Ecosystem Antifragility: Beyond Integrity and Resilience

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

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

Miguel Equihua Zamora 1 , Mariana Espinosa 2 , Carlos Gershenson 3, 4, 5 , Oliver López-Corona Corresp., 4, 6, 7 , Mariana Munguia 8 , Octavio Pérez-Maqueo 1 , Elvia Ramírez-Carrillo 9

1 Red ambiente y sostenibilidad, Instituto de Ecología A.C, Xalapa, Veracruz, Mexico 2 Doctorado en Ciencias Sociales y Humanidades, UAM-Cuajimalpa., cdmx, cdmx, Mexico 3 IIMAS, Universidad Nacional Autónoma de México, cdmx, Mexico 4 Centro de Ciencias de la Complejidad (C3), Universidad Nacional Autónoma de México, cdmx, cdmx, Mexico 5 ITMO University, St. Petersburg, 199034, Russian Federation, St. Petersburg, Russia 6 Cátedras CONACyT, Comisión Nacional para el Conocimiento y Uso de la Biodiversidad (CONABIO), cdmx, Mexico 7 Red ambiente y sostenibilidad, Instituto de Ecología A.C., Xalapa, Veracruz, Mexico 8 Comisión Nacional para el Conocimiento y Uso de la Biodiversidad (CONABIO), CDMX, Mexico 9 Facultad de Psicología, Universidad Nacional Autónoma de México, cdmx, Mexico

Corresponding Author: Oliver López-Corona Email address: [email protected]

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.

PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.27813v1 | CC BY 4.0 Open Access | rec: 21 Jun 2019, publ: 21 Jun 2019 1 Ecosystem Antifragility: Beyond Integrity

2 and Resilience. 2,* 8,* 3,5,6,* 1,2,3,* 3 M. Equihua , M. Espinosa , C.Gershenson , O. Lopez-Corona´ , 7,* 2,* 4,* 4 M. Mungu´ıa , O. Perez-Maqueo´ , and E. Ram´ırez-Carrillo

1 5 Catedras´ CONACyT, Comision´ Nacional para el Conocimiento y Uso de la 6 Biodiversidad (CONABIO), CDMX, Mexico´ 2 7 Red ambiente y sostenibilidad, Instituto de Ecolog´ıaA.C., Xalapa, Mexico´ 3 8 Centro de Ciencias de la Complejidad (C3), Universidad Nacional Autonoma´ de 9 Mexico,´ CDMX, Mexico´ 4 10 Facultad de Psicolog´ıa,Universidad Nacional Autonoma´ de Mexico,´ CDMX, Mexico´ 5 11 IIMAS, Universidad Nacional Autonoma´ de Mexico,´ CDMX, Mexico´ 6 12 ITMO University, St. Petersburg, 199034, Russian Federation. 7 13 Comision´ Nacional para el Conocimiento y Uso de la Biodiversidad (CONABIO), CDMX, 14 Mexico´ 8 15 Doctorado en Ciencias Sociales y Humanidades, UAM-Cuajimalpa.

16 Corresponding author:

17 Following the Hardy-Littleton rule, all authors will appear in alphabetical order. And all are ∗ 18 corresponding authors

19 Email address: [email protected]; [email protected]; [email protected]; 20 [email protected]; [email protected]; [email protected]; 21 [email protected]

22 ABSTRACT

23 We review the concept of ecosystem resilience in its relation to ecosystem integrity from an information 24 theory approach. We summarize the literature on the subject identifying three main narratives: ecosystem 25 properties that enable them to be more resilient; ecosystem response to perturbations; and complexity. 26 We also include original ideas with theoretical and quantitative developments with application examples. 27 The main contribution is a new way to rethink resilience, that is mathematically formal and easy to 28 evaluate heuristically in real-world applications: ecosystem antifragility. An ecosystem is antifragile if 29 it beneﬁts from environmental variability. Antifragility therefore goes beyond robustness or resilience 30 because while resilient/robust systems are merely perturbation-resistant, antifragile structures not only 31 withstand stress but also beneﬁt from it.

32 1 INTRODUCTION

33 Sustainable development needs to preserve the structure and functioning of natural ecosystems, i.e. their 34 integrity, as a Sine qua non condition. In previous work [25] an operational framework has been 35 developed to quantify ecosystem integrity as well as viable standards useful for managing the way people 36 intervene ecosystems, promoting development along sustainable avenues.

37 Humans are starting to be recognized as an overwhelming forcing factor modulating biosphere 38 dynamics. In this view, the Earth system can be interpreted as entering in a geological era that can 39 be called the Anthropocene [99] the Technocene [62] or even Capitalocene [41], depending on the 40 conceptual stance adopted. Because of this driving influence of human decisions, there has been long 41 interest in understanding and measure the way ecosystems recover (or not) from human perturbations. To 42 operationalize sustainability, we require working definitions of this ecosystem ability and metrics to asses 43 it, in addition to ecosystem integrity (as a component of a likely state variable accounting for the amount 44 of natural capital assets: condition × extension). The recovery property has already been encapsulated 45 into the ecosystem resilience concept. Here, we propose that information theory is a suitable framework 46 to encompass both ecosystem integrity and resilience. 47 According to [25], numerous studies have aimed to find a suitable and inclusive definition of ecological 48 integrity; however, no general consensus has been achieved to date. In their work, Equihua and co- 49 workers embrace a complex systems approach in which ecosystems are considered as self-organized 50 entities constrained in their structure (including biological composition or biodiversity), and function by 51 thermodynamic dissipative system properties [52, 87, 67, 72] and evolutionary processes [61].

