The tradeoff between division of labor and robustness in complex, adaptive systems is shaped by environmental stability Authors: Cole Mathis1, Yu Liu2, José Aguilar-Rodríguez3,4, Stojan Davidovic5, Rohan Mehta6, Emília Garcia-Casademont7, Zhi Qiao 8, Ali Kharrazi9, Renske Vroomans10, Sean M. Gibbons11,12,* Author Affiliations: 1 Physics Department, Arizona State University, U.S.A. 2 Mathematics Department, Uppsala University, Sweden 3 University of Zurich, Institute of Evolutionary Biology and Environmental Studies, Zurich, Switzerland 4 Swiss Institute of Bioinformatics, Lausanne, Switzerland 5 Center for Adaptive Behavior and Cognition, Max Plank Institute, Germany 6 Biology Department, Stanford University, U.S.A. 7 Institute of Evolutionary Biology, CSIC-UPF, Barcelona, Spain 8 NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 9 Graduate School of Public Policy, University of Tokyo, Japan 10 Theoretical Biology Group, Utrecht University, Holland 11 Graduate Program in Biophysical Sciences, University of Chicago, U.S.A. 12 Institute for Genomics and Systems Biology, Argonne National Laboratory, U.S.A. *Corresponding author: sean.gibbons@fulbrightmail.org Abstract We propose the existence of a fundamental tradeoff between division of labor (i.e. cooperation, lack of redundancy, or interconnectedness) and robustness in complex, adaptive systems. It is unlikely that a universal optimum for this tradeoff exists, and we suggest that context-dependent optima are likely tuned by the stability of the environment in which a system evolves. Here, we describe the mechanistic basis for this tradeoff in both natural and social systems, and show how these systems should self-organize under differing levels of environmental noise. We use agent-based modeling to explore the way simple, generalized systems optimize along this tradeoff curve under different perturbation regimes. We then investigate real-world data sets from qualitatively different environments, with differing levels of stability. We also look at more quantitative relationships between environmental disturbances and biological network structure, using microbial mesocosms under controlled laboratory conditions. Finally, we compare structural properties of natural systems to our model outputs. Overall, we find that complex networks maintain a high degree of connectivity under a large range of environmental variability, but the prevalence of high-degree nodes (agents/organisms with large numbers of connections) drops to zero above a noise threshold.

I. Introduction

Division of labor is a common emergent property in complex systems, and is inversely

proportional to the self-sufficiency of the actors. This phenomenon is driven by

individual and systems-level cost reduction, leading to improved efficiency; for example,

partitioning tasks minimizes the costs associated with task switching (1) and loss of

individual traits reduces the cost of maintaining expensive genetic or metabolic

infrastructure (2). However, the more entities rely upon their surroundings, the more

vulnerable they are to fluctuations in the environment. On one end of the tradeoff

spectrum, a system is minimally redundant and maximally efficient, but very sensitive to

perturbations. On the other end, a system is highly redundant/inefficient, but robust to

disturbances. Thus, when controlling for the size of a network (number of nodes), two

parameters govern the placement of a system in this space: 1) the magnitude of efficiency

gain achieved by labor division, and 2) the rate at which perturbations disrupt the system

and overwhelm the deterministic progression towards higher efficiency. In this report, we

explore this cost curve in different types of environments, ranging from stable to noisy

(Fig. 1).

We hypothesize that natural systems will evolve to maximize their efficiency up

to a fragility threshold, and that this threshold will be determined by the level of

background noise in the environment. In this paper we elaborate on the nature of this

fundamental tradeoff in a few distinct types of complex systems, then we describe a

generalized model for this tradeoff and compare the results of in silico experiments with

data collected from naturally occurring microbial communities and microbial mesocosms

in controlled laboratory studies.

Figure 1. Putative relationship between division of labor and environmental noise, based on the proposed tradeoff.

Biological Systems

Metabolic Networks

Free-living organisms are sustained by metabolic networks: complex chemical reaction

networks that contain hundreds to thousands of reactions catalyzed by protein enzymes

(3). These networks convert nutrients into biomass or energy and produce chemicals that

play a role in different biological processes: energy storage, defense, communication, etc.

