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Environmental Modelling & Software xx (2006) 1e8www.elsevier.com/locate/envsoft

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Ecosystems as evolutionary complex systems: Network analysis offitness models

Brian D. Fath a,*, W.E. Grant b

a Biology Department, Towson University, Towson, MD 21252, USAb Department of Wildlife and Fisheries Sciences, Texas A&M University, College Station, TX 77843, USA

Received 25 September 2005; received in revised form 24 October 2005; accepted 15 December 2005

Abstract

Understanding and managing ecosystems as biocomplex wholes is the compelling scientific challenge of our times. Several different system-theoretic approaches have been proposed to study biocomplexity and two in particular, Kauffman’s NK networks and Patten’s ecologicalnetwork analysis, have shown promising results. This research investigates the similarities between these two approaches, which to datehave developed separately and independently. Kauffman (1993) has demonstrated that networks of non-equilibrium, open thermodynamicsystems can exhibit profound order (subcritical complexity) or profound chaos (fundamental complexity). He uses Boolean NK networks todescribe system behavior, where N is the number of nodes in the network and K the number of connections at each node. Ecological networkanalysis uses a different Boolean network approach in that the pair-wise node interactions in an ecosystem food web are scaled by the through-flow (or storage) to determine the probability of flow along each pathway in the web. These flow probabilities are used to determine system-wide properties of ecosystems such as cycling index, indirect-to-direct effects ratio, and synergism. Here we modify the NK model slightlyto develop a fitness landscape of interacting species and calculate how the network analysis properties change as the model’s species coevolve.We find that, of the parameters considered, network synergism increases modestly during the simulation whereas the other properties generallydecrease. Furthermore, we calculate several ecosystem level goal functions and compare their progression during increasing fitness and deter-mine that at least at this stage there is not a good correspondence between the reductionistic and holistic drivers for the system. This research islargely a proof of concept test and will lay the foundation for future integration and model scenario analysis between two important networktechniques.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Boolean networks; Coevolution; Ecological modeling; Fitness landscapes; Network analysis

1. Introduction

One goal of theoretical ecosystem ecology is to identify andquantify system-level concepts and find general patterns ofecosystem organization. One promising method has been toconceptualize ecosystems as networks connected by theirtransfer and exchange of energy and matter within and acrosssystem boundaries. Several different developments of this

* Corresponding author. Fax: þ1 410 704 2405.

E-mail address: bfath@towson.edu (B.D. Fath).

1364-8152/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.envsoft.2005.12.023

conceptualization have been realized. Independently, theyhave added significantly to our understanding of ecosystemsyet there has been a lack of integration with these methodsbecause of the different terminology, notation, history, disci-plinary genesis, emphasis, and application. The main goal ofthis project is to find linkages between two commonly usedBoolean representations of ecological networks. In particular,we link ecosystem theory based on network analysis to Kauff-man’s theory of self-organized systems in order to test thehypothesis that network properties of homogenization, ampli-fication, indirect effects, and synergism increase as an ecosys-tem coevolves to higher fitness levels.

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2. Background

2.1. Ecological network analysis

Ecological network analysis is a mathematical methodol-ogy to represent material, energy and information within anecosystem as a network of nodes (compartments, components,storages, etc.) and connections (links, flows, etc.). Networkanalysis provides a system-oriented perspective to identifyand quantify direct and indirect relations between all theobjects in the system. After the first application of inputeout-put analysis to investigate flow distribution in ecosystems(Hannon, 1973), several formulizations of ecological networkanalysis have arisen including Ascendency Analysis (Ulano-wicz, 1980, 1986, 1997) and Network Environ Analysis (Pat-ten, 1978, 1981, 1982, 1985).

