Community assembly in a model ecosystem

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Ecological Modelling 103 (1997) 267 285

Community assembly in a model ecosystem Peter T. Hraber *, Bruce T. Milne Department o[ Biology, University of New Mexico, Albuquerque, NM 87131, USA Received 31 December 1996; accepted 13 May 1997

Abstract Environmental and biotic factors regulate species abundance and the composition of ecological communities. However, it is difficult to demonstrate principles of community assembly in nature due to observational limitations and historical effects. Numerical simulation can be used to evaluate whether hypothesized mechanisms produce a pattern seen in natural communities. Most models of many-species communities assume similar behavior by all individuals in a population. Here we describe an agent-based model of many-species communities in which interactions among individual agents are subjected to Darwinian selection (Holland's Echo model). To evaluate the response of the model to evolutionary mechanisms, we present a simple version of the model, and describe an experiment which evaluated community assembly patterns with and without selection for genome-mediated interactions, at various levels of invasion. Increased invasion rates decreased population size, but increased species richness and evenness. Interactions acted upon by natural selection resulted in greater population sizes and lower species richness and evenness than random interactions. Genotype abundances deviated from the expectation of equal abundances, and were affected by both the invasion rate and nature of inter-agent interactions. Species abundance patterns indicate that communities formed by random and selective interactions follow different assembly rules, and communities formed under high invasion rates showed no coherent community-level pattern of genotype abundances. Thus, both invasion rate and type of interspecific interaction regulate the assembly of species into communities. © 1997 Elsevier Science B.V. Keywords: Adaptive agents; Community ecology; Genetic algorithms; Invasion; Neutral evolution

1. Introduction

* Corresponding author. Tel.: + 1 505 2773431; fax: + l 505 2770304; email: [email protected]

Identifying and understanding mechanisms which regulate the c o m p o s i t i o n o f ecological c o m munities is a p r i m a r y focus o f t h e o r e t i c a l a n d e m p i r i c a l ecologists. F i e l d a n d l a b o r a t o r y studies

0304-3800/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI1 S0304-3800(97)001 1 1-7

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indicate that biotic (e.g. interspecific interaction) and abi0tic (e.g. climate) factors both conspire to determine which species can coexist in a habitat (Hutchinson, 1959; Kodric-Brown et al., 1984; Drake, 1991; Tilman, 1996). However, several considerations make it difficult to characterize general principles which govern community assembly. Observations of extant communities are limited in their temporal scale of observation, and presume the community is in equilibrium or represents an optimal partitioning of available resources (Brown, 1981; May, 1986; Ricklefs and Schluter, 1993). Moreover, timing and rate of colonization confer an integral role to historical contingency, or 'frozen accidents', in determining the dominant species in a community (Paine, 1974; Drake, 1991; Tilman, 1996). Additionally, interactions between species occur in a pairwise manner, but the presence or absence of a third species may mediate the outcome of any two-species interaction (Ricklefs and Schluter, 1993; Brown, 1994). Empirical efforts to derive general rules of community assembly are limited by these considerations. Despite these difficulties, ecologists have identified several processes at work in the organization of species into communities: 1. Niche partitioning/competitive exclusion. Similarity of resource use by two species may result in displacement of either species through competitive interactions. The result may be evolutionary adaptation through selection for differences in resource use, or competitive exclusion of either species from the community (Gause, 1934; Hutchinson, 1959; Roughgarden, 1976; Brown, 1981; Tilman and Pacala, 1993). 2. Multiple domains of attraction. Communities may have multiple stationary distributions, or 'alternative stable states' which represent different species assemblages possible under similar conditions (Gilpin and Case, 1974; Sutherland, 1974; Kolasa and Pickett, 1989). History plays a principal role in deciding which alternative state is manifest (Drake, 1991; Ricklefs and Schluter, 1993; Brown, 1994). 3. Spatial extent. Spatial heterogeneity, such as resource patchiness or gradients, can amelio-

rate effects of competition or predation by providing local refugia and opportunity for refined adaptation (Huffaker, 1958; Turner, 1989; Kareiva, 1990; Milne, 1992; Milne et al., 1992; Levin, 1993; Keitt and Johnson, 1995). 4. Open, driven systems. Ecological communities are adaptive systems which respond to continuous environmental and population fluctuations. Rescue effects, colonization, and invasion may also be involved, whether on an island or continent. Assumptions of equilibrium are often not tenable (MacArthur and Wilson, 1967; Wiens et al., 1986; Brown, 1994; Tilman, 1996). We will investigate primarily points 1 and 2 with the model system presented here. A small change of model parameters would make it possible to consider point 3, but we will reserve this issue for future work. Point 4 warrants comment, and is the motivation for us to compare the effects of chance and competition on community structure. Much debate has focused on the vagaries of population fluctuations and uncertainty about whether any observable community pattern results primarily from interspecific interaction (point 1) or random fluctuations (Diamond, 1975; Caswell, 1976; Connor and Simberloff, 1979). To demonstrate that a biotic process regulates an observed pattern it must be shown, by reason of parsimony, that chance alone cannot produce the same pattern. Demonstrating such an effect is difficult in field observations, but can be done rigorously by means of experimental manipulation and numerical simulation.