(4). At present, in modern living systems, there is an “efficient division of metabolic

labor between enzymes” (5). Enzymes tend to be highly specialized in catalyzing a given

reaction. However, it is very likely that the ancestral metabolism was different. In 1976,

Jensen proposed that primitive enzymes had much broader specificities and were not

specialized to one substrate or reaction (6). Therefore, a relatively few generalist enzymes

of low efficiency, acting on several different substrates, were enough to confer a wide

Environmental Noise

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range of metabolic capabilities. If we measure the efficiency of a metabolism in terms of

the flux of matter trough a metabolic network or in terms of the cell growth that it can

sustain (7), the primordial metabolism envisioned by Jensen was not very efficient.

However, we can hypothesize that the ancient metabolism was robust to genetic

perturbations. In a metabolism where enzymes tend to be multifunctional, the chance that

an enzyme could compensate for the activity of another knocked-out enzyme is higher

than in a metabolism where all enzymes are highly specific. It is known that in

Escherichia coli, the overexpression of promiscuous enzymes can compensate for the

loss of otherwise essential enzymes (8). If it were not for gene duplication, primordial

enzymes would have been trapped in a situation in which mutational improvements in the

specificity (and efficiency) of a reaction would diminish the specificity of the other

promiscuous activities catalyzed by the same enzyme (6, 9). However, gene duplication,

followed by the sub-functionalization of the two copies of the enzyme (10), allowed the

selective divergence of generalist enzymes into the current specialist enzymes and

increased metabolic efficiency.

Therefore, a tradeoff between division of metabolic labor and robustness could

have shaped the evolution of metabolism. The predictions of this hypothesis remain to be

tested empirically. For example, given that even in modern metabolism we find specialist

and generalists enzymes (11, 12), the fraction of promiscuous or generalist enzymes

should be higher in less efficient but more robust metabolic networks. A theoretical study

has shown how metabolic networks that evolve in fluctuating environments acquire

greater robustness to single-gene deletions than networks evolved in stable environments

(12). Networks computationally evolved in changing environments become more robust

partially through the acquisition of multifunctional enzymes. This metabolic study seems

to support our thesis that the tradeoff between division of labor and robustness in

complex networks is controlled by environmental pressure.

Ecology and Evolution

In evolutionary biology, the rise of cooperation has been a controversial topic due to the

implication of group-level selection (13, 14). Equivocal evidence has been found for

strong group selection in biological systems (15, 16). In order to circumvent group

selection, kin selection (cooperation between related entities) has often been invoked as a

selfish mechanism for altruistic behavior between similar 'types' (17). However,

cooperation often occurs between unrelated organisms in real-world ecosystems (e.g.

flowering plants and their pollinators, Egyptian plovers and crocodiles, mycorrhizal fungi

and land-plants). A recent theory in ecology called the Black Queen Hypothesis (BQH)

proposed a novel ecological mechanism for how interdependencies may evolve between

unrelated entities due to adaptive trait loss in the presence of 'leaky' ecological functions

(2, 18, 19). The BQH (18) describes the evolution of functional dependencies in

microbial ecosystems through adaptive gene loss. The BQH relies on two basic

assumptions: 1) organisms unavoidably produce public goods (e.g. the excretion of

metabolic by-products); and 2) organisms that make use of these public goods will delete

their own costly metabolic infrastructure needed for producing them (19). This results in

a subset of organisms getting stuck with providing necessary services for the rest of the

community. Organisms that can outsource essential functions escape the cost of

performing these activities and expand the number of resources available for their growth

and reproduction. Although this process of adaptive gene loss starts out as selfish, and

may appear unstable, it has been shown to lead to long-term ecotype co-existence (20).

However, this process should make the ecosystem more delicate, as the destruction of

key species will eliminate their crucial service(s) and potentially lead to system collapse

(19).

Figure 2. Panel A: the largest connected component (bold red edges) grows as the number of randomly placed edges is increased in a graph with 150 nodes (p = percentage of all possible non-redundant connections between nodes). Panel B shows a critical transition in the size of the largest component when p = 0.009.

Apart from the mechanism by which division of labor evolves, the degree to

which division of labor is possible in the face of environmental variability is still an open

question. There is work in many fields showing that the connectivity of a network is

inversely proportional to network resilience, and that different network architectures are

optimal for maintaining maximal connectivity and resilience in an unpredictable world

(21). Ecological networks have been shown to be ‘robust, yet fragile’, meaning that they

are resistant to random shocks, but vulnerable to targeted attacks on key species (22, 23).

