A peer-reviewed version of this preprint was published in PeerJ on 11February 2020.

View the peer-reviewed version (peerj.com/articles/8533), which is thepreferred citable publication unless you specifically need to cite this preprint.

Equihua M, Espinosa Aldama M, Gershenson C, López-Corona O, Munguía M,Pérez-Maqueo O, Ramírez-Carrillo E. 2020. Ecosystem antifragility: beyondintegrity and resilience. PeerJ 8:e8533 https://doi.org/10.7717/peerj.8533

Ecosystem antifragility: Beyond integrity and resilienceMiguel Equihua Zamora 1 , Mariana Espinosa 2 , Carlos Gershenson 3, 4, 5 , Oliver López-Corona Corresp., 4, 6, 7 , MarianaMunguia 8 , Octavio Pérez-Maqueo 1 , Elvia Ramírez-Carrillo 9

1 Red ambiente y sostenibilidad, Instituto de Ecología A.C, Xalapa, Veracruz, Mexico2 Doctorado en Ciencias Sociales y Humanidades, UAM-Cuajimalpa., cdmx, cdmx, Mexico3 IIMAS, Universidad Nacional Autónoma de México, cdmx, Mexico4 Centro de Ciencias de la Complejidad (C3), Universidad Nacional Autónoma de México, cdmx, cdmx, Mexico5 ITMO University, St. Petersburg, 199034, Russian Federation, St. Petersburg, Russia6 Cátedras CONACyT, Comisión Nacional para el Conocimiento y Uso de la Biodiversidad (CONABIO), cdmx, Mexico7 Red ambiente y sostenibilidad, Instituto de Ecología A.C., Xalapa, Veracruz, Mexico8 Comisión Nacional para el Conocimiento y Uso de la Biodiversidad (CONABIO), CDMX, Mexico9 Facultad de Psicología, Universidad Nacional Autónoma de México, cdmx, Mexico

Corresponding Author: Oliver López-CoronaEmail address: lopezoliverx@otrasenda.org

We review the concept of ecosystem resilience in its relation to ecosystem integrity froman information theory approach. We summarize the literature on the subject identifyingthree main narratives: ecosystem properties that enable them to be more resilient;ecosystem response to perturbations; and complexity. We also include original ideas withtheoretical and quantitative developments with application examples. The maincontribution is a new way to rethink resilience, that is mathematically formal and easy toevaluate heuristically in real-world applications: ecosystem antifragility. An ecosystem isantifragile if it benefits from environmental variability. Antifragility therefore goes beyondrobustness or resilience because while resilient/robust systems are merely perturbation-resistant, antifragile structures not only withstand stress but also benefit from it.

PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.27813v1 | CC BY 4.0 Open Access | rec: 21 Jun 2019, publ: 21 Jun 2019

Ecosystem Antifragility: Beyond Integrity1

and Resilience.2

M. Equihua2,*, M. Espinosa8,*, C.Gershenson3,5,6,*, O. Lopez-Corona1,2,3,*,3

M. Munguıa7,*, O. Perez-Maqueo2,*, and E. Ramırez-Carrillo4,*4

1Catedras CONACyT, Comision Nacional para el Conocimiento y Uso de la5

Biodiversidad (CONABIO), CDMX, Mexico6

2Red ambiente y sostenibilidad, Instituto de Ecologıa A.C., Xalapa, Mexico7

3Centro de Ciencias de la Complejidad (C3), Universidad Nacional Autonoma de8

Mexico, CDMX, Mexico9

4Facultad de Psicologıa, Universidad Nacional Autonoma de Mexico, CDMX, Mexico10

5IIMAS, Universidad Nacional Autonoma de Mexico, CDMX, Mexico11

6ITMO University, St. Petersburg, 199034, Russian Federation.12

7Comision Nacional para el Conocimiento y Uso de la Biodiversidad (CONABIO), CDMX,13

Mexico14

8Doctorado en Ciencias Sociales y Humanidades, UAM-Cuajimalpa.15

Corresponding author:16

Following the Hardy-Littleton rule, all authors will appear in alphabetical order. And all are17

corresponding authors∗18

Email address: equihuam@gmail.com; mariana.espinosa.aldama@gmail.com; cgg@unam.mx;19

olopez@conacyt.mx; mariana.munguia@conabio.gob.mx; octavio.maqueo@gmail.com;20

elviarc@gmail.com21

ABSTRACT22

We review the concept of ecosystem resilience in its relation to ecosystem integrity from an information

theory approach. We summarize the literature on the subject identifying three main narratives: ecosystem

properties that enable them to be more resilient; ecosystem response to perturbations; and complexity.

We also include original ideas with theoretical and quantitative developments with application examples.

The main contribution is a new way to rethink resilience, that is mathematically formal and easy to

evaluate heuristically in real-world applications: ecosystem antifragility. An ecosystem is antifragile if

it benefits from environmental variability. Antifragility therefore goes beyond robustness or resilience

because while resilient/robust systems are merely perturbation-resistant, antifragile structures not only

withstand stress but also benefit from it.

23

24

25

26

27

28

29

30

31

1 INTRODUCTION32

Sustainable development needs to preserve the structure and functioning of natural ecosystems, i.e. their33

integrity, as a Sine qua non condition. In previous work [25] an operational framework has been34

developed to quantify ecosystem integrity as well as viable standards useful for managing the way people35

intervene ecosystems, promoting development along sustainable avenues.36

Humans are starting to be recognized as an overwhelming forcing factor modulating biosphere37

dynamics. In this view, the Earth system can be interpreted as entering in a geological era that can38

be called the Anthropocene [99] the Technocene [62] or even Capitalocene [41], depending on the39

conceptual stance adopted. Because of this driving influence of human decisions, there has been long40

interest in understanding and measure the way ecosystems recover (or not) from human perturbations. To41

operationalize sustainability, we require working definitions of this ecosystem ability and metrics to asses42

it, in addition to ecosystem integrity (as a component of a likely state variable accounting for the amount43

of natural capital assets: condition× extension). The recovery property has already been encapsulated44

into the ecosystem resilience concept. Here, we propose that information theory is a suitable framework45

to encompass both ecosystem integrity and resilience.46

According to [25], numerous studies have aimed to find a suitable and inclusive definition of ecological47

integrity; however, no general consensus has been achieved to date. In their work, Equihua and co-48

workers embrace a complex systems approach in which ecosystems are considered as self-organized49

entities constrained in their structure (including biological composition or biodiversity), and function by50

thermodynamic dissipative system properties [52, 87, 67, 72] and evolutionary processes [61].51

The concept of ecosystem resilience was first introduced by Holling [46] to portray the persistence of52

natural structures in the presence of environmental stressors due to natural or anthropogenic triggers. In53

his seminal work, Holling follows a system dynamics analysis. There, resilience concept is defined as:54

“the capacity of a system to absorb disturbance and reorganize while undergoing a change so as to still55

retain essentially the same function, structure, identity, and feedbacks” [109]. Nevertheless, resilience56

or stability, which is commonly used as synonym, have at least 163 different definitions (grouped in at57

least 70 concepts of stability/resilience) [37]. For example, Saint-Beat and co-workers [91] summarize58

resilience and others related concepts often used interchangeably as follows:59

• Resilience is the rate at which a system returns after a disturbance to the equilibrium state [20,60

Pimm]. Long return time is equivalent to low resilience. A community’s resilience relies on the61

least resilient species (the slowest to return to equilibrium). This definition of resilience corresponds62

to the ‘engineering resilience’ defined by [47] And assumes that there is only one balance or a63

stable state [38].64

• Persistence is the time for a variable to remain in the same state before changing to a different one65

[Pimm]. Persistence is a measure of a system’s capacity to preserve itself over time. [63]66

• Resistance is described as the capacity of an ecosystem in the presence of external disturbance to67

preserve its initial state. [43]. Only small changes (in amount and intensity) within an ecosystem68

correspond to high resistance. This concept is similar to the ‘ecological resilience’ defined by [47]69

and Suggests that various stable states exist.70

• Robustness relates to the durability of the stability of the environment. Robustness is then a measure71

of the amount of disturbance an ecosystem can endure before it changes to a different state. [63].The72

more robust the food web is, the more stable it is.73

In this paper, we ascertain that both integrity and resilience frameworks can be formalized using74

information theory. The second law of thermodynamics is probabilistic by its very nature, because its75

formulation involves a probabilistic description of the state of a system [78]. It is customary in information76

theory to relate the Shannon information of a random variable X (microscopic state of a physical system)77

with the thermodynamic entropy of the system. This relation can be expressed as78

H(X) =−k∑x

[p(x) log p(x)], (1)

where p(x) is the probability density and assuming k as the Boltzmann’s constant. Even more, once79

the connection between entropy (thermodynamics) and Shannon information (information theory) is80

made, we are able to calculate self-organization as the complement (1−H(X)) of Shannon information81

[26]. Following [25, 15], an ecosystem that is in a high level of integrity is at full capacity to develop as82

a natural consequence of the continued operation of the processes of self-organization to dynamically83

incorporate the original species set at its location, which can be interpreted as the mechanistic basis of84

what is known as ecosystem resilience.85

It is clear then that in order to construct a unified narrative for ecosystem integrity and resilience,86

information theory is a suitable framework candidate. With that aim, we present in this paper a critical87

review of the literature and then develop a novel proposal related to what we consider a suitable concept88

not incorporated in dynamic ecosystem understanding so far: antifragility, which is the non linear (convex)89

response of a system to profit from variability in the payoffs space, or in simple words the ability of a90

system to benefit from surrounding randomness.91

2/25

Figure 1. Ecosystem Integrity three-tier model. Ecological integrity is understood to

be an underlying attribute in the constitution of ecosystems that produce specific

manifestations in their structural characteristics, development processes, and acquired

composition. In short, ecological integrity arises from processes of self-organization

derived from thermodynamic mechanisms that operate through the locally existing

biota, as well as the energy and materials at their disposition, until attaining “optimal”

operational points which are not fixed, but rather vary according to variations in the

physical conditions or changes produced in the biota or the environment. In the figure

we show the three-tier model of ecosystem integrity (3TEI), the inner tier is hidden to

the observer, but its status can be inferred by the information available at the

instrumental or observational tier where measurements on structure (including

composition or other biodiversity features) and function are obtained, of course

considering the context where the ecosystem is developing. Arrow tips indicate the

direction of assumed mechanistic influence, although information can go either way.