1.1. Models of community assembly Theoretical approaches to community assembly often invoke models which capture essential features of interspecific interactions. The construction of any model requires assumptions about the system it emulates, and must therefore make a trade-off between realism and generality (Levins, 1968; May, 1973; Holland et al., 1986). A model with many empirically-based assumptions sacrifices generality for increased accuracy of predictions. General models, on the other hand, sacrifice accuracy and detail for abstraction. The crucial

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difference between the two is their utility. Specific models are used to forecast the future behavior of a system, such as the expected effects of increased temperature or CO2 concentration on species distributions. General models are most effective at helping to characterize the consequences of universal mechanisms and to formulate general principles. This is achieved through a process of formalizing intuition and evaluating the effects of defined mechanisms on system behavior. In this context, general models serve as a 'macroscope' (Brown, 1994), by which the researcher may explore many alternative hypotheses prior to (or in conjunction with) designing and executing an empirical study. May (1973) has suggested that the entire modeling enterprise is best served by feedback between general and specific approaches. To illustrate the utility of general models in ecology, consider the debate over the relation between ecosystem stability and species richness, a measure of ecological complexity (May, 1973; Pimm, 1984; Tilman, 1996). By evaluating the effects of random species assemblages on community stability, May (1973) found that increased complexity yields decreased stability. The fact that this result seemingly contradicts observations of field ecologists leads to the hypothesis that empirical communities must be assembled by different processes than in May's simulated communities. Insights gained from modeling populations of one or two species with differential or finite-difference equations led to differential equation models of many-species assemblages (Levins, 1968; May, 1973; Roughgarden, 1976; Pimm, 1984). Such models characterize interspecific interactions with pairwise interaction coefficients a/s and aji, which specify the effects of species j on species i, and vice versa. Dual negative a.. values yield a competitive effect; dual positive values yield a mutualistic effect. When modeling multispecies interactions this way, species interaction coefficients are typically generated randomly, and properties of the matrix are evaluated, e.g. May (1973). In differential equation models, all species are assumed to interact with equal probability. This property is characteristic of a well-mixed, or 'mean-field', system of interactions (Durrett and

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Levin, 1994a). Scrutiny of this assumption reveals that a spatial context for interactions can have a great effect on system dynamics. One way to provide a spatial context for interactions is by assigning locations to individuals on a lattice. Durrett and Levin (1994a) compared differential equation and lattice-based models of two-species dynamics (hawks and doves), and showed that local interactions predict different outcomes for competitive interactions than the mean-field approach. Durrett and Levin's study suggests that further revisions of model assumptions can likewise affect system dynamics. In particular, lattice-based models have been extended to consider differences in identity among individuals (De Angelis and Gross, 1992; Hiebeler, 1994; Langton, 1994; O1son and Sequeira, 1995). Individual-based models have been implemented in two ways: by approximating speciation and extinction as random branching processes (Aldous, 1995; Durrett and Levin, 1996), and by modeling genetic variation with a genetic string which encodes some aspects of the individuals' behavior (Langton, 1989; Holland, 1992). In the latter approach, the string is analogous to an individual's genome, and the modeler determines what mapping of genome to genes is most relevant to the phenomena being studied. The first approach models neutral evolution of a community, because interactions between individuals are not influenced by species' identities. The second approach emulates natural selection, because of a relationship between genotype and reproductive success. The genetic approach is exemplified by the Hartviggsen and Starmer (1995) model of plantherbivore interactions. In their study, individual plants have genes which confer resistance to herbivory, and herbivores have genes which overcome the plants' defenses. The gene-for-gene coevolution made possible by this scheme is an attempt to emulate the genetic mechanisms of coevolution in natural plant-herbivore relations. This approach allowed an investigation of density-dependent effects on a defense-resistance 'arms race'. The greatest potential benefit of genetically explicit modeling is that it may yield different results

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than a model which assumes no differences or random selection of differences among individuals, making it possible to study different issues than possible when all individuals behave identically (Hiebeler, 1994; Olson and Sequeira, 1995). Allowing Darwinian selection among individuals allows us to study the evolutionary dynamics of a model system. Thus, rather than studying static species interaction matrices or neutral evolution among individuals with arbitrary reproductive success, selection in genetically explicit models affords analysis of the emergent properties of an adaptive dynamical system. Though this term has many meanings, the principle of emergence has received considerable attention in the light of individual-based, genetically explicit modeling (Green, 1993; Langton, 1994; Olson and Sequeira, 1995). Here we are interested in determining whether selection among genotypes yields different functional associations of genotypes than random (neutral) selection. We will test the hypothesis that genome-mediated interactions yield different emergent community-level patterns than random interactions by comparing properties of communities evolved under both interaction scenarios. Modeling adaptive system dynamics via individualistic mechanisms of selection is a fundamentally different approach than modeling with differential equations. The parameters in differential equation models approximate phenomenological attributes of populations without identifying mechanisms (Schmitz and Booth, in press) or considering variation among individuals. For example, the population growth rate r in logistic growth models represents the combined population-level consequences of mate choice, clutch size, juvenile survivorship, etc. of the individuals in the population. At present, most specific models are based on differential or finite-difference equations. In forest modeling for instance, the user parameterizes terms in the growth equations, then allows the system to iterate (Urban and Shugart, 1992). Competitive interactions may or may not be considered. These models have been made spatially explicit through the definition of a biotic interaction neighborhood and incorporation of local abi-

otic conditions. This approach can be used to predict which species will dominate in a community over time, but is a phenomenological, rather than process-based, approach. By simulating processes believed responsible for a pattern, we can evaluate what mechanisms are most influential in producing the pattern. 1.2. The Echo model