52 The concept of ecosystem resilience was first introduced by Holling [46] to portray the persistence of 53 natural structures in the presence of environmental stressors due to natural or anthropogenic triggers. In 54 his seminal work, Holling follows a system dynamics analysis. There, resilience concept is defined as: 55 “the capacity of a system to absorb disturbance and reorganize while undergoing a change so as to still 56 retain essentially the same function, structure, identity, and feedbacks” [109]. Nevertheless, resilience 57 or stability, which is commonly used as synonym, have at least 163 different definitions (grouped in at 58 least 70 concepts of stability/resilience) [37]. For example, Saint-Beat´ and co-workers [91] summarize 59 resilience and others related concepts often used interchangeably as follows:

60 • Resilience is the rate at which a system returns after a disturbance to the equilibrium state [20, 61 Pimm]. Long return time is equivalent to low resilience. A community’s resilience relies on the 62 least resilient species (the slowest to return to equilibrium). This deﬁnition of resilience corresponds 63 to the ‘engineering resilience’ deﬁned by [47] And assumes that there is only one balance or a 64 stable state [38].

65 • Persistence is the time for a variable to remain in the same state before changing to a different one 66 [Pimm]. Persistence is a measure of a system’s capacity to preserve itself over time. [63]

67 • Resistance is described as the capacity of an ecosystem in the presence of external disturbance to 68 preserve its initial state. [43]. Only small changes (in amount and intensity) within an ecosystem 69 correspond to high resistance. This concept is similar to the ‘ecological resilience’ deﬁned by [47] 70 and Suggests that various stable states exist.

71 • Robustness relates to the durability of the stability of the environment. Robustness is then a measure 72 of the amount of disturbance an ecosystem can endure before it changes to a different state. [63].The 73 more robust the food web is, the more stable it is.

74 In this paper, we ascertain that both integrity and resilience frameworks can be formalized using 75 information theory. The second law of thermodynamics is probabilistic by its very nature, because its 76 formulation involves a probabilistic description of the state of a system [78]. It is customary in information 77 theory to relate the Shannon information of a random variable X (microscopic state of a physical system) 78 with the thermodynamic entropy of the system. This relation can be expressed as

H(X) = −k∑[p(x)log p(x)], (1) x

79 where p(x) is the probability density and assuming k as the Boltzmann’s constant. Even more, once 80 the connection between entropy (thermodynamics) and Shannon information (information theory) is 81 made, we are able to calculate self-organization as the complement (1 − H(X)) of Shannon information 82 [26]. Following [25, 15], an ecosystem that is in a high level of integrity is at full capacity to develop as 83 a natural consequence of the continued operation of the processes of self-organization to dynamically 84 incorporate the original species set at its location, which can be interpreted as the mechanistic basis of 85 what is known as ecosystem resilience.

86 It is clear then that in order to construct a unified narrative for ecosystem integrity and resilience, 87 information theory is a suitable framework candidate. With that aim, we present in this paper a critical 88 review of the literature and then develop a novel proposal related to what we consider a suitable concept 89 not incorporated in dynamic ecosystem understanding so far: antifragility, which is the non linear (convex) 90 response of a system to profit from variability in the payoffs space, or in simple words the ability of a 91 system to benefit from surrounding randomness.

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.

92 2 SURVEY METHODOLOGY

93 It has recently been shown that Web of Science and Scopus is invisible to a big proportion of highly-cited 94 papers in the social sciences and humanities. And even when the percentage of missing highly cited 95 papers in Web of Science (WOS) and Scopus is in the natural, lives and social sciences. The Spearman 96 quotation correlation coefficients in Google Scholar are more powerful in all fields compared to Web of 97 Science and Scopus. The researchers conclude that highly cited papers available in the inclusive Google 98 Scholar database actually show important deficiencies in the coverage of the Web of Science and Scopus 99 in certain study fields. Consequently, using these selective databases to calculate bibliometric indices 100 based on the number of highly cited papers could generate partial evaluations in poorly covered fields 101 [68].

102 For these reasons we choose Google Scholar as the search engine in which we use the term: “Ecosys- 103 tem Resilience” AND “Information theory” AND “ “Ecosystem Integrity”. After excluding patents and 104 citations we end up with 20 items. Of those 20 co-occurrences, books and patents were discarded, leaving 105 10 entries that where fully read. Finally, only 7 items were selected to be analyzed and included in the 106 present review. However, Google Scholar, does not provide easy access to cited references, which indicate 107 the knowledge background of the selected items. For that reason, the analysis was completed with data 108 from the Web of Science and the Astrophysics Data System to visualise their cited references network 109 using Science of Science (Sci2) Tool and Gephi. To further analyze the literature set, a text corpus was 110 assembled taking the text of the title and abstract of target documents as a bag of words. This small corpus 111 was then processed with latent Dirichlet allocation (LDA) and latent semantic analysis (LSA) techniques.