2 SURVEY METHODOLOGY92

It has recently been shown that Web of Science and Scopus is invisible to a big proportion of highly-cited93

papers in the social sciences and humanities. And even when the percentage of missing highly cited94

papers in Web of Science (WOS) and Scopus is in the natural, lives and social sciences. The Spearman95

quotation correlation coefficients in Google Scholar are more powerful in all fields compared to Web of96

Science and Scopus. The researchers conclude that highly cited papers available in the inclusive Google97

Scholar database actually show important deficiencies in the coverage of the Web of Science and Scopus98

in certain study fields. Consequently, using these selective databases to calculate bibliometric indices99

based on the number of highly cited papers could generate partial evaluations in poorly covered fields100

[68].101

For these reasons we choose Google Scholar as the search engine in which we use the term: “Ecosys-102

tem Resilience” AND “Information theory” AND “ “Ecosystem Integrity”. After excluding patents and103

citations we end up with 20 items. Of those 20 co-occurrences, books and patents were discarded, leaving104

10 entries that where fully read. Finally, only 7 items were selected to be analyzed and included in the105

present review. However, Google Scholar, does not provide easy access to cited references, which indicate106

the knowledge background of the selected items. For that reason, the analysis was completed with data107

from the Web of Science and the Astrophysics Data System to visualise their cited references network108

using Science of Science (Sci2) Tool and Gephi. To further analyze the literature set, a text corpus was109

assembled taking the text of the title and abstract of target documents as a bag of words. This small corpus110

was then processed with latent Dirichlet allocation (LDA) and latent semantic analysis (LSA) techniques.111

3/25

FINDINGS112

Basic scientometrics i.e word clouds113

In Table 1 we show basic metrics for papers selected and in Fig. 2 the results of applying LDA analysis114

to this corpus, based on [95] with an interactive viz that can be opened in any browser. A video explaining115

the use of this kind of viz could be found in here. According with LDA, papers can be allocated to the116

topics as indicated in Table 2; although, topics can be concurrently present in any paper.117

Source Google Scholar

Papers 7

Citations 454

Years 23

Cites Year 19.74

Cites Paper 64.86

Cites Author 149.78

Papers Author 1.95

Authors Paper 4

h index 7

g index 7

hc index 6

hI index 1.75

hI norm 6

AWCR 71.34

AW index 8.45

AWCRpA 17.93

e index 20.12

hm index 1.95

QueryDate 2019-02-04

Cites Author Year 6.51

hI annual 0.26

h coverage 100

g coverage 100

star count 3

year first 1996

year last 2018

ECC 454

Table 1. Table shows basic metrics for the Google Scholar search.

In Fig. 3 we present a main axes plot based in a latent semantic analysis (LSA). Both LDA and LSA118

suggest it is possible to recognize four groups in the corpus, which are further discussed in the section119

analyzing perceived literature narratives below. In addition, the analysis of the citation network reveals120

several unconnected sub-groups, while the cited references in general are very poorly connected. We121

interpret this findings as evidence of poor interdisciplinary crossover on the conceptual development of122

ecological resilience and integrity, which prompted our interest in the issues we discuss in this paper.123

LDA Topic Paper

1 Cabezas 2005

1 Sidle 2013

2 Aronson 1996

2 Gustavson 2002

3 Saint-Beat 2015

4 Filotas 2014

4 Schmeller 2018

Table 2. Allocation of paper to dominant topic found by LDA.

4/25

Figure 2. Latent Dirichlet Allocation (LDA) analysis based on [95] with an interactive

viz to be opened in any browser. In short, the interface has two main panels. Topic

pattern on the left and terms frequencies on the right. The left panel shows a general

perspective of the discovered subjects indicating how common each is in the corpus (the

set of papers) and how they relate to each other ; the subjects are plotted as circles

whose centers are characterized by the computed range between the subjects (projected

into 2 dimensions). The prevalence in the corpus of each topic is proportional to the

circle size. The right panel has a bar chart showing the meaning of terms, informative of

the possibly interpretation of the topics essence. You can pick any subject interactively

and find out the function of terms in it. Two overlaid bars are shown at each place when

pointing to a subject, displaying the topic-specific frequency of each word (in red) and

the corpus-wide frequency (in blue gray). When no topic is selected, the right panel

displays the top 30 most salient terms for the dataset. A video explaining the use of this

kind of viz could be found in here

In Fig.5 we show a TreeMap for the type of documents that cite the set of reviewed ones; in Fig6 The124

organizations of origin for the documents that cite the set of reviewed ones; in Fig.7 we show the number125

of documents that cite the set of reviewed ones; and in Fig.8 the distribution of documents that cite the set126

of reviewed ones in terms of research field.127

2.1 Literature narratives128

We consider that in the context of the relation of resilience with ecological integrity under the lens of129

information theory, narratives has gone from trying to identify (with information theory tools) ecosystem130

features that allow them to be more resilient and hence maintain their integrity, to a more technical131

approach using times series or network analysis in addition to new mathematical concepts (see fig. 9).132

The importance of interactions and a complex system approach is highlighted. Finally, the field completes133

a circle getting again back into ecology and refining resilience feature and properties of ecosystems.134

2.2 Resilience features and properties I135

Noble and Slayter [? ] have described several categories of essential life history characteristics that are136

helpful in determining a species ’ reaction to recurrent disturbances. In the revised paper by Aronson137

and co-workers [4], the authors make reference to a previous work [3] where they modified the Noble138

and Slayter’s concept [77], defining Vital Ecosystem Attributes (VEAs). The VEAs attempt to capture139

5/25

Figure 3. Latent semantic analysis of papers

those features or attributes that are correlated with and that can serve as ecosystem structure and function140

indicators, the same attributes in the Ecosystem Integrity Model (3TEI) instrument layer from Equihua141

and colleagues [25]. In that way, one may interpret that a resilient ecosystem is one for which VEAs fall142

into an optimal range, represented as ecosystem integrity. But unlike the 3TEI model, VEAs requires143

intense (often cost and not scalable) fieldwork that most likely make VEAs not such a good option for a144

national assessment of ecosystem condition trends.145

Structure Function

Perennial species richness Biomas productivity

Annual species richness Soil organic matter

Total plant cover Maximum available water reserves

Aboveground phytomass Coefficient of rain off efficacy

Beta diversity Rain use efficacy

Life form spectrum Length of water availability period

Keystone species Nitrogen use efficacy

Microbial biomass Microsymbiont effectiveness

Soil biota diversity Cyclic indexes

Table 3. Vital Ecosystem Attributes according to [3]

6/25

Figure 4. The citation network for the selected paper including them

Figure 5. TreeMap for the type of documents that cite the set of reviewed ones

From the idea of VEAs (see Table3) Aronson and his colleagues [4] extracted the insight into146

considering 16 quantifiable characteristics for use on a more particular spatial scale: the landscape as a147

multifunctional environment. The writers intend to use the fresh Vital Landscape Attributes (VLAs) to148

evaluate the outcomes of ecological restoration or rehabilitation conducted from the view of the landscape.149

Most interesting is the chance that as VEAs relate to the integrity of the ecosystem, VLAs (see Table??150

could give way to a 3TLI model.151

In a different line of thought, the reviewed paper by Gustavson and co-workers [39] develops a152

general index that may serve as a proxy of ecosystem resilience from an information theory perspective.153

The authors report that attempts have been made to describe and evaluate resilience, but an overall154

predictive or theoretical connection between resilience characteristics and ecosystem dynamics has yet155

7/25

Figure 6. Organizations of origin for the documents that cite the set of reviewed ones

Figure 7. Number of documents that cite the set of reviewed ones

Figure 8. Distribution of documents that cite the set of reviewed ones in terms of

research field

to be advanced. Similarly, much has been discussed about possible interactions between stability and156

structure and, generally speaking, predictable interactions between resilience characteristics and how157

8/25

Figure 9. Summary of concepts and narratives in selected papers

ecosystems work are not intuitively evident and may not exist.158

To this end, they turn to Ulanowicz’s Ascendancy Theory [? ] which is a measure of the magnitude of159

the information flow through an ecosystem’s network framework. Some constraints for its use are that it160

requires a comparatively full description of the nature and magnitude of all species interactions.161

Ascendancy is defined by the average mutual information as presented [105] between component a to162

b is:163

As = K∑i∑

j

p(ai,b j)log

[

p⟨

b j|ai

⟩

p⟨

b j

⟩

]

, (2)

where, p⟨

b j|ai

⟩

, the probability of b j given that ai has occurred; and p(b j), the probability that b j164

will occur.165

The upper limit of ascendancy is the capacity for growth, and the distinction between ability and ascen-166

dancy is called the overhead system, which represents a multiplicity of paths and can therefore eventually167

be linked with the complexity of the ecosystem.[116]. A complex structure is key for sustainability [45]168

because the diversity of processes plays a crucial role on system survives [106]. In particular, to enhance169

ecosystem’s long-term sustainability, a particular densely connected network structure is advantageous.170

Such a scheme is sufficiently effective and sufficiently varied. Efficiency-diversity equilibrium is essential171

to ecosystem resilience. Ultimately, ascendancy captures in a single index the capacity of an ecosystem to172

prevail against disruption by virtue of its combined organisation and size, it was suggested that, in order173

to attain sustainability, it should always be possible for systems of human use to regain their ascendancy174

within culture and ecosystems [88].175

2.3 Ecosystems response to perturbations176

The main interest in the selected paper form Cabezas and co-workers [12] is not ecosystem integrity nor177

resilience per se but sustainability. A concept they relate to integrated systems comprising humans and the178

rest of nature (probably a socioecosystem). They emphasize that the structures and operation of the human179

element (in terms of culture, economy, law, etc.) must be such that they strengthen or encourage the180

persistence of the natural component’s structures and operation (in terms of ecosystem trophic connections,181

biodiversity, biogeochemical cycles, etc.), and vice versa. It is in the idea that ”persistence” and ”operation182

of the natural component” that the connection is made. From their perspective a sine qua non condition to183

achieve sustainability [83] is ecosystem stability which they conceptualize using Fisher information [11],184

view that is further developed in several papers [24, 50, 70, 69, 115, 2, 51, 35].185

Following [30] and [69] Consider the central problem of estimating the actual value θ forstate variable.