Echo was developed as a general model of the dynamics of adaptive systems. Adaptive systems are ensembles of individuals with the capacity to process information about their environment and modify their behavior according to some goal (e.g. reproduction). We call this capacity agency, and adaptive individuals agents. Properties of adaptive systems cannot be inferred from the individual agents, because the interactions among agents determine the emergent dynamics of the entire system. Canonical examples of such systems are the nervous system, economies, and ecosystems. While Echo is intended to embody mechanisms common to all adaptive systems, we are concerned here with the plausibility of Echo as a model of ecological systems. Holland (1995) gives an overview of properties and mechanisms common to a range of adaptive systems, and builds a model whose mechanisms embody these tenets. The resulting model has similarities to extant modeling approaches, and also provides some innovative mechanisms which make it of particular value in the study of many-species dynamics. Echo is intended as a class of models whose members represent increasing complexity (Holland, 1994, 1995). This specification calls for an integrated environment, in which model mechanisms can be added, deleted or modified according to experimental goals. One benefit of this approach is that mechanisms can be evaluated by studying their effects, as in the macroscope notion described above. While it is easy to list mechanisms believed important in adaptive systems, it is much more difficult to show they are both necessary and sufficient to produce empirical patterns. We use this integrated modeling approach to ask whether genome-mediated biotic interactions result in different species assemblages than random

P.T. Hraber, B.T. Milne / Ecological Modelling 103 (1997)267-285

interactions, and to evaluate the effects of invasion rate on community assembly. As with the lattice-based models described above, Echo is an individual-based, spatially explicit model (Fig. 1). Unlike a lattice-based model with one individual per node, populations of

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many agents interact within a site, and may migrate to other sites in a world of user-defined geometry. In addition to spatial and individual based bookkeeping, Echo individuals have a genetic component. Agents have their own haploid genomes, which are subject to heritable variation and differential reproductive success. Thus, an analogy to evolution is possible, along the lines of genetic algorithm simulations of evolutionary processes (Holland, 1992; Mitchell, 1996). (It is no accident that Holland was a key figure in the advancement of genetic algorithms as models of evolutionary dynamics; Echo is an extension of the genetic algorithm). As they evolve from initial conditions, populations tend to be dominated by genotypes with greatest reproductive success. It is also possible to create novel genotypes, or to model formation of novel species. This is a stark contrast to extant many-species models, which either list all species and determine their interactive attributes a priori, or model evolution as a random branching process (Aldous, 1995; Durrett and Levin, 1996). Echo genotypes are subject neither to random selection, as in stochastic spatial 'voting' models (Durrett and Levin, 1994b), nor to a user-defined fitness function, as in genetic algorithms (Holland, 1992; Mitchell, 1996), but rather to an endogenous fitness function. Endogenous fitness is made possible by resource-limited reproduction and genetically mediated behavior (Jones and Forrest, 1993, Holland, 1994, 1995). Resources constitute the basic currency of Echo, and are used to construct agent genotypes. An agent can obtain resources from the environment or by interacting with other agents. An agent may self-reproduce only when it has gathered sufficient resources to copy its genome. Genetically-mediated behavior determines whether two agents can interact. The Echo agent genome has two main regions, each with three genes that code for a particular interaction (Fig. 2). The tag region of the genome codes for traits which are visible to other agents. The conditions region represents traits which are internal 'detector' states, known only to the agent itself. Matching a condition gene against a tag gene allows the interaction coded by those genes to occur. This

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Fig. 2. The Echo agent genome. Tag genes code for traits visible to other agents; condition genes represent internal states. Matching one agent's tag gene against another agent's condition gene allows the interaction encoded by those genes to occur. The trading resource specifies what an agent will offer if trade occurs. In simple Echo, only two alleles (a or b) are permitted per gene. abstraction is an obvious oversimplification of biological reality, but is intended to capture the essentials of signaling mechanisms between individuals. The clearest analogy is that the tag gene codes for a phenotypic trait, such as a banner or marking, and the condition gene recognizes the preferred phenotype for interaction, resulting in political alliance or mate choice (Holland, 1994, 1995). The three possible types of biotic interaction are combat, trade, and mating. Combat and trade allow for unilateral or bilateral resource exchange, respectively. They are intended to mimic predator-prey and mutualistic interactions. Mating allows for genetic recombination by forming daughter agents from the parents. Implementation of interaction criteria is described in Section 2.1. The innovative mechanisms described above make Echo a promising source of inspiration for more specific modeling approaches. Like other general models, Echo has some clear drawbacks which limit the specificity with which predictions can be made (Forrest and Jones, 1994; Hiebeler, 1994; Hraber et al., in press). Chief among these is the high level of abstraction involved. It is not clear what a resource or agent corresponds to in the real world. A resource could be some energetic or chemical component of the system, or some metabolic product. Likewise, an agent could be a simple or complex individual organism, but there is no variation in body size or organizational complexity among agents. It is therefore unclear what we mean by a species. Because Echo agents can reproduce both sexually and asexually, and can hybridize readily, we cannot apply the

biological species concept of Mayr (1963). Rather, we define a unique genotype as a species, though alternative definitions have been considered (Forrest and Jones, 1994). Further, there are no explicit trophic relationships in Echo, no explicit flow of resources from abiotic sources to primary producers to herbivores to carnivores, nor any nutrient cycling by detritivores. Rather, any trophic structures must be designed by the user or allowed to evolve. For instance, Schmitz and Booth (in press) examine the stability of three-species food chains using a modified version of Echo in which evolutionary mechanisms are disabled. Even when such atrophic network has been carefully designed, there is no guarantee it will persist if evolutionary mechanisms are operating. Another potential drawback to using general models like Echo is the number of decisions which must be made prior to running a simulation (Hiebeler, 1994; Forrest and Jones, 1994; Hraber et al., in press). In Echo, the user must specify the initial populations and system parameters (such as mutation rate and world topology). Model sensitivity in large parameter spaces is difficult to evaluate, because parameter interactions may greatly affect system dynamics. It is also very difficult to interpret the structure in populations generated by evolution, as genomes are allowed to evolve in a 'messy' manner, in which gene lengths vary. Infinite allele models are not amenable to standard analysis tools in community ecology (Levins, 1968). 1.3. Goals o f this study

