On Steady-State Modelling of *

V. CHRISTENSEN and D. PAULY International Center for Living Aquatic Resources Management MCPO Box 2631, 0718 Makati Metro Manila, Philippines

CHRISTENSEN, V. and D. PAULY. 1993. On steady-state modelling ofecosystems, p.14-19. In V. Christensen and D. Pauly (eds.) Trophic models ofaquatic ecosystems. ICLARM Conf. Proc. 26, 390 p.

Abstract

This paperprovides a briefdescription oftherationalebehind steady-statemodelling, andoftheimplementation ofthe II software system, a system for straightforward construction, parametrization, and balancing of steady-state trophicmodels of(aquatic) ecosystems. ECOPATH II iswrittenfor MS DOS computers and is available as a public domain software from the ICLARM Software Project. ECOPATH IIis structured around a systemoflinearequations initiallyproposed by J.J. Polovina and coworkers. Also, it incorporates routines for computation of several maturity and network flow indices proposed by various theoretical ecologists, notably the Odum brothers and R.E. Ulanowicz.

Modelling of Ecosystems without"crashing" or"exploding",wherepopulations go eitherextinctorgrowwithoutbound, respectively. The word "model" has several meanings; for This is one reason why most aquatic biologists shy scientists and, more specifically, for biologists away from constructing such models, or even from working at the level, "models" may be interacting with "modellers" (who, often being defined as consistent descriptions, emphasizing nonbiologists, may have scant knowledge of the certainaspectsofthesysteminvestigated, asrequired intricate interactions between living organisms). to understand their function. Another reason is that one needs to be able to Thus, models mayconsistofa text("wordmodels") describe the dynamics ofall key biological processes or a graph showing the interrelationships of the (growth, reproduction, mortality, etc.) to build various components of a system. Models may also realistic dynamic models. Obtaining sufficient consist of equations, whose parameters describe knowledge to do this is difficult for most ecosystems. "states" (the elements included in the models) and However, "modelling" does not necessarily imply "rates" (ofgrowth, mortality, food consumption, etc.) "simulation modelling". There are various ways of ofthe elements ofthe model. constructing quantitative models of ecosystems . The behavior ofmathematical models is difficult which avoid the intricacies of dynamic simulation (often impossible) to explore without computers. modelling, yet still provide many Of the benefits of This is especially the case for "simulation models", fully-fledged modelling, viz: i.e., thoserepresentations ofecosystemswhichfollow, • requiring the biologist/ecologist to review and throughtime, the interactive behavior ofthe (major) standardize all available data on a given components ofan ecosystem. ecosystem and identify information gaps; Simulationmodels are difficult to build, and even • requiring the would-be modeller to identify more difficult to get to simulate realistically the estimates (of states and/or rates) that are behavior of a system over a long period of time, mutually incompatible, and which, if true, would prevent the system from maintaining itself (e.g., prey productions that ar~ too low relative to assumed food requirements of *Includes extracts from the ECOPATH II manual ofChristensen and Pauly (l992a). ICLARM Contribution No. 831. predators);

14 15 In many cases, the period considered will be one A (typical)year,withthestate andrate estimates used for model construction pertainingto differentyears. lJ) lJ) o Suchmodels mayrepresent a decade ormore, during E which little changes have occurred. o (6 When ecosystems have undergone massive changes, two or more models may be needed, representing the ecosystem before, (during) and after the changes (Fig. 1). AB an example, three Time models of the Peruvian upwelling ecosystem were constructed, covering different periods before and afterthe collapse ofthe anchovetafisheries (Jarre et