The estimation comes from an inference process from imperfect observation y = θ + x in the presence

of some random noise x. This kind of measurement-inference process will hence be called “smart

9/25

Vital Landscape Attributes (VLAs)

Type, number and range of landform

The number of ecosystems

Type, number and range of land units

Diversity, length and intensity of former human uses

Diversity of present human uses

Number and proportions of land use types

Number and variety of ecotunes-zones

Number and types of corridors

Diversity of selected critical groups of organisms (functional groups)

Range and modalities of organisms

regularly crossing ecotunes

Cycling indexes of the flow and exchanges of water,

nutrients, and energy within and among ecosystems

Pattern and tempo water and nutrient movement

Level of anthropogenic transformation of landscape

Spread of disturbance

Number and importance of biological invasions

Nature and intensity of the different sources of

degradation, whether legal or illegal

Table 4. Vital Landscape Attributes as proposed by [4]

measurement” of θ whose result is an estimator θ that is a function of an imperfect observation θ(y). This

is a closed system, meaning that it’s well described by {y, θ ,x} without the need to consider additional

sources of noise. Consider also that the estimator is unbiased in terms of being a good estimator on

average⟨

θ(y)⟩

= θ . In this case, the mean-square error obeys the Cramer-Rao inequality

e2I ≥ 1, (3)

where I is the Fisher Information of the system, calculated as186

I =∫

dy

P0(y|θ)

[

dP0(y|θ)

dθ

]2

, (4)

in which P0(y|θ) is the probability density function of measuring a particular value of y given the true187

value θ of the state variable in question. Then, since the error decreases as information increases, Fisher188

information may be understood as the quality of the estimation θ from a smart measurement. Then,189

if the system is characterized by a phase space with m state variables xi that define the phase vector190

s = (x1, ...,xi,...,xm) associated with a smart measurement y, then we can prove that191

I (s) =1

T

T∫

0

s′′2

s′4dt, (5)

where T is the time period required for one cycle of the system; s′(t) is the tangential speed and s′′(t) is192

scalar acceleration tangential to the system path in phase space. Both are calculated in terms of the state193

variables xi as194

s′ (t) =

√

m

∑i

(

dxi

dt

)2

, (6)

s′′ (t) =1

s′ (t)

m

∑i

(

dxi

dt

d2xi

dt2

)

. (7)

10/25

A simple and robust approach to calculating tangential velocity and acceleration uses the three-point195

difference scheme196

dxi

dt=

αxi(t +∆ta)− (α2 −1)xi(t)− xi(t −α∆ta)

α(α +1)∆ta, (8)

d2xi

dt=

αxi(t +∆ta)− (α +1)xi(t)− xi(t −α∆ta)

α(α +1)∆t2a/2

, (9)

where xi(t) is a central data point, xi(t −∆ta) is the next point following the center xi and xi(t −∆tp) is197

the previous point to it. For evenly-spaced points ∆ta = ∆tp and α = ∆tp/∆ta is the ratio of the previous198

and following time space.199

The thesis suggested by [? ] is that a shift in Fisher information may signal a change of regime in a200

dynamic system to [? ]:201

• Fisher information is a function of measurement variability. Low variability results in high Fisher202

information and low Fisher information results in high variability.203

• Systems in stable regime tends to exhibit constant Fisher information. Then, organization losses204

points to greater variability and a decrease of Fisher information.205

• Self-organizing systems reduce their variability and gain Fisher information.206

• ”If resilience is defined by the intensity, frequency, and duration of a perturbation that a system can207

withstand before fundamentally changing in function and structure, then we would hypothesize that208

Fisher information would return to the same value or higher in more resilient systems.” [12]209

We found this last point of much interest because not only it provides a formal definition of the210

resilience concept, but it also provides a specific way to measure it via Fisher information. In order to test211

this idea, we analyze NDVI data for ”US-Me1: Metolius - Eyerly burn” Ameriflux site [40] site in Oregon212

for which is documented as an intermediately aged ponderosa pine forest that was severely burned in213

the 2002 Eyerly wildfire. The AmeriFlux network of approximately 100 research stations is the main214

research group and information supplier for big terrestrial carbon cycling syntheses in the Americas and215

has established a database for micrometeorological, meteorological and biological information.216

Data of NDVI was downloaded using the application for Extracting and Exploring Analysis Ready217

Samples (AppEEARS) that enables users to subset geospatial data-set using spatial, temporal, and218

band/layer parameters (https://lpdaacsvc.cr.usgs.gov/appeears/). In particular, we219

used MOD13A3.006 1km2 resolution monthly data of NDVI from 01-01-1990 to 01-01-2018.220

Focusing into the 2002 wildfire, we show in Figure10 that as expected, with the wildfire disturbance221

the system experience both great changes in NDVI and its corresponding Fisher information. We found222

that Fisher information returns to previous values after 18 months approximately but not the NDVI values.223

On the one hand, recovering Fisher information could be related to the way Filotas and co-workers224

[27] understand ecological resilience as ”the amount of change that an ecosystem can absorb before it225

loses its ability to maintain its original function and structure”. After a disturbance, the authors claim that226

a resilient system has the ability to recover its initial structure, features, and feedback ; in other words, its227

integrity.228

On the other hand, it seems then that the criterion of Fisher information is necessary but not sufficient229

to ensure ecosystem resilience. For example one should expect that after a disturbance, essential variables230

as the ones proposed by Schmeller and co-workers [92], which could be seen as a modern version of231

Aronson’s Vital Ecological Attributes [4], return to previous values. In a recent work Dutrieux [23]232

combine into one index signaling from TM and ETM+ B4 band, corresponding to Near Infra Red233

(NIR) with wavelength of 770−900nm which provide information about canopy biomass; and B5 band234

corresponding to Short Wave Infra-Red (SWIR1) with wavelength of 1550− 1750nm which provide235

information about canopy moisture content:236

NDMI =NIR−SWIR1

NIR+SWIR1. (10)

11/25

Figure 10. In red the normalized NDVI time series for the 1km2 pixel corresponding

to the coordinates of the US-Me1 site of Ameriflux with a monthly sampling. In blue,

the corresponding values of Fisher’s information using the Cabezas and collaborators

algorithm (https://github.com/csunlab/fisher-information)

Low NDMI values for bare soils and thin forest canopies are anticipated, while greater values237

correspond to thicker, completely developed forest canopies. [111].238

The author created the following harmonic model to compare values before and after a disturbance:239

yt = αi+3

∑1

γisin

(

2π jt

f+δ j

)

+ εt , (11)

where the dependent variable y at a given time t is expressed as the sum of an intercept αi, a sum240

of different frequency harmonic components representing seasonality and an error εt . In the model j241

corresponds to the harmonic order, 1 being the annual cycle, γ j and δ j correspond respectively to the242

amplitude and phase of the harmonic order j, and f is the known frequency of the time-series (i.e., number243

of observations per year).244

New values are then estimated for each spectral band and each time series observation using the245

corresponding matched model, enabling Euclidean distance to be calculated with the following formula.:246

Dt =

√

√

√

√

k

∑i=1

(yit − yit)2. (12)

The author then applies it to spectral recovery time for a set of 3596 Landsat time-series sampled247

from regrowing forests across the Amazon basin, thus producing estimates of recovery time in spectral248

properties, which he calls spectral resilience. On average, he found that spectral resilience takes about 7.8249

years, with a large variability (sd =5.3 years) for disturbed forests to recover their spectral properties. Now250

we have a new problem, how to determine the thresholds for (a) distance between initial and final values251

for both state (essential/vital) variables and their Fisher information; (b) the time scale these recovering252

should occur. In principle we believe (a) could be determined from ecological integrity measurements,253

but it is currently an open research question we are not addressing here.254

12/25

In another line of thoughts, Sidle and co-workers [94] focus on ascertaining under what circumstances255

ecosystems exhibit resilience, tipping points or episodic resetting. They point out that while ecosystem256

resilience originated from ecological perspective, latest debates concentrated on geophysical characteris-257

tics and that it is acknowledged that dynamic system properties may not return to their former state after258

disturbances (see for example [16, 36, 60, 85, 74, 100]). Tipping points generally arise when chronic259

(typically anthropogenic but sometimes natural) changes push ecosystems to thresholds that cause process260

and function collapse even in a permanent way. Resetting ecosystems happens when episodic natural261

disasters break thresholds with little or no warning resulting in long-term modifications in environmental262

characteristics or functioning of the ecosystem. Of special interest is the work of Steffen and co-workers263

[100] who consider earth biosphere as a whole system and study its possible trajectories under the current264

planetary crisis. In particular, they explore the risk of self-reinforcing feedback that could eventually push265

the Earth’s biosphere system to a planetary threshold that, if crossed, could prevent climate stabilization266

near the Holocene temperature regime (the pre-industrial conditions set out in the Paris Agreement). In267

the worst case scenario Earth could be driven into the ongoing warming track of a ”Hothouse Earth” path,268

even though human emissions were lowered.269

As in other papers reviewed, Sidle and co-workers [94], state that ”if a system is viewed as resilient, it270

is generally perceived as remaining within specified bounds, probably close to the optimal operational271

points” mentioned in [25]. Which sets again the question of which should be the variables under the272

“bounded ecosystem” and how to determine the range of values to consider the ecosystem as resilient.273

More to the point, how much time should be spanned between an ecosystem perturbation for the resilience274

variable returning to their bound limits? In principle, we consider that this should be in the same order of275

magnitude that the -natural characteristic time scale of the ecosystem. But once again, the measurement276

of characteristic time scale for an arbitrary state variable of the ecosystem is an open question. The277

main problem is that in most cases we will not have a mechanistic model for the variable in question278

but time-series only. In [1] the authors use the Wigner function to explore if there is an special time279

scale under which the system reaches an optimal representation. For multiple time series observed, they280

contrasted entropy values covering a variety of distinct time domains. For their natural characteristic time,281

they found that entropy is highly likely to be minimal, implying minimum uncertainty in time-frequency282

space. Another alternative might be to consider the τ0 time in which the system’s memory tends to283

be zero, defined by the absolute τ time value for which the C(τ) auto-correlation function crosses the284

horizontal axis [? ].285

2.4 Complexity perspective286

The Filotas and co-workers reviewed paper [27] provides a remarkable introduction to complexity. The287

authors decompose complexity into eight features an then goes to relate them into a new narrative for288

forests, making as a result an interesting connection with resilience and integrity. Generally speaking, a289

system is complex either it presents a sufficiently number of components with strong enough interaction290

or it exhibits changes in the configuration space comparable to the observer’s time scale, and in most291

cases both. Forests as a system and forest management, certainly occupy a high position in the complexity292

gradient.293

The authors focus on forests, but clearly what they describe is applicable to all types of ecosystems.294