Despite the numerous limitations to using Echo to emulate community assembly processes, we believe the abstractions in the model are worth investigation. We have been evaluating the ecological implications of the current model implementation. The remaining sections of this paper describe an experiment that used a simplified version of Echo. Reducing the mechanisms and the size of the genotype space (by restricting the number of alleles) made simulated evolutionary dynamics amenable to study by multivariate ordination techniques. By analyzing evolved communities, we identify principles of community

P.T. Hraber, B.T. Milne /Ecological Modelling 103 (1997)267-285

assembly, and discuss the correspondence of assembly rules between artificial and natural communities. Neutral models are used in biology to evaluate what mechanisms are necessary to produce a phenomenon of interest (Caswell, 1976; Nitecki and Hoffman, 1987; Gardner and O'Neill, 1991). Assortative mating, competition and predation, and direct or indirect mutualisms are all believed to regulate species diversity in ecological communities. The main issue for both Echo and the real world is whether these mechanisms are necessary to produce community assembly patterns. To ask whether assembly patterns are a result of biotic processes (interspecific interactions), or random selection, we compare effects of genome-mediated interactions with those of random interactions. Caswell (1976) presents arguments from niche theory and systems theory, supported by numerical simulations, that communities regulated by biotic factors should be more productive and complex than communities limited only by abiotic processes. If random interactions should yield fewer species and less community structure than genome-mediated interactions, we may infer that biotic interactions yield non-random organizations of species. We also examine effects of invasion rate on the ability of the model to evolve to alternative community states. The magnitude of invasions from a regional species pool is known to dampen local population fluctuations (MacArthur and Wilson, 1967; Ricklefs and Schluter, 1993; Brown, 1995). We therefore expect that a high invasion rate (as modeled by mutation) will reduce effects of local extinctions. Our ultimate goal is to obtain insights about the current version of Echo to inform future attempts to model specific systems with similar mechanisms, as well as to identify where the current model needs the most refinement.

2. Methods The simulations described in this study were conducted using a public domain release of Echo, version 1.2 (Jones and Forrest, 1993), which is a

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subset of the class of models described by Holland (1994, 1995). The source code is written in C and has been compiled on UNIX, Macintosh, and PC architectures. The graphical user interface is written with X library functions and runs on U N I X and L I N U X operating systems. As in Holland's original scheme, Echo is intended to capture mechanisms essential to ecological and evolutionary dynamics in an abstract form. However, the current implementation of Echo proved intractable to analytic techniques used to infer community pattern in ecology (Hraber et al., in press). To make the model amenable to analysis, we created a version of Echo with a minimal set of mechanisms. We call the reduced version of Echo simple Echo. In simple Echo, almost all stochastic mechanisms are disabled. These mechanisms include resource input (agents must interact for resources), taxation, random death, and migration. Our simulations were conducted with a two-letter resource alphabet (a,b). Rather than allow agent genotypes to evolve in an open-ended manner, genotypes were constrained to a finite set of possibilities via a simple mutation operator, which substitutes one allele value for another. Because agent genomes have n = 7 genes and m = 2 possible alleles, the net size of the genotype space is m n = 27, or 128 genotypes. To minimize effects of genetic drift on the final distribution of genotypes, one copy of each genotype was present in the initial population of each site. Each agent was given an initial supply of ten of each resource in its resource reservoir. No other resources were added to the system. While it is possible to examine effects of local interactions among sites in Echo by enabling migration, agents in our experiments were not allowed to migrate. Simulations were conducted in square worlds with 16 sites on a side. This yielded 256 independent replicates with the same initial conditions. Because there is no spatial context by which to create local interactions, the present study is a mean-field model of interactions among genetically distinct individuals. An iteration of simple Echo has only two steps. At each site in the world: (1) agents interact. The number of interactions in a site equals the interac-

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tion fraction (1/8) multiplied by the population size. All agents in a site interact with equal probability. Interactions are described in detail below; (2) agents with enough resources to copy their genomes self-replicate. Mutation is applied at a per-gene rate at the time of mating and self-replication. Several mutation rates were studied to examine effects of invasion on community assembly. Here we model invasion as genetic mutation. We note that modeling invasion as an effect of mutation is more akin to reducing reproductive fidelity than to geographic spread of invasive novelties. The mutation rate determines the rate at which reproducing individuals are replaced by a different genotype in the population. Because we are using genotypes to define species, reproduction with mutation is analogous to invasion by another species. In another study (Hraber et al., in press), we considered effects of migration on species richness in worlds of varied size. Future work will examine effects of invasion as migration between cells of the lattice is enabled. Since fitness is endogenous in Echo, we have no a priori expectations for the fitness of a genotype. Thus, we expect all genotypes to have similar reproductive ass success. A naive hypothesis about genotype abundances is that each of the S species present should be represented in equal amounts of N/S, where N is the total population size. Further, we naively expect genotype abundance to be unaffected by the nature of interagent interactions.