Bernard Patten used mathematical systems theory as a foun-dation for studying ecosystems (Patten et al., 1976; Patten,1978, 1981, 1982). He stressed the utility of the inclusioneexclusion principle of set theory as a way to formalize thetransactions that naturally occur in food webs. A binary inter-action exists in ecological networks, simplified often as a ques-tion of ‘‘who eats whom,’’ but more broadly as the transfer ofconservative energyematter between any two entities in thesystem. Much of the subsequent work in network ecologybuilds on this basic premise of direct energyematter transac-tions between coupled binary pairs. These transactions formthe basis of both direct and indirect ecological relations, suchas predation (direct), neutralism (direct), altruism (direct), mu-tualism (indirect) and competition (indirect) that are of impor-tance to community ecology. Some of the primary findings ofthis research include the importance of indirect effects as theypropagate through the myriad of network connections (Higashiand Patten, 1989) and synergism, individual compartments inan ecosystem gaining positive value from being embedded ina larger network (Patten, 1992; Fath and Patten, 1998).

2.2. Ecological network properties

Several network properties have been developed with fourin particular: amplification, indirect effects, homogenization,and synergism, used most regularly to investigate ecosystembehavior. Since they have been described elsewhere, onlya brief description is provided here (see Fath and Patten,1999a for details). The four properties relate the distributionand contribution of conservative energyematter flow throughthe network’s many direct and indirect pathways. One measureof resource distribution is given in the direct flow intensity, ortransfer efficiency, matrix G, whose values, gij ¼ fij/Tj, repre-sent the likelihood of flow along a given path, where fij corre-sponds to the flow from compartment j to compartment i, andTj ¼ Sj(si) ¼ 0,n fij is the total sum of flow through compart-ment j including input and output boundary flows ( fi0 andf0j, respectively). Tj

in ¼ Tjout at steady state. In the direct flow

intensity matrix, G, all elements have a non-negative valueless than one (0 � gij < 1) and can be interpreted as a probabil-ity of flow along each pathway. Using standard inputeoutput

analysis techniques, an integral flow intensity matrix, N, iscomputed from the convergent power series:

N¼G0þG1 þG2þG3þ.¼ ðI�GÞ�1 ð1Þ

where I is the multiplicative identity matrix.Elements nij in the matrix N include the contribution of di-

rect (m ¼ 1) and indirect (m > 1) pathways, and therefore arealways greater than or equal than the values of G. The G andN matrices are used to define the amplification, homogeniza-tion, and indirect effects properties. A specific quantitativetest exists to determine each property (Table 1).

Amplification occurs whenever an off-diagonal element ofthe integral flow matrix is greater than one (nij > 1). The inte-gral flow from j to i, can exceed one when cycling drives morethan the equivalent of one unit of input flow over the pathway.This property was observed in several of the small-scalemodels but is rare in large-scale models (Fath, 2004).

The homogenization property compares the resource distri-bution between the direct and integral flow intensity matrices.It was observed that, due to the contribution of indirect path-ways, flow in the integral matrix was more evenly distributedor more homogenized than that in the direct matrix, meaningthat flow is comprised of contributions from many parts of thenetwork. Network homogenization occurs when the coefficientof variation N, CV(N), is less than the coefficient of variationof G, CV(G), because this indicates that the network flow ismore evenly distributed in the integral matrix (Fath and Patten1999b).

Indirect effects are calculated as the integral contributionsminus the direct and initial boundary input (Indi-rect ¼ N � I � G). The indirect to direct effects ratio is a mea-sure of the relative strength of these two factors. When theratio is greater than one, then indirect effects are greaterthan direct effects.

The fourth property, network synergism is based on a netflow intensity matrix, D, where dij ¼ ( fij � fji)/Ti. Unlike theother series in which the elements are non-negative, entries

Table 1

Four network properties

Property Test

1. Amplification nij > 1 for isj

2. Homogenization

CVðGÞCVðNÞ> 1

3. Ratio of indirect to

direct effects Pni¼1

Pnj¼1

�nij � iij � gij

�

Pni¼1

Pnj¼1

gij

> 1

4. Synergism

Pþutility

jP�utilityj> 1

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in D can be positive or negative (�1 � dij � 1). The elementsof D represent the relative utility between that (i,j ) pairing. Anintegral utility matrix U, is obtained from the power series as:

U¼ D0þD1þD2 þD3þ.ðI�DÞ�1 ð2Þ

This methodology is used to determine qualitative relationsbetween any two components in the network such as preda-tion, mutualism, competition, etc. Quantitatively, synergismarises when the integral positive utility exceeds negative utilitybecause of mutualistic relations in the system and is calculatedas the ratio of the magnitude of the positive and negativeutilities.