3/25 112 FINDINGS

113 Basic scientometrics i.e word clouds 114 In Table 1 we show basic metrics for papers selected and in Fig. 2 the results of applying LDA analysis 115 to this corpus, based on [95] with an interactive viz that can be opened in any browser. A video explaining 116 the use of this kind of viz could be found in here. According with LDA, papers can be allocated to the 117 topics as indicated in Table 2; although, topics can be concurrently present in any paper.

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.

118 In Fig. 3 we present a main axes plot based in a latent semantic analysis (LSA). Both LDA and LSA 119 suggest it is possible to recognize four groups in the corpus, which are further discussed in the section 120 analyzing perceived literature narratives below. In addition, the analysis of the citation network reveals 121 several unconnected sub-groups, while the cited references in general are very poorly connected. We 122 interpret this ﬁndings as evidence of poor interdisciplinary crossover on the conceptual development of 123 ecological resilience and integrity, which prompted our interest in the issues we discuss in this paper.

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

124 In Fig.5 we show a TreeMap for the type of documents that cite the set of reviewed ones; in Fig6 The 125 organizations of origin for the documents that cite the set of reviewed ones; in Fig.7 we show the number 126 of documents that cite the set of reviewed ones; and in Fig.8 the distribution of documents that cite the set 127 of reviewed ones in terms of research ﬁeld.

128 2.1 Literature narratives 129 We consider that in the context of the relation of resilience with ecological integrity under the lens of 130 information theory, narratives has gone from trying to identify (with information theory tools) ecosystem 131 features that allow them to be more resilient and hence maintain their integrity, to a more technical 132 approach using times series or network analysis in addition to new mathematical concepts (see fig. 9). 133 The importance of interactions and a complex system approach is highlighted. Finally, the field completes 134 a circle getting again back into ecology and refining resilience feature and properties of ecosystems.

135 2.2 Resilience features and properties I 136 Noble and Slayter [? ] have described several categories of essential life history characteristics that are 137 helpful in determining a species ’ reaction to recurrent disturbances. In the revised paper by Aronson 138 and co-workers [4], the authors make reference to a previous work [3] where they modiﬁed the Noble 139 and Slayter’s concept [77], deﬁning Vital Ecosystem Attributes (VEAs). The VEAs attempt to capture

5/25 Figure 3. Latent semantic analysis of papers

140 those features or attributes that are correlated with and that can serve as ecosystem structure and function 141 indicators, the same attributes in the Ecosystem Integrity Model (3TEI) instrument layer from Equihua 142 and colleagues [25]. In that way, one may interpret that a resilient ecosystem is one for which VEAs fall 143 into an optimal range, represented as ecosystem integrity. But unlike the 3TEI model, VEAs requires 144 intense (often cost and not scalable) ﬁeldwork that most likely make VEAs not such a good option for a 145 national assessment of ecosystem condition trends.

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

146 From the idea of VEAs (see Table3) Aronson and his colleagues [4] extracted the insight into 147 considering 16 quantiﬁable characteristics for use on a more particular spatial scale: the landscape as a 148 multifunctional environment. The writers intend to use the fresh Vital Landscape Attributes (VLAs) to 149 evaluate the outcomes of ecological restoration or rehabilitation conducted from the view of the landscape. 150 Most interesting is the chance that as VEAs relate to the integrity of the ecosystem, VLAs (see Table?? 151 could give way to a 3TLI model.

152 In a different line of thought, the reviewed paper by Gustavson and co-workers [39] develops a 153 general index that may serve as a proxy of ecosystem resilience from an information theory perspective. 154 The authors report that attempts have been made to describe and evaluate resilience, but an overall 155 predictive or theoretical connection between resilience characteristics and ecosystem dynamics has yet

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

156 to be advanced. Similarly, much has been discussed about possible interactions between stability and 157 structure and, generally speaking, predictable interactions between resilience characteristics and how

8/25 Figure 9. Summary of concepts and narratives in selected papers

158 ecosystems work are not intuitively evident and may not exist. 159 To this end, they turn to Ulanowicz’s Ascendancy Theory [? ] which is a measure of the magnitude of 160 the information ﬂow through an ecosystem’s network framework. Some constraints for its use are that it 161 requires a comparatively full description of the nature and magnitude of all species interactions. 162 Ascendancy is deﬁned by the average mutual information as presented [105] between component a to 163 b is:

p b j|ai As = K∑∑p(ai,b j)log , (2) i j " p b j #

164 where, p b j|ai , the probability of b j given that ai has occurred; and p(b j), the probability that b j 165 will occur.