2.1. Structured and random interactions Echo agents can interact by combat, trade, and mating. Combat and trade allow for predatory and mutualistic exchange of resources, respectively. Mating allows for the creation of new genotypes via recombination. Whether two individuals will interact is determined by comparing tag and condition genes for that interaction (Fig. 2). To examine effects of genetic interactions on population dynamics, we studied two versions of interspecific interactions: selective and neutral versions of Echo. In the selective model, matching alleles in tag and condition genes allows the inter-

action coded by those genes. In the neutral model, tag-based interaction criteria are replaced with small probabilities, thereby enabling interactions regardless of alleles. Tests for interactions are conducted sequentially: first for combat, then for trade, and finally for mating. When testing for combat, agent A will attack agent B if A's 'combat condition' (com gene) matches B's 'offense tag' (off gene) (Fig. 2). Agent B is given a chance to escape which is equal to its probability of losing. The probability of defeat in combat is calculated using B's 'defense tag' (def gene). In simple Echo, this probability always equals 1/2. If the attacked agent does not escape and is not defeated, it is given a chance to attack by the same criteria. If either agent attacks, no other interactions can occur. Whenever combat occurs, the winner kills the loser and obtains all of its resources. If no combat occurs, agents may trade or mate. Agents A and B will trade if A's 'trading condition' (tra gene) matches B's 'offense tag' (off gene), and vice versa. Unlike combat, trade occurs only by mutual consent. If trade occurs, the 'trading resource' (res locus) determines what resource each agent offers in trade. If an agent has any of these resources stored in its reservoir, it will give half of them to the agent with whom it is trading. Agents who trade have no way of knowing in advance what they will receive in exchange, or whether they will indeed receive anything. Finally, if combat does not occur, and whether or not trading happens, the prospect of a mating interaction is tested. Agents A and B will recombine if A's 'mating condition' (mac gene) matches B's 'mating tag' (mat gene), and vice versa. Like trade, mating occurs only by mutual consent. During mating, two crossover points in the agents' genomes are randomly selected for recombination. The parents' resources are divided equally between the daughters, and the parents are replaced by their daughters in the population. In the neutral model, when two agents are chosen to interact, the decision of whether they will fight, trade, or mate is based on random tosses of a biased coin, rather than the stringmatching criteria described above. This neutral model results in a model ecology in which species

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interact with one another randomly. Aside from this modification, all other bookkeeping is performed identically in both models. Neutral model parameter values were chosen to ensure that, on average, the expected number of interactions per generation would be the same as in the selective model (Table 1). For neutral model simulations in this study, the probabilities of combat, trade and mating were 1/2,1/4, and 1/4, respectively. These values correspond to the chance that two agents with randomly chosen alleles will meet the criteria to interact.

2.2. Experimental design and data analysis To test whether selectively structured interactions and invasion rate affect system dynamics, we used a 2 x 3 factorial design, with treatments for model type (selective versus neutral) and three mutation regimes (IL = 0.0025, 0.025, 0.25, where kt is the per-gene mutation rate). Each combination of treatments was replicated 256 times with initial conditions set as described above. In order to avoid temporal pseudo-replication in statistical inference, we gathered response variables for replicate simulations only at the 10000th iteration. Abundances of each of the 27 genotypes were recorded for each replicate, as were population size (N), number of species (S), and evenness (J), the probability that two randomly selected individuals represent different genotypes, (Pielou, 1977). We used multivariate analysis of variance (MANOVA) to test for significant effects of treatments on response variables N, S and J, using the Table 1 Comparison of expected interaction and neutral models Interaction Pcombat = P ,,¢j- ,.o.,

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ANOVA procedure in SAS (1990). Both treatment effects and their interaction were tested for significance. Pillai's trace was used as a test statistic, as it is relatively insensitive to violations of test assumptions and a precise F statistic can be computed in SAS. Aside from slightly unequal variances, test assumptions were not violated. Standardized correlation coefficients were used to evaluate which response variables contributed most to the differences between experimental treatments. Standardization corrects for differences in the magnitude of response variables by normalizing the variance of each variable to one. To evaluate reproductive success of agent genotypes, we calculated the relative abundance of each genotype for each treatment. Relative abundance of a genotype is the net abundance of agents with that genotype over all 256 replicates (the sum of a column of the abundance data matrix), divided by the net number of agents in all replicates (sum over all columns in the abundance data matrix). Finally, we analyzed species associations using principal component analysis (PCA) on community abundance matrices. A community abundance matrix is a representation of species abundances for each replicate. Rows correspond to replicate sites (256), and columns to genotypes (128). Separate analyses were conducted on matrices from each treatment. Principal components were determined using the covariance matrix with no pre-transformation, and were performed using the P R I N C O M P procedure (SAS Institute, 1990). PCA gives an ordination of sample units such that each successive axis of the ordination is orthogonal and axes are ordered so that they describe decreasing proportions of the overall variance. This is achieved by decomposing the species covariance matrix into a latent matrix of eigenvectors and eigenvalues. Eigenvalues (A) indicate the amount of variation described by each axis. Eigenvectors give the loadings of each species on a particular axis. Multiplying an eigenvector by the abundance matrix gives a projection of sample units in which sites most disparate in species composition are most distant. We use this ordination to infer community structure and alternative community states.

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3. Results

For populations in single sites with a low mutation rate, the selective model stabilized between 2500 and 10 000 iterations, while the neutral model continued to fluctuate (Fig. 3). The selective model exhibited logistic-type population growth, which asymptoted at carrying capacity, while the neutral model exhibited sustained fluctuations in population size (Fig. 3a). Species richness and evenness decreased over time for both models, though the magnitude of fluctuations was greater for the neutral than for the selective model (Fig. 3b, c). Increasing mutation made both models behave similarly, with all response variables fluctuating consistently throughout a simulation (Fig. 4). While neither model was in static equilibrium for species richness or evenness at 10000 iterations, we considered this sufficient time for transient dynamics to settle into predictable behavior.