2.3. Ecological goal functions

Objective functions, goal functions, optimization criteria,extremal principles, and orientors are examples of criteriathat have been used to track ecosystem growth and develop-ment at a system level (e.g., Muller and Leupelt, 1998). Holis-tic ecological goal functions typically track certainthermodynamic or informational aspects of the ecosystem.This holistic approach to monitoring, assessing, even orientingat the ecosystem level contrasts sharply with the reductionisticapproach which measures the fitness of an organism in its en-vironment. In fact this schism between evolutionary ecologyand ecosystem ecology is a major challenge to modern ecolog-ical theory (Odling-Smee et al., 2003). Odum (1969) proposedseveral trends to be expected in the growth and development ofecosystems from early to mature stages (Table 2).

Five such goal functions, minimize specific entropy pro-duction, maximize energy throughflow, maximize exergy stor-age, maximize retention time, and maximize cycling, areconsidered here (Table 3).

Table 2

Trends to be expected in ecosystem development (after Odum, 1969)

Ecosystem attribute Developmental

stage

Mature

stage

Community energetics

Gross production/community

respiration (P/R ratio)

<1> w1

Gross Production/standing crop

biomass (P/B ratio)

High Low

Biomass supported/unit energy

flow (B/E ratio)

Low High

Food chains Linear Weblike

Nutrient cycling

Mineral cycles Open Closed

Nutrient exchange rate Rapid Slow

Overall homeostasis

Internal symbiosis Undeveloped Developed

Nutrient conservation Poor Good

Stability (resistance to

external perturbations)

Poor Good

Entropy High Low

Information Low High

The least specific entropy production principle, which orig-inates from Prigogine and Wiame (1946) and Prigogine (1980,1947), is the ratio of entropy production and the mass of systemproducing the entropy. In ecological terms, minimizing specificentropy means minimizing the entropy production (mainte-nance cost) for a given biomass (structure). In other words, anecosystem tends to become more energetically efficient andthe respiration to biomass ratio decreases over time. Maximumpower principle is taken as a second thermodynamic goal func-tion. Lotka and Odum have hypothesized that an ecosystem de-velops towards maximizing power (Lotka, 1922; Odum andPinkerton, 1955), which is interpreted here as the highest pos-sible throughflow of energy. A third perspective states that ex-ergy storage increases during ecosystem development(Jørgensen et al., 2000; Jørgensen, 2002; Fath et al., 2001).This property, maximizing exergy storage becomes the thirdhypothesis considered herein. Exergy is the maximum amountof work the system can perform relative to an environmentalreference state; therefore, a system with greater exergy ismoved further from its reference state and further from thermo-dynamic equilibrium. Jørgensen and Mejer (1977, 1979) haveintroduced the application of exergy to account for ecosystemdevelopment by using as a reference the same system at thermo-dynamic equilibrium at the same temperature and pressure. Inecosystems an increase in exergy corresponds to an increasein the overall biomass and an increase in the species’ internalorganizationdthis informational component is currently mea-sured as genetic complexity (Marques et al., 1997; Jørgensenet al., 2005). Throughflow and exergy storage are related bythe retention time of the system, so a fourth hypothesis, maxi-mum retention time, is also considered here. Finally, the systemcycling (Finn, 1976) described above has also been consideredas an ecological goal function.(Morowitz, 1968)

The goal is to find a pattern between how these variousthermodynamic-based ecological goal functions (Fath et al.,2004) respond to changes in fitness in the coevolutionary fit-ness models described below.