Nevertheless, forests are a good model because they are both widely and intensively managed, and also295

because they are deeply coupled with human systems. The designed approach can thus assist forest296

scientists and managers in conceptualizing forests as integrated socio-ecological systems and provide297

concrete examples of how to manage forests as complex adaptive systems.298

There are at least 800 different definitions of a forest. Some of them are used simultaneously in the299

same country for different purposes or scales [64]. This is in part due to the fact that forest types differ300

widely, depending on factors such as latitude, climate patterns, soil properties, and human interactions. It301

also depends on who is defining it. An economist could describe a forest in a very distinct manner to a302

forester or a farmer, in accordance with their specific interests. One of the most widely used definitions is303

that by FAO (1998), that defines a forest as ”the track of land with area over 0.5 ha, tree canopy cover304

larger than 10%, which is not primarily subject to agricultural or other specific non-forest uses”. For young305

forests or regions where tree growth is suppressed by climatic factors, trees should be capable of reaching a306

height of at least 5 m in situ while meeting the requirement for canopy cover. In general, forest definitions307

are based on two different perspectives. One, associated with quantitative cover/density variables such as308

13/25

minimum area cover, minimum tree height, or minimum crown size. The other, relates to characteristic309

spatial features of the territory such as the presence of plantations, agricultural activities or non-forest310

trees within the forest itself [56, 57, 58, 64, 107]. The issue is that when natural forests are significantly311

degraded or superseded by plantations, essential ecosystem services such as CO2 sequestration may be312

lost ; but they may technically continue to be classified as forests under many definitions.313

Thus, a fundamental key characteristic of ecosystems is essentially missing in most forest definitions,314

its complexity. The authors point out that complex systems science provides a transdisciplinary framework315

to study systems characterized by (1) heterogeneity, (2) hierarchy, (3) self-organization, (4) openness, (5)316

adaptation, (6) memory (homeostasis?), (7) non-linearity, and (8) uncertainty. These eight characteristics317

are shared by complex systems regardless of the nature of their constituents and the article exemplifies318

them in ”forest terms”. The conclusion they reach in terms that complex systems approach has inspired319

both theory and applied approaches to improve ecosystem resilience and adaptability is most relevant320

to our article. While forests are prime examples of complex systems (Perry 1994), forest ecology and321

management approaches are only starting to emerge and complex systems are seldom invoked in the field.322

Heterogeneity or how ecosystem components are distributed in space is important from several points323

of view. For example, the spatial distribution of resources imposes restrictions on animal foraging and324

ultimately significant patterns. It has widely been proved that the foraging patterns of a variety of animals325

involve many spatio-temporal scales, as described by Levy walks [108, 84, 73]. This statistical behavior326

is present even in human movement patterns [10], and has been linked to evolutionary advantages in terms327

of search strategies in complex environments [6]. Further more, resources distribution patterns induce328

foraging behaviours linked to seed dispersal. Those patterns feedback into the ecosystem dynamics and329

influence the distribution of resources in time [8]. In the reviewed paper, the authors point out that human330

influence on ecosystems may reduce ecosystem complexity by altering spatial heterogeneity. For instance,331

forest cover and resident biota have been homogenized by intensifying and standardizing cultivation332

methods within and between woodlots. Changing these patterns can significantly influence the capacity333

of the landscape to maintain materials and energy effectively processed and host the region’s biota; this334

change in turn decreases its integrity, resulting in a loss of resilience. Even more, in the context of the335

Anthropocene [99] or Technocene [62], Human impact in extreme cases may modify forest heterogeneity336

generating new ecological patterns (niche construction) and interactions, without historical equivalents337

[93].338

Spatial heterogeneity can be altered by invasive species threaten biodiversity through predation339

[21, 86], competition [42], disease transmission [114], and facilitation of the establishment of further340

invasive species [97]).It has been reported that the decrease and extinction of native species due to invasive341

predators can generate cascade effects that extend through the whole ecosystem and beyond. [14]. In342

particular depredation effects resulting from human introduced species can be severe [96, 22]. Both rats343

(Rattus rattus), cats (Felis catus) and dogs (Canis lupus familiaris) are recognized as the worst threat344

species following recent studies [48]. In natural areas worldwide, dogs are threatening some 200 species,345

some of them even included in IUCN threat categories. Likewize, Feral cats and red fox (Vulpes vulpes)346

predation processes has been documented as a cause of the decline or extinction of two thirds of Australia’s347

digging mammal species [28, 112]. Reduced disturbance to soil in the absence of digging mammals has348

led to impoverished landscapes where little organic matter incorporates into the soil and rates of seed349

germination is low [28]. The predation of seabirds through introduced Arctic foxes (Alopex lagopus) in350

the Aleutian archipelago has reduced nutrient input and soil fertility, eventually causing vegetation to shift351

from grassland to dwarf shrubland.352

In a recent work [18], a deep relation between Levy walks and resilience has been shown. In this353

work, Danneman and coworkers unveil an essential yet unexplored multi-scale movement property of354

Levy walks, how it play an important role in the stability of populations dynamics.Using Lotka-Volterra355

models, they predict that generally diffusing foragers tend to become extinct in fragile fragmented habitats,356

while their populations become resilient to degraded circumstances and have maximized abundance when357

individuals undertake Levy flights. Their analytical and simulated findings, change the scope of multi-358

scale foraging from individual to population level, making it of major value to a wide scope of applications359

in biology of conservation.Their findings indicate that Levy flights reach a balance between exploration360

and exploitation, which in turn will benefit the stability and resilience of the population. In that way,361

modern forest management is becoming more compatible with this complexity feature (heterogeneity) by362

promoting it through strategic cuts that emulate natural disturbances; leave intact some structures and363

14/25

organisms, Including dead and living trees and intact patches of forests ; and promote mixtures of tree364

species. These methods are similar to the comparatively recent strategy of using biodiversity to boost365

yield and resilience in natural and managed ecosystems [27]. Generalizing these ideas we reckon there is366

important evidence suggesting that in order to preserve ecosystem integrity and resilience, management367

systems should consider maintaining minimum levels of ecosystem complexity.368

Of course, this poses a new challenge: How to measure complexity? Following Gershenson and369

co-authors [31] one may measure complexity using again the Shannon information. In this information370

theory framework, in order to have new information, the old one has to be transformed. Thus, we can371

define information emergence E as the rate of information transformation. therefor emergence is identified372

directly with Shannon’s information H or I. In addition, self-organization (S), a key feature of complex373

systems, has been correlated with an increase in order (i.e. as a reduction of entropy) [31]. Thus, if374

emergence implies an increase in information, which is analogous to entropy; self-organization should be375

anti-correlated with emergence in such a way that376

S = 1− I = 1−E. (13)

In this way, following [31, 26] complexity can be measured as377

C = 4∗E ∗S. (14)

378

Under the complex systems perspective ecosystems are not systems that can simplistically be managed379

top-down. We must explicitly consider that the interactions take place in multiple and hierarchical levels.380

This is a general feature of complex systems, components are organized hierarchically in such a way that381

elements at different levels interact to form an architecture that characterizes the system. In this way,382

complexity asserts that a phenomenon occurring at one scale cannot be understood without considering383

cross-scale interactions. But it also means that environmental policy, management and intervention needs384

to be rethought in terms of scale. In this respect Taleb is assembling ”Principles of policy under complexity”385

(draft version available at: http://www.academia.edu/38433249/Fractal_Localism_386

Political_Clarity_under_Complexity) which include the understanding of policies as scale387

dependent, and so we should consider that instead of aiming at one monolithic policy for managing388

ecosystem, we should go on to develop a range of them linked to different levels of application Such389

approach will be required to reduce the risk of catastrophic hidden effects.390

Understanding the coupling of natural and human sub-systems provide a whole new narrative that391

challenges management. Ecosystems management is the outcome of collective actions among different392

agents such as decision makers, scientists, managers, concerned citizens and so on. As complexity, key393

for ecosystem integrity and resilience, is at dynamic balance between emergence and self-organization394

(S) (Eq.2.4), some (and the correct type of) self-organization is necessary to be fostered, but too much395

of it is bad. Too much (form the wrong type) of S may sustain unwanted feedbacks with detrimental396

consequences. For instance, illegal logging in Borneo can be seen as a self-organizing phenomenon397

supported by interactions among all levels in the stakeholder hierarchy [82]. The mechanism is explained398

by Filotas and co-workers, starting with pit sawyers taking out livings and pirate loggers taking advantage399

of governance failures. This alone could not generate such a great impact, unless it couples with400

unscrupulous timber buyers and corrupt governmental officials laundering the illegal wood. Experience401

in Mexico suggests that corruption might be the common link in practically all-important ecosystem402

degradation processes. Finally, they say that savvy international traders are the higher link that provides403

lucrative outlets for ill-gotten goods. According to the authors narrative, where illegal logging occurs,404

wood markets are flooded, wood prices are depressed, and standing trees are undervalued. Under such405

conditions, community forest managers are not motivated to implement sustainable forest management406

practices, which often involve short-term investments for only long-term returns. These conditions result407

in a self-organized feedback that sustains illegal logging.408

The characteristic of complex systems not having a unique description scale is related to one of the409

most omnipresent system-wide phenomena, the 1/ f behavior on frequency space for the fluctuation time410

series.This so-called pink noise is one of a family of 1/ f β colored or fractal noises defined by the β411

15/25

scaling exponent and deemed to be a criticality fingerprint. It is common to comparing and classifying412

fluctuation dynamics according to their resemblance to three archetypal noise groups: white (β ∼ 0),pink413

(β ∼−1)and Brownian (β ∼−2) [5, 59, 55, 79]. The universality of criticality is still under consideration414

and is known as the ’ ’ criticality hypothesis ” which states that systems in a dynamic system that shifts415

between order and disorder reach the greatest level of computing capacities when reaches a balance416

between robustness and flexibility(see [38] and references therein). This idea of criticality was recently417

used in an information theory approach to defining ecosystem health and sustainability [83]. In this paper,418

Ramırez-Carrillo et al. Consider that an ecosystem is healthy if it is in criticality, as a mixture of scale419

invariance (as power laws in power spectra) and a balance between adaptability and robustness.420

These power laws appear in numerous phenomena including earthquake statistics, solar flares, epi-421

demic outbreaks, etc., as summarized by [83]. [66, 76, 98]. They also are a common theme in biology422