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tion rate, and vanished at the highest mutation rate (Fig. 6a). We observed a similar pattern for the neutral model, though the fitness differential was not as great, and differences in abundance vanished at the medium mutation rate. Different genotypes dominated in the neutral model than in the selective model (Fig. 6b).

P.T. Hraber, B.T. Milne / Ecological Modelling 103 (1997) 267-285

Genotypes with the greatest abundance in both models had several interesting properties (Table 4). Successful genotypes were comprised of half a's and half b's 480 (the r e s gene was reproduced at no resource cost). Genotypes which differed only by r e s alleles were almost equally successful; we therefore considered these genotypes equivalent in further analysis. A genotype and its complement (substituting a's for b's and vice versa) 300

(a)

.

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had about the same abundance. Successful genotypes in the neutral model were not the same as those in the selective model. Genotypes with greater abundance in the neutral model with low mutation were comprised of half a's and half b's, but did not have the same properties as successful

P.T. Hraber, B.T. Milne /Ecological Modelling 103 (1997)267-285

278

Table 2 M A N O V A test statistics and values for Pillai's trace for response variables N, S and J Source

Value

F

N u m D.F.

Den D.F.

Pr>F

Type Mutation Type x mutation

0.840 1.665 1.006

2671.7 2534.5 515,9

3 6 6

1528 3058 3058

0.0001 0.0001 0.0001

genotypes in the selective model. The least successful genotypes were those least balanced in resource composition, that is, all a's or b's, or five of one resource and one of the other (Fig. 6). Principal component analyses on simulations with low mutation gave three dominant eigenvectors which described 45 and 17% of the variation in species abundances from selective and neutral models, respectively (Table 5). For the selective model, the first 17 principal component axes described greater than random amounts of variation (as defined by the ratio of sample standard deviation tr 2 to the number of variables S). Taken together, these axes described 92% of the variation. For the neutral model, more PCA axes were required to explain the variation in species abundances: 38 principal component axes described greater than random proportions of variation, and together accounted for 98% of the total variation. Differences between model types were less evident with increased mutation. The first three axes accounted for 10 and 7% of the variation in Table 3 Standardized correlation coefficients from canonical comparison of response variables in M A N O V A procedure

selective and neutral models (Table 6). Approximately 50 principal component axes described more than random amounts of the variation (72%) in both model types. Replicate sites in the selective model demonstrated dominance by particular genotypes (a, fl, 7, ~), or sets of genotypes (ct and fl, or 7 and g), but never by other sets of genotypes (e.g. • or fl with y or ~, Fig. 7a, c, e). The neutral model also showed dominance by particular genotypes (v, ~, re, p, Fig. 7b, d, f), but never by combinations of these genotypes as in the selective model. Thus, in the selective model, populations evolved into guilds of compatible genotypes, while neutral model populations evolved to dominance by an exclusive genotype. Selective Model (a)

- - ~-o.oo25

10%

32

64

96

128

genotype Neutral Model

Source

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10.462

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--0.409 0.500

9.438 5.946

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0.062

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3.970

-2.358

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Type Mutation Type x mutation

S

J

Pairwise comparisons are made between low and medium, and medium and high mutation rates.

genotype Fig. 6. Relative abundance of genotypes for selective and neutral models. Relative abundances were means from 256 replicates at 10000 iterations. Genotypes were converted to decimal n u m b e r s from binary strings (a = 0, b = 1) with genes ordered as in Fig. 2. Greek letters indicate the eight most a b u n d a n t genotypes in each model.

P.T. Hraber, B.T. Milne/Ecological Modelling 103 (1997)267-285

Table 4 Most common genotypesin selectiveand neutral model simulations Genotype Decimal

Percent abundance Binary

p = 0.0025

p = 0.25

Selective model :~ 26 27 [] 44 45 ;' 82 83 6 100 101

aabbaba aabbabb ababbaa ababbab babaaba babaabb bbaabaa bbaabab

9.3 8.2 5.7 9.0 9.2 8.6 9.1 10.9

1.2 1.3 1.2 1.2 1.1 1.2 1.2 1.2

Neutral model v 22 23 28 29 56 57 p 84 85

aababba aababbb aabbbaa aabbbab abbbaaa abbbaab bababaa bababab

3.3 2.2 3.9 1.4 2.5 3.1 5.0 2.9

0.8 0.8 0.8 0.8 0.7 0.8 0.8 0.7

With increased mutation, populations within a site did not evolve to dominance by a small set of genotypes. Rather, populations were unpredictable assemblages of many genotypes, as shown by the spherical cloud of sample units in Fig. 8. The selective model showed slightly more variation than the neutral model on the first principal component axis (Fig. 8a, b). This variation corresponded to dominance by genotypes with one allele in the o f f gene and the complementary allele in the corn gene (i.e. genotypes which do not undergo combat).

