Page 1

Catastrophic regime shifts in model ecological

communities are true phase transitions

J. A. Capit´ an‡ and J. A. Cuesta§

Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matem´ aticas, Escuela

Polit´ ecnica Superior, Universidad Carlos III de Madrid, E-28911 Legan´ es, Madrid, Spain

Abstract.

changes. These shifts are theoretically understood as a regime switch between alternative stable

states of the ecosystem dynamical response to smooth changes in external conditions.

models introduce nonlinearities in the macroscopic dynamics of the ecosystem that lead to different

stable attractors among which the shift takes place. Here we propose an alternative explanation of

catastrophic regime shifts based on a recent model that pictures ecological communities as systems

in continuous fluctuation, according to certain transition probabilities, between different micro-

states in the phase space of viable communities. We introduce a spontaneous extinction rate that

accounts for gradual changes in external conditions, and upon variations on this control parameter

the system undergoes a regime shift with similar features to those previously reported. Under our

microscopic viewpoint we recover the main results obtained in previous theoretical and empirical

work (anomalous variance, hysteresis cycles, trophic cascades). The model predicts a gradual loss

of species in trophic levels from bottom to top near the transition. But more importantly, the

spectral analysis of the transition probability matrix allows us to rigorously establish that we are

observing the fingerprints, in a finite size system, of a true phase transition driven by background

extinctions.

Ecosystems often undergo abrupt regime shifts in response to gradual external

Usual

Submitted to: J. Stat. Mech.

‡ jcapitan@math.uc3m.es

§ cuesta@math.uc3m.es

arXiv:1008.0335v2 [q-bio.PE] 7 Sep 2010

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Catastrophic regime shifts in model ecological communities are true phase transitions2

1. Introduction

Ecosystems are exposed to continuous changes in external conditions.

environmental conditions, climate oscillations, variations in the amount of resources and nutrient

loading, habitat fragmentation, harvest or loss of species diversity are a few examples of these

gradual changes. They often change slowly, even linearly, with time [1]. It is usually assumed

that the response of the system to external changes is smooth most of the times. However,

occasionally sudden changes can occur.For example, the sudden loss of transparency and

vegetation observed in shallow lakes due to human-induced effects [2]; corals overgrown by macro-

algae in the Caribbean reef seem to shift between two stable states rather than responding

smoothly to external conditions [3, 4]; or in savannahs, sparse trees with a grass layer can switch

to a dense woody state as a result of the alternation in fire and grazing regimes [5, 6]. All

these phenomena share the feature that ecosystems seem to change between two different stable

states. Sudden changes between two regimes are the so-called catastrophic shifts [7]. Hence, when

subjected to a slowly changing external control variable, ecological communities may show little

change until a critical point is reached. Then a sudden switch to a contrasting state can occur.

The simplest theoretical explanation to catastrophic regime shifts comes from the existence

of alternative stable states in the dynamical ecosystem response to gradual changes. The shift

between two alternative stable states is responsible for the transition.

of different stable states is associated to nonlinearities.

smooth external variations allows for the existence of alternative stable states [7, 8].

effects of nonlinearities have also been observed in natural communities. For example, it has

been established that the non-linear dynamics of overexploited marine ecosystems magnifies the

variability in the abundance of exploited species [9, 10].

In these models, ecosystems are described from a macroscopic viewpoint. Usually a global

magnitude, representative of the whole community, is used as fundamental variable (for instance,

the total biomass density). Models are basically devised to describe the time evolution of this

magnitude by means of non-linear functional responses [7, 11].

been applied to spatially extended interacting communities [12, 13]. This theoretical approach

is conceptually very similar to the traditional thermodynamic explanation of phase transitions

in physical systems, such as the liquid-vapor transition. The lack of convexity of theoretical

thermodynamic potentials —such as free energy— leads to alternative stable states corresponding

to liquid and vapor and both phases coexist below a certain critical temperature. In [11, 12, 13]

the analogy is very clear. The dynamics of biomass density follows a logistic growth with carrying

capacity K and a density-dependent consumption term modelled as a sigmoidal (Holling’s type III)

functional response. It is precisely this type of functional response which allows for the existence

of two separate, stable equilibrium points in the dynamics above a certain critical value of the

carrying capacity.

However, the current understanding of phase transitions in physical systems goes beyond

phenomenological, macroscopic models.The microscopic approach of Statistical Mechanics

represents a more fundamental way of understanding phase transitions, and many elaborated

Seasonal changes of

Often the existence

A non-linear ecosystem response to

The

Recently this paradigm has

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Catastrophic regime shifts in model ecological communities are true phase transitions3

theories have been developed to account for abrupt shifts in physical systems. Our aim in this

paper is to propose an alternative explanation, based mainly in a microscopic approach, to these

catastrophic regime shifts in ecosystems. In fact, we will show that a linear functional response in

the dynamics of individual species can lead to a global shift in the ecosystem between a species-rich

attractor and a state with low species richness. The early-warnings that are usually mentioned

as precursors of catastrophic regimes, like the increasing fluctuations near the transition [9, 14] or

the appearance of trophic cascades [15], will be recovered within our framework.

Based on an assembly model introduced recently [16, 17, 18], that pictures ecosystems

as complex, stable entities which keep on fluctuating between different micro-states through

successional invasions of rare species, we will introduce a background species extinction rate

accounting for the gradual, external variations to which natural communities are subjected. We

will find a threshold rate above which the system undergoes a phase transition, and the rate will

play the role of the shift control parameter. We will show that fluctuations become critical in

the vicinity of the transition, and that the ultimate collapse of the ecosystem correspond to a

gradual loss of species from bottom to top. It is worth remarking that trophic cascades have been

recognized as signals of overexploitation in marine communities [15].

