such as linear growth required by Hynes' method (Hamilton, 1969) and replace these with additional data requirements on growth. When the size spectrum covers several species, a size-specific, species-independent growth rate is assumed. No assumptions are made about food intake or conversion efficiencies. The formulae are applied to estimate total production in those parts of the oceans for which size spectrum is readily available.

36 The present paper contains only an outline of the proposed method. Numerical procedures required to apply the method generally, apart from such situations where size spectrum is readily available as a smooth, differentiable function, will be dealt with in a separate paper (Paloheimo, in preparation). DEFINITION OF PRODUCTION By production we understand the total biomass elaborated by the cohort, population, or community per unit time (e.g. Chapman, 1971). It can be measured either as total production in a unit time (e.g. 1 year) or as production at some specific time t, expressed on a per unit time basis. If P(t) is the production at time t then the total production equals the integral of it. In a population at equilibrium total production equals the total biomass losses due to all sources of mortality. Production can be decomposed into production due to growth and production due to recruitment. Both components should be included in the estimate of the total production even though in practice they may be difficult to separate. Many species carry their eggs or neonates in their body cavities during the reproductive period. The production due to recruitment is thus included in the production due to growth till such time as the eggs or neonates are released when the production due to growth suddenly becomes negative for a period of time. Strictly speaking the production of eggs or young should not be included in production due to growth but the amount of biomass released at spawning or at the time of molting should be included in the reproductive component of the production. In some cases the production due to growth that includes the egg or neonate production may well approximate the total production. BASIC FORMULAE In the general case, the density function of numbers at age a at time t (per volume, per area, or per population) can be represented by n(a, t). This satisfies a simple partial differential equation known as the M'Kendrick or Foester equation (e.g. Rubinow, 1980):

an(a, t) aa

+

On(a, t) 8t

-

Z ( a , t) n(a, t)

(1)

where Z(a, t) is the age-specific instantaneous mortality rate at time t. Recruitment enters as a boundary condition and depends to some extent on the manner of reproduction. Let re(a, t) be the age-specific fecundity. In

37 the case of continually reproducing populations with nondormants eggs we have:

n(O, t)=

:/mre(a, t) n(a, t) da

(2)

For a cohort of a univoltine population we can in the present context simply put:

,,(0, t) = g(t)

(3)

where g(t) expresses the rate of hatching of eggs produced by the previous generation. For algae and other such organisms that produce by fission the effect of fission must be included among the factors that determine the age composition. Rubinow (1980) has suggested following change in our basic equation:

n(a, t) 8a

+

an(a, t) at

=

-(Z(a, t) + X(a, t)),,(a, t)

where )~(a, t) is the rate of fission. The boundary condition is now:

n(O, t)= 2

n(a, t) )~(a, t) da

However, the above two equations do not necessarily result in a stable weight composition even when all the rates are time-independent, nor are they compatible with a simple growth model dw(a)/da = G(a) w(a) or more complex one shown in equation (5). Till such time when an adequate algal growth model is available, we use a somewhat simplistic model. We assume that algae divide whenever they reach the maximum weight or age and that w(a m) = 2 W(0). The boundary conditions are now: n(0, t) : 2 n ( a m , t)

n(a, l)=0

for

a> a m

(4)

Growth of individuals, if no stochastic variability is permitted, can be described by an equation:

Ow(a,oa t) +-Ow(a'3t t) -G(a, t) w(a, t)

(5)

where G(a, t) is the age-specific rate of growth at time t, and w(a, t) the weight at age a, time t. Total production can now be estimated either as the net biomass accumulation plus the biomass losses or as the biomass accumulation due to

38 recruitment plus the biomass accumulation due to growth. Hence the following identity must hold:

P(t)= foamow(a' t) n(a' =n(O,t) w(O,t)+

t) On(a, da- foamw(a, t) [ On(a, 3a + ~t

foamn(a,t) [Ow(a't) 3a

+

Ow(a, 0t

t)]

da

t)]

] da (6)

where P(t) is the total production per unit time at time t. Each term in equation (6) can be written in readily interpretable form, e.g. the first term fight of the first equality sign is the rate of change in the population biomass, rB(t) B(t), where B(t) is the total biomass at time t, and rB(t) biomass weighted average intrinsic rate of increase. Altogether we get:

P(t)

Zn(t)]B(t ) =n(O, t) w(O, t) + GB(t ) B(t) Zn(t), GB(t) are the biomass weighted average of the