[33, 34, 32, 110]. Several researchers reported evidence of dynamic criticality in physiological pro-423

cesses such as heart activity and suggested that it could be a main characteristic of a healthy state.424

[54, 49, 89]. Some studies [34, 90] found compelling evidence of a relationship between healthy hearts425

and scale-invariant noise, around 1/ f regime, backed by medical evidence.426

This complexity approach is clearly complementary to the ecosystem integrity narrative, and we427

should consider that an ecosystem is resilient if, in addition to maintaining its EVLs values inside a ”safe”428

range, it also keeps them within a critical dynamic region (scale invariance and “1/ f ” fluctuations) [83].429

2.5 Resilience features and properties II430

The main purpose of the Saint-Beat and co-workers reviewed paper [91] is to know how distinct ecosystems431

react to global change in terms of composition and dynamics and eventually, how persistence, strength, or432

resilience of the ecosystem can be assessed. The authors show that ecological network assessment (ENA)433

offers an effective approach for describing local stability, combining both quantitative and qualitative434

elements. They warn, however, that describing real conservation cases combining local and global stability435

remains an incomplete task. The authors focus on three resilience-related results that emerge from their436

ENA: (a) the role of species diversity in the structure and functioning of the ecosystem ; (b) the number of437

trophic links and strength of interactions ; (c) the stability of the ecosystem in terms of cycling capacity,438

omnivory spread and ascendancy.439

Native species richness generally improves ecosystem resistance. High biodiversity (without con-440

sidering non-native species) was proposed to contribute to minimize the threat of major ecosystem441

modifications in reaction to environmental disturbances [71]. Experiments on species invasion in grass-442

land parcels indicate that local biodiversity decreases the settlement and success of a variety of invaders443

[53].Similarly, studies on manipulation of grassland diversity indicate that elevated diversity improves444

inter-specific competition and thus decreases the danger of invasion [75, 44].In this line of thought, the445

authors proceed to summarize the proof. For instance, Brose [9] demonstrates that herbivorous existence in446

a society leads to the coexistence of manufacturers in a mixed predator-prey model and producer-nutrient447

model with a structural model of food-webs. Species diversity also contributes to reduce the impact on448

resistance and resilience for example in coastal ecosystem [113]. Similarly, the maintenance of integral449

ecosystems and the services they provide is critical to human societies and the preservation of species450

diversity seems crucial to achieving this goal [13].451

However, the authors warns that the consideration of a single variable such as diversity can not452

be sufficiently thorough to evaluate the stability of ecosystems due to the complexity of ecosystems.453

Therefore, they suggest that building holistic indexes in the ENA framework is a better approach for a454

thorough knowledge of the food web structure and its role in the functioning of the ecosystem.455

If one would be able to construct a sufficiently detailed trophic network (something very difficult to do456

in general), one could use standard network analysis tools to understand, for example, the topology of the457

network, such as connectance. In that sense, the authors summarize evidence from the literature to show458

that an increase in links dissipates the impact of variability in species distribution and increases stability459

[65]. Higher connectivity thus improves the strength of the ecosystem as well as resilience [20] . If so,460

connectivity seems to be a useful measure of the robustness of food webs and indirectly of ecosystem461

stability.462

In addition to connectivity, they show the importance of interaction strength diversity. Following463

ideas of [104], They claim that ecosystem stability requires a balanced presence of weak and strong464

interactions. Suppression of weak interactions destabilizes the system for a certain amount of species.465

16/25

Moreover, the food web would be stable if and only if main predator-prey interactions are combined466

with weak interactions in the context of high diversity. Thus, due to their ability to fluctuate and adapt467

within ecosystems, weak interactions function as a stabilizing force in food webs and consequently the468

ecosystem.469

To gain a deeper understanding of stability, Saint-Beat and colleagues examine the effect of cycling,470

the presence of omnivorous and ascending.471

For instance, the presence of omnivory gives an ecosystem trophic flexibility, a clear beneficial feature472

that reflects integrity and resilience of the ecosystem. The researchers claim that omnivory provides to473

the ecosystem a superior buffer to deal with environmental disturbances Because omnivorous species474

enable faster ecosystem reaction by rapidly moving trophic routes following disturbance.For example, if a475

disturbance impacts low trophic levels, omnivorous species that are directly linked to it, would respond476

rapidly. In comparison, a particular predator must wait until the disturbance reaches its own level ;477

therefore, the response time will be longer.478

As in the reviewed paper by Gustavson and co-workers [39], Saint-Beat and co-workers show how479

ascendency could be used as a key indicator to evaluate ecosystems functioning. The authors indicate that480

to understand the function of ”ascendancy” two kinds of stability must be differentiated. A system with481

elevated inner stability is a system with adequate inner limitations to enable a strongly organized structure,482

corresponding to a high ascendency (high mutual information). Typically under this condition, ecosystems483

are some how protected against internal perturbations but leave them vulnerable to external ones. On the484

other hand, since low ascendancy is linked to redundancy, ecosystems become more resilient to external485

disturbances. Interestingly enough, too high level of ascendancy is recognized as a a characteristic of486

stress and may indicate a decrease ecosystem resilience.487

Summarizing, in the dynamic response of ecosystems under the criticality framework a healthy488

ecosystem is found where a balance between robustness and adaptation develops. In the case of network489

topology, the ecosystems need to develop a good balance between strong and weak interactions in order490

to be stable. In the case of the ascendancy narrative, a stable ecosystem should develop a good balance491

between ascendancy and overhead, which seems to give resistance and resilience to ecosystems. This492

leads us to think what we develop in the next section, in which we will ascertain whether all these three493

kinds of balance could be particular cases of a more general evolutionary strategy of living systems: the494

antifragility.495

BEYOND RESILIENCE, ANTIFRAGILITY496

Living systems can and must do much more than merely react to the environment’s variability through497

random mutations followed by selection; they must certainty have built-in characteristics that enable them498

to discover alternatives to cope with adversity, variability and uncertainty. Anti-fragility is one of these499

characteristics [17, 101].500

If one considers what does really mean that something is fragile, the key property is that it gets501

damaged by environmental variability. Now if we ask our nearest colleague at random, about the exact502

opposite of fragile, most likely we would get concepts such as robustness or resilience. But at close503

inspection it is clear that none of them are the exact opposite of fragile. Both represent systems that are504

insensitive to environmental variability or get affected only momentarily, quickly returning to its initial505

state.506

The exact opposite of fragility as defined by Taleb is antifragility, which is a property that enhances507

the system’s functional capacity to response to external perturbations[102]. In other words, a system508

is antifragile if it benefits from environmental variability, works better after being disturbed. Then509

antifragility is beyond robustness or resilience. While the robust / resilient systems tolerate stress and510

remain the same, antifragile structures not only withstand stress but also gain from it. The immune system511

provide significant illustration of antifragile systems. When subjected to various germs at a young age, our512

immune system will improve and gain different capabilities to overcome new illnesses in the future[81].513

A formal definition of antifragility as convexity in the payoffs space is found in [103, 102]. Lets

consider a two times continuously differentiable “response” or payoff function f (x). Then the function’s

convexity will be defined by the relation∂ 2 f

∂x2 ≥ 0 which can be simplified under the right conditions

to 12[ f (x+∆x)+ f (x−∆x)] ≥ f (x). Then the response function f will exhibit non-linearity to dose,

which means that a dose increase will have a much higher impact in relation to this increase. Taleb

17/25

generalizes this result to a linear combination for which ∑αi = 1, 0 ≤ αi ≤ 1 in such a way that

∑ [αi f (xi)]≥ f [∑(αixi)]. Again simplifying the argument, under the correct conditions we end up with

f (nx) ≥ n f (x). This way, if X is a random variable with support in [a,b] where the function f is well

behaved, and f is convex, we get Jensen’s Inequality [6],

E( f (x))≥ f (E(x)). (15)

Without loss of generality, if its continuous distribution with density ϕ(x) and support in [a,b] belongs to

the location scale family distribution, with ϕ(x/σ) and σ > 0, then, with Eσ , the mapping representing

the expectation under a probability distribution indexed by the scale σ , we have:

∀σ2 > σ1,Eσ2[ f (x)]≥ Eσ1

[ f (x)] . (16)

514

This way, Taleb defines local antifragility as ”a situation in which, over a specific interval [a,b], either515

the expectation increases with the scale of the distribution as in Eq 2.5, or the dose-response is convex516

over the same interval”.517

Although Antifragility framework was developed by Taleb in the context of financial risk analysis, duo518

to its universal mathematical formalism it has track attention and has been applied far away its original519

scope. There are applications of the Antifraglity concept from molecular biology to urban planning520

(see [81] and references inside). In their work, Pineda and co-workers [81] proposed a straightforward521

implementation of antifragility by defining as payoff function the complexity of the system. which makes522

a lot of sense in the context of our review because complexity is highly related with critically and hence523

with these these trade-off balance between robustness and adaptability.524

The authors defined fragility as525

∮

=−∆C |∆x| , (17)

where ∆C is the change in system complexity due to a perturbation of degree |∆x|. As complexity can526

always be normalized to [0,1], then positive values of∮

define fragile systems; when∮

is zero the system527

is robust/resilient; and for negative values of∮

the system is antifragile.528

Then, Pineda and co-workers [81] apply it to random Boolean networks (RBNs) of a model of genetic529

regulatory works. They found that ordered RBNs are the most antifragile and demonstrated that, as530

expected, seven biological well studied networks such as CD4+ T cell differentiation and plasticity or531

Arabidopsis thaliana cell-cycle, are antifragile.532

We know, from Central Limit Theory, that normal distributions can only emerge from (simple)533

systems without interactions (probabilistic independence). When we take into account interactions (no534

probabilistic independence) then the corresponding probability distribution will have fat-tails. In that535

sense complexity is related with fat-tails and fat-tails with fragility/antifragility [101]. In Taleb’s narrative,536

normal distribution in the response function characterize robust systems; whereas left fat-tailed are fragile,537

and right fat-tailed are antifragile systems. Most interestingly, Fossion and co-workers [29] have related538

homeostasis (physiological resilience?) to pairs of physiological variables, one to be controlled (the539

one that remains in homeostasis) and another one that controls the former. The main idea is that in540

order to have a homeostatic physiological variable (normal), the body must use other variables (right541

fat-tailed) to absorb a random injection of matter, energy, information or any combination of them from542

the environment. In Figure 4 of their paper, they present results for variability analysis of heart rate543