lJ) al. 1991). Otherexamples ofthis canbe found inthis lJ) o volume for Lake Tanganyika, Lake Victoria, and E Lake Turkana, all in Africa. .Q (D When seasonal changes are to be emphasized, differentmodels maybe constructedfor eachseason, orfor extreme situations ("summer" us. "winter"). AB Time anexample,BairdandUlanowicz (1989) constructed c four models describing the seasons in Chesapeake Bay and one "average" model to represent the whole year. Likewise, Jarre and Pauly (this vol.) describe the dynamics of the annual cycle of the Peruvian upwellingsystemusing 12 steady-statemodels, each representing a monthly period. The same idea can be applied to aquaculture situations, where a pond and its producers and Time consumers can be described for instance at the beginning, midpoint and end of a growing season. Fig. 1. Schematic representation ofpossible trends in an Alternatively, a pond canbe modelled as the average ecosystem. (A) Strong, regularchangesas, e.g., due to the succession of such states. Ruddle and Christensen (this vol.) ofseasons, notwell represented by an annual mean (B). (B) Rapid illustrate this approach. transition between two stable states, of which each is well Judicious identification of periods long enough represented byits ownmean(Bl' B ). (C) ExampIe ofa biomass that 2 for sufficient data to be available, but short enough does notreach equilibrium. Duringa briefperiod (te), this biomass canbe represented by a singlevalue (Be) whose confidence interval for massive changes not to have occurred, will thus will usually bracket the change ofbiomass during the interval t • e solve most problems associated with the lack of a • requiringthesamewould-be modellertointeract time dimension in "steady-state" models. with specialities other than herlhis own, e.g., a planktonspecialistwill haveto eithercooperate The ECOPATH II Model withfishbiologists andothercolleaguesworking onvarious consumergroups, oratleastreadthe The ECOPATH II system combines an approach literature they produce. by Polovina (1984a) for estimation of biomass and To avail of these and other related advantages food consumption ofthevarious elements (species or without having to get involved in simulation groups of species) of an with an modelling, one's models can be limited to describing approach proposed byUlanowicz (1986) for analysis "average" (or "steady-state") states and rates. This of flows between the elements of ecosystems limitation, as we shall see, is not as constraining as (Christensen and Pauly 1992b). it may appear at first sight. Itis consistent with the AB described by Pauly et al. (this vo!.), the core workofmost aquaticbiologists, whose stateandrate routine of ECOPATH II is derived from the estimates also represent "averages", applying to a ECOPATH program ofPolovina and Ow (1983) and certain period (although this generally is neither Polovina (1984b, 1985). stated by the authors, nor realized by the readers). The ecosystem is modelled using a set of The approach we propose is to use states and simultaneous linear equations (one for each group i rates estimated for single species in a multispecies in the system), i.e., context to describe aquatic ecosystems in rigorous, quantitative terms, during the (arbitrary) period to Production by (i) - all on (i) - nonpredation losses of (i) ­ which the state and rate estimates apply (Fig. 1). export of (i) =0, for all (i). 16 This can also be put as In a "steady-state" model, the energy input and output ofall living groups mustbe (or are) balanced, = p.1- M2. 1- p. 1 (I-EE-)1 - EX.1 0 ••• 1) by definition. The basic ECOPATH equation (1) includes only (i), where p.1 is the production of M2.1 is the total the production of a box. Here production equals predation mortality of (i), EE; is the ecotrophic predationmortalityplus exportplus othermortality. efficiency of (i) or the proportion of the production Whenbalancingthe energyflow ofa box, otherflows that is either exported or predated upon, (1 - EE) is should be included. Thus, the "other mortality", and EX; is the export of (i). Consumption =production + respiration + unassimilated food Equation (1) can be re-expressed as n From this the respiration can be estimated as a B.*PB. -:LB.*QB.*DC.. - PB.*B. (I-EE) - EX; =0 difference (but see below). 1 1 j=l) ) )1 1 1 or Parametrization n B.*PB.*EE. -:L B.*QB.*DC.. - EX; = 0 ...2) 1 1 1 j=l) ) )1 The data requirements of steady-state models areverylimitedincomparisonto those ofsimulation whereB. is the biomass of(i), PB. is the production! 1 1 models. At the same time, steady-state models are biomass ratio, QB; is the consumptionlbiomass ratio very useful for making summaries ofavailable data and DC .. is the fraction ofprey (i) in the average diet )1 and trophic flows in a system. Also, and quite ofpredator (j). importantly, thesemodelshelpidentifygaps in one's Based on (2), for a system with n groups, n linear knowledge aboutanecosystem. Together, thismakes equations can be given in explicit terms, steady-state models a good starting point for ecosystem modelling.

Consumption

There are various approaches for obtaining estimates of consumptionlbiomass ratio (QB); they This system of simultaneous linear equations may be split into (i) analytical methods and (ii) canbesolvedthroughmatrixinversion. InECOPATH holistic methods. II, this is done using the generalizedinverse method (i) The analytical methods involve estimation of described by Mackay (1981), which has features ration, pertaining to one or several size/age making it generally more versatile than standard classes, and their subsequentextrapolation to inverse methods. a wide range ofsize/age classes, representing For example, if the set of equations is an age-structured population exposed to a overdetermined (more equations than unknowns) constant or variable mortality. and the equations are not consistent with each Therequiredestimatesofrationare obtained other, the generalizedinverse methodprovidesleast from laboratory experiments, from studies of squaresestimateswhichminimizethediscrepancies. the dynamics of stomach contents in nature If, on the other hand, the system is (Jarre et aI. 1991; see Fig. 2), or by combining underdetermined (more unknowns than equations), laboratory and field data (Pauly 1986). an answerthatis consistentwith the data (although (ii) The existingholistic methods for estimationof not unique) will still be output. QB are empirical regressions for prediction of Generally only one of the parameters Bi' PBi' QB from some easyto quantify characteristics QB. or EE. may be unknown for any group i. In l' 1 . of the animals for which the QB values are special cases, however, QB; may be unknown III required (Palomares and Pauly 1989; Pauly et additionto one ofthe otherparameters(Christensen aI. 1990; Palomares 1991; Pauly et aI., this andPauly1992a). Exports anddietcompositions are vol.). always required for all groups. Production The Energy Balance of a Box Production includes all matter elaborated by a A box,in an ECOPATH II model, may be a group group (whetheritis ultimately eaten, fished or dies of of(ecologically) related species, a single species, or a other causes) over the period considered. Total single size/age group of a given species. 17 Respiration A