The main feature of the assembly model presented in [16] is to provide a complete

characterization of the phase space of the system. The time evolution of the system is fully

described by means of a Markov chain. Under the effect of increased spontaneous extinctions, an

initially species-rich ecosystem can move to a region of the phase space where the species richness

decreases. Through a spectral analysis of the transition probability matrix we will be able to

show rigorously that the shift we observe corresponds to the “trace” of a true phase transition

—according to the definition of Statistical Mechanics— in finite size.

The paper is organized as follows. In Section 2 we will review the main features of the

assembly model and discuss the effect of a background species extinction rate in the ecosystem.

We will find that the Markov process is ergodic and that there is an increasing probability of

finding the ecosystem close to extinction as the rate grows. In Section 3 we will study some

signals with anomalous behavior near the shift, and show under which conditions hysteresis in

the average number of species arise. In Section 4 we will show that the observed behavior would

correspond to a true phase transition were the system infinitely large. So we can conclude that

what we are actually observing are the fingerprints of this transition in systems of finite size.

2. Background species extinction

Our starting point is a simple assembly model that we have recently introduced [16]. Let us

briefly summarize its main features. We describe ecological communities introducing three basic

simplifications.First, species are arranged in a finite number of trophic levels, and feeding

relationships take place exclusively between contiguous trophic levels. Second, the strength of

interactions is averaged over the whole community (mean-field), thus all species at the same

level are trophically equivalent. And third, population dynamics is modelled by Lotka-Volterra

equations. In spite of these oversimplifications, the resulting communities reproduce all results

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Catastrophic regime shifts in model ecological communities are true phase transitions4

previously found in earlier assembly models [16, 19, 20].

The mean-field assumption allows us to represent each community by the set of occupancy

numbers of each trophic level, {s?}L

of species at level ?. All species are considered as consumers, with uniform intrinsic average

mortality rate α. Predation between adjacent levels is modelled by constants γ+and γ−. The

former represents the per-capita rate of increase in population due to feeding, the latter accounts

for the mean damage caused by being predated. Communities are sustained by a primary, abiotic

resource characterized by its intrinsic growth rate R. This represents the saturation value that the

resource reaches at equilibrium in absence of predation, and measures the energy influx available

for each community. Species at the first trophic level predate on this resource and transfer the

energy upwards in the food web. Increasing R has the effect of allowing more trophic levels and

more species in each level as well (see [16]).

Population dynamics is ruled by Lotka-Volterra equations, which guarantees the existence

of a unique interior equilibrium point. Equilibrium communities whose populations are above of

a given extinction threshold nc are considered as viable. For a full account on the population

dynamics and the implications of the invasion dynamics in this model, we refer the reader to [17].

Our model represents a substantial improvement with respect to former assembly models in

that it provides a complete characterization of all possible invasion pathways within the set of

viable communities. Equivalently, an assembly graph G, i.e. the graph whose nodes are viable

communities and whose links connect communities through invasions, can be fully obtained under

these assumptions. The existence of a extinction threshold renders this graph finite. We assigned

to each link a certain transition probability dependent on an invasion rate ξ. The mean time

between consecutive invasions, ξ−1, is assumed to be large compared to the time that each

community needs to restore its dynamical equilibrium.

Weighting the links of G with probabilities defines a finite Markov chain over the set of viable

communities. In [18] we showed that this chain is aperiodic, hence communities can be either

transient or recurrent. We always found a single connected set of recurrent communities upon

increasing the resource saturation R. This set was either a single absorbing community or a more

complex end state of recurrent communities, the Markov process being ergodic over those complex

sets. The existence of complex end states is probably one of the most remarkable results of the

model (for a detailed account of results see [18]).

According to this picture, ecosystems evolve though successional invasions until reaching

a final end state, either a single absorbent community or a complex, closed set of recurrent

communities. When the process reaches a complex end state, successional invasions transform

the ecosystem into some other of the communities belonging to that set, so the process visits all

communities in this set, albeit with different frequencies —given by the asymptotic probability

distribution of the Markov process within this set. In terms of species this means that communities

keep continuously changing and eventually, after enough time has elapsed, all the original species

in the ecosystem will be replaced by new ones. Therefore the ecosystem keeps on fluctuating

between the different communities comprising the closed set, which persists as a robust, stable

entity.

?=1, L being the total number of levels and s? the number

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Catastrophic regime shifts in model ecological communities are true phase transitions5

Let us now introduce a rate of spontaneous extinctions. Species in natural communities are

often subject to overexploitation. Intensive hunting in terrestrial communities or the increasing

fishing pressure in marine ecosystems are good examples of this. Sometimes the species population

is seriously altered due to habitat destruction, in other cases due to exposure to epidemics or

diseases. Many effects like these ones can effectively decrease to critical levels the number of

individuals of a certain species or even cause its extinction. We will represent these situations by

means of a probability rate, η, which accounts for the probability per unit time for a species to

go extinct for reasons other than being eaten. Actually this probability rate should depend on

the species and its environment but, for the sake of simplicity, we will assume it uniform for all

species.

Our model is amenable to introduce background extinctions in a simple way. Elementary

processes in the original model for η = 0 were transitions between viable communities carried out

by single-species invasions. Now two different processes, either invasion or extinction, can connect

two communities. Thus, for a given community E with L trophic levels, we need to determine all

the possible transitions carried out by invasions and spontaneous extinctions at each level (and

by invasions at level L + 1 as well). The graph G will now contain both types of transitions.