166 The upper limit of ascendancy is the capacity for growth, and the distinction between ability and ascen- 167 dancy is called the overhead system, which represents a multiplicity of paths and can therefore eventually 168 be linked with the complexity of the ecosystem.[116]. A complex structure is key for sustainability [45] 169 because the diversity of processes plays a crucial role on system survives [106]. In particular, to enhance 170 ecosystem’s long-term sustainability, a particular densely connected network structure is advantageous. 171 Such a scheme is sufficiently effective and sufficiently varied. Efficiency-diversity equilibrium is essential 172 to ecosystem resilience. Ultimately, ascendancy captures in a single index the capacity of an ecosystem to 173 prevail against disruption by virtue of its combined organisation and size, it was suggested that, in order 174 to attain sustainability, it should always be possible for systems of human use to regain their ascendancy 175 within culture and ecosystems [88].

176 2.3 Ecosystems response to perturbations 177 The main interest in the selected paper form Cabezas and co-workers [12] is not ecosystem integrity nor 178 resilience per se but sustainability. A concept they relate to integrated systems comprising humans and the 179 rest of nature (probably a socioecosystem). They emphasize that the structures and operation of the human 180 element (in terms of culture, economy, law, etc.) must be such that they strengthen or encourage the 181 persistence of the natural component’s structures and operation (in terms of ecosystem trophic connections, 182 biodiversity, biogeochemical cycles, etc.), and vice versa. It is in the idea that ”persistence” and ”operation 183 of the natural component” that the connection is made. From their perspective a sine qua non condition to 184 achieve sustainability [83] is ecosystem stability which they conceptualize using Fisher information [11], 185 view that is further developed in several papers [24, 50, 70, 69, 115, 2, 51, 35]. 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 θˆ that is a function of an imperfect observation θˆ(y). This is a closed system, meaning that it’s well described by {y,θˆ,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 θˆ(y) = θ. In this case, the mean-square error obeys the Cramer-Rao inequality

e2 I ≥ 1, (3)

186 where I is the Fisher Information of the system, calculated as

dy dP (y|θ) 2 I = 0 , (4) P (y|θ) dθ Z 0 187 in which P0(y|θ) is the probability density function of measuring a particular value of y given the true 188 value θ of the state variable in question. Then, since the error decreases as information increases, Fisher 189 information may be understood as the quality of the estimation θ from a smart measurement. Then, 190 if the system is characterized by a phase space with m state variables xi that deﬁne the phase vector 191 s = (x1,...,xi,...,xm) associated with a smart measurement y, then we can prove that

T 1 s′′2 I (s) = dt, (5) T s′4 Z0 ′ ′′ 192 where T is the time period required for one cycle of the system; s (t) is the tangential speed and s (t) is 193 scalar acceleration tangential to the system path in phase space. Both are calculated in terms of the state 194 variables xi as

m dx 2 s′ (t) = ∑ i , (6) dt s i 1 m dx d2x s′′ (t) = ∑ i i . (7) s′ (t) dt dt2 i

10/25 195 A simple and robust approach to calculating tangential velocity and acceleration uses the three-point 196 difference scheme dx αx (t + ∆t ) − (α2 − 1)x (t) − x (t − α∆t ) i = i a i i a , (8) dt α(α + 1)∆ta 2 d xi αxi(t + ∆ta) − (α + 1)xi(t) − xi(t − α∆ta) = 2 , (9) dt α(α + 1)∆ta /2

197 where xi(t) is a central data point, xi(t − ∆ta) is the next point following the center xi and xi(t − ∆tp) is 198 the previous point to it. For evenly-spaced points ∆ta = ∆tp and α = ∆tp/∆ta is the ratio of the previous 199 and following time space.

200 The thesis suggested by [? ] is that a shift in Fisher information may signal a change of regime in a 201 dynamic system to [? ]:

202 • Fisher information is a function of measurement variability. Low variability results in high Fisher 203 information and low Fisher information results in high variability.

204 • Systems in stable regime tends to exhibit constant Fisher information. Then, organization losses 205 points to greater variability and a decrease of Fisher information.

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

207 • ”If resilience is deﬁned by the intensity, frequency, and duration of a perturbation that a system can 208 withstand before fundamentally changing in function and structure, then we would hypothesize that 209 Fisher information would return to the same value or higher in more resilient systems.” [12]

210 We found this last point of much interest because not only it provides a formal definition of the 211 resilience concept, but it also provides a specific way to measure it via Fisher information. In order to test 212 this idea, we analyze NDVI data for ”US-Me1: Metolius - Eyerly burn” Ameriflux site [40] site in Oregon 213 for which is documented as an intermediately aged ponderosa pine forest that was severely burned in 214 the 2002 Eyerly wildfire. The AmeriFlux network of approximately 100 research stations is the main 215 research group and information supplier for big terrestrial carbon cycling syntheses in the Americas and 216 has established a database for micrometeorological, meteorological and biological information.

217 Data of NDVI was downloaded using the application for Extracting and Exploring Analysis Ready 218 Samples (AppEEARS) that enables users to subset geospatial data-set using spatial, temporal, and 219 band/layer parameters (https://lpdaacsvc.cr.usgs.gov/appeears/). In particular, we 2 220 used MOD13A3.006 1km resolution monthly data of NDVI from 01-01-1990 to 01-01-2018.