This suggests that the evolved network structure can support a rather large number of

connections before there is a significant reduction in robustness. This type of effect is

A. B.

reminiscent of percolation theory (24), and the growth of giant components in random

graphs (Fig. 2). In fact, most ecological networks are highly connected (25) and have

likely evolved to be adjacent to such a tradeoff threshold (26), allowing them to achieve

high levels of efficiency without incurring a large cost to their robustness.

Social Systems

Economics

Issues of competition and cooperation have been particularly important in economics.

Similar to the role of selfish gene in biological evolution, the founder of modern

economics, Adam Smith (1776), identified self-interest as the key generator of a

functioning economic system. From that point on economics has become, primarily, a

study of competition among selfish individuals. However, Smith recognized the

importance of division of labor and its benefits for the economy. He observed that

division of labor leads to qualitative increases in productivity, which naturally emerge

from economic interactions. Smith marked avoidance of task switching costs and

development of expertise as main reasons for improvement in labor productivity. Smith’s

arguments have at least two direct parallels in biology. First, recent evidence supports the

minimization of task switching costs in the evolution of division of labor in natural

systems (1). Another analog to biology, which sparked significant attention in economics,

is Smith’s observation that the power of division of labor is limited by the size of the

market (27-29). In biological systems, it has been shown that division of labor emerges

only when group (colony) size exceeds a certain point (30). This mirrors established work

in population genetics and recent ecological theory, which shows selective (deterministic)

forces (i.e. propensity of a system to evolve towards higher efficiency) are overwhelmed

by stochastic effects below a certain threshold in the system’s size (31, 32). This scaling

idea also relates to the economic literature, which has considered division of labor as the

main force behind economies of scale (33, 34).

More recently economists have devoted considerable thought to questioning

whether or not the integration of international markets leads to more specialization (35).

Even though division of labor has been recognized as a factor in market structuring and

increasing interconnectedness, there was little evidence about its impact on stability of

the markets. The financial crisis of 2007-2009 is an extreme example for how

specialization is a very important factor in the organization of global economy. This crisis

has revealed how entire economies depend on particular functions provided by a few

specialized financial institutions (36). The failure of a specialized, but essential, function

in a financial system - like interbank markets - can lead to the collapse of the global

economy (37).

II. Results and Discussion

Model and Results

We define a kinetic agent-based model to explore the effects of environmental

noise and perturbations on network division of labor and robustness. We assume an

initially well-mixed population of interacting agents. Each agent has three vector

properties: 𝑃 is the resource production capability of an agent, 𝑁 represents the

consumption needs of the agent (both normalized to unity), and 𝑅, which represents the

resources currently in the agent’s possession. There is a fixed rate of raw material influx,

which is uniformly distributed amongst the agents. Agents produce resources from this

raw material based on their production capabilities at a rate 𝑘!. Agents have a

consumption rate, 𝑘! ,  which defines the minimum amount of resources they require to

sustain function and increase fitness. When an agent has more resources than its

consumption needs (𝑅! > 𝑘!𝑁! for all i), it consumes its resources first to maintain itself,

and then to increase its fitness (f). After agents maximize their own fitness, (i.e. 𝑅! < 𝑘!𝑁!

for any i), they have a certain probability of donating any excess resources they have to

other agents who need that resource. This donation is biased by the history of the agent,

where they are more likely to donate to agents that have received resources from them in

the past, and less likely to give it to any other agent. Thus, as the system evolves, it

transitions from ‘well-mixed’ to ‘spatially-distributed,’ or from homogenous to

heterogeneous.

When agents are unable to meet their consumption needs, they may be removed

from the system. The probability of being removed from the system is proportional to

time since the agent has met its consumption needs. New agents are added to the model

as mutated copies of the fittest individuals. The propensity of an agent being added to the

system is inversely proportional the death rate, thus the total number of agents in the

system is free to vary and typically fluctuates. Agents that are more specialized in their

production (i.e., produce one resource more than others) have a lower consumption rate

than other agents, which reflects the fact that they have a lower ‘genetic overhead’ to

maintain. When an exchange occurs, there is a certain probability that it will be lost from

the system, and we refer to this probability as environmental noise, which is simply due

to small-scale stochastic fluctuations. There can also be externally applied perturbations

in which a fraction of the population is removed from the system. These shocks can be

random, harming all agents indiscriminately, or targeted harming only those agents

whose are specialized in production of a specific resource.