15 Oreochromis niloticus Lake Awasa, Ethiopia As mentioned above, respiration is estimated by ~20 L= 23cm;W=265g c> ECOPATH II as a difference, and hence is not a required parameter. If, however, explicit estimates t 15 ·iii of respiration are available, these can be used for :3 10 Qi "calibration", Le., a model's inputs can be modified !l ..s 5 until, for any given box, the computed respiration c: matchesthe available estimate;this approachmakes ~ 0 O~18~IO---'12!<-1-,1;14~16~I""8--'!~""0-:2bI2--'!~4~0e-::12-;!-04-:--:100 it possible for another parameter of that box, e.g., ~ Time (hours) PB, to be unknown.

o Bagrus dogmac (jj Lake Victoria, Kenya Network Flow Indices [=31.3cm,W= 300g 6 B ~ 5 • c> The ECOPATH II software links concepts ·iii :3 4 developed by theoretical ecologists, especially the >. -0 theory of Ulanowicz (1986), with those used by 0 3 • .0 biologists involved with fisheries and aquaculture . -0 2 management. The following section gives a brief e..~ account of some of the concepts from theoretical that are included in ECOPATH II. !! 1 [ I ! I 1 I D Q ~ E B ~ ~ ~ ~ Ascendency is a measure ofthe average mutual Time (hours) information in a system, scaled by system Fig. 2. Two dailycycles ofstomachcontents ofAfrican fishes (from throughput. These quantities are derived from Palomares 1991), fitted bymeans ofthe MAXIMS softwareofJarre informationtheory(Ulanowicz 1986; Ulanowicz and et al. (1990). (A) Oreochromis niloticus (Cichlidae), based on data Norden 1990). Ifone knows the location of a unit of in Getachew (1987). Note single feeding period, from 7 to 16 hours. energy, the uncertainty of where it will go next is (B) Bagrus dogmac (Bagridae), based on data in Okach andDadzie (1988). Note two feeding periods per day, at dawn and dusk, as reducedbyanamountknownasthe"averagemutual often occurs in piscivores (Hobson et al. 1981). information", mortality, when constant, is equal to production over biomass. Therefore,insteady-statemodels, itis safeto where, ifTr is a measure ofthe fromj to treat estimates oftotal mortality (Z) as equivalent to i, f is the 1fraction of the total flow from j that is the production/biomass ratio (PIB) (Allen 1971). ij represented by Tij, or Predation fij = Tij~ T kj·