221 Focusing into the 2002 wildﬁre, we show in Figure10 that as expected, with the wildﬁre disturbance 222 the system experience both great changes in NDVI and its corresponding Fisher information. We found 223 that Fisher information returns to previous values after 18 months approximately but not the NDVI values.

224 On the one hand, recovering Fisher information could be related to the way Filotas and co-workers 225 [27] understand ecological resilience as ”the amount of change that an ecosystem can absorb before it 226 loses its ability to maintain its original function and structure”. After a disturbance, the authors claim that 227 a resilient system has the ability to recover its initial structure, features, and feedback ; in other words, its 228 integrity.

229 On the other hand, it seems then that the criterion of Fisher information is necessary but not sufﬁcient 230 to ensure ecosystem resilience. For example one should expect that after a disturbance, essential variables 231 as the ones proposed by Schmeller and co-workers [92], which could be seen as a modern version of 232 Aronson’s Vital Ecological Attributes [4], return to previous values. In a recent work Dutrieux [23] 233 combine into one index signaling from TM and ETM+ B4 band, corresponding to Near Infra Red 234 (NIR) with wavelength of 770 − 900nm which provide information about canopy biomass; and B5 band 235 corresponding to Short Wave Infra-Red (SWIR1) with wavelength of 1550 − 1750nm which provide 236 information about canopy moisture content:

NIR − SWIR1 NDMI = . (10) NIR + SWIR1

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)

237 Low NDMI values for bare soils and thin forest canopies are anticipated, while greater values 238 correspond to thicker, completely developed forest canopies. [111]. 239 The author created the following harmonic model to compare values before and after a disturbance:

3 2π jt yt = αi+ ∑ γisin + δ j + εt , (11) f 1

240 where the dependent variable y at a given time t is expressed as the sum of an intercept αi, a sum 241 of different frequency harmonic components representing seasonality and an error εt . In the model j 242 corresponds to the harmonic order, 1 being the annual cycle, γ j and δ j correspond respectively to the 243 amplitude and phase of the harmonic order j, and f is the known frequency of the time-series (i.e., number 244 of observations per year). 245 New values are then estimated for each spectral band and each time series observation using the 246 corresponding matched model, enabling Euclidean distance to be calculated with the following formula.:

k D = (yˆ − y )2. (12) t v∑ it it ui=1 u t 247 The author then applies it to spectral recovery time for a set of 3596 Landsat time-series sampled 248 from regrowing forests across the Amazon basin, thus producing estimates of recovery time in spectral 249 properties, which he calls spectral resilience. On average, he found that spectral resilience takes about 7.8 250 years, with a large variability (sd =5.3 years) for disturbed forests to recover their spectral properties. Now 251 we have a new problem, how to determine the thresholds for (a) distance between initial and ﬁnal values 252 for both state (essential/vital) variables and their Fisher information; (b) the time scale these recovering 253 should occur. In principle we believe (a) could be determined from ecological integrity measurements, 254 but it is currently an open research question we are not addressing here.

12/25 255 In another line of thoughts, Sidle and co-workers [94] focus on ascertaining under what circumstances 256 ecosystems exhibit resilience, tipping points or episodic resetting. They point out that while ecosystem 257 resilience originated from ecological perspective, latest debates concentrated on geophysical characteris- 258 tics and that it is acknowledged that dynamic system properties may not return to their former state after 259 disturbances (see for example [16, 36, 60, 85, 74, 100]). Tipping points generally arise when chronic 260 (typically anthropogenic but sometimes natural) changes push ecosystems to thresholds that cause process 261 and function collapse even in a permanent way. Resetting ecosystems happens when episodic natural 262 disasters break thresholds with little or no warning resulting in long-term modiﬁcations in environmental 263 characteristics or functioning of the ecosystem. Of special interest is the work of Steffen and co-workers 264 [100] who consider earth biosphere as a whole system and study its possible trajectories under the current 265 planetary crisis. In particular, they explore the risk of self-reinforcing feedback that could eventually push 266 the Earth’s biosphere system to a planetary threshold that, if crossed, could prevent climate stabilization 267 near the Holocene temperature regime (the pre-industrial conditions set out in the Paris Agreement). In 268 the worst case scenario Earth could be driven into the ongoing warming track of a ”Hothouse Earth” path, 269 even though human emissions were lowered.

