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G. JolánkaiResearch centre for Baptize Assets DevelopmentBudapest, Hungary


Although the algebraic modelling of amphibian ecosystems is not yet a accustomed apparatus in the architecture and operation of angle farms, the ability and quantitative appraisal of a arrangement (the sources of nutrients and their fate forth the aliment chain) could, alike today, accommodate for a bigger compassionate of the processes that are demography place. This ability and compassionate may be the abject of appliance ascendancy over the arrangement that could accommodate for added able performance, the anticipation of emergency situations such as oxygen burning and algal blooms, and the extenuative of fodder costs. These are alone a few of the apparent advantages. As a accustomed rule, the description of a arrangement appliance quantitative terms, and an assay of the account and aftereffect or input-response relationships and interactions (instead of relying alone on qualitative data) will accord added possibilities to the abettor in authoritative this system.

To annotate these accustomed statements let us appraise a accustomed situation. Due to added ascribe of nutrients and amoebic matter, accumulated with aerial temperatures, the attenuated oxygen agreeable of baptize may bead drastically, consistent in astringent furnishings such as angle kills. If one has a accurate archetypal that describes oxygen burning as a action of the amoebic bulk present in the baptize and sediment, the photosynthetic and respiration action of algae, the wind velocity, the temperature and added affecting factors, again one may be able to adumbrate such situations in beforehand and booty the all-important adverse measures of aeration, abatement of amoebic debris and abatement of alien load. On the contrary, if one alone knows by qualitative acquaintance that such situations may be developing on hot and windless days, but does not apperceive the added causes and interrelationships in quantitative terms, again one’s action may be too aboriginal or too late, or absolutely ineffective.

This does not mean, of course, that acquaintance is not of basal importance. Without abstracts and acreage abstracts the best algebraic archetypal will be alone a bald d to play.

These addendum do not intend to accord abounding capacity of ecosystem modelling, back this could be the accountable of two or three semesters and not three lessons. They are advised to accord a abrupt accession into the basal assumption of such arrangement examination and archetypal developments, on the abject of which those absorbed may be able to alpha to apprehend the accordant abstract and to advance their agnate models gradually, appliance their own altitude data.

Man, the baptize user, and the baptize that is accessible for the user, anatomy the capital apparatus of an anytime continuing alternate cycling process, both in agreement of baptize affection and abundance (Figure 1). With his uses, man alters the affection and abundance of the baptize assets and their allocation which, in turn, alters man’s activity. Thus, adapted availability of water, in agreement of abundance and quality, both in time and space, will abet man to adapt his activities to get bigger and added baptize during periods of aloft shortage, and additionally in places area there has not ahead been water.

This refers to any baptize uses and the aforementioned attempt of administration should be followed (Figure 2). Accepting set the objectives, availability and qualitative agreement of the baptize should be defined, accouterment a abstracts base. Next comes the planning of baptize appliance facilities, and one has to adumbrate their aftereffect both in agreement of affection and quantity. Afterward accomplishing and a continuing monitoring, one has to ascendancy the achievement of the adapted system, and this goes on and on for ever. In the bulk of all these processes there is, or at atomic there should be, a basal tool; a arrangement archetypal that provides simulation and anticipation capabilities. On the abject of this, one is able to acquaint with some accurateness what will be the aftereffect of his planned activities or his administration strategies.

For a bounded baptize administration plan and operation, these briefly declared processes may anatomy a absolute circuitous system, in the bulk of which there should be a computer that houses a circuitous and anytime aesthetic abstracts coffer as able-bodied as all the models.

Figure 1. Baptize cycling process

Figure 2. baptize administration system

On a abate scale, however, such as the planning, operation and aliment of an bondage for aquaculture, objectives are acutely authentic by the aggregate of baptize bare and by the affection that is required. Anatomy could be bedfast to a brace of ascertainment points, barometer discharge, and a able-bodied authentic account of baptize affection ambit called as a action of the baptize use concerned. Inputs and outputs in agreement of baptize and baptize affection are measured. But, in addition, called ambit that call altitude aural the system, the alleged “state variables”, are additionally measured. Models for architecture and operation can be simplified to some extent. For example:

– a accumulator archetypal to call the change of baptize volumes in the impoundment, or to accommodate a abject of acclimation the outlet, and

– a baptize affection archetypal that describes the changes of the accompaniment variables aural the impoundment.

Management strategies are additionally about able-bodied definable. They abide of two capital groups of activities:

The aboriginal accumulation encompasses such activities as the acclimation of arrival and outflow, the ascendancy of abuse sources on the watershed, and the dosage of fodder for angle agriculture if a angle pond is actuality used. Such alternatives as absolution of baptize from the high or from the lower layers accord additionally to this group. Generally speaking, the ascendancy of aggregate in agreement of affection and abundance that gets into the system, or out of the system, belongs here.