In a trophic model such as constructed by Qi is the probability that a unit ofenergy passes ECOPATH II, itis predationthatlinksthe groups in through i, or a system. Thus, what is consumption for one group is mortality (production) for its prey. Therefore, information on predation is important for understanding the dynamics of ecosystems. I is a probability and is scaled by multiplication Unfortunately, properly presented information on with the total throughput ofthe system, T, where diet compositionis sparse -fish populationdynamics T= has traditionally treated fish populations as ifthey LI)..T I)... were independent, and a large part ofthe available information on diet compositions is expressed on a Thus, "per .cent occurrence" or "point" basis or as "", all of which are of little use for A = T * I, quantificationofdiets. Whatareneededaremeasures based on energy, weight or volume. where A is called "ascendency". There is an upper For quantified ecosystem models such as limit for the size ofthe ascendency, estimated from· ECOPATH II, the diet compositions should be expressed as the proportion (weight, volume or C=H *T, energy) each prey constitutes to the overall diet. 18 where C is called "development capacity" and H is Conclusion called "statistical entropy" and is estimated from We hope thattherationalepresentedinthispaper, together with the other contributions in this volume, will help establish the potential of steady-state The difference betweencapacityandascendencyis modelling as a tool to improve our understanding of called"system overhead". This provides a limitfor the ecosystems, especially for data-sparse areas. increase of ascendency and reflects the system's ECOPATH II, andforthcoming newdevelopments "strength in reserve" from which it can draw to meet (Christensen 1991), will, we hope, build a bridge unexpected perturbations (Ulanowicz 1986). between methodologies commonly used by fisheries Ascendency, overheads and capacity can all be biologists and by theoretical ecologists. split into contributions from imports, internal flow, exports and dissipations (respiration). These Aclm.owledgement contributions are additive; examples can be found in several ofthe contributions in this volume. The development and dissemination of the The unit for these measures is "flowbits", or the ECOPATHIIsystemwasmadepossiblebythe"Global product of flow (e.g., t'km-2year-1) and bits, an comparisons of aquatic ecosystems" project, financed information. unit corresponding to the amount of through a grant to ICLARM from the Danish uncertainty associated with a single binary decision. International Development Agency (DANIDA). Trophic Aggregation References In addition to including a routine for calculating group-specific fractional trophic levels, as suggested Allen, K.R 1971. Relation between production and biomass. J. by Odum and Heald (1975), we have included a Fish. Res. Board Can. 28:1573-1581. routine in the ECOPATH II system that aggregates Baird, D. and RE. IDanowicz. 1989. The seasonal dynamics ofthe the entire system into discrete trophic levels sensu Chesapeake Bay ecosystem. Eco!. Monogr. 59(4):329-364. Christensen, V. 1991. On ECOPATH, Fishbyte, and fisheries Lindeman (1942). This routine is used by a number of management. Fishbyte 9(2):62-66. theauthorsinthisvolume, andisbasedonanapproach Christensen, V. and D. Pauly. 1992a. A guide to the ECOPATH II suggested by Ulanowicz (in press) which reverses the software system (version 2.1). ICLARM Software 6, 72 p. routine for calculation offractional trophic levels. For Christensen, V. and D. Pauly. 1992b. ECOPATH II - a system for balancing steady-state ecosystem models and calculating example, if a group obtains 40% of its food as a network characteristics. Eco!. Modelling 61: 169-185. and 60% as a first-order , 40% and Getachew, T. 1987. Food, nutrition and digestive efficiency in 60% oftheflow throughthegroup areattributedtothe Oreochromis niloticusLinn. (Pisces, Cichlidae)inLakeAwasa, herbivore level and the first level, Ethiopia. UniversityofWaterloo, Canada. 189 p. Ph.D. thesis. Hannon, B. 1973. The structure of ecosystems. J. Theor. Bio!. respectively. 41:535-546. Based on these computations, the efficiency of Hannon, B. and C. J oiris. 1989.A seasonal analysis ofthe southem transfer between discrete trophic levels can be North Sea ecosystem. Ecology 70(6):1916-1934. calculated as the ratio ofthe flow that is transferred Hobson, E.S., W.N. McFarland and J.R Chess. 1981. Crepuscular and nocturnal activities ofCalifornian nearshore fishes, with from one trophiclevelto thenext(orto thefishery) and consideration of their scotopic visual pigments. U.S. Fish. the throughput at the . Bul!. 79(1):1-30. Jarre,A., P. Muck and D. Pauly. 1991. Two approachesfor modelling fish stock interactions in the Peruvian upwelling ecosystem. Mixed Trophic Impacts ICES Mar. Sci. Symp.193:178-184. Jarre, A., M.L. Palomares, M.L. Soriano, V.C. Sambilay, Jr. and D. Leontief(1951) developed a methodto quantifythe Pauly. 1990. A user's manual for MAXIMS, a computer direct and indirect interactions ofvarious sectors of program for estimating the food consumption of fishes from diet stomach contents data and population parameters. the economy of the USA, using what has since been ICLARM Software 4, 27 p. called the Leontief matrix. This was first used in Leontief, W.W. 1951. The structure of the U.S. economy. 2nd ed. ecology by Hannon (1973) and Hannon and Joiris Oxford University Press, New York. (1989) to assess theimpactofanygroup in a systemon Lindeman, RL. 1942. The trophic-dynamic aspect of ecology. Ecology 23:399-418. all other groups. Mackay, A. 1981. The generalized inverse. Pract. Comput. Ulanowicz and Puccia (1990) developed a similar (September):108-110. approach, ~d a routine based on their method has Odum, W.E. and E.J. Heald. 1975. The -based of been incorporated in the ECOPATH II system. an estuarinemangrovecommunity, p. 265-286. In L.E. Cronin (ed.) Estuarine research. Vo!. 1. 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