270 As in other papers reviewed, Sidle and co-workers [94], state that ”if a system is viewed as resilient, it 271 is generally perceived as remaining within speciﬁed bounds, probably close to the optimal operational 272 points” mentioned in [25]. Which sets again the question of which should be the variables under the 273 “bounded ecosystem” and how to determine the range of values to consider the ecosystem as resilient. 274 More to the point, how much time should be spanned between an ecosystem perturbation for the resilience 275 variable returning to their bound limits? In principle, we consider that this should be in the same order of 276 magnitude that the -natural characteristic time scale of the ecosystem. But once again, the measurement 277 of characteristic time scale for an arbitrary state variable of the ecosystem is an open question. The 278 main problem is that in most cases we will not have a mechanistic model for the variable in question 279 but time-series only. In [1] the authors use the Wigner function to explore if there is an special time 280 scale under which the system reaches an optimal representation. For multiple time series observed, they 281 contrasted entropy values covering a variety of distinct time domains. For their natural characteristic time, 282 they found that entropy is highly likely to be minimal, implying minimum uncertainty in time-frequency 283 space. Another alternative might be to consider the τ0 time in which the system’s memory tends to 284 be zero, deﬁned by the absolute τ time value for which the C(τ) auto-correlation function crosses the 285 horizontal axis [? ].

286 2.4 Complexity perspective

287 The Filotas and co-workers reviewed paper [27] provides a remarkable introduction to complexity. The 288 authors decompose complexity into eight features an then goes to relate them into a new narrative for 289 forests, making as a result an interesting connection with resilience and integrity. Generally speaking, a 290 system is complex either it presents a sufﬁciently number of components with strong enough interaction 291 or it exhibits changes in the conﬁguration space comparable to the observer’s time scale, and in most 292 cases both. Forests as a system and forest management, certainly occupy a high position in the complexity 293 gradient.

294 The authors focus on forests, but clearly what they describe is applicable to all types of ecosystems. 295 Nevertheless, forests are a good model because they are both widely and intensively managed, and also 296 because they are deeply coupled with human systems. The designed approach can thus assist forest 297 scientists and managers in conceptualizing forests as integrated socio-ecological systems and provide 298 concrete examples of how to manage forests as complex adaptive systems.

299 There are at least 800 different definitions of a forest. Some of them are used simultaneously in the 300 same country for different purposes or scales [64]. This is in part due to the fact that forest types differ 301 widely, depending on factors such as latitude, climate patterns, soil properties, and human interactions. It 302 also depends on who is defining it. An economist could describe a forest in a very distinct manner to a 303 forester or a farmer, in accordance with their specific interests. One of the most widely used definitions is 304 that by FAO (1998), that defines a forest as ”the track of land with area over 0.5 ha, tree canopy cover 305 larger than 10%, which is not primarily subject to agricultural or other specific non-forest uses”. For young 306 forests or regions where tree growth is suppressed by climatic factors, trees should be capable of reaching a 307 height of at least 5 m in situ while meeting the requirement for canopy cover. In general, forest definitions 308 are based on two different perspectives. One, associated with quantitative cover/density variables such as

13/25 309 minimum area cover, minimum tree height, or minimum crown size. The other, relates to characteristic 310 spatial features of the territory such as the presence of plantations, agricultural activities or non-forest 311 trees within the forest itself [56, 57, 58, 64, 107]. The issue is that when natural forests are significantly 312 degraded or superseded by plantations, essential ecosystem services such as CO2 sequestration may be 313 lost ; but they may technically continue to be classified as forests under many definitions.

314 Thus, a fundamental key characteristic of ecosystems is essentially missing in most forest definitions, 315 its complexity. The authors point out that complex systems science provides a transdisciplinary framework 316 to study systems characterized by (1) heterogeneity, (2) hierarchy, (3) self-organization, (4) openness, (5) 317 adaptation, (6) memory (homeostasis?), (7) non-linearity, and (8) uncertainty. These eight characteristics 318 are shared by complex systems regardless of the nature of their constituents and the article exemplifies 319 them in ”forest terms”. The conclusion they reach in terms that complex systems approach has inspired 320 both theory and applied approaches to improve ecosystem resilience and adaptability is most relevant 321 to our article. While forests are prime examples of complex systems (Perry 1994), forest ecology and 322 management approaches are only starting to emerge and complex systems are seldom invoked in the field.

323 Heterogeneity or how ecosystem components are distributed in space is important from several points 324 of view. For example, the spatial distribution of resources imposes restrictions on animal foraging and 325 ultimately significant patterns. It has widely been proved that the foraging patterns of a variety of animals 326 involve many spatio-temporal scales, as described by Levy´ walks [108, 84, 73]. This statistical behavior 327 is present even in human movement patterns [10], and has been linked to evolutionary advantages in terms 328 of search strategies in complex environments [6]. Further more, resources distribution patterns induce 329 foraging behaviours linked to seed dispersal. Those patterns feedback into the ecosystem dynamics and 330 influence the distribution of resources in time [8]. In the reviewed paper, the authors point out that human 331 influence on ecosystems may reduce ecosystem complexity by altering spatial heterogeneity. For instance, 332 forest cover and resident biota have been homogenized by intensifying and standardizing cultivation 333 methods within and between woodlots. Changing these patterns can significantly influence the capacity 334 of the landscape to maintain materials and energy effectively processed and host the region’s biota; this 335 change in turn decreases its integrity, resulting in a loss of resilience. Even more, in the context of the 336 Anthropocene [99] or Technocene [62], Human impact in extreme cases may modify forest heterogeneity 337 generating new ecological patterns (niche construction) and interactions, without historical equivalents 338 [93].