The links corresponding to extinction transitions can be obtained just as the invasion ones [16].

Given a community E ∈ G, we randomly remove one of its species and calculate the equilibrium

population densities of the resulting community E?. If the community is viable, then we establish

a transition between E and E?. If some species go below the extinction level nc, then we apply

the same sequential extinction procedure that was followed to obtain the invasion graph (for a

detailed discussion of this procedure see [17]). We repeat these sequential extinctions until the

final community E??is viable.

The transition probability pij for the transition from community i to community j can be

written as

pij= δij+ ξqij+ ηuij,(1)

where δijis the Kronecker delta, and matrices Q = (qij) and U = (uij) account for the relative

frequency of invasions and extinctions, respectively. For i ?= j we set qij= nij/(L + 1), nijbeing

the number of different invasions of i that lead to j and L+1 the total number of possible invasions

of i, provided it has L trophic levels. For i ?= j we define uij= mij/Si, where mijis the number

of different extinctions in i that lead to j and Si=?L

yields

?

When η = 0 we recover the original transition matrix of our model (see [18]). This is quite a

singular case, though, not representative of what happens for any η > 0 —no matter how small.

In fact, there is a major difference between the cases η = 0 and η > 0, regarding the

properties of the Markov chain. For any η > 0, there is a non-zero probability for all the S species

in a community to go extinct. Let ∆t be the time unit between consecutive iterations of the

?=1s(i)

? is the number of possible extinctions

of i. We set uii= 0 and calculate the diagonal of Q so that P = (pij) is a stochastic matrix. This

qii= −

i?=j

?

qij+η

ξuij

?

.(2)

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Catastrophic regime shifts in model ecological communities are true phase transitions6

(a)

{0 0}

{1 0}

{2 0}

{1 1}

{2 1}

{1 2}

{2 2}

{1 3}

{2 3}

{1 4}

{2 4}

{1 5}

{2 5}

{1 6}

{2 6}

{3 6}

{2 7}

{3 7}

{4 7}

{3 3}

{4 3}

{3 4}

{3 2}

{4 2}

{3 1}

{4 1}

{3 0}

{4 0}

{5 0}

{6 0}

{5 1}

{6 1}

{5 2}

{6 2}

{5 3}

{6 3}

{5 4}

{6 4}

{4 4}

{5 5}

{6 5}

{4 5}

{5 6}

{6 6}

{4 6}

{5 7}

{6 7}

{7 7}

{7 6}

{7 3}

{8 3}

{7 4}

{7 2}

{8 2}

{7 1}

{8 1}

{7 0}

{8 0}

{9 0}

{10 0}

{9 1}

{10 1}

{9 2}{9 3}

{8 4}

{8 5}

{7 5}

{11 1}{11 0}

{12 0}

{13 0}

{14 0}{15 0}{16 0}

{17 0}

{18 0}

{19 0}

{20 0}

{3 5}

(b)

{0 0}

{1 0}

{2 0}

{1 1}

{2 1}

{1 2}

{2 2}

{1 3}

{2 3}

{1 4}

{2 4}

{1 5}

{2 5}

{1 6}

{2 6}

{3 6}

{2 7}

{3 7}

{4 7}

{3 3}

{4 3}

{3 4}

{3 2}

{4 2}

{3 1}

{4 1}

{3 0}

{4 0}

{5 0}

{6 0}

{5 1}

{6 1}

{5 2}

{6 2}

{5 3}

{6 3}

{5 4}

{6 4}

{4 4}

{5 5}

{6 5}

{4 5}

{5 6}

{6 6}

{4 6}

{5 7}

{6 7}

{7 7}

{7 6}

{7 3}

{8 3}

{7 4}

{7 2}

{8 2}

{7 1}

{8 1}

{7 0}

{8 0}

{9 0}

{10 0}

{9 1}

{10 1}

{9 2} {9 3}

{8 4}

{8 5}

{7 5}

{11 1}{11 0}

{12 0}

{13 0}

{14 0}{15 0}{16 0}

{17 0}

{18 0}

{19 0}

{20 0}

{3 5}

(c)

{0 0}

{1 0}

{2 0}

{1 1}

{2 1}

{1 2}

{2 2}

{1 3}

{2 3}

{1 4}

{2 4}

{1 5}

{2 5}

{1 6}

{2 6}

{3 6}

{2 7}

{3 7}

{4 7}

{3 3}

{4 3}

{3 4}

{3 2}

{4 2}

{3 1}

{4 1}

{3 0}

{4 0}

{5 0}

{6 0}

{5 1}

{6 1}

{5 2}

{6 2}

{5 3}

{6 3}

{5 4}

{6 4}

{4 4}

{5 5}

{6 5}

{4 5}

{5 6}

{6 6}

{4 6}

{5 7}

{6 7}

{7 7}

{7 6}

{7 3}

{8 3}

{7 4}

{7 2}

{8 2}

{7 1}

{8 1}

{7 0}

{8 0}

{9 0}

{10 0}

{9 1}

{10 1}

{9 2}{9 3}

{8 4}

{8 5}

{7 5}

{11 1}{11 0}

{12 0}

{13 0}

{14 0}{15 0}{16 0}

{17 0}

{18 0}

{19 0}

{20 0}

{3 5}

(d)