= [rB(t ) +

where age specific rates; for instance, GB(t) is defined by:

(7) corresponding

G.(t)-- f G(a, t)w(a, t)n(a, t)da/fw(a, t) n(a, t)da = fG(a, t) w(a, t) n(a, t)da/B(t) It follows that when the population is at equilibrium (i.e. rn(t ) = 0) the average (biomass weighted) mortality, ZB(t), is equal to the biomass turnover rate or to the production to biomass ratio. To equate the average mortality rate (i.e. numbers weighted) with the production to biomass ratio is incorrect although fairly common (e.g. Waters, 1969; Cushing, 1975; pp. 189-190). Simplifying both sides of our basic equation (6) and assuming that the growth is more or less independent of time, G( a, t) = G(a) and w(a, t) -w(a), we arrive at an equation for estimating total production:

foam

3n(a't)da=n(O,t)w(O,t)+ foW(am)n(a,t) dw(a) (8)

Integrating over time this can further be simphfied, and an estimate of the total production per unit time can be based on the annual or per unit time average figures: foN'"')w(a) d N ( a )

N(0) w ( 0 ) + fo

w(am)N(a)dw(a)

(9)

39

N(a) is the total number of organisms of age a, i.e.

where

n(a, t) dt

N(a)=

Periodically reproducing populations such as fishes require a somewhat different treatment. Such populations consist of discrete age groups or cohorts. These may be denoted by: N, (t) is number of i year olds at time t, i > 0 An estimate of production due to growth at time t is now: Pg(t)=

i=am 8w(i, Y'~ Ni(t ) 3t

t)

i=0 i=a

m

= E Ni(t) Gi(t) w(i, t)=GB(t ) B(t)

(10)

i=0

This is in fact the Ricker's original formula (1946) with the difference that GB(t ) is the biomass weighted average growth rate. The production due to recruitment is given as before by the estimated cohort size multiplied by the initial weight of recruits.

Production estimates The above formulae may be used to approximate the total production in world oceans. Both the size spectrum and growth rate seem to be related to the average body size of organisms, and at least the size spectrum seems to be relatively stable in those parts of the oceans where it has been studied. Platt et al. (1984) give the density function of organisms per volume of water as b(w) (our notation), termed by them as normalized size spectrum, where:

b(w) dw = flow" dw is the number of particles in size range w, w + dw (11) The exponent m has the value of m = - 1 . 2 2 for the north central gyre of the Pacific in Platt et al (1984) and m - - - 1 . 2 3 for microplankton in Rodriguez and Mullin (1986). The total size spectrum may be divided into octave classes with upper boundary of each class being twice the lower, w i = 2wi+ 1 The size spectrum as given in equation (11) covers some 12-15 such classes for particles over 1-2 ~m in spherical diameter. For smaller autotrophic particles in the 0.25-2 g m range, Platt et al. postulate a value of m in the north central gyre of the Pacific in the range - 0 . 8 9 to - 0 . 8 6 depending on the depth zone.

40 Since the number of particles as given by equation (11) must be the same as the numbers in a corresponding age group, one may set: n(a) da=b(w)dw

(11')

To get hold of n (a) for the purpose of evaluating the production integral in equation (8), d w / d a needs still to be specified. I will now turn to evaluating the growth rate. Individual growth of unicellular organisms is related to population growth when growing in conditions that exclude the usual sources of mortality (i.e. losses due to to convection and and grazing). Fenchel (1974) specifies the maximum intrinsic rate of increase both for unicellular and multicellular forms as: r = otw -v

(12)

where 7 = 0.25. Blasco et al (1982) determined intrinsic rate of increase for a number of marine diatoms in laboratory. While the functional form in equation (12) was verified, their y was different and depended on whether the total volume (3' = 0.11) or the carbon content (3' = 0.14) was used as a surrogate measure of the organism's weight. Both values are appreciably lower than Fenchel's. I have no idea why the discrepancy. Assume now that the individual growth within each octave class is given by: dw = Gw t~ da

(13)

In the absence of mortality the time to doubling of initial weight of an organisms may be equated to the population doubling time. Taking the mean weight within an octave class (w i, 2wi) as 1.5w. solving equation (13) and equating the doubling times from equations (12) and (13), we get: /3 = 1 - V G = a ( 2 ' - 1)/(3, ln(2)(3/2) ~)

(14) (14')