HR and blood pressure BP for (a) healthy control(s), (b) recently diagnosed diabetic patient(s) and (c)544

long-standing diabetic patient(s). They show that for healthy patients BP is normal and HR is right545

fat-tailed. In the case of recently diagnosed diabetic patients, BP start to lose normality and develop a left546

tail and HR tends to normality. Eventually, long-standing diabetic patients, BP has a clearly left fat-tailed547

behavior and HR has become normal. This is very compelling evidence of the role of antifragility in548

human health.549

In a general manner, Taleb [101] Suggests the so-called barbell (or bimodal) approach as an archetypal550

strategy for achieving antifragility. The first step towards antifragility is to reduce downsides instead551

of increasing upsides. In other words, by reducing exposure to adverse low probability but elevated552

18/25

adverse payoff occurrences (i.e. ’ ’ black swans ” events) and allowing natural antifragility to function on553

its own. We follow Taleb with a vulgar finance instance, where the idea is easiest to explain, although554

most of them are misunderstood. The barbel approach in finances comprises of placing 90% of your555

resources in safe instruments (provided that you are protected against inflation) or what is referred to as the556

”value repository number,” and 10% in very risky, maximal risky bonds, exposing yourself to unpredicted557

massive gains in a convex way. In this way, one ends up with some sort of bimodal optimization taking558

advantage of at the same time of the robustness of safe inversion and on the other hand the adaptability of559

high risky ones. Anyone who has a 100% stake in so-called ”medium” securities (unimodal optimization)560

is at danger of ”complete risk ruin”. This Barbel Strategy addresses the issue of incomputability and561

fragility in the assessment of the hazards of unusual occurrences.562

As in the barbell strategy, a basic mechanism to achieve antifragility, is a thorough strategy to risk563

management under fat-tailed distributions, and those are widely present in nature, then it should be very564

ubiquitous in natural systems. We identify this barbell risk strategy as the ”good balance” property in565

network topology by means of the relation of strong and weak interactions or between ascendancy and566

overhead; and balance robustness and adaptability, identified as fingerprint of critically (scale invariant567

and i/ f type of noise), in the dynamic of system’s fluctuations. All these three ”good balances” are568

related, as we showed with ecosystem integrity and resilience.569

Figure 11. Basic characteristics of systems in terms of antifragility, which is the

property of a system to respond in a convex way to perturbations or variability.

DISCUSSION AND CONCLUSIONS570

From the analysis of the literature, we found that the citation network from reviewed network is not571

percolated, what we interpret as a lack of unification in the research field and an opportunity for inter-572

19/25

disciplinary work. We found (see Figure 9) three main narratives (a) Ecosystem properties that enable573

them to be more resilient; (b) Ecosystem response to perturbations; and (c) Complexity. From this and574

complementary literature consulted we have identified 11 possible indicators for ecosystem resilience575

(See Table 5). In particular we show how to apply Fisher information in a study case which we consider a576

very promising proxy of resilience, since it has a solid formal framework, it is easy to implement and it577

can be applied to any kind of system.578

Key Indicator Measure/proxy Requires Resilience

FI Fisher Information Stability Time series

First, more stable ecosystem are more resilient

and according to Cabezas et al. for a system to be

resilient, after a disturbance the FI values prior to

it must be recovered

Div DiversityOptional / use of

resource spacePresence field data

In general to greater diversity, greater resilience

But there are exceptions related to changes of

composition and use of resources

Co Network Conectance Stability

Knowing the networks and being

able to quantify the intensity of the

connections, Gustavson proposes ways

to deal with the lack of information about it

Increase in the number of connections dissipates the effect of

variation in distribution of species and enhances stability species

Omn Presence of omnivore species Communication between different scales Presence of omnivore species Presence of omnivore species enhance stability and resilience

NC Network Criticality

Balance between

robustness (strong Interactions) and

adaptability (Weak Interactions)

Knowing the networks and being

able to quantify the intensity of the

connections, Gustavson proposes ways

to deal with the lack of information about it

Observations show that ecosystems are more resilient when

there is a good balance between the number of strong and weak

connections

L-VCLotka-Volterra

Coefficients

Given a community matrix,

if all the real parts of its

eigenvalues are negative the

ecosystem is stable

Community matrix More stable ecosystem are more resilient

As Ascendency Mean mutual information

Given a network of interactions (i.e trophic

network) it measures how well, on average,

the network articulates a flow event between

any two nodes.

Capture in a single index the ability of an ecosystem

to prevail against disturbance by virtue of its

combined organization and size.

Levy Levy Flights

Scaling coefficient of foraging

patterns for key species such

as puma or jaguar

It is a proxy of resources spatial complexityIt has been shown that Levy flights foraging patterns

are related and enhance ecosystems resilience

Frac Fractality Spatial complexity High resolution satellite images More complex ecosystems should be more resilient

AF AntifragilityChange in the complexity of a biotic (i.e trophic)

network, in the face of disturbances

Network of interactions, can be a

Boolean network of co-occurrences

of a key species such as puma or jaguar

with its prey for example

Resilience would be an intermediate state between

fragility and antifragility

H Homeostasis System Homeostasis Time Series Equivalent of resilience

Table 5. Resilience measure found in the literature review and complementary papers

Nevertheless a new way to reinterpret resilience emerged from this critical literature review: an-579

tifragility. This novel framework developed by N.N. Taleb [101, 103, 102] is based on fat-tailed, non-580

linear responses of the system to variability (see Figure 11). In a simple way, if a system has a concave581

(non-linear) payoff function dependent of certain variable, then the system is fragile to it. On the contrary,582

if the payoff in convex then it is antifragile and if the system is essentially insensible to variability, then583

is robust /resilient. In Taleb’s work, antifragility is associated with bimodal risk strategy called “The584

Barbell” which we believe manifest itself in the narratives as “a good balance” between (i) strong and585

weak interactions in network topology; (ii) adaptability and robustness (criticality); and (iii) ascendancy586

and overhead.587

In the long term, considering the coupling of ecosystem with human systems (i.e. via climate change)588

we consider that antifragility is a more desirable feature than resilience. Thinking in socioecosystems, we589

can see that they usually not only keep on living, but they do flourish and evolve, even in the presence590

of great stressors such as climate crisis or land change. In fact, in a recent work [19], It has been shown591

that the outcome of using antifragility as a design criterion is that the scheme being studied demonstrates592

a more favorable behavior than a ”simply” robust model in a setting that is susceptible to black swans593

(unpredictable, very low frequency of ocurrancebut vey high impact events) . Then, for socioecosystem594

governance, planning or in general, any decision making perspective, antifragility might be a valuable and595

more desirable goal to achieve than a resilience aspiration [7].596

REFERENCES597

[1] Abe, S., Sarlis, N., Skordas, E., Tanaka, H., and Varotsos, P. (2005). Origin of the usefulness of the598

natural-time representation of complex time series. Physical review letters, 94(17):170601.599

[2] Ahmad, N., Derrible, S., Eason, T., and Cabezas, H. (2016). Using fisher information to track stability600

in multivariate systems. Royal Society open science, 3(11):160582.601

[3] Aronson, J., Floret, C., Le Floc’h, E., Ovalle, C., and Pontanier, R. (1993). Restoration and rehabilita-602

tion of degraded ecosystems in arid and semi-arid lands. ii. case studies in southern tunisia, central603

chile and northern cameroon. Restoration ecology, 1(3):168–187.604

20/25

[4] Aronson, J. and Le Floc’h, E. (1996). Vital landscape attributes: missing tools for restoration ecology.605

Restoration Ecology, 4(4):377–387.606

[5] Bak, P., Tang, C., and Wiesenfeld, K. (1988). Self-organized criticality. Physical review A, 38(1):364.607

[6] Bartumeus, F. (2007). Levy processes in animal movement: an evolutionary hypothesis. Fractals,608

15(02):151–162.609

[7] Blecic, I. and Cecchini, A. (2018). Planning for antifragility and antifragility for planning. In610

International Symposium on New Metropolitan Perspectives, pages 489–498. Springer.611

[8] Boyer, D. and Lopez-Corona, O. (2009). Self-organization, scaling and collapse in a coupled automaton612

model of foragers and vegetation resources with seed dispersal. Journal of Physics A: Mathematical613

and Theoretical, 42(43):434014.614

[9] Brose, U. (2008). Complex food webs prevent competitive exclusion among producer species.615

Proceedings of the Royal Society B: Biological Sciences, 275(1650):2507–2514.616

[10] Brown, C. T., Liebovitch, L. S., and Glendon, R. (2007). Levy flights in dobe ju/’hoansi foraging617

patterns. Human Ecology, 35(1):129–138.618

[11] Cabezas, H. and Fath, B. D. (2002). Towards a theory of sustainable systems. Fluid phase equilibria,619

194:3–14.620

[12] Cabezas, H., Pawlowski, C. W., Mayer, A. L., and Hoagland, N. T. (2005). Sustainable systems621

theory: ecological and other aspects. Journal of Cleaner Production, 13(5):455–467.622

[13] Chapin Iii, F. S., Zavaleta, E. S., Eviner, V. T., Naylor, R. L., Vitousek, P. M., Reynolds, H. L., Hooper,623

D. U., Lavorel, S., Sala, O. E., Hobbie, S. E., et al. (2000). Consequences of changing biodiversity.624

Nature, 405(6783):234.625

[14] Courchamp, F., Chapuis, J.-L., and Pascal, M. (2003). Mammal invaders on islands: impact, control626

and control impact. Biological Reviews, 78(3):347–383.627

[15] Crabbe, P., Holland, A. J., Ryszkowski, L., and Westra, L. (2000). Implementing Ecological628

Integrity: restoring regional and global enivronmental and human health:[proceedings of the NATO629

Advanced Research Workshop on Implementing Ecological Integrity: Restoring Regional and Global630

Environmental and Human Health, Budapest, Hungary, June 26-July 1, 1999], volume 1. Springer631

Science & Business Media.632

[16] Dakos, V., Matthews, B., Hendry, A., Levine, J., Loeuille, N., Norberg, J., Nosil, P., Scheffer, M., and633

De Meester, L. (2018). Ecosystem tipping points in an evolving world. bioRxiv, page 447227.634

[17] Danchin, A., Binder, P. M., and Noria, S. (2011). Antifragility and tinkering in biology (and in635

business) flexibility provides an efficient epigenetic way to manage risk. Genes, 2(4):998–1016.636

[18] Dannemann, T., Boyer, D., and Miramontes, O. (2018). Levy flight movements prevent extinctions637

and maximize population abundances in fragile lotka–volterra systems. Proceedings of the National638