{0 0}

{1 0}

{2 0}

{1 1}

{2 1}

{1 2}

{2 2}

{1 3}

{2 3}

{1 4}

{2 4}

{1 5}

{2 5}

{1 6}

{2 6}

{3 6}

{2 7}

{3 7}

{4 7}

{3 3}

{4 3}

{3 4}

{3 2}

{4 2}

{3 1}

{4 1}

{3 0}

{4 0}

{5 0}

{6 0}

{5 1}

{6 1}

{5 2}

{6 2}

{5 3}

{6 3}

{5 4}

{6 4}

{4 4}

{5 5}

{6 5}

{4 5}

{5 6}

{6 6}

{4 6}

{5 7}

{6 7}

{7 7}

{7 6}

{7 3}

{8 3}

{7 4}

{7 2}

{8 2}

{7 1}

{8 1}

{7 0}

{8 0}

{9 0}

{10 0}

{9 1}

{10 1}

{9 2}{9 3}

{8 4}

{8 5}

{7 5}

{11 1}{11 0}

{12 0}

{13 0}

{14 0}{15 0}{16 0}

{17 0}

{18 0}

{19 0}

{20 0}

{3 5}

Figure 1. Assembly graph G obtained for R = 120, comprising 79 communities with up to 2

trophic levels, for 4 increasing values of η/ξ. Diameter of nodes is proportional to their asymptotic

probability. Labels of nodes represent species occupancies {s1,s2}. For the sake of clarity, only

transitions carried out through invasions are shown. For η = 0, the recurrent set contains 9

nodes (colored in blue). Red nodes represent transient states for η = 0. (a) The most probable

communities are those in the recurrent set (η/ξ = 0.05). (b) Some communities, close to this

set, are visited with high frequency (η/ξ = 0.3). (c) Almost none of the 9 originally recurrent

communities are visited (η/ξ = 0.6). (d) The most probable community corresponds to the total

extinction state (η/ξ = 1).

Markov chain. Thus the removal of all species caused by sequential spontaneous extinctions has a

probability at least equal to (η∆t)S. The non-vanishing probability of total extinction implies that

the process can return to the initial state (the empty community ∅) and therefore the Markov

chain becomes ergodic. This has to be compared with the former model (η = 0), for which we just

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Catastrophic regime shifts in model ecological communities are true phase transitions7

0.0

0.5

1.0

η/ξ

1.5

2.0

0.0

0.2

0.4

0.6

0.8

1.0

P

Pr

Pφ

Figure 2. Probability Prof finding the process in one of the communities of the recurrent set for

η = 0 (dashed line), and probability P∅of the empty community (full line), as functions of the

ratio η/ξ, for R = 1340.

found a tiny fraction of recurrent states, and almost all the communities in the assembly graph

were transient [16, 18].

Ergodicity implies that any possible state of the ecosystem can be reached with a non-zero

—albeit sometimes small— asymptotic probability. According to (1) we simply need to solve the

linear system

?

to obtain the (row) vector π of asymptotic probabilities πifor all i ∈ G. Therefore the asymptotic

distribution depends on the relative strength between rates. We expect that, when this ratio is

small enough, the subset of communities with the highest probability coincides with the recurrent

subset found for η = 0. However, as this ratio increases, the probability of finding the process

within this subset should decrease. This effect can be observed in Figure 1, where we plot G

with its asymptotic distribution for R = 120 and 4 values of the quotient η/ξ. The remaining

parameters of the model have been set as in our previous work: α = 1, γ+ = 0.5, γ− = 5,

ρ = 0.3 and nc= 1. (We will use this set of parameters throughout this paper.) As η/ξ increases,

communities that were recurrent for η = 0 are visited with decreasing asymptotic probabilities.

Eventually, when the ratio is large enough, these communities are hardly visited and the process

stays with high probability in communities close to the empty ecosystem, ∅.

This effect is more clearly seen in terms of the dependence of Pr—the probability of finding

the process in any of the communities of the recurrent set for η = 0— and P∅—the probability

of finding the ecosystem extinct— on η/ξ. A typical behavior of these probabilities is depicted in

0 = πQ +η

ξU

?

(3)

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Catastrophic regime shifts in model ecological communities are true phase transitions8

0.0 0.0

0

0.2 0.20.4 0.4

0.6 0.6

0.8 0.8

1.0 1.01.2 1.2

0.0

0.2

0.4

0.6

0.8

1.0

P

Pr

Pφ

1.4

η/ξ

60

120

180

240

300

S , σS

2

S

σS

2

η2/ξ

η1/ξ

Figure 3. Average number of species (black dashed line) and its variance (black full line) as

functions of η/ξ, for R = 1340. The average number of species decreases monotonically whereas

its variance exhibits a double peak at values η1/ξ ≈ 0.33 and η2/ξ ≈ 0.46. The first one coincides

with the abrupt drop of Pr(red dashed line) but the second one precedes the increase of P∅(red

full line) announcing it.

Figure 2. This plot corresponds to a resource saturation R = 1340, for which G has 397698 nodes

and 539 recurrent communities. In Figure 2 we can observe an abrupt decrease of Prat η/ξ ≈ 0.33,

and P∅increases abruptly as well when η/ξ ≈ 0.65. Needless to say, these two magnitudes resemble

the typical behavior of order parameters in the vicinity of a phase transition. A small increase in

η causes a shift from the stable, recurrent set at η = 0 to communities close to extinction. In this

sense, increasing background extinctions drive the system from a stable, species-rich attractor to

a species-poor region of the phase space. The system thus undergoes a catastrophic regime shift

analogous to those commonly observed in overexploited ecological communities [2]–[6].