4. Discussion 4.1. Assembly processes in simple Echo

Our results show that genome-mediated interactions can yield different temporal dynamics, response variables, and patterns of genotype abundance than random interactions. This demonstrates that our simulated communities

279

were not random ensembles of genotypes, but rather groups of agents with specialized adaptations for coexistence within a habitat. Because population sizes were larger for simulations with non-random interactions, we conclude that populations in the selective model had properties which enabled the communities to attain greater productivity. We will discuss these properties, then discuss the case where the selective and neutral models behaved similarly. Besides having properties described in Section 3.2, the genotypes which were selected to form guilds partake in some interactions but not in others. Specifically, genotypes • and fl, as well as 7 and ~, cannot engage in combat (off and com genes have different alleles). Some of these genotypes (~ and fi) can trade with their own kind, but the others (fl and 7) cannot engage in trade within a guild. Finally, all of these genotypes can mate with their own kind, but not with other members of the guild. Because these guild-level interactions were exhibited by the selective and not the neutral model, we consider them as rules for community assembly in the model system we have described. In addition to exhibiting community assembly patterns, the selective model demonstrates alternative community states. That is, due to the equal abundance of a and b resources, and the symmetrical distribution of genotype abundances, the two guilds of species are functionally equivalent, differing only by having complementary alleles at each locus. From initial conditions at which all alleles are equiprobable, agents interact until a particular allele outcompetes the other within a site. For example, the a and b alleles for the com gene occur with equal frequency until one allele dominates within a site. After dominance is established, the minority allele decreases in abundance until it becomes extinct. Because of the functional linkage between off and corn genes in testing for a combat interaction, the same happens at the corn locus. Similar allele fixation dynamics occur at other loci. We refer to this process of gradual allele fixation as canalization. Once a locus has been fixed to a particular allele, it is highly unlikely that the locus can be invaded by the complementary allele. A site with a fixed allele can

280

P.T. Hraber, B.T. Milne//Ecological Modelling 103 (1997) 267-285

Table 5 Eigenvalues (A) of the covariance matrix and variation explained by principal component axes for s~mulations at ~ = 0.0025 Axis

Eigenvalue (A)

Proportion of variation (%)

Cumulative variation (%)

Selectivemodel PC1 5785.9 PC2 4309.7 PC3 3506.1 Axes with A > tr2/S

19.1 14.2 11.6 17

19.1 33.3 44.9 91.8

Neutral model PCI 2555.3 PC2 2065.4 PC3 1761.4 Axes with A > trz/s

6.8 5.5 4.7 38

6.8 12.3 17.0 98.4

only be invaded by imposing a sufficiently high mutation rate. When a system loses invariance to translations for a portion of its state space, we say that the system exhibits broken symmetry (Nicolas and Prigogine, 1989; Palmer, 1989). When symmetrybreaking occurs, a large portion of state space becomes inaccessible. Further, more information is required to describe the set of alternative system states after symmetry has been broken than beforehand. Thus, symmetry-breaking increases the algorithmic information content, and therefore the complexity, of the system (Nicolas and Prigogine, 1989). To illustrate, consider the issue of chirality in molecules produced via biosynthesis. Presumably, either the L- or D-form of an amino acid, nucleotide, or monosaccharide should be able to fulfill the requisite biochemical function. However, biochemistry has evolved to preferentially use one form of a molecule over its symmetric twin in metabolism. In ecological terms, it may be that alternative community states are manifest through symmetry-breaking processes. The challenge for operationalizing this notion is to provide a description of biotic communities which considers the potential for symmetries in state space. Simulations with a high mutation rate do not exhibit assembly into guilds, because symmetrybreaking via allele fixation does not occur. Rather, sufficiently high mutation provides a sufficient counter-balance to selection to ensure that both alleles occur with nearly equal frequency. Given this observation, we infer there are

two possible phases for community assembly: (1) if the mutation rate is sufficiently low, allele fixation is inevitable, and populations evolve to an orderly, canalized state; (2) if mutation is greater than the strength of selection, both alleles are present with equal likelihood, resulting in a disorded, 'gaseous' state. The existence of distinct phases for the macroscopic dynamics of a system is well known in physics, as many systems are capable of different behavior as some driving parameter is adjusted (Langton, 1990; Kauffman, 1993; Green, 1994; Milne et al., 1996). To make this analogy clear, consider the behavior of water as temperature is varied and pressure held constant. At low temperatures, water forms crystalline ice, and we require little effort to describe the motion of individual water molecules over time. At high temperatures, molecules move about much more freely and are less predictable. In the case of water, varying temperature varies the kinetic energy in the system. This is analogous to varying mutation rate in our model system, and the change from canalized to gaseous state is analogous to the change of phase from solid ice to liquid water or water vapor. 4.2. Limitations and implications

We have identified several processes which regulate community assembly in an abstract model ecosystem. These processes are: evolution of alternative community states by symmetry-breaking;

P.T. Hraber, B.T. Milne /Ecological Modelling 103 (1997) 267-285

281

Table 6 Eigenvalues (A) of the covariance matrix and variation explained by principal component axes for simulations at/~ = 0.25 Axis

Eigenvalue (A)

Proportion of variation (%)

Cumulative variation (%)

Selective model PC 1 12.49 PC2 6.76 PC3 6.53 Axes with A > o2/S

4.7 2.6 2.5 50

4.7 7.3 9.8 72.5

Neutral model PC 1 6.04 PC2 5.70 PC3 5.66 Axes with A > o2/S

2.4 2.3 2.3 53

2.4 4.7 7.0 72.1

selection for functional guilds due to non-random interactions among agents, and sensitivity to mutation rate. What does our artificial ecosystem tell us about community assembly processes in nature? We will answer this by discussing the assumptions and shortcomings of our model, then identifying correspondences between the model and reality. Both suggest directions for further inquiry. We discussed difficulties of using abstract models in Section 1.2. In particular, we note the artificiality of modeling population dynamics with a discrete, finite-state model. The simplifications used to make Echo amenable to analysis may divest the model of some subtler dynamics. Increasing the number of alleles, as well as the alphabet size, would allow for more potential stable equilibria, though analytic techniques other than PCA should be used. Implementation of interaction criteria and the genome of the Echo agent could strongly affect simulation outcomes (Hiebeler, 1994; Hraber et al., in press). Genetically explicit models require some judgments in mapping genotypes to individual behavior. The interaction scheme used here is only one of a large set of possibilities. We hope the current version is a plausible metaphor for the spectrum of ecologically relevant behavior. In placing an extreme emphasis on inter-agent interactions, we have eliminated the effects of environmental variation, or abiotic effects, on assembly processes. We will examine effects of spatial heterogeneity in the extrinsic abiotic