2.4. Kauffman’s NK model

Stuart Kauffman uses binary Boolean networks to find gen-eral laws of system self-organization (Kauffman, 1993, 1996,2000). His main thesis is that biological systems are composed

Table 3

Ecological goal functions tracked during evolution of model

1. Minimize specific entropy production (SPD) (Prigogine and Wiame,

1946). Decrease in the respiration to biomass ratio

2. Maximize energy throughflow (TST) (Odum, 1983). Increase in the

internal energy flow

3. Maximize exergy storage (TSS) (Jørgensen and Mejer, 1977). Exergy

storage (biomass) and information increase due to shift to more

complex species composition.

4. Maximize retention time (RET) (Cheslak and Lamarra, 1981).

Biological components develop mechanisms to increase time lags to

maintain the energy stores longer.

5. Maximize cycling (TSC) (Morowitz, 1968). Energy and matter are

retained in the system through network feedbacks.

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of autonomous agents, or self-replicating systems that performwork, which are ‘‘co-constructing and propagating organiza-tion’’ (Kauffman, 2000, p. 5). An emphasis is placed onco-construction and coevolution because of the cyberneticfeedback that makes agents adapt to other agents at thesame time they modify their own environment. There recentlyhas been renewed interest in the impact species have on eachother and on their environment (e.g., Jones et al., 1997; Od-ling-Smee et al., 2003). Coevolution and indirect effects areboth manifestations of interacting networks.

In his NK model, Kauffman (2000) addresses species co-evolution by coupling the influence from genes of one speciesto genes of another species. The basic module of the NKmodel represents an organism with N genes, each havingtwo alleles, 0 and 1. The contribution of each gene to the fit-ness of the organism depends on the allele of that gene and thealleles of K other genes in its genome, called ‘‘epistatic’’ in-puts. In this simple model there are 2N combinations of allelesthat influence fitness. Each allele is randomly assigned a fitnesscontribution value for each of the 2N combinations in which itoccurs. The overall fitness of each combination of alleles (i.e.,each genotype) is taken as the mean of the fitness values of theconstituent alleles. The result is a fitness landscape, such thatevery allele combination has a specific fitness value (Table 4shows an example for N ¼ 3).

When there is a flip in one allele from 0 to 1 or vice versa,the fitness contribution of the gene changes. If the result ishigher fitness, then the allele shift is accepted, if not, then itis not accepted. Kauffman found that when the number of con-nections to other genes, K, is low the system quickly evolvesto a global fitness maximum. As the number of connections in-creases there are more local peaks until the point when the sys-tem is completely interconnected (K ¼ N � 1) and theresulting fitness landscape is fully random. The more localpeaks that occur, then the more improbable it is to ‘‘climb’’to the global peak, resulting, on average, in an overall lowerfitness. However, Kauffman maintains that fitness landscapesare not random but instead are generated by the coevolutionaryinteractions of the various species. Therefore, the next step isto link NK models of various species.

Table 4

Example of fitness landscape

1 2 3 Fitness

value w1

Fitness

value w2

Fitness

value w3

Average

fitness �w

0 0 0 0.2 0.5 0.4 0.37

0 0 1 0.7 0.1 0.2 0.33

0 1 0 0.5 0.9 0.8 0.73

0 1 1 0.3 0.3 0.1 0.23

1 0 0 0.5 0.4 0.4 0.43

1 0 1 0.1 0.5 0.3 0.30

1 1 0 0.9 0.2 0.8 0.63

1 1 1 0.6 0.8 0.4 0.60

There are eight possible binary combinations of three genes. Each is assigned

a random fitness value between 0 and 1, and the fitness for each allele combi-

nation is the mean of the three values. This procedure is used to construct a fit-

ness landscape. For example, starting with each gene expressing a 0, the fitness

is 0.37. If the allele on the first gene flips to ‘‘1’’ then fitness increases to 0.43.

This simple model has only one fitness peak at (0,1,0).

In the multi-species version of Kauffman’s NK model, thefitness value of each allele depends not only on the allele ofthat gene and on the alleles of K epistatic genes, but also onthe alleles of C other genes in each of S other species. If thereare two species coupled together, then each gene has K þ Cinputs, and a table of random fitness contributions is generatedthat has 2(KþC ) combinations. A model in which each speciesis connected with S other species has 2(KþCS ) possible states,so the number of possible states grows combinatorically. Thefitness of the species is calculated as the mean of the fitnessvalues of the alleles in its current genotype; each species isassumed to be isogenic. Now, when one species evolves (aflipping of an allele on a gene) this likely has ramificationsfor the other species by deforming the overall fitness land-scape. Kauffman found that in general coevolving systemscoupled in this manner behave either in an ordered or chaoticregime, separated by a phase transition depending on the num-ber of couplings.