3. Signals of catastrophic regime shifts

From the perspective of conservation and management of ecosystems, it is very important to

determine some signals that may alert of the proximity of a catastrophic transition. These are the

so-called early-warnings of catastrophic regime shifts [14], and act as flags for the approach of a

critical threshold. Although our model is minimalistic, the phenomenology of several magnitudes

reveal a critical behavior near the shift described in the previous section. Now, upon gradually

increasing the external “stress” (i.e., the ratio η/ξ) on our system, we will observe abrupt changes

in these magnitudes close to the regime shift.

We shall begin assuming very slow variations in η/ξ, i.e., the ecosystem undergoes very many

invasions before changes in the control parameter are noticeable. In this situation we can assume

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Catastrophic regime shifts in model ecological communities are true phase transitions9

0.0

0

0.0

0.5 0.5

1.0 1.0

η/ξ

1.51.5

2.02.0

30

60

90

120

150

σT

2/T2

0.0

0.2

0.4

0.6

0.8

1.0

Pφ

η3/ξ

Figure 4. Relative variance σ2

ecosystem (full line), compared to the probability of total extinction P∅(dashed line), for R = 1340.

The maximum of the relative variance roughly coincides with the point at which P∅ starts to

increase. This maximum is reached at η3/ξ ≈ 0.67.

T/T2of the first-return time (average T = P−1

∅) to the empty

that the community is always at is steady state. At the end of this section we will analyze the

effect of relaxing this assumption and allowing for a mixing of these two time-scales: the scale of

variation of the stress and the scale of invasion.

A first precursor of the shift is the fluctuation of the mean number of species in the

ecosystem. In [12] fluctuations were measured by the spatial heterogeneity of a single magnitude

(a biomass density) representing the community as a whole, whereas the kind of fluctuations we

are considering here are due to changes in the average number of species induced by invasions

and spontaneous extinctions. In Figure 3 we plot the average number of species S =?

variance

?

The rapid growth of fluctuations provides an alert of the proximity of the catastrophic shift

[7, 12, 14] Fluctuations for R = 1340 exhibit a double peak at η1≈ 0.33ξ and η2≈ 0.46ξ. The

first one is related to the abrupt drop of the probability Prthat we showed in Section 2. In addition,

these two peaks correspond to a gradual decrease in the number of species at each trophic level,

as we will show below.

The second peak at η2/ξ announces the abrupt increase of P∅, but does not coincides with it.

The increase of this probability is connected to the fluctuations of the time of first return to the

empty community, whose mean value is given by T = P−1

the Markov chain [21], we can calculate the relative variance σ2

i∈GπiSi,

where Siis the total species richness of the i-th community. Its fluctuations are measured by the

σ2

S=

i∈G

πiS2

i− S2. (4)

∅. Using the first-passage distribution of

T/T2(see Appendix for details) and

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Catastrophic regime shifts in model ecological communities are true phase transitions10

(a)

0 606030 30

(b)

0 60603030

(c)

0 606030 30

(d)

0 6060 3030

Figure 5. Ecosystem profiles (mean number of species in each trophic level) for increasing η/ξ

ratios. This plot corresponds to a resource saturation R = 1340, for which communities have up

to 4 trophic levels. Lower levels are shown with darker color. (a) The ecosystem maintain its

pyramidal structure (η/ξ = 0.05). (b) The first trophic level starts to collapse (η/ξ = 0.3). (c)

The second level starts to loose species (η/ξ = 0.43). (d) For large values of the ratio, the system

is close to extinction (η/ξ = 0.6).

the result is shown in Figure 4. The maximum relative fluctuation occurs nearly at the onset of

increase of P∅, η3≈ 0.67ξ. We thus expect that relative fluctuations in the average return time to

any state i close to ∅ will be amplified close to the extinction transition. This notwithstanding, it

is hard to figure out how this fluctuation could be used in practice as a signal of the catastrophe.

A second signal of the transition in this model is a gradual loss of species in trophic levels from

bottom to top. This effect can be qualitatively observed in the ecosystem profile (see Figure 5),

where the average number of species at each trophic level is shown. When the quotient of rates

increases from η/ξ = 0.3 [panel (b)] to η/ξ = 0.43 [panel (c)], the number of species in the first

level decreases considerably, but the rest of levels remain almost unaltered. After that [panel (d)],

a simultaneous loss of species in the first and second levels takes place.

Figure 6 shows the number of species at each level averaged over G versus η/ξ. We observe

that the decrease of s1approximately coincides with the first peak of σ2

s2corresponds to the second peak at η2. After that, species at lower levels are unable to sustain

upper levels and a trophic cascade occurs. The third and fourth levels start to be emptied near η3.

There is a clear correspondence between the values at which trophic levels start to collapse and

the location of the maxima of σ2

as the extinction rate increases is a clear signal of the catastrophic regime shift. Besides, trophic

cascades have been recognized empirically as signals of over-fishing in marine communities [15].

Sat η1, and the decrease of

Sand σ2

T/T2. In any case, the loss of species from bottom to top

Page 11

Catastrophic regime shifts in model ecological communities are true phase transitions11

0.0

0.5

1.0

η/ξ

1.5

2.0

0.0

0.2

0.4

0.6

0.8

1.0

sl/sl(0)

l = 1

l = 2

l = 3

l = 4

η3/ξη1/ξη2/ξ

Figure 6. Average number of species at each trophic level, s?=?

points η1and η2corresponding to the maxima of σ2

which the first and second levels start to collapse. The species loss in the third and fourth levels

starts when the probability P∅becomes appreciable (by η3, the maximum of σ2

i∈Gπis(i)

?, for ? = 1,2,3,4 and

R = 1340, normalized by the average number of species for η = 0, s?(0). We have marked the

S, which roughly coincide with the points at

T/T2).