environment on Echo population dynamics elsewhere, but have not yet determined effects of local interactions on community assembly. Environmental variation is known to play an integral role in limiting species' abundances (Kareiva, 1990; Levin, 1993; Tilman and Pacala, 1993; Brown, 1995; Tilman, 1996; Milne et al., 1996), and may lead to assembly rules additional to those seen here. Though we considered many alternative implementations for the neutral model, and did observe different behavior between selective and neutral models, there may be better ways to emulate random interactions and neutral evolution. In particular, we note that the transition point between canalized and gaseous phases of assembly was lower in the neutral than in the selective model. This could be a result of selection acting as a weaker force in the neutral model, or could be a more subtle artifact of our implementation. That is, effective population size (number of interacting agents) is different in the two models over time. An alternative neutral model implementation could adjust for random allele fixation over time, by assuming a modified functional form. Yet another alternative neutral model could be derived from a closed-form solution to Echo population dynamics, and may be similar in functional form to the neutral models studied by Caswell (1976). The real world is far more complicated than the model presented here. The space of possible genotypes is vastly greater than the 128 possibilities considered here. Given the wealth of models with

P.T. Hraber, B.T. Milne /Ecological Modelling 103 (1997) 267-285

282

simplifying assumptions, we conclude this study with numerous caveats, but hope to summarize what we have learned. We have evaluated the genetically explicit modeling approach by comparing community structure formed by selective and neutral evolutionary processes. The selective model yields different results than a neutral version with random, non-selective interactions. As in natural systems, variation in the nature of individual interactions does make a difference at the population level. In the absence of environmental variation and at sufficiently low mutation rates, populations evolve into guilds. Such guilds do not undergo predatory interactions (combat), and produce more individuals than unstructured communities. Increased productivity is a result of resources being distributed among individuals in Selective Model

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the population, rather than being concentrated in particular individuals. The combat interaction resuits in accumulating resources, but trade and mating result in resources being shared. Variation in timing and magnitude of mutation or perturbation could either disrupt or attenuate this trend for predatory interactions to be selected against. As seen in natural communities, dominance of the community by a particular species is partly a result of the legacy of history (Drake, 1991; Ricklefs and Schluter, 1993; Brown, 1995; Tilman, 1996). One implication of our results is that natural systems may exhibit phases of assembly and disorder. As was observed in our simulations, the rate at which new genotypes are introduced is either greater or less than the rate at which selection can act. A high invasion rate results in a relatively

P.T. Hraber, B.T. Milne /Ecological Modelling 103 (1997) 267-285

disordered assemblage of genotypes, while a lower invasion rate allows canalization of genotypes into discrete communities. Thus, the invasion rate m a y affect the phase of assembly over time, or across the landscape. In empirical communities, this phenomenon might be manifest as a level of gene flow above which selection for geographic differentiation is not possible. Identifying the critical invasion rate for community canalization may help develop ways to manage landscapes to maintain a particular ecological function, or to make comparisons between different ecosystems. The formalism of the theory of spatial phase transitions may provide a means to generalize our results. The phase transition framework has proven applicable in a wide range of biological processes (Green, 1994; Milne et al., 1996), and m a y help formalize the notion of critical phenomena in the development of many-species assemblages. As a closing consideration for the implications of our findings, we reiterate the formalism of the modeling relation from Holland et al. (1986) and Rosen (1991). The modeling relation recognizes the duality between observations of nature and in the model, and describes a formalism for model verification. For a model to be useful, it must be predictive in some capacity. Over time, behavior of the model must reasonably approximate behavior in nature. The state mapping of a natural system onto itself over time is subject to natural principles. The goal of constructing a model is to identify these natural principles. We do this by mapping the state of the natural system onto the model, iterating the model, then mapping back onto the state of the natural system. Numerous simplifications (coarse-grainings) are applied at each mapping, but useful mappings will ensure that the end-products (from nature mapping onto itself, and from translating into the model system and back) agree. I f they do, we can demonstrate 'commutativity of the diagram', or that mechanisms in our formal system are sufficient to emulate natural laws, at a reasonable level of coarse-graining. If the model does not agree with nature, or the coarse-graining is too severe, we can refine the model by revising assumptions and mechanisms therein.

283

The modeling relation and commutativity of the diagram are relevant here, as they help justify generalizations from highly abstract models. We have simplified tremendously in our approach, but have identified components in our model necessary to produce ecologically plausible behavior. Our genetically explicit ecosystem model, while far from nature, helps identify principles and mechanisms for assembly of many-species communities, and ways to refine our modeling approach.

Acknowledgements The authors are indebted to J.H. Brown, S. Forrest, S. Fraser, J.H. Holland, T. Jones, and A. Kosoresow for stimulating conversations about Echo. This research has been done at the Santa Fe Institute with the support of G r a n t No. N00014-951-1000 from the Office of Naval Research, acting in cooperation with the Defense Advanced Research Projects Agency. Sevilleta L T E R publication no. 97.

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