We have recreated Kauffman’s multi-species NK modelhere to investigate the fitness of coevolving species with a par-ticular interest in understanding how coevolutionary processesaffect ecosystem properties. A few modifications to the origi-nal model as presented above are noted. Each time step duringthe simulation, any one of four events, randomly chosen, mayhappen. (1) A randomly chosen species may evolve to a newgenotype via recombination, if the randomly chosen new ge-notype has a higher fitness value than the current genotype.A randomly chosen species may be replaced by a new speciesthat (2) may have a different K than the current species, buthas the same C and S, (3) may have a different C, but hasthe same K and S, or (4) may have a different S, but has thesame K and C, if the new species has a higher fitness valuethan the current species. Thus, as species evolve or arereplaced by invading species, they change their own fitnesslandscape (Kauffman, 1996, 2000) as well as the fitness land-scape of the other species. The above restrictions could be re-laxed in future research to study more general cases, but fornow the model was used to generate a time series of connec-tance matrices. We apply ecological network analysis to eachmatrix. Eventually, it would be useful to look at models thathave more realistic ecosystem structures by using the method-ologies developed in Fath (2004) or perhaps to see if over timespecies in the models naturally evolve into a configurationsimilar to a trophic structure. However, that is beyond thescope of this paper. Here we present the initial results fromthis research, which uses a five-species model coevolving for10 time steps under two different species coupling regimes.

3. Integrated model

In the first simulation, all species were initialized withS ¼ 1 (i.e., each species is connected with one other species),and in the second simulation all species were initialized withS ¼ 4 (connected to all other species). Every time interval dur-ing the simulation, we generate a connectance matrix based onthe current fitness and S values of the set of species. Elementsof the connectance matrix are equal to 0 if the fitness values of

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0.232

0.138

0.138

0.138

0.232

0.232

0.454 0.454

0.225

0.2250.2250.454

0.417 0.417

0.227

0.227

0.227

0.427

0.227

0.656

0.227

0.227Sp4

Sp0

Sp3 Sp2

Sp1

Sp4

Sp0

Sp3 Sp2

Sp1 Sp4

Sp0

Sp3 Sp2

Sp1 Sp4

Sp0

Sp3 Sp2

Sp1

Sp4

Sp0

Sp3 Sp2

Sp1 Sp4

Sp0

Sp3 Sp2

Sp1

0.520

0.657 0.6570.657

0.417 0.417 0.417

0.415

0.607 0.607

(a)

(f)(e)(d)

(c)(b)

Fig. 1. Model evolution during ten time steps. New figures are given when adaptations are selected: (a) initial configuration and weighting factors. Configuration at

(b) t ¼ 3, (c) t ¼ 5, (d) t ¼ 6, (e) t ¼ 9, (f) t ¼ 10.

the genes of the ‘‘to’’ species are not affected by the genes ofthe ‘‘from’’ species, and diagonal elements are equal to 0, thatis, species are not connected to themselves. Values of the otherelements of the connectance matrix are calculated as the fit-ness value of the ‘‘to’’ species divided by its S value, that is,the sum of all elements ‘‘to’’ a given species is equal to its fit-ness value.

The elements of the connectance matrix represent the fit-ness contribution among connected species. In order to applyecological network analysis to these matrices, we assume thatelements of the connectance matrix represent relative rates ofenergy flow among the set of species. Obviously, fitness is notflow, but in a more general sense the fitness represents a mea-sure of influence between species. The flow probabilitybetween two compartments is the proportion of flow to totalthroughflow ( gij ¼ fij/Tj) where Tj is the total throughflowinto compartment j. This could also be interpreted as the prob-ability of influence between two compartments (Patten et al.,1976). Here we assume that the fitness contribution (from0 to 1) can be used as a measure of the weighted influence.This allows us to apply ecological network analysis to eachmatrix and calculate the four ecological network propertiesdescribed above.