In the remaining of the section we will consider the variation of η as a non-equilibrium process.

Now we shall assume that, although the variation of η is still not faster than ξ−1, the two scales

are comparable in the sense that the process is not able to remain in the steady state anymore.

An estimate of the time scale for the convergence to the steady state is provided by

1

log|λ2|,

where λ2is the second largest eigenvalue (in modulus) of the stochastic matrix P (see Figure 7).

The distance from the probability distribution after n iterations of the Markov chain to the steady

state is proportional to |λ2/λ1|n, hence the definition of tc(note that the maximum eigenvalue

λ1= 1 since P is a stochastic matrix). For R = 1340, the number of iterations needed to reach

equilibrium near the shift are around 103.

The faster variation of η is implemented by producing a small change ∆η every ∆n < ξtc

iterations of the Markov chain. We start by increasing η in these increments until reaching an

arbitrary value beyond the regime shift. Then we repeat the process by decreasing η in the same

increments. This way we can track any observable along the cycle, by computing its averages after

k = 0,1,... increments ∆η using the probability distribution

ξtc= −

(5)

π(k∆η/ξ) = π(0)P∆n(0)P∆n(∆η/ξ)···P∆n(k∆η/ξ),

given any initial distribution π(0) at η = 0 (reverse order in matrix products applies for decreasing

η). In Figure 8 the average species richness exhibits a hysteresis cycle. As ∆n increases this cycle

(6)

Page 12

Catastrophic regime shifts in model ecological communities are true phase transitions 12

0.00.2 0.4

0.6

0.8

1.01.2 1.4

η/ξ

200

400

600

800

1000

ξtc

η2/ξ

η1/ξ

Figure 7. Convergence time ξtc= −(log|λ2|)−1needed to reach the steady state of the Markov

chain, λ2being the second largest eigenvalue of matrix P. Two peaks appear in ξtccorresponding

to the two maxima of σ2

S.

narrows, recovering the quasi-stationary process in the limit ∆n → ∞. In this limit the process

is reversible and the cycle collapses to the curve of mean number of species shown in Figure 3.

The existence of hysteresis loops in overexploited systems has been reported in the literature

as another signal of catastrophic shifts in ecological communities [7, 12, 14]. We have obtained it

for the mean number of species, but a similar behavior will be observed in any other magnitude

(like the total biomass density, as in [12]). In spite of the simplicity of our model, hysteresis cycles

appear as well as other usual precursors of catastrophic regimes, such as anomalous variance.

But, unlike previous models, our model allows a deeper understanding of the transition, as we will

discuss below. An important insight this model provides is that hysteresis can arise as a result of

a time-scale mixing that keeps the system out of equilibrium rather than to non-linearities of the

underlying population model.

4. Phase transition in finite size

As we have mentioned, our model provides a full description of the phase space of the system by

means of a transition probability matrix. We are going to take advantage of this fact to show

rigorously that the phenomenology that we have described in Section 3 corresponds to a phase

transition —in the sense of Statistical Mechanics— in finite size.

In Statistical Mechanics, phase transitions are associated to non-analyticities of the free energy

of a physical system. Whenever the system is described by a transfer matrix, the free energy

is obtained from its largest eigenvalue. A true phase transition would then be associated to the

crossing of the leading eigenvalue with the second largest (in modulus) one, because such a crossing

Page 13

Catastrophic regime shifts in model ecological communities are true phase transitions13

0.0

0

0.2 0.4

0.6

η/ξ

0.8

1.0 1.2

50

100

150

200

S

∆n = 500

∆n = 200

∆n = 100

Figure 8. Hysteresis cycles for ∆η/ξ = 0.005 and three values of ∆n (see text). Dashed line

represents the average number of species in the quasi-stationary process of variation of η (for

∆n → ∞). The larger ∆n the further from equilibrium the system and the narrower the cycle.

causes a non-analytic behavior of the largest eigenvalue as a function of the control parameter

[22]. The counterpart of a transfer matrix in a Markov chain is the transition probability matrix

P. Thus a crossing of eigenvalues of P would rigorously prove that the system undergoes a phase

transition.

Strictly speaking, the described shift can not be a true phase transition because of the

finiteness of our system. According to the Perron-Frobenius theorem, an irreducible matrix†

with non-negative entries has a unique largest real eigenvalue and its corresponding eigenvector

has strictly positive components [23]. Reducible matrices are related to processes with transient

states. Since the Markov chain is ergodic for η > 0, its stochastic matrix P it is irreducible.

Then the theorem implies that its maximum eigenvalue λ1= 1 is simple for any value of η/ξ (its

corresponding positive left eigenvector is precisely the asymptotic probability distribution). This

excludes any eigenvalue crossing, therefore any phase transition. True phase transitions can only

occur in infinite states Markov chains. When the limiting chain of a sequence of finite Markov

chains develops an eigenvalue crossing —hence a phase transition— the second largest eigenvalue

of the elements of this sequence approaches the largest one near the location of the true phase

transition, the more so the larger the size (number of states) of the Markov chain. This is the

fingerprint of the phase transition in finite systems. It is also associated to a qualitative change

in the eigenvector associated to the leading eigenvalue.

