Post on 05-Jul-2020
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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or
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
Hig
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Nod
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10
20
30
40
30 40 50 60 70 80
E. degree degree
coun
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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.