339 Spatial heterogeneity can be altered by invasive species threaten biodiversity through predation 340 [21, 86], competition [42], disease transmission [114], and facilitation of the establishment of further 341 invasive species [97]).It has been reported that the decrease and extinction of native species due to invasive 342 predators can generate cascade effects that extend through the whole ecosystem and beyond. [14]. In 343 particular depredation effects resulting from human introduced species can be severe [96, 22]. Both rats 344 (Rattus rattus), cats (Felis catus) and dogs (Canis lupus familiaris) are recognized as the worst threat 345 species following recent studies [48]. In natural areas worldwide, dogs are threatening some 200 species, 346 some of them even included in IUCN threat categories. Likewize, Feral cats and red fox (Vulpes vulpes) 347 predation processes has been documented as a cause of the decline or extinction of two thirds of Australia’s 348 digging mammal species [28, 112]. Reduced disturbance to soil in the absence of digging mammals has 349 led to impoverished landscapes where little organic matter incorporates into the soil and rates of seed 350 germination is low [28]. The predation of seabirds through introduced Arctic foxes (Alopex lagopus) in 351 the Aleutian archipelago has reduced nutrient input and soil fertility, eventually causing vegetation to shift 352 from grassland to dwarf shrubland.

353 In a recent work [18], a deep relation between Levy´ walks and resilience has been shown. In this 354 work, Danneman and coworkers unveil an essential yet unexplored multi-scale movement property of 355 Levy´ walks, how it play an important role in the stability of populations dynamics.Using Lotka-Volterra 356 models, they predict that generally diffusing foragers tend to become extinct in fragile fragmented habitats, 357 while their populations become resilient to degraded circumstances and have maximized abundance when 358 individuals undertake Levy flights. Their analytical and simulated findings, change the scope of multi- 359 scale foraging from individual to population level, making it of major value to a wide scope of applications 360 in biology of conservation.Their findings indicate that Levy´ flights reach a balance between exploration 361 and exploitation, which in turn will benefit the stability and resilience of the population. In that way, 362 modern forest management is becoming more compatible with this complexity feature (heterogeneity) by 363 promoting it through strategic cuts that emulate natural disturbances; leave intact some structures and

14/25 364 organisms, Including dead and living trees and intact patches of forests ; and promote mixtures of tree 365 species. These methods are similar to the comparatively recent strategy of using biodiversity to boost 366 yield and resilience in natural and managed ecosystems [27]. Generalizing these ideas we reckon there is 367 important evidence suggesting that in order to preserve ecosystem integrity and resilience, management 368 systems should consider maintaining minimum levels of ecosystem complexity.

369 Of course, this poses a new challenge: How to measure complexity? Following Gershenson and 370 co-authors [31] one may measure complexity using again the Shannon information. In this information 371 theory framework, in order to have new information, the old one has to be transformed. Thus, we can 372 deﬁne information emergence E as the rate of information transformation. therefor emergence is identiﬁed 373 directly with Shannon’s information H or I. In addition, self-organization (S), a key feature of complex 374 systems, has been correlated with an increase in order (i.e. as a reduction of entropy) [31]. Thus, if 375 emergence implies an increase in information, which is analogous to entropy; self-organization should be 376 anti-correlated with emergence in such a way that

S = 1 − I = 1 − E. (13)

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

C = 4 ∗ E ∗ S. (14)

378

379 Under the complex systems perspective ecosystems are not systems that can simplistically be managed 380 top-down. We must explicitly consider that the interactions take place in multiple and hierarchical levels. 381 This is a general feature of complex systems, components are organized hierarchically in such a way that 382 elements at different levels interact to form an architecture that characterizes the system. In this way, 383 complexity asserts that a phenomenon occurring at one scale cannot be understood without considering 384 cross-scale interactions. But it also means that environmental policy, management and intervention needs 385 to be rethought in terms of scale. In this respect Taleb is assembling ”Principles of policy under complexity” 386 (draft version available at: http://www.academia.edu/38433249/Fractal_Localism_ 387 Political_Clarity_under_Complexity) which include the understanding of policies as scale 388 dependent, and so we should consider that instead of aiming at one monolithic policy for managing 389 ecosystem, we should go on to develop a range of them linked to different levels of application Such 390 approach will be required to reduce the risk of catastrophic hidden effects.