We have computed the 10 largest eigenvalues of P (in modulus) using Arnoldi iteration [24]

† A matrix P is reducible if there exists a permutation matrix W such that WTPW =

associated chain is not ergodic.

?

X

Y

0

Z

?

. Hence the

Page 14

Catastrophic regime shifts in model ecological communities are true phase transitions14

0.0

0.5

1.0

η/ξ

1.5

2.0

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

(λ−1)/ξ

0.20.4

0.6

-0.010

-0.005

0.000

η2/ξη1/ξ

Figure 9. First (dashed line) and second (full line) eigenvalues of P for a resource saturation

R = 1340. Inset shows a zoom of the region for which λ1= 1 and λ2are closest, and two maxima

appear at η1/ξ = 0.330520... and η2/ξ = 0.462633.... These points coincide with the maxima

observed in σ2

Sand tc(see Figure 3 and Figure 7).

(useful for computing a few eigenvalues of large sparse matrices). In all cases the numerical

method provides a real second eigenvalue. Figure 9 shows the dependence of λ2as a function of

η/ξ for R = 1340. Not surprisingly, we identify the transition points as those of closest approach

to the first eigenvalue (i.e., the two maxima observed in λ2). Those maxima are reached at

η1/ξ = 0.330520... and η2/ξ = 0.462633..., which coincide with the values observed for the

peaks in σ2

cascade in the system, whose levels get emptied from bottom to top.

We have varied the system size (controlled by the amount of resource, R [18]) in order to

check that the second eigenvalue gets closer to the first one as size increases. The system size

is measured by the average number of species in the recurrent set for η = 0, and we do observe

that the larger the system size the closer is λ2to 1 (see Figure 10). It is difficult to increase the

size beyond S ≈ 300 because the number of nodes in G grows approximately as eκ√R[18]. For

S ≈ 300 the number of viable communities is larger than 106, and the eigenvalue computation

becomes very demanding.

To conclude this section, we have obtained the phase diagram of this system. In Figure 11

we show in a η/ξ vs. R diagram the points at which λ2reaches its maxima, yielding the transition

lines corresponding to the different trophic cascades of the system. Basically transition lines are

horizontal, and the number of undergone trophic cascades appears to be related to the number

of trophic levels of the communities. We believe that, at some point in the region in which

communities have 5 trophic levels, a third maximum would appear related to a separate trophic

cascade at the third level. Bearing this hypothesis out is difficult, though, because it would require

Sand tc (see Section 3). As we have shown before, each transition yields a trophic

Page 15

Catastrophic regime shifts in model ecological communities are true phase transitions15

0.20 0.30 0.40

η/ξ

0.50

0.60

-0.008

-0.006

-0.004

-0.002

0.000

(λ2−1)/ξ

203.3

175.4

146.9

121.0

66.2

Figure 10. Second eigenvalue of P for several values of R (legend shows the average number of

species for each R at η = 0). Dotted curve corresponds to R = 465, for which the system only

allows three trophic levels and only a single maximum is observed. For higher values of R (which

allow up to 4 levels), a second maximum appears, related to the trophic cascade in the second

level. As the system size increases, λ2gets closer to 1.

eigenvalue calculations with too large matrices.

All this analytical evidence allows us to claim that these model ecosystems undergo a

catastrophic transition driven by the relative extinction rate. The dependence of λ2 with η/ξ

actually exhibits two maxima and so does the variance in species number. This points towards

the existence of a double phase transition, each one associated to a trophic cascade that collapses

the lowest and next-to-lowest trophic level in the ecosystem. Increasing the external stress over

the system above these values increases the probability of driving the ecosystem to extinction.

Consequently there is a threshold in η/ξ above which the extinction of the ecosystem is the most

likely event. Small variations in that region can cause the collapse of stable ecosystems.

5. Conclusions

In this paper we have used a recently introduced model [16] to propose an alternative explanation

to the observed catastrophic regime shifts in overexploited ecosystems [7, 12]. Previous models

postulate nonlinearities in the dynamics of a magnitude representative of the whole ecosystem,

like for instance the total biomass density, and this non-linear behavior leads to bi-stability. The

system undergoes a transition due to a change of regime that leads the system from one stable state

to another. This viewpoint is almost the same used in the classical explanations of the liquid-vapor

transition in thermodynamics, for which the potential of the system has two alternative stable

states.

Our model, however, is inspired in a microscopic description, more related to the perspective

Page 16

Catastrophic regime shifts in model ecological communities are true phase transitions16

500

1000

R

1500

0.0

0.2

0.4

0.6

0.8

1.0

η/ξ

2nd cascade

1st cascade

L=3 L=4

deficient (1st level)

extinct

extant

Figure 11. Phase diagram of the system: black circles correspond to the abscissa of the first

maximum of λ2(trophic cascade at the first level), and red ones to the second maximum (trophic

cascade at the second level). Above the second transition, the process stays in communities close

to the extinct ecosystem.

of Statistical Mechanics. Our viable communities are micro-states in a finite phase space and

represent different states of the ecosystem. The transition between a species-rich attractor (the

recurrent set at η = 0) and an attractor with low number of species (communities close to the

empty ecosystem) is explained as the crossing of two eigenvalues of the transition matrix of

the Markov chain. This transition is driven by an increasing external force represented by a

background species extinction rate, that takes into account the overall mortality for reasons other

than predation (overexploitation, habitat destruction, epidemics...).

In spite of its being minimalistic, the model reproduces qualitatively the phenomenology

observed in overexploited ecosystems. We have studied for this model the behavior of several

early-warnings that announce the catastrophic shift, and we have found the same behavior as that

obtained both in previous models [7, 12] and empirically [14]. These features are shared by many

systems under the framework of the elementary catastrophe theory [25].