We then examined the temporal dynamics of these proper-ties as the set of species co-evolve through different fitnesslandscapes to test the hypothesis that cycling index, homoge-nization, amplification, indirect effects, and synergism in-crease as the ecosystem coevolves. Note that ecologicalnetwork analysis is a steady-state analysis; however, we treatthe model generated from each time step as a snapshot intime. As the system changes over time, we can determinethe network properties of the system in that particular state.One other assumption is needed to run the analysis, which isthat the model ecosystems, as open systems, receive externalinput. Energy enters the system largely through primary

producer and lower trophic level species. Usually, for a modelthis size (5 compartments) external input into one compart-ment is enough, but in some of these simulations the first com-partment is eliminated after which time there would be nofurther input available to higher trophic levels. Therefore,a unit of input is given to each of the five compartments.

4. Numerical simulation results

In the first simulation, each species is connected to oneother species. The connectance values can change at eachtime step given the occurrence of a randomly chosen event,as described above. Fig. 1 shows the digraph changes for thefirst simulation in which all species are originally connected

Table 5

Example of two connectance matrices generated by the first simulation model

Sp 1 Sp 2 Sp 3 Sp 4 Sp 5

T ¼ 2

Sp 1 0 0.52 0 0 0

Sp 2 0 0 0.42 0 0

Sp 3 0 0 0 0.42 0

Sp 4 0 0 0 0 0.66

Sp 5 0.61 0 0 0 0

T ¼ 3

Sp 1 0 0.66 0 0 0

Sp 2 0 0 0.23 0.23 0

Sp 3 0 0 0 0.42 0

Sp 4 0 0 0 0 0.66

Sp 5 0.61 0 0 0 0

Reading from columns to rows, at time 2, Sp 2 affects Sp 1 (0.52), Sp 3 affects

Sp 2 (0.42), Sp 4 affects Sp 3 (0.42), Sp 5 affects Sp 4 (0.66), and Sp 1 affects

Sp 5 (0.61). At T¼ 3, Sp 2 is replaced by a new Sp 2 that is affected by Sp 3

and Sp 4 (overall fitness is higher (0.46 versus 0.42). The new Sp 2 also has

caused a change in the fitness value of Sp 1.

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to one other species. Table 5 shows two matrices generated bythe model at time steps 2 and 3. For example, we see that inthe third time step a new species 2 appears which is alsodependent on species 4 and the overall connectance or fitnessfrom species 2 to species 1 increases. Changes such as thesecontinue through to the end of the simulation after 10 timesteps. When the ecological network properties of these con-nectance matrices from each time step are calculated we findthe following: amplification does not occur at any time step;the cycling index, homogenization, and ratio of indirect to di-rect effects all decrease over time; and the synergism parame-ter rises until T¼ 9 at which time it starts to drop by the end ofthe simulation (Fig. 2).

In the second simulation, each species is initially linked tofour other species (complete graph minus self-loops). Severalchanges occur immediately, most notably, the connectionbetween Sp 4 and Sp 5 is lost. During the 10-step simulationthe system becomes more articulated, meaning there are fewerconnections between species, but these changes would only beaccepted if the overall fitness of the species increases. Onesimple measure to consider is the total number of connectionsin the system during each time step (Table 6). We see a similarpattern in the network parameters in the second simulation aswell. Amplification does not occur at any time step. Cyclingindex and indirect effects ratio decrease, while in this simula-tion homogenization bounces around but is fairly flat. Syner-gism also oscillates reaching a peak in the middle of the

Table 6

Connections in Simulation A (species initially connected to one species) and

Simulation B (species initially connected to four species)

T A: # links B: # links

0 5 20

1 5 19

2 5 19

3 6 19

4 6 19

5 5 19

6 4 19

7 4 18

8 4 18

9 5 18

10 7 14

0

1

2

3

4

5

6

0 2 4 6 8 10time step

param

eter valu

e

Cycling index homogenization synergism i/d ratio

Fig. 2. Behavior of network properties for first simulation.

simulation and dropping again near the end (Fig. 3; note inthe figure that synergism is plotted on the alternate y-axis).