391 Understanding the coupling of natural and human sub-systems provide a whole new narrative that 392 challenges management. Ecosystems management is the outcome of collective actions among different 393 agents such as decision makers, scientists, managers, concerned citizens and so on. As complexity, key 394 for ecosystem integrity and resilience, is at dynamic balance between emergence and self-organization 395 (S) (Eq.2.4), some (and the correct type of) self-organization is necessary to be fostered, but too much 396 of it is bad. Too much (form the wrong type) of S may sustain unwanted feedbacks with detrimental 397 consequences. For instance, illegal logging in Borneo can be seen as a self-organizing phenomenon 398 supported by interactions among all levels in the stakeholder hierarchy [82]. The mechanism is explained 399 by Filotas and co-workers, starting with pit sawyers taking out livings and pirate loggers taking advantage 400 of governance failures. This alone could not generate such a great impact, unless it couples with 401 unscrupulous timber buyers and corrupt governmental ofﬁcials laundering the illegal wood. Experience 402 in Mexico suggests that corruption might be the common link in practically all-important ecosystem 403 degradation processes. Finally, they say that savvy international traders are the higher link that provides 404 lucrative outlets for ill-gotten goods. According to the authors narrative, where illegal logging occurs, 405 wood markets are ﬂooded, wood prices are depressed, and standing trees are undervalued. Under such 406 conditions, community forest managers are not motivated to implement sustainable forest management 407 practices, which often involve short-term investments for only long-term returns. These conditions result 408 in a self-organized feedback that sustains illegal logging.

409 The characteristic of complex systems not having a unique description scale is related to one of the 410 most omnipresent system-wide phenomena, the 1/ f behavior on frequency space for the ﬂuctuation time β 411 series.This so-called pink noise is one of a family of 1/ f colored or fractal noises deﬁned by the β

15/25 412 scaling exponent and deemed to be a criticality fingerprint. It is common to comparing and classifying 413 fluctuation dynamics according to their resemblance to three archetypal noise groups: white (β ∼ 0),pink 414 (β ∼ −1)and Brownian (β ∼ −2)[5, 59, 55, 79]. The universality of criticality is still under consideration 415 and is known as the ’ ’ criticality hypothesis ” which states that systems in a dynamic system that shifts 416 between order and disorder reach the greatest level of computing capacities when reaches a balance 417 between robustness and flexibility(see [38] and references therein). This idea of criticality was recently 418 used in an information theory approach to defining ecosystem health and sustainability [83]. In this paper, 419 Ram´ırez-Carrillo et al. Consider that an ecosystem is healthy if it is in criticality, as a mixture of scale 420 invariance (as power laws in power spectra) and a balance between adaptability and robustness.

421 These power laws appear in numerous phenomena including earthquake statistics, solar ﬂares, epi- 422 demic outbreaks, etc., as summarized by [83]. [66, 76, 98]. They also are a common theme in biology 423 [33, 34, 32, 110]. Several researchers reported evidence of dynamic criticality in physiological pro- 424 cesses such as heart activity and suggested that it could be a main characteristic of a healthy state. 425 [54, 49, 89]. Some studies [34, 90] found compelling evidence of a relationship between healthy hearts 426 and scale-invariant noise, around 1/ f regime, backed by medical evidence.

427 This complexity approach is clearly complementary to the ecosystem integrity narrative, and we 428 should consider that an ecosystem is resilient if, in addition to maintaining its EVLs values inside a ”safe” 429 range, it also keeps them within a critical dynamic region (scale invariance and “1/ f ” ﬂuctuations) [83].

430 2.5 Resilience features and properties II

431 The main purpose of the Saint-Beat´ and co-workers reviewed paper [91] is to know how distinct ecosystems 432 react to global change in terms of composition and dynamics and eventually, how persistence, strength, or 433 resilience of the ecosystem can be assessed. The authors show that ecological network assessment (ENA) 434 offers an effective approach for describing local stability, combining both quantitative and qualitative 435 elements. They warn, however, that describing real conservation cases combining local and global stability 436 remains an incomplete task. The authors focus on three resilience-related results that emerge from their 437 ENA: (a) the role of species diversity in the structure and functioning of the ecosystem ; (b) the number of 438 trophic links and strength of interactions ; (c) the stability of the ecosystem in terms of cycling capacity, 439 omnivory spread and ascendancy.

440 Native species richness generally improves ecosystem resistance. High biodiversity (without con- 441 sidering non-native species) was proposed to contribute to minimize the threat of major ecosystem 442 modiﬁcations in reaction to environmental disturbances [71]. Experiments on species invasion in grass- 443 land parcels indicate that local biodiversity decreases the settlement and success of a variety of invaders 444 [53].Similarly, studies on manipulation of grassland diversity indicate that elevated diversity improves 445 inter-speciﬁc competition and thus decreases the danger of invasion [75, 44].In this line of thought, the 446 authors proceed to summarize the proof. For instance, Brose [9] demonstrates that herbivorous existence in 447 a society leads to the coexistence of manufacturers in a mixed predator-prey model and producer-nutrient 448 model with a structural model of food-webs. Species diversity also contributes to reduce the impact on 449 resistance and resilience for example in coastal ecosystem [113]. Similarly, the maintenance of integral 450 ecosystems and the services they provide is critical to human societies and the preservation of species 451 diversity seems crucial to achieving this goal [13].

452 However, the authors warns that the consideration of a single variable such as diversity can not 453 be sufﬁciently thorough to evaluate the stability of ecosystems due to the complexity of ecosystems. 454 Therefore, they suggest that building holistic indexes in the ENA framework is a better approach for a 455 thorough knowledge of the food web structure and its role in the functioning of the ecosystem.

456 If one would be able to construct a sufﬁciently