Catastrophes have characteristic fingerprints. Some of the standard flags of catastrophic

regimes are modality, anomalous variance and hysteresis [26]. These are precisely the signals we

find in our model. Our system is bimodal because it undergoes transitions between an attractor of

high species richness to another stable state with low species richness. Fluctuations in the mean

number of species exhibit peaks at the transition points, and take very large values compared to

their values far away from the transition. We thus have anomalous behavior of variances. And the

average number of species exhibits hysteresis cycles when the system is kept out of equilibrium,

as we have shown in Section 3. This explanation of hysteresis is alternative to the existence of

non-linearities in the population model, and points towards the speed of variation of the external

Page 17

Catastrophic regime shifts in model ecological communities are true phase transitions17

stress. The difference with the usual explanation is that in this case the ecosystem can recover its

initial state after releasing the external stress, provided that we wait long enough and that there

is availability of invaders.

The main advantage of our model is the microscopic description that it provides. The full

characterization of the phase space of the system has allowed us to show rigorously that the

system undergoes a true phase transition by computing the second eigenvalue of the transition

matrix, which gets closest to the first eigenvalue in the vicinity of the transition. This provides a

theoretical support to the critical behavior exhibited by magnitudes like the fluctuation of species

richness.

We have found evidence for a double phase transition in the system, associated to the gradual

loss of species from bottom to top. This effect is new to former models and could be used as early

warning for the catastrophic shift by monitoring the species abundance at low trophic levels

in overexploited ecosystems. Trophic cascades have been revealed as possible mechanisms of

catastrophic shifts in natural communities, though [15]. Thus, with the caveat that this is just a

very simplified picture of real communities, our analysis of this model predicts that overexploited

systems will begin to collapse first at lower levels.

Assuming a different extinction rate at each trophic level would be more realistic.

example, in overexploited marine ecosystems, the impact of fishing pressure is stronger in higher

trophic levels. A simple model like ours could shed light in determining whether a strong extinction

pressure in higher levels is more harmful than in lower levels. Refined versions of the model could

allow investigating this kind of effects.

Most theoretical explanations of the catastrophic phenomena observed in ecological systems

subject to high exploitation pressure rely on nonlinearities in the macroscopic dynamics of the

system. The take-home message of this paper is that a microscopic model like ours, based in

ecologically reasonable assumptions and in a simple, linear population dynamic models, can exhibit

the same phenomenology as non-linear, macroscopic models. This way, our approach can serve as

an alternative explanation of the catastrophic regime shifts observed in ecological communities.

For

Acknowledgements

This work is funded by projects MOSAICO, from Ministerio de Educaci´ on y Ciencia (Spain)

and MODELICO-CM, from Comunidad Aut´ onoma de Madrid (Spain). The first author also

acknowledges financial support through a contract from Consejer´ ıa de Educaci´ on of Comunidad

de Madrid and Fondo Social Europeo.

Appendix

This appendix is devoted to calculate the fluctuation of the average time of first return to the

empty ecosystem. Throughout this section f(n)

ij

starting form the state i the first entry to j occurs after n steps. The distribution {f(n)

as the first-passage distribution for the state j (in particular, {f(n)

will stand for the probability that in a process

ij} is known

jj} represents the distribution

Page 18

Catastrophic regime shifts in model ecological communities are true phase transitions18

of the recurrence times for j). This distribution is related to the probabilities p(n)

from i to j in exactly n steps [21] according to

?

Let us study the case i = j = ∅. We are interested in calculating the variance of the recurrence

time for the empty community, whose mean value is given by T =?∞

(f(0)

1

V (z)

and T = F?(1). It can be easily shown that the radius of convergence of V (z) is equal to 1, and

that V (1) diverges by definition. By imposing that F?(1) is finite we find that, near z = 1,

V (z) ≈ a(1 − z)−1and a = limn→∞p(n)

T = F?(1) = a−1= P−1

The variance of the recurrence time is obtained as

∞

?

In order to calculate F??(1), we need to obtain the next-to-leading (constant) term in the series

expansion of V (z) in powers of 1 − z, V (z) = P∅(1 − z)−1+ b + O(1 − z). Using (A.2) we get

F??(1) = 2b/P2

σ2

T

T2= 2b + P∅− 1.

To conclude with this calculation, we need to obtain a way to compute numerically the constant

b. This is an easy task, because

∞

?

Hence, in the limit z → 1,

∞

?

Therefore we simply need to iterate the matrix P and truncate the series up to certain error

tolerance to compute b.

ij of a transition

p(n)

ij=

n

ν=1

f(ν)

ijp(n−ν)

jj

. (A.1)

n=1nf(n)

∅∅. To this purpose

we introduce the generating functions V (z) =?∞

F(z) = 1 −

n=0p(n)

∅∅zn(p(0)

∅∅= 1) and F(z) =?∞

n=0f(n)

∅∅zn

∅∅= 0). Then (A.1) is equivalent [21] to

(A.2)

∅∅= P∅.Hence the mean recurrence time for ∅ is

∅.

σ2

T=

n=0

n2f(n)

∅∅− T2= F??(1) − F?(1)[1 − F?(1)].(A.3)

∅and

(A.4)

b + O(1 − z) = V (z) −

P∅

1 − z=

n=0

?

p(n)

∅∅− P∅

?

zn. (A.5)

b =

n=0

?

p(n)

∅∅− P∅

?

.(A.6)

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