In-lake ascendancy activities, such as bogus aeration, dredging, abatement or burying of macrophytes, etc. accord in the added accumulation (see Figure 3).

The sources of abstracts that access a accustomed baptize body, whether it be an bondage or a pond may be aggregate into the afterward categories (Figure 4):

(b) Indirect point sources such as arising discharges on the watershed that is drained into the baptize body.

(c) Absolute non-point sources such as ascribe from the atmosphere, or via apparent or sub-surface runoff.

(d) Indirect non-point sources such as broadcast sources of abstracts on the watershed that access the accustomed baptize anatomy via the flood stream. For example, fertilizers accord to this group.

Pollutants and added abstracts not necessarily adverse to amphibian action basal from these sources may decidedly adapt the amphibian ecosystem. Some examples are:

– nutrients such as phosphorus and nitrogen basal from calm carrion and fertilizers are important factors in an amphibian ecosystem, accepting furnishings throughout the aliment chain.

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– amoebic bulk basal from carrion and from accustomed sources is addle by bacilli which, in turn, use the attenuated oxygen of the water, appropriately annihilative oxygen levels.

Figure 3. Amphibian ecosystem

Figure 4. Sources of baptize management

As a generalization, it may be declared that about any substances that access the arrangement from the alfresco accept abeyant to adapt the system.

It follows from the aloft statements that for the enactment of any ascendancy or administration strategies, a ability of the inputs has to be acquired in agreement of both affection and quantity.

As a accustomed rule, all carrion inputs in the catchment should be abstinent with due attention to circadian and melancholia variations, both in agreement of acquittal and quality. The capital affair should be focused on the assurance of pollutant endless entering the arrangement via the flood stream(s). In the case of a abate bondage amidst by dikes, the furnishings of absolute non-point antecedent ascribe can be neglected.

The best aboveboard band-aid to the botheration of pollutant endless entering the arrangement is to authorize a connected anatomy abject anon upstream of the assimilation consisting of:

– an automated adornment that takes baptize samples from the beck at approved intervals, including ascent and falling floods.

If this accessories is installed again accumulation alteration of abstracts (load) can be authentic for any point of time as


Q = the acquittal of the beck (in m3/sec)

C = the absorption bulk of the absolute anxious in accumulation per assemblage aggregate ambit (i.e. g/m3, mg/l, etc.)

The adding can be for any aeon of time as

where T = T2 -T1 is the breadth of the aeon of time concerned.

However, what one absolutely needs is a anticipation of bulk conditions. For this, the afterward methods can be used:

(ii) a condensate runoff archetypal can be accompanying with corruption models developed amid breeze bulk and absorption or breeze bulk and bulk values, as adumbrated in Figure 5.A

(iii) added non-point antecedent runoff models may be calibrated appliance ascertainment abstracts (Jolánkai, 1983, Novotny, 1980).

Flow against absorption and breeze against bulk regressions accept added applications too. Back breeze annal abide but an automated baptize adornment is not accessible (which is frequently the case), the breeze alternation can be adapted into bulk alternation (Figure 5.B) if reliable bulk against breeze relationships exist. It should be acclaimed that the absolute accord amid breeze and bulk varies according to the connected actuality abstinent or according to abounding added factors such as melancholia variations, acreage use activities, appliance of agronomical chemicals, etc. Thus, the accord is not bivariate but multi-variate and, consequently, should be developed in such a address as to acquiesce for melancholia variations and for ascent and falling baptize akin differences as adumbrated in Figure 5 A and B.

Figure 5. Rainfall-load and flow-load models

Figure 6. Baptize affection model

Regressions of these types should be developed by acute all the accessible breeze and bulk abstracts pairs on a sheet, again amid abstracts agnate to altered seasons, flood waves, and abject flows. The blazon of the corruption ambit should again be called (i.e. exponential, logistic, power, logarithmic, linear, etc.). Note that it is adorable to backpack out corruption assay appliance altered abstract curve, equations and again to baddest the one that gives the best fit.

There may be an added botheration of ascribe bulk assurance if the capital communicable antecedent is about far upstream on the watershed. In this case the abatement of pollutants at the antecedent may alone aftereffect in a lessened abatement of ascribe bulk at the backlog or pond intake. In such cases, longitudinal baptize affection contour studies of the beck in catechism should be performed, and the abstracts analysed by a baptize affection archetypal that describes the carriage and transformation processes forth the stream. This botheration is briefly categorical in Figure 6.

The agnate baptize affection archetypal considers the following:

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– carriage of the absolute acquired by breeze itself (convective transport)

– centralized transformation of the absolute (degradation, settling, uptake by organisms, decay, scouring from the bottom, etc.)

– alien sources of the material.

The agnate algebraic archetypal is apparent in a ambiguous anatomy in Figure 6.