Note that neither of the growth parameters G a n d / 3 depend on the octave class. Having specified the growth, the density function of numbers at age n (a) can now be calculated. From equations (11), (11') and (13) it now follows that: n ( a ) = /3oGw ''+#

d n ( a ) =/30 G ( m + /3) w re+o-' d w

(15) (15')

Total biomass within each octave class follows directly from the biomass

41 spectrum in equation (11). The production can be estimated using equations (8) and (15'). The results are: B i =

fw~'w,W b ( w ) dw flo (2"+2 , m+2

-

1') w i"+2

mf + Pi = - fw w,w d n ( a ) = -floG(2 re+o+1- 1) m T 1 7fl 1 w~+B+ l

(16) (17)

i

Note that the term m + fl is negative and that the production given above is in fact positive. Total production over k octave classes covering the size range from w 0 to w k = 2kWo can also be readily calculated:

k m+fl (Wn~+fl+a__wr~+k+l) P = Y'~Pi= floGm + fl + 1

(17')

i

Since the size of the smallest organisms is somewhat indeterminate and since there is also an uncertainty about the size spectrum in the first two or three size classes in the north Pacific equation (17') may need to be modified. Both the biomass and the production increase with each octave class. On the other hand the biomass turnover rate, or P / B ratio decreases. From above it follows that:

P,/B, = a w 7 r

(18)

where A= -

G(2 m+o+l- 1) ( m + f l ) ( m + 2) (2 " + 2 - 1 ) ( m + f l + 1)

(18')

Due to the nature of our derivation, the above equation is only valid for unicellular organisms that reproduce by fission. However, conditional on the validity of Fenchel's relationship, equation (12) and the form of the growth curve (13), the production estimate should be valid for multicellular organisms as well. The numerical value of A depends on the actual values that we assign to m, fl and y. Note that change in P / B ratio with size or with octal class depends only on the growth parameter y (or fl, see equation 14) of the organism. DISCUSSION

It is known that production can be estimated from the cohort size spectrum. We have shown that production can also be estimated from the current size spectrum. The growth curve is implicit in the cohort size spectrum and hence need not be specified explicitly. When the production is

42 estimated using the current size spectrum, the growth rate at each point of the spectrum is required. To be more precise, to estimate the production due to growth from the current size spectrum, the rate at which the organisms go through each size class is required. To estimate to total production, the rate at which the number at age change is required. The necessary information can in both cases be obtained if the size spectrum and the growth rate are known. Whenever the size spectrum can be specified by a smooth differentiable function as e.g. in Platt et al. (1984), both the total production (left side of equation 8) and the production due to growth (right side of equation 8) can be readily calculated given the growth rate. When the size spectrum is specified by discrete data points, numerical difficulties arise in estimating the production, more so in estimating the total production than in estimating the production due to growth. These difficulties are dealt with in more detail in Paloheimo (in preparation). The differences in numerical difficulties in estimating the total production and the production due to growth can be appreciated by noting that the total production estimate (left side of equation 8) requires the second-order differential (or difference) of the accumulated numbers at age function, while the production due to growth (right side of equation 8) requires only the first-order differential (or difference) of the accumulated numbers at age or size. Size composition or size spectrum is routinely collected especially for many aquatic organisms (Sheldon et al., 1972; Paloheimo and Fulthorpe, 1987). It has been proposed as a major characteristic of the community structure as whole (Platt et al., 1981). In particular, we have shown here that if the growth is in fact size-dependent as assumed here, the size spectrum does characterize the pattern of productivity in an ecosystem as well. Because of uncertainty about size spectrum at very small particle sizes as well as because of my unfamiliarity with the marine data, I have refrained from placing actual numerical values on the expressions for total production by octave classes (equation 17) or for the overall production (equation 17'). Because of the ease at which the size spectrum can be determined, it is interesting to speculate whether the technique such as presented here can actually be used in place of in situ determination of primary pelagic production. The main difficulty will be determination of size-specific growth rates. Here we have circumvented the problem by assuming the validity of Fenchel's relationship. Production to biomass ratios are commonly used to specify differences among populations and trophic levels (Dickie et al., 1987). We have shown that for non-equilibrium populations the P/B ratio may be equated to the sum of the prevailing intrinsic rate of increase and the average mortality rate, both averages being biomass weighted. This in turn can be equated to