Next we observed the progression of certain goal functionsto see how they also responded to changes in the fitness model(Figs. 4 and 5). In both simulations, the total system through-flow and total system storage increase. Retention timeincreases initially and then decreases and the overall trend forspecific dissipation is to increase. Cycling, which was plottedon Figs. 2 and 3 as one of the network properties, decreases dur-ing the simulations. Based on these results only the maximizeexergy storage and maximize power goal functions behavedas expected; however, the models used in this proof of conceptstage were not ecologically realistic. Future research willinvolve further refinement of the model by considering otherinitial structural patterns, using different sized models, andallowing the model to run longer.

5. Discussion

One theme of the International Environmental Modellingand Software Society (iEMSs) Conference, which the papersin this issue represent, is how to acquire through science andmodeling a better understanding of complexity in environmen-tal systems, primarily the sustainability of coupled ecologicaland social systems. One of the main motivations for Kauff-man’s Boolean networks was to identify fundamental

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10time step

param

eter valu

e

02468101214161820

Cycling index homogenization i/d ratio synergism

Fig. 3. Behavior of network properties for second simulation. Synergism is

plotted on the alternate y-axis.

0

5

10

15

20

25

30

35

0 2 4 6 8 10time step

pro

perty valu

e

0

0.05

0.1

TSS RET TST SDP

Fig. 4. Behavior of ecological goal functions for first simulation. TSS and RET

are plotted on the alternate y-axis.

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principles relating network structure and behavior. He foundthat system behavior ranged from very stable (‘‘deeply or-dered’’ in Kauffman’s terminology) to chaotic depending onthe number of nodes and connections, and that systems tendedto evolve toward an intermediate level of connectivity (‘‘theedge of chaos’’). These results indicate that there is an optimalcomplexity for system design, which balances the opposingevolutionary tendencies toward increased stability (and henceless adaptability) and toward increased flexibility (with theconcomitant risk of losing system integrity). In the ecologicalexamples given above, we can see this tendency for systems toevolve toward an intermediate level of connectivity (Table 6).

The network models described herein, although theoreticalin nature, provide insight into system organization and due totheir reliance on first principles of Boolean analysis are applica-ble to many different system types. Therefore, they providea good platform for modeling integrated systems. This paperpresents an initial foray into two different important networkapproaches with the aim of gaining better confidence in howthey may be linked or they complement each other. The ecolog-ical network properties inform about structure, function, andrelations, but they have traditionally been applied to static net-works. Kauffman’s NK model gives a way to let the modelsevolve over time, and thus see how the properties changeover time. Since the ecological goal functions are assumed totrack ecosystem development, they have already been usedfor structural dynamic models (Nielsen, 1992) and model cali-bration (Jørgensen and Fath, 2004). It is also important to findother useful management applications for these methodologies.In the case of managing biological and societal systems inter-acting together and coevolving on correlated landscapes, a hier-archical management approach dealing with the subsystemsshows potential to find a compromise solution between com-plexity levels of the different subsystems (Alexander, 1964).

6. Conclusions

In conclusion, we have recreated Kauffman’s multi-speciesNK model and used it to investigate the coevolution of a simplemodel ecosystem. Furthermore, we have used the fitnessvalues generated by the model as surrogates for the probability

0

5

10

15

20

25

30

35

0 2 4 6 8 10time step

pro

perty valu

e

0

0.05

0.1

TSS RET TST SDP

Fig. 5. Behavior of ecological goal functions for second simulation. TSS and

RET are plotted on the alternate y-axis.

of influence between the compartments. This allows the appli-cation of network analysis techniques to determine the valuesof specific network properties. In particular, we found that net-work synergism appears to respond positively as fitnessincreases, and the other properties respond negatively. This pa-per represents the first attempt to integrate the two Booleantechniques; further research is needed to more deeply under-stand the interrelation between them.

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