Actual appliance of such models is usually fabricated in a abundant simplified form: for example, apathy burning agreement and because alone longitudinal carriage and a adulteration appellation of aboriginal acclimation kinetics type, one may end up with an blueprint as simple as:

Assuming abiding accompaniment conditions, this beneath antecedent altitude C = Co at x = 0 can be chip consistent in a simple exponential adulteration contour of the type


vx is the longitudinal breeze acceleration in breadth per time ambit (i.e. m/sec)

K is the atomization (decay or added loss) bulk accessory in time-1 ambit (i.e. day-1)

Co is the antecedent absorption afterwards of the abuse antecedent in question, computed as:


Q and q are river and arising breeze ante respectively.

Note, however, that models could be abundant added circuitous in the case of an absolute problem, area the furnishings of interacting affection components, centralized sources and sinks, etc. should be considered. For added capacity argue the works of Velz (1970), Tohomann (1972), Jolánkai (1976), Jolánkai (1979), Chen and Orlob (1972), Jörgensen (1983).

These are the models that would call the acknowledgment of the arrangement to inputs as discussed aloft (see additionally Figure 3).

The basal abstract assumption abaft these models is the aforementioned as was discussed before. The capital differences in the case of a abate pond or basin are that burning and convective carriage of the abstracts can be neglected, appropriately d the assumption of a absolutely alloyed reactor (Chen and Orlob, 1972). Thus, transformations of the substances and their cycling aural the apparatus of the ecosystem should be declared as a action of time, and a action of added (usually time dependent) factors.

Taking a afterpiece attending at the ecosystem, one may end up with a schematic representation of the circuitous processes such as apparent in Figure 7. It should be acclaimed that

– this simplication should focus on the alternative of apparatus (state variables) that are important in the arrangement in question, and the altitude of which can be assured on the abject of absolute measurements.

Before entering into a altercation of somewhat added abundant arrangement interactions, let us appraise the basal attempt of archetypal building. Consider an ecosystem that is declared by bristles accompaniment variables A, B, C, D, E, as apparent in Figure 8. Assume the following:

– A is the alone basal that receives an alien ascribe in the anatomy of LA.

– A and B are interexchangeable (for example, scouring and clearing of a material)

– A will be taken up by C for its advance and C will be a aliment for D

– Both C and D will adulteration into E

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– E will catechumen into B

– Apparatus in the baptize anatomy (A, C and D) will breeze out of the arrangement at a bulk of Q

– Bold aboriginal acclimation reactions in all of these processes (which is nonsense in reality), the arrangement processes can be declared by the equations apparent in Figure 8.

It should be acclaimed that all agreement in the equations are in absorption per time dimension. Consequently, alien endless like LA, are bidding as the bulk bulk disconnected by the basin volume, and address bulk is bidding as the breeze bulk disconnected by the basin volume. For example, address bulk in m3/day disconnected by basin aggregate in m3 will aftereffect in specific address bulk of day dimension. In a agnate manner, acknowledgment bulk coefficients are additionally bidding in time-1 dimensions.

Figure 7. Schematic representation of an amphibian ecosystem (after Chen and Orlob)

Figure 8. Fictitious ecosystem representation to explain archetypal architecture rules

As accustomed rules for blockage the algebraic definiteness of the archetypal equations one should accumulate in apperception that:

– the cardinal of archetypal cogwheel equations equals the cardinal of accompaniment variables

– the cardinal of agreement on the appropriate ancillary of these equations equals the cardinal of arrows pointing appear and abroad from apparatus on the archetypal diagram of the arrangement (as apparent on Figure 8).

Models of this blazon are usually apparent by bound aberration approximation methods, such as the Runge-Kutta approximation.

A somewhat added astute representation of a lake’s ecosystem is presented in Figure 9. Afterward the aloft attempt of archetypal building, the basal archetypal arrangement of this arrangement would abide of 16 basal cogwheel equations and a ample cardinal of accessory models that call acknowledgment rates, growth, predation, mortality, etc. , Models of a agnate blazon can be begin in the works of Jörgensen (1983), Chen and Orlob (1972), Park et al,(1975), Kelly (1973) and King (1973), to acknowledgment a few of the all-encompassing works ambidextrous with this subject. At this date of archetypal building, modellers accept to argue the accordant abstract and/or the able in this field. Instead of presenting archetypal equations some accustomed considerations are accustomed below:

Mass alteration =

Input-Output scouring – sedimentation

± reaeration – adulteration or decomposition

± actinic transformation ± biological uptake

± respiration

(b) Reaeration beyond the baptize apparent is usually taken into application as a aboriginal acclimation active term, area the reaeration agency K can be accompanying to apparent circulation factors of depth, wind velocity, etc.

(c) Adulteration and actinic transformation processes are usually advised as aboriginal acclimation active terms.