43 the s u m o f the b i o m a s s r a t e o f r e c r u i t m e n t a n d the b i o m a s s w e i g h t e d a v e r a g e g r o w t h rate. R e m a r k a b l e regularities in the b i o m a s s to p r o d u c t i o n r a t i o s o b s e r v e d in h i g h e r o r g a n i s m s ( D i c k i e et al., op. cit.) s e e m to b e valid f o r u n c e l l u l a r o r g a n i s m s as well ( e q u a t i o n 18). H o w e v e r , the w e i g h t e x p o n e n t ~, for m a r i n e u n i c e l l u l a r pelagic o r g a n i s m s a p p e a r s to b e s o m e w h a t lower than documented for multi-ceUular organisms. ACKNOWLEDGEMENTS I wish to t h a n k L . M . D i c k i e a n d T r e v o r P l a t t of B e d f o r d I n s t i t u t e o f O c e a n o g r a p h y , D a r t m o u t h , N.S. C a n a d a f o r their interest a n d for editorial c o m m e n t s . I also wish to a c k n o w l e d g e S t e w a r t B l a k e for t e c h n i c a l assistance. REFERENCES Allen, K.R., 1950. The computation of production in fish populations. N.Z. Sci. Rev., 8: 89. Allen, K.R., 1951. The Horokiwi stream. Bull. Mar. Dep. N.Z. Fish. 10, 238 pp. Blasco, D., Packard, T.T. and Garfield, P.C., 1982. Sice dependence of growth rate, respiratory electron transport system activity, and chemical composition in marine diatoms in the laboratory. J. Phycal., 18: 58-63. Borgmann, U., 1982. Particle-size-conversion efficiency and total animal production in pelagic ecosystems. Can. J. Fish. Aquat. Sci., 39: 668-674. Chapman, D.W., 1971. Production. In: W.E. Ricker (Editor), Methods for Assessment of Fish Production in Fresh Waters. IBP Handbook, 3. Blackwell, Oxford, 348 pp. Cushing, D.H., 1975. Marine Ecology and Fisheries. Cambridge University Press, Cambridge, 278 pp. Dickie, L.M., Kerr, S.R. and Broudreu, P.R., 1987. Size-dependent processes underlying regularities in ecosystem structure. Ecol Monogr., 57: 233-250. Fenchel, T., 1974. Intrinsic rate of natural increase: the relationship with body size. Oecologia, 14: 317-326. Hamilton, A.L., 1969. On estimating annual production. Limnol. Oceanogr., 14: 771-782. Mann, K.H., 1969. The dynamics of aquatic ecosystems. Adv. Ecol. Res., 6: 1-81. Nees, J. and Dugdale, R.C., 1959. Computation of production for populations of aquatic midge larvae. Ecology, 40: 425-430. Paloheimo, J.E. and Fulthorpe, R.R., 1987. Factors influencing plankton community structure and production in fresh water lakes. Can. J. Fish. Aquat. Sci, 44: 650-657. Platt, T., Mann, K.H. and Ulanowicz, R.E. (Editors), 1981. Mathematical Models in Biological Oceanography. Unesco, Paris, 157 pp. Platt, T., Lewis, M. and Geider, R., 1984. Thermodynamics of the Pelagic ecosystems: Elementary closure conditions for biological production in the the open oceans. In: M.J.R. Fasham (Editor), Flows of Energy and Materials in Marine Ecosystems. Plenum, New York, pp. 49-84. Ricker, W.E., 1946 Production and utilization of fish populations. Ecol. Monogr., 16: 374-391. Rodriguez, J. and Mullin, M.M., 1986. Relation between biomass and body weight of plankton in a steady state oceanic ecosystem. Linmol. Oceanogr., 31: 361-370.

44 Rubinow, S.I., 1980. Cell kinetics. In: L.A. Segel (Editor), Mathematical Models in Molecular and Cellular Biology. Cambridge University Press, Cambridge, pp. 502-522. Sheldon, R.W., Prakash, A. and Sutcliffe, W.H., 1972. The size distribution of particles in ocean. Limnol. Oceanogr., 17: 327-340. Steemann-Nielsen, E., 1963. Fertility of the oceans: productivity definition and measurement. In: M.N. Hill (Editor), The Sea, Vol. 2. WHey, New York, pp. 129-164. Waters, T.F., 1969. The turnover ratio in production ecology of fresh water invertebrates. Am. Nat., 103: 173-185. Waters, T.F., 1977. Secondary production in inland waters. Adv. Ecol. Res., 10: 91-164.