Academy of Sciences, 115(15):3794–3799.639

[19] de Bruijn, H., Großler, A., and Videira, N. (2019). Antifragility as a design criterion for modelling640

dynamic systems. Systems Research and Behavioral Science.641

[20] DeAngelis, D. (1980). Energy flow, nutrient cycling, and ecosystem resilience. Ecology, 61(4):764–642

771.643

[21] Doherty, T. S., Davis, R. A., van Etten, E. J., Algar, D., Collier, N., Dickman, C. R., Edwards, G.,644

Masters, P., Palmer, R., and Robinson, S. (2015). A continental-scale analysis of feral cat diet in645

australia. Journal of Biogeography, 42(5):964–975.646

[22] Doherty, T. S., Glen, A. S., Nimmo, D. G., Ritchie, E. G., and Dickman, C. R. (2016). Invasive preda-647

tors and global biodiversity loss. Proceedings of the National Academy of Sciences, 113(40):11261–648

11265.649

[23] Dutrieux, L. P. (2016). Multidimensional remote sensing based mapping of tropical forests and their650

dynamics. PhD thesis, Wageningen UR.651

[24] Eason, T. and Cabezas, H. (2012). Evaluating the sustainability of a regional system using fisher652

information in the san luis basin, colorado. Journal of environmental management, 94(1):41–49.653

[25] Equihua Zamora, M., Garcıa Alaniz, N., Perez-Maqueo, O., Benıtez Badillo, G., Kolb, M., Schmidt,654

M., Equihua Benıtez, J., Maeda, P., and Alvarez Palacios, J. (2014). Integridad ecologica como655

indicador de la calidad ambiental. Bioindicadores: Guardianes de nuestro futuro ambiental, pages656

695–718.657

[26] Fernandez, N., Maldonado, C., and Gershenson, C. (2014). Information measures of complexity,658

emergence, self-organization, homeostasis, and autopoiesis. In Guided self-organization: Inception,659

21/25

pages 19–51. Springer.660

[27] Filotas, E., Parrott, L., Burton, P. J., Chazdon, R. L., Coates, K. D., Coll, L., Haeussler, S., Martin,661

K., Nocentini, S., Puettmann, K. J., et al. (2014). Viewing forests through the lens of complex systems662

science. Ecosphere, 5(1):1–23.663

[28] Fleming, P. A., Anderson, H., Prendergast, A. S., Bretz, M. R., Valentine, L. E., and Hardy, G.664

E. S. (2014). Is the loss of a ustralian digging mammals contributing to a deterioration in ecosystem665

function? Mammal Review, 44(2):94–108.666

[29] Fossion, R., Rivera, A. L., and Estanol, B. (2018). A physicist’s view of homeostasis: how time series667

of continuous monitoring reflect the function of physiological variables in regulatory mechanisms.668

Physiological measurement, 39(8):084007.669

[30] Frieden, B. R. (2000). Physics from fisher information: a unification.670

[31] Gershenson, C. and Heylighen, F. (2003). When can we call a system self-organizing? In European671

Conference on Artificial Life, pages 606–614. Springer.672

[32] Gisiger, T. (2001). Scale invariance in biology: coincidence or footprint of a universal mechanism?673

Biological Reviews, 76(2):161–209.674

[33] Goldberger, A. L. (1992). Fractal mechanisms in the electrophysiology of the heart. IEEE Engineering675

in Medicine and Biology Magazine, 11(2):47–52.676

[34] Goldberger, A. L., Peng, C.-K., and Lipsitz, L. A. (2002). What is physiologic complexity and how677

does it change with aging and disease? Neurobiology of aging, 23(1):23–26.678

[35] Gonzalez-Mejia, A. M., Eason, T., Cabezas, H., and Suidan, M. T. (2012). Computing and interpreting679

fisher information as a metric of sustainability: regime changes in the united states air quality. Clean680

Technologies and Environmental Policy, 14(5):775–788.681

[36] Gough, C. M., Bond-Lamberty, B., Stuart-Haentjens, E., Atkins, J., Haber, L., and Fahey, R. (2017).682

Carbon cycling at the tipping point: Does ecosystem structure predict resistance to disturbance? In683

AGU Fall Meeting Abstracts.684

[37] Grimm, V. and Calabrese, J. M. (2011). What is resilience? a short introduction. In Viability and685

Resilience of Complex Systems, pages 3–13. Springer.686

[38] Gunderson, L. H. (2000). Ecological resilience—in theory and application. Annual review of ecology687

and systematics, 31(1):425–439.688

[39] Gustavson, K., Lonergan, S. C., and Ruitenbeek, J. (2002). Measuring contributions to economic689

production—use of an index of captured ecosystem value. Ecological Economics, 41(3):479–490.690

[40] Guy, G. A., Kosugi, T., and Sulzman, E. (2007). Ameriflux network aids global synthesis. Eos,691

88(28).692

[41] Haraway, D. (2015). Anthropocene, Capitalocene, Plantationocene, Chthulucene: Making Kin.693

Environmental Humanities, 6(1):159–165.694

[42] Harris, D. B. and Macdonald, D. W. (2007). Interference competition between introduced black rats695

and endemic galapagos rice rats. Ecology, 88(9):2330–2344.696

[43] Harrison, G. W. (1979). Stability under environmental stress: resistance, resilience, persistence, and697

variability. The American Naturalist, 113(5):659–669.698

[44] Hector, A., Dobson, K., Minns, A., Bazeley-White, E., and Lawton, J. H. (2001). Community699

diversity and invasion resistance: an experimental test in a grassland ecosystem and a review of700

comparable studies. Ecological Research, 16(5):819–831.701

[45] Ho, M.-W. and Ulanowicz, R. (2005). Sustainable systems as organisms? BioSystems, 82(1):39–51.702

[46] Holling, C. S. (1973). Resilience and stability of ecological systems. Annual review of ecology and703

systematics, 4(1):1–23.704

[47] Holling, C. S. (1996). Engineering resilience versus ecological resilience. Engineering within705

ecological constraints, 31(1996):32.706

[48] Hughes, J. and Macdonald, D. W. (2013). A review of the interactions between free-roaming domestic707

dogs and wildlife. Biological Conservation, 157:341–351.708

[49] Ivanov, P. C., Rosenblum, M. G., Peng, C.-K., Mietus, J., Havlin, S., Stanley, H. E., and Goldberger,709

A. L. (1996). Scaling behaviour of heartbeat intervals obtained by wavelet-based time-series analysis.710

Nature, 383(6598):323.711

[50] Karunanithi, A. T., Cabezas, H., Frieden, B. R., and Pawlowski, C. W. (2008). Detection and712

assessment of ecosystem regime shifts from fisher information. Ecology and society, 13(1).713

[51] Karunanithi, A. T., Garmestani, A. S., Eason, T., and Cabezas, H. (2011). The characterization of714

22/25

socio-political instability, development and sustainability with fisher information. Global Environmental715

Change, 21(1):77–84.716

[52] Kay, J. J. (1991). A nonequilibrium thermodynamic framework for discussing ecosystem integrity.717

Environmental Management, 15(4):483–495.718

[53] Kennedy, T. A., Naeem, S., Howe, K. M., Knops, J. M., Tilman, D., and Reich, P. (2002). Biodiversity719

as a barrier to ecological invasion. Nature, 417(6889):636.720

[54] Kiyono, K., Struzik, Z. R., Aoyagi, N., Togo, F., and Yamamoto, Y. (2005). Phase transition in a721

healthy human heart rate. Physical review letters, 95(5):058101.722

[55] Kleinen, T., Held, H., and Petschel-Held, G. (2003). The potential role of spectral properties in723

detecting thresholds in the earth system: application to the thermohaline circulation. Ocean Dynamics,724

53(2):53–63.725

[56] Kleinn, C. (1991). Zum waldbegriff in forstlichen großrauminventuren (on forest definitions in large726

area forest inventories. in german with english abstract).727

[57] Kleinn, C. (1992). On the compatibility of forest inventory results. In Conference on integreating728

forest management over space and time, pages 278–285.729

[58] Kleinn, C. (2001). A cautionary note on the minimum crown cover criterion in forest definitions.730

Canadian Journal of Forest Research, 31(2):350–356.731

[59] Landa, E., Morales, I. O., Fossion, R., Stransky, P., Velazquez, V., Vieyra, J. L., and Frank, A. (2011).732

Criticality and long-range correlations in time series in classical and quantum systems. Physical Review733

E, 84(1):016224.734

[60] Langdon, P. G., Dearing, J., Dyke, J., and Wang, R. (2016). Identifying and anticipating tipping735

points in lake ecosystems. Past Global Changes Magazine, 24:16–17.736

[61] Levin, S. A. (2005). Self-organization and the emergence of complexity in ecological systems.737

Bioscience, 55(12):1075–1079.738

[62] Lopez-Corona, O., Ramırez-Carrillo, E., and Magallanes-Guijon, G. (2019). The rise of the techno-739

bionts: toward a new ontology to understand current planetary crisis. Researchers One.740

[63] Loreau, M., Downing, A., Emmerson, M., Gonzalez, A., Hughes, J., Inchausti, P., Joshi, J., Norberg,741

J., and Sala, O. (2002). A new look at the relationship between diversity and stability. Biodiversity and742

ecosystem functioning: Synthesis and perspectives, pages 79–91.743

[64] Lund, H. G. (2006). Definitions of forest, deforestation, afforestation, and reforestation. Forest744

Information Services Gainesville, VA.745

[65] MacArthur, R. (1955). Fluctuations of animal populations and a measure of community stability.746

Ecology, 36(3):533–536.747

[66] Mandelbrot, B. B. (1982). The fractal geometry of nature, volume 2. WH freeman New York.748

[67] Manuel-Navarrete, D., Kay, J., and Dolderman, D. (2007). Evolution of the ecological integrity749

debate. Sustaining Life on Earth: Environmental and Human Health Through Global Governance,750

pages 127–138.751

[68] Martın-Martın, A., Orduna-Malea, E., and Lopez-Cozar, E. D. (2018). Coverage of highly-cited752

documents in google scholar, web of science, and scopus: a multidisciplinary comparison. arXiv753

preprint arXiv:1804.09479.754

[69] Mayer, A. L., Pawlowski, C., Fath, B. D., and Cabezas, H. (2007). Applications of fisher information755

to the management of sustainable environmental systems. In Exploratory data analysis using Fisher756

information, pages 217–244. Springer.757

[70] Mayer, A. L., Pawlowski, C. W., and Cabezas, H. (2006). Fisher information and dynamic regime758

changes in ecological systems. Ecological modelling, 195(1-2):72–82.759

[71] McNaughton, S. J. (1977). Diversity and stability of ecological communities: a comment on the role760

of empiricism in ecology. The American Naturalist, 111(979):515–525.761

[72] Michaelian, K. (2005). Thermodynamic stability of ecosystems. Journal of theoretical biology,762