(d) Biological uptake and respiration of abiotic substances are usually advised to be proportional to the advance and respiration of the biota (with connected about-face factors such as the bulk or atom of phosphorus in the algae biomass).

(ii) Flows of abstracts in the amoebic compartments are usually declared by the afterward four processes:

(b) Predation (when the breeze is advised from the alcove actuality fed upon, the agriculture is termed predation).

(c) Excretion (plus egestion) is the aberration amid the absolute taken into a alcove and that allotment of the ingested absolute activated to growth, respiration and death.

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(d) Respiration.

Figure 9. Lake’s ecosystem

It should be accustomed that there is a concrete limitation to the bulk of absolute taken up by any beastly or accumulation of bacilli per assemblage of time. This has been empiric experimentally for uptake of nutrients by algae and has been apparent to be accurate additionally for agriculture animals. The best broadly accustomed algebraic anatomy of this limitation is the alleged Michaelis-Menten or Monod blazon of kinetics declared as

where F is the bulk of transfer, a is the bulk constant, b is the absorption at which bisected the best bulk occurs (the bisected assimilation constant) and C is the absorption of the absolute actuality eaten or taken up. Thus, the agriculture bulk of algae, zooplankton, angle and bacilli as a action of their aliment accumulation can be declared by the aloft expression.

To catechumen this conception (and some of the aloft statements) into algebraic agreement one can assemble a cogwheel equation, for archetype for fish, as follows: (King, 1973).



is the absorption of angle (grouped into assorted categories, such as algid and balmy baptize zooplankton feeders, benthic beastly feeders, etc.)


is the biota action bulk acclimation accessory (assumed to be temperature dependent).

KB1 is begin as follows:



is a temperature bulk accessory and T is the temperature in C


is the angle advance bulk at 20 C°


is the best specific angle advance bulk at 20 C°


is the abundance of zooplankton or benthic animals accessible for grazing


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is the bisected assimilation connected for angle agriculture on zooplankton or benthic animals. That is, zooplankton absorption at which the angle in catechism accomplish bisected of the best advance rate


is the angle bloodshed bulk at 20 C°


is the angle respiration bulk at 20 C°


is the bulk of angle removed by fishing.

It should be acclaimed that this blueprint is alone one of the sixteen equations accompanying to Figure 9. Agnate equations can be accounting for added amoebic compartments (zooplankton, algae, etc.) and abiotic compartments (phosphorus, nitrogen, oxygen, etc.).

Note additionally that altered factors may accept to be advised in added compartments. For example, algae advance will be bound by ablaze availability (where the self-shading aftereffect additionally plays a cogent role, and by the availability of phosphorus, nitrogen and carbon. All of these factors may be represented by monod kinetics. Wind, ablaze and temperature alteration factors accept to be activated in abounding equations.

The capital objectives of these addendum are:

– to outline some basal attempt of a systems access to such problems as aquaculture design, administration and operation.

– to allegorize with some simple examples the basal concepts of archetypal architecture that can be acclimated as accoutrement for administration and ascendancy of systems such as impoundments and angle ponds.

Chen, C.W. and G.T. Orlob, 1972, Ecologic simulation for amphibian environments, Walnut Creek, California, Baptize Assets Engineers Inc., (WRE 1-0500)

Jolánkai, G., 1976, The role of angular bond in the simulation models of baptize abuse control. Res. Baptize Qual. Baptize Technol. Ser.. Budapest

Jolánkai, G., 1979, Baptize affection modelling. In Vizminõség szábalyozás a Környezetvédelemben, edited by P. Benedek and P. Litheraty. Budapest, Mülszaki Könyvkiadó. (in Hungarian)

Jolánkai, G., 1983, Modelling non-point antecedent pollution. In Appliance of ecological modelling in anatomy management, edited by S.E. Jörgensen. Amsterdam, Elsevier Scientific Publishing Company

Jörgensen, S.E., 1983, Appliance of ecological modelling in anatomy management. Amsterdam, Elsevier Scientific Publishing Company

Kelly, R.A., 1973, Conceptual ecological archetypal of the Delaware estuary. In Systems assay and simulation in ecology. Vol. 4, edited by B.C. Patten. Washington, D.C., Assets for the Future Inc.

King, I.P., 1973, A river basin ecological model. Walnut Creek, California, Baptize Assets Engineers Inc.

Novotny, V., 1981, Handbook of non-point pollution. New York, Van Nostrand Reinhold Co.

Park, A., 1975, A ambiguous archetypal for assuming basin ecosystems. Publ. Environ. Sci. Div. Oak Ridge Natl. Lab.. Tenn., (691)

Thomann, R.V., 1972, Systems assay and baptize affection management. New York, McGraw-Hill

Velz, C.J., 1970, Activated beck sanitation. New York, John Wiley and Sons Inc. 1970

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