It is important to note that this is an open system with a constant flux of energy

coming in (energy used to manufacture resources), so that mean fitness is always

increasing within the system. However, energy flux is fixed in this model, meaning that if

more agents are added, then every agent produces fewer resources in a fixed time

interval. We investigated the case where all agents produced resources from the

environment at a uniform rate, 𝑘! = 1, for all agents. However 𝑘! was a linearly

decreasing function in the agent’s specialization, so more specialized agents require

fewer resources to sustain themselves.

Our model results indicate that the degree of random environmental noise has

little impact on the network structure of the community (i.e., the proportion of generalists

vs. specialists), up to a critical threshold (Fig. 3). Beyond this threshold, the system loses

its highest-degree nodes (i.e., ‘hub’ species, with large numbers of connections; Fig 3E).

Even at the highest noise level tested, the largest connected sub-graph spans almost the

entire network. Thus, we find that in our model system, rather than a phase transition in

the size of the giant component (e.g. Fig. 2), we see a switch in the presence or absence

of highly connected hub-species at high levels of environmental noise. These results,

however, are rather tenuous, as they are based on a small number of simulations. We will

reinforce these results by running replicate simulations, with a denser sampling of

parameter space.

Figure 3. Agent-based model results. Panels A-D show the network structures for model simulations under varying levels of noise (noise defined as probability that a resource is lost during an exchange between actors: A = 30%, B = 40%, C = 60%, and D = 80%). Colors denote the three different resource types included in the model. Bar graphs show degree distribution for each network. Panel E shows the magnitude of the highest degree node as environmental noise increases. The dashed line is not a fit, but a qualitative estimate of the relationship between highest degree and environmental noise. This qualitative relationship is very preliminary, and will require many replicate simulations with a denser sampling of parameter space.

Microbial Time Series Analysis

We investigated the relationship between ambient environmental noise levels and the

network structure of microbial interactions by analyzing time series data from four

qualitatively different natural environments: human skin, human feces, freshwater lake,

and the English Channel. Direct or indirect species-species interactions were estimated by

A. C.

B. D.

Environmental Variability

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10

20

30

40

30 40 50 60 70 80

E. degree degree

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calculating significant pairwise correlations between Operational Taxonomic Units

(OTUs, i.e., ‘species’; see Methods). Networks were then built with OTUs as nodes and

correlations as edges (Fig. 3A-D; inset plots). We hypothesized that human skin would

be the most variable environment due to the behavior of the host (i.e., showering,

application of lotion, sweating, exposure to sunlight, etc.) and the ocean would be the

most stable. Further, we suspected that the fecal and lake data sets would be intermediate

to these two extremes. Analysis of the correlation structure of these environments showed

that the ocean environment had the largest proportion of high-degree nodes (OTUs with

many connections), while the skin environment had a degree distribution weighted

toward lower-degree nodes (Fig. 3).

Overall, our correlation matrices show a high degree of connectivity across all

environments, but edges are more evenly distributed across nodes in noisier environments

(Fig. 3). This result mirrors prior work in ecology, showing that ecological networks are

highly connected (25, 38). However, our results indicate that individual organisms with

large numbers of connections are less favored in noisy environments. The loss of these

species could have a disproportionately negative effect on the food web (22, 23, 25, 39).

Reliance of the ecological network on these hub species might be too costly in a highly

variable environment, with greater risks of stochastic extinctions in noisy systems. On the

other hand, these highly connected organisms likely increase the efficiency, productivity,

and functionality of stable ecosystems, which is why they should arise under permissive

conditions (39-41). Individual connections with a hub species may be subject to different

tradeoff optima, where higher gains in efficiency may tolerate stronger fluctuations than

minor gains. Further, the proportion of obligate vs. facultative interrelationships between

taxa would also determine a network’s ability to restructure itself as noise levels change.

Figure 3. Histograms showing degree distributions for correlation networks from four different environmental time series: human skin (A); human feces (B); freshwater lake (C); and the English Channel (D). Network structures derived from the environmental correlation matrices are shown in each panel.

Microbial Mesocosm Experiment

We ran a series of experiments on replicate microbial mesocosms in order to explicitly

test how perturbations influence microbial correlation network structure. Replicate

communities, inoculated from the same source enrichment, were grown up in 96 well

plates. The control communities receive no applied disturbance for 40 days, while

disturbance replicates were subjected to different regimes of periodic disturbance

(ranging in magnitude: low, medium, and high; and in type: biomass removal and

ultraviolet radiation). At the end of the experiment, the control replicates were subjected

coun

t co

unt

degree degree

A. B.

C. D.

to a single point disturbance (heat shock at 50° C for 2 hrs). Independent of whether the

disturbance was applied periodically over the growth period or as a single high-intensity

disturbance at the end of the experiment, and independent of the type or magnitude of

disturbance, the effect was the same (Fig. 4). In the control community, the highest

degree node had 19 connections, and the average ‘disturbed’ community’s highest degree

node had only 8.2 connections (± 1.3, SE). Thus, our experimental results mirror our

environmental data, showing that high levels of environmental noise reduce the number

of high-degree nodes in an ecological interaction network. Interestingly, the largest

connected component in all of these treatments spanned almost the entire network,

showing again that these ecological networks are highly connected, and that the degree

distribution is the primary network measure constrained by environmental noise.

Figure 4. Bar plot showing the highest degree node from OTU correlation networks derived from mesocosm treatment replicate treatments. For the control treatments, ‘before’ and ‘after’ refer to before and after the application of a single point disturbance (heat shock at 50° C for 2 hrs). For the remaining biomass removal (BR) and ultraviolet (UV) treatments, ‘low’, ‘medium’, and ‘high’ refer to the magnitudes of periodically applied disturbance treatments. Dotted line shows average value over all disturbance replicates (excluding the ‘before’ control).

III. Conclusion

Our original hypotheses were that a fundamental tradeoff between division of labor and

system robustness exists, and that this tradeoff is tuned to the amount of environmental

variability that a system experiences during its development. It is clear from the literature

that such a tradeoff exists across a number of systems (e.g. social, biological, physical),

and our model simulations and empirical results verify that this tradeoff is tuned to the

level of environmental variance that a system is exposed to. Specifically, we show the

occurrence of high-degree nodes (i.e., hubs) in a network has a non-linear dependence

upon the level of environmental noise. Contrary to our original hypothesis, biological

networks - and the networks produced by our model - are able to maintain high overall

connectivity under a large range of environmental conditions. Thus, the size of the giant

component (largest connected sub-graph, normalized to the number of nodes in the

system) does not change. However, beyond an upper environmental noise threshold, the

frequency of high degree nodes quickly drops to zero. Therefore, we find that natural

systems are able to maintain a large amount of low-degree of connectivity independent of

environmental fluctuations, but highly specialized entities with many connections will

not persist in noisy environments. Future work will focus on in-depth characterization of

the seemingly non-linear transition in the degree distribution as a function environmental

noise. Our results can be applied broadly to many different fields involving complex,

adaptive systems: e.g., environmental conservation (i.e. conservation of keystone species

in the face of climate change), evolutionary theory (i.e. efficiency, robustness, and

evolvability in noisy environments), metabolic engineering (i.e. generalist vs. specialist

enzymes in different environmental regimes), or policy making (i.e. division of labor and

robustness in the economies of stable vs. war-torn countries). We hope our work will help

spark new avenues of research focused on identifying general laws that shape complex,

adaptive systems.

IV. Methods

Models

Model code is available upon request from cole.mathis@asu.edu and supplemental model

code is available upon request from yu.liu@math.uu.se.  

Microbial data analysis

Microbial time series data were obtained from the Earth Microbiome Project (EMP) data

portal: www.microbio.me/emp. For each data set, we ran closed reference OTU picking

(≥ 97% sequence identity) against the Greengenes database (May, 2013 release) to build

the phylotype-abundance matrix, with default setting in the Quantitative Insights into

Microbial Ecology (QIIME) analysis package (42-44). Taxonomic identities were

determined using the RDP classifier (45). Only abundant taxa were retained for each

analysis (at least 0.1% relative abundance across a data set). OTU matrices containing

abundant taxa were normalized to an even sequence depth across samples. The resulting

matrices were then processed using SparCC, to calculate all pairwise linear correlations

between OTUs across the time series (46). Correlation matrices were filtered to only

include significant correlations (p < 0.05) with rho-values ≥ 0.3. This filtered correlation

matrix was then used to construct a network, where each OTU represents a node and each

correlation an edge. Network statistics were then calculated using the NetworkX package

in python, and plotting was done using Matplotlib (47, 48).

Mesocosm Experiments

Microbial communities were enriched from the surface water of Lake Ontario, and grown

on BG11 medium (49) for several months. Initially, cyclohexamide was added to the

medium (0.5 µg mL-1) to prevent the growth of eukaryotes. Mesocosm experiments were

inoculated into 96-well plates from this enrichment culture (1:1 culture to fresh medium),

and grown over the course of a month with varying regimes of disturbance. Biomass

removal disturbances were applied by removing a percentage of the culture from a well

(5%, 10%, or 15%) and replacing it with an equivalent volume of fresh medium. UV

disturbances were applied by allowing wells to sit uncovered beneath a UV sterilization

lamp for varying lengths of time (5, 10, or 15 minutes; G15T8 bulb; emission peak at 254

nm, 5 W). Samples were harvested by spinning down cells for 15 minutes at 5,000 RCUs.

DNA extractions and PCR amplification of the V4 region of the 16S gene were carried

out using EMP standard protocols (http://www.earthmicrobiome.org/emp-standard-

protocols/). Samples were sequenced at the Argonne National Laboratory Illumina

sequencing facility. Data was processed using QIIME and SparCC, as described in the

previous section.

V. Acknowledgements

This study was made possible by the 2014 Complex Systems Summer School (CSSS) at

the Santa Fe Institute, NM, USA. Thanks to the Santa Fe Institute faculty and staff for

helpful advice and discussion. Special thanks to the CSSS 2014 summer school

organizers and to our fellow attendees for an inspirational month of interdisciplinary

science.

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VII. Supplemental Materials An Alternative Model Introduction Specialization and division of labor are everywhere in nature. As described above,

division of labor can arise when species lose their ability to perform a particular function

and must rely on their environment to supplement this deficit (BQH).

In this model, we assume that all the individuals have the same needs to survive

(e.g. 1 A + 1 B + 1 C) and these needs never change. However, the ability to gather a

given resource (A, B, or C) can mutate.

General Description of the Model

This model is agent-based, and does not take spatial effects into account. Every

individual has a 'genome' which can be inherited and mutated. The genome records an

agent’s ability to gather resources, and its consumption rate for each resource. Each agent

is able to store a certain number of unused resources, which can be exchanged between

individuals. At every time step, individuals have the opportunity to trade their inventory

with other agents. The rule for trading is that both individuals attempt to maximize their

lifespans by asking for resources that they need to increase their fitness, but neither party

can lose out in the trade (neither agent comes away from the trade with lower fitness). In

addition, every time step, individuals have a set of probabilities for gathering resources,

reproducing (with every reproduction event there is a probability of mutating the resource

production rates of the offspring) or dying. If an individual runs out of a limiting resource

then it dies.

Definitions

We define the stability of the system as the magnitude of its response to an external

perturbation. For the same perturbation, a larger response denotes lower stability.

Specifically, we consider the change in population size as the response variable. There

are two kinds of perturbation we will investigate: 1) perturbation of resource inputs, and

2) perturbation of species abundances (i.e., mortality rate). Interestingly, these

perturbation types could have different effects. We consider two systems: one uniquely

composed of self-sufficient generalists, called system X, another uniquely composed of

interdependent specialists, called system Y.

Resource input perturbations

In both systems X and Y, if we decrease resource A input by 10%, the size of the entire

population will drop by 10%, rendering 10% of resources B and C unusable, as each

agent needs an equal amount of each resource. The effect of this disturbance is the same

whether or not the system is comprised of generalist or specialists.

Species perturbation

In system X, if we remove 10% of the individuals, the size of the whole population

decreases by 10%. However, in system Y, if we remove 10% (with respect to the whole

population) of only specialists that gather resource A (i.e. a ‘targeted attack’), other

specialists who gather B and C will also decrease in abundance by 10%, as they require

A, which can only be obtained through trading with the A-gathering specialists. Thus, the

overall population size of system Y will decline by 30%. Therefore, the specialist-

community is more fragile than the generalist-community when faced with targeted

attacks (targeted attacks are not possible in a generalist system, where all agents are

identical).

Results and Discussion

Perfect specialization will evolve if 1) specialists are more efficient at gathering their

resource than a generalist, and 2) the whole system has a carrying capacity, which means

it cannot support infinite number of individuals (carrying capacity depends on resource

inputs, physical dimension, etc.). Thus, specialization will reduce the maintenance cost of

individuals, and increase the overall carrying capacity of the system. However, this

process may be disrupted by perturbations.

Figure S1. Shows the progression of our model system toward perfect specialization over time.

Figure S2 shows how more specialized systems are more sensitive to

disturbances. If we remove a certain number of individuals (denoted as m) from a non-

specialized system, this perturbation will not propagate any further throughout the

system, because the agents are self-sufficient and do not depend on one other. However,

if we kill m individuals in a perfectly specialized system (like in Fig. S2 at T = 250), it is

likely that more individuals would die afterwards since they are dependent on trading

with each other (however, if an equal number of each specialist type is removed, then the

perturbation will not propagate). To illustrate this point, we can see that in Figure S2, at

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T = 250, when we remove ~20 randomly chosen specialists, the total population size

drops by ~60 individuals (3-fold amplification of the original disturbance).

Figure S2. Following shocks, specialization continues to increase, and rarely decreases. Red vertical lines represent random species removal events.

We hypothesized that our model system would eventually stabilize at a certain

degree of specialization, which would depend on the noisiness of the environment. The

larger the number of perturbations, the lower the steady-state specialization level.

However, our simulations show that specialization almost always increases, and will

continue to increase until perfect specialization is attained. Large disturbances can crash

the system, or slow/pause the development of specialists, but as long as the system is not

destroyed it continues to advance towards perfect specialization (see Figure S3-S6).

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In summary, our model system always moves towards perfect specialization, but

perturbations will temporarily disrupt this process. The more specialized the system is

when perturbations occur, the longer the system needs to recover, but perturbations

almost never force specialization go backwards.

Figure S3. The system is collapsed by a large perturbation at T = 320.

Figure S4. The population size drops following a perturbation at T=300.

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Figure S5. High-frequency perturbations do not prevent the attainment of perfect specialization, although they do slow the process down.

Figure S6. Low-frequency perturbations do not slow the attainment of perfect specialization as much as high-frequency perturbations do.

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In the prior simulations, we only discussed systems where individuals can

efficiently exchange with each other. The exchange efficiency is determined by the

parameters ‘Trialpercent’ and ‘IntelligenceDegree’. If we set the two parameters to be

low, then there is a higher chance that an attempted exchange is aborted. This reduced

efficiency could be considered as another kind of perturbation (i.e. an exchange

perturbation). This exchange perturbation would be a major cost to specialization, and we

hypothesized that perfect specialization would be impossible in the presence of this

perturbation. However, simulation results seem to invalidate our initial hypothesis. The

higher the exchange perturbation, the more time is needed to reach perfect specialization,

but if the system evolves for infinite time then perfect specialization will always be

reached. Thus, in this model, we find that perfect specialization is the only global

equilibrium, although there may be several sub-equilibria.

Figures S1-S6 above have the parameters ‘Trialpercent = 1’ and

‘IntelligenceDegree = 60’, in most cases, perfect specialization can be reached before T =

500. However, if we set these parameters lower (see Figure S7), more time is required to

reach perfect specialization.

Figure S7. Trialpercent = 0.1 and IntelligenceDegree = 30. 1500 time steps are needed to reach perfect specialization. When keep Trialpercent as 0.1, and set IntelligenceDegree lower, e.g. 20 as

shown in Figure S8-S10, we can see that the degree of specialization is still low at T =

2000. However, given enough time, we predict that specialists will eventually displace

generalists.

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Figure S8. Trialpercent = 0.1 and IntelligenceDegree = 20. Perfect specialization is not yet achieved by 2000th time step.

Figure S9. Trialpercent = 0.1 and IntelligenceDegree = 10. Perfect specialization is not yet achieved by the 2000th time step.

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There are many reasons why real-world systems do not exhibit perfect

specializations. First, the time that the system has had to evolve may not be long enough

for it to have reached equilibrium. Second, highly specialized systems are very fragile,

and it is possible that they are destroyed regularly by stochastic perturbations and are thus

forced to restart the specialization process over and over again (i.e., an evolutionary

successional process). Third, heterogeneity in the consumptive needs and production

capabilities of evolving entities could give rise to partial specialization, as niche space

expands and the number of ways to make a living continuously grows. Finally,

environmental heterogeneity likely plays a more nuanced role in specialization,

preventing trading partners from interacting and imposing an adaptive cost on

specialization, but this simplified model is somehow not capturing this process.

Thus, in this model, perfect specialization will always happen if specialization is

adaptive. Specifically, selection pressure will be applied if the following conditions are

satisfied: specialization has higher efficiency, exchanges are possible, and resources are

limited. Different levels of environmental noise only affect the time needed to reach

perfect specialization.