237(3):323–335.763

[73] Miramontes, O., Boyer, D., and Bartumeus, F. (2012). The effects of spatially heterogeneous prey764

distributions on detection patterns in foraging seabirds. PloS one, 7(4):e34317.765

[74] Moore, J. C. (2018). Predicting tipping points in complex environmental systems. Proceedings of the766

National Academy of Sciences, 115(4):635–636.767

[75] Naeem, S., Knops, J. M., Tilman, D., Howe, K. M., Kennedy, T., and Gale, S. (2000). Plant diversity768

increases resistance to invasion in the absence of covarying extrinsic factors. Oikos, 91(1):97–108.769

23/25

[76] Newman, M. E. (2005). Power laws, pareto distributions and zipf’s law. Contemporary physics,770

46(5):323–351.771

[77] Noble, I. R. and Slatyer, R. (1980). The use of vital attributes to predict successional changes in plant772

communities subject to recurrent disturbances. Vegetatio, 43(1-2):5–21.773

[78] Parrondo, J. M., Horowitz, J. M., and Sagawa, T. (2015). Thermodynamics of information. Nature774

physics, 11(2):131.775

[79] Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E., and Goldberger, A. L. (1994).776

Mosaic organization of dna nucleotides. Physical review e, 49(2):1685.777

[Pimm] Pimm, S. L. The balance of nature?: ecological issues in the conservation of species and778

communities/stuart l. pimm.779

[81] Pineda, O. K., Kim, H., and Gershenson, C. (2018). Antifragility of random boolean networks. arXiv780

preprint arXiv:1812.06760.781

[82] Putz, M. V. and Chattaraj, P. K. (2013). Electrophilicity kernel and its hierarchy through softness in782

conceptual density functional theory. International Journal of Quantum Chemistry, 113(18):2163–2171.783

[83] Ramırez-Carrillo, E., Lopez-Corona, O., Toledo-Roy, J. C., Lovett, J. C., de Leon-Gonzalez, F.,784

Osorio-Olvera, L., Equihua, J., Robredo, E., Frank, A., Dirzo, R., et al. (2018). Assessing sustainability785

in north america’s ecosystems using criticality and information theory. PloS one, 13(7):e0200382.786

[84] Ramos-Fernandez, G., Mateos, J. L., Miramontes, O., Cocho, G., Larralde, H., and Ayala-Orozco, B.787

(2004). Levy walk patterns in the foraging movements of spider monkeys (ateles geoffroyi). Behavioral788

ecology and Sociobiology, 55(3):223–230.789

[85] Ravindran, S. (2016). Inner workings: Coral reefs at a tipping point. Proceedings of the National790

Academy of Sciences, 113(19):5140–5141.791

[86] Rayner, M. J., Hauber, M. E., Imber, M. J., Stamp, R. K., and Clout, M. N. (2007). Spatial792

heterogeneity of mesopredator release within an oceanic island system. Proceedings of the National793

Academy of Sciences, 104(52):20862–20865.794

[87] Regier, H. A. (1995). Ecosystem integrity in a context of ecostudies as related to the great lakes795

region. In Perspectives on Ecological Integrity, pages 88–101. Springer.796

[88] Reynolds, C. (2002). Resilience in aquatic ecosystems–hysteresis, homeostasis, and health. Aquatic797

Ecosystem Health & Management, 5(1):3–17.798

[89] Rivera, A. L., Estanol, B., Fossion, R., Toledo-Roy, J. C., Callejas-Rojas, J. A., Gien-Lopez, J. A.,799

Delgado-Garcia, G. R., and Frank, A. (2016a). Loss of breathing modulation of heart rate variability in800

patients with recent and long standing diabetes mellitus type ii. PloS one, 11(11):e0165904.801

[90] Rivera, A. L., Estanol, B., Senties-Madrid, H., Fossion, R., Toledo-Roy, J. C., Mendoza-Temis, J.,802

Morales, I. O., Landa, E., Robles-Cabrera, A., Moreno, R., et al. (2016b). Heart rate and systolic blood803

pressure variability in the time domain in patients with recent and long-standing diabetes mellitus. PloS804

one, 11(2):e0148378.805

[91] Saint-Beat, B., Baird, D., Asmus, H., Asmus, R., Bacher, C., Pacella, S. R., Johnson, G. A., David,806

V., Vezina, A. F., and Niquil, N. (2015). Trophic networks: how do theories link ecosystem structure807

and functioning to stability properties? a review. Ecological indicators, 52:458–471.808

[92] Schmeller, D. S., Weatherdon, L. V., Loyau, A., Bondeau, A., Brotons, L., Brummitt, N., Geijzen-809

dorffer, I. R., Haase, P., Kuemmerlen, M., Martin, C. S., et al. (2018). A suite of essential biodiversity810

variables for detecting critical biodiversity change. Biological Reviews, 93(1):55–71.811

[93] Seastedt, T. R., Hobbs, R. J., and Suding, K. N. (2008). Management of novel ecosystems: are novel812

approaches required? Frontiers in Ecology and the Environment, 6(10):547–553.813

[94] Sidle, R. C., Benson, W. H., Carriger, J. F., and Kamai, T. (2013). Broader perspective on ecosystem814

sustainability: Consequences for decision making. Proceedings of the National Academy of Sciences,815

110(23):9201–9208.816

[95] Sievert, C. and Shirley, K. (2014). Ldavis: A method for visualizing and interpreting topics. In817

Proceedings of the workshop on interactive language learning, visualization, and interfaces, pages818

63–70.819

[96] SILVA-RODRIGUEZ, E. A. and Sieving, K. E. (2011). Influence of care of domestic carnivores on820

their predation on vertebrates. Conservation Biology, 25(4):808–815.821

[97] Simberloff, D. (2011). How common are invasion-induced ecosystem impacts? Biological invasions,822

13(5):1255–1268.823

[98] Sornette, D. (2006). Critical phenomena in natural sciences: chaos, fractals, selforganization and824

24/25

disorder: concepts and tools (springer series in synergetics).825

[99] Steffen, W., Crutzen, P. J., and McNeill, J. R. (2007). The anthropocene: are humans now overwhelm-826

ing the great forces of nature. AMBIO: A Journal of the Human Environment, 36(8):614–621.827

[100] Steffen, W., Rockstrom, J., Richardson, K., Lenton, T. M., Folke, C., Liverman, D., Summerhayes,828

C. P., Barnosky, A. D., Cornell, S. E., Crucifix, M., et al. (2018). Trajectories of the earth system in the829

anthropocene. Proceedings of the National Academy of Sciences, 115(33):8252–8259.830

[101] Taleb, N. N. (2012). Antifragile: how to live in a world we don’t understand, volume 3. Allen Lane831

London.832

[102] Taleb, N. N. (2018). (anti) fragility and convex responses in medicine. In International Conference833

on Complex Systems, pages 299–325. Springer.834

[103] Taleb, N. N. and Douady, R. (2013). Mathematical definition, mapping, and detection of (anti)835

fragility. Quantitative Finance, 13(11):1677–1689.836

[104] Ulanowicz, R. E. (1983). Identifying the structure of cycling in ecosystems. Mathematical Bio-837

sciences, 65(2):219–237.838

[105] Ulanowicz, R. E. (1986). A phenomenological perspective of ecological development. In Aquatic839

Toxicology and Environmental Fate: Ninth Volume. ASTM International.840

[106] Ulanowicz, R. E., Goerner, S. J., Lietaer, B., and Gomez, R. (2009). Quantifying sustainability:841

resilience, efficiency and the return of information theory. Ecological complexity, 6(1):27–36.842

[107] Vidal, C., Lanz, A., Tomppo, E., Schadauer, K., Gschwantner, T., di Cosmo, L., Robert, N., et al.843

(2008). Establishing forest inventory reference definitions for forest and growing stock: a study towards844

common reporting.845

[108] Viswanathan, G. M., Buldyrev, S. V., Havlin, S., Da Luz, M., Raposo, E., and Stanley, H. E. (1999).846

Optimizing the success of random searches. nature, 401(6756):911.847

[109] Walker, B., Holling, C. S., Carpenter, S. R., and Kinzig, A. (2004). Resilience, adaptability and848

transformability in social–ecological systems. Ecology and society, 9(2).849

[110] West, B. J. (2010). Fractal physiology and the fractional calculus: a perspective. Frontiers in850

physiology, 1:12.851

[111] Wilson, E. H. and Sader, S. A. (2002). Detection of forest harvest type using multiple dates of852

landsat tm imagery. Remote Sensing of Environment, 80(3):385–396.853

[112] Woinarski, J. C., Burbidge, A. A., and Harrison, P. L. (2015). Ongoing unraveling of a continental854

fauna: decline and extinction of australian mammals since european settlement. Proceedings of the855

National Academy of Sciences, 112(15):4531–4540.856

[113] Worm, B., Barbier, E. B., Beaumont, N., Duffy, J. E., Folke, C., Halpern, B. S., Jackson, J. B.,857

Lotze, H. K., Micheli, F., Palumbi, S. R., et al. (2006). Impacts of biodiversity loss on ocean ecosystem858

services. science, 314(5800):787–790.859

[114] Wyatt, K. B., Campos, P. F., Gilbert, M. T. P., Kolokotronis, S.-O., Hynes, W. H., DeSalle, R.,860

Daszak, P., MacPhee, R. D., and Greenwood, A. D. (2008). Historical mammal extinction on christmas861

island (indian ocean) correlates with introduced infectious disease. PloS one, 3(11):e3602.862

[115] Zellner, M. L., Theis, T. L., Karunanithi, A. T., Garmestani, A. S., and Cabezas, H. (2008). A863

new framework for urban sustainability assessments: Linking complexity, information and policy.864

Computers, environment and urban systems, 32(6):474–488.865

[116] Zorach, A. C. and Ulanowicz, R. E. (2003). Quantifying the complexity of flow networks: how866

many roles are there? Complexity, 8(3):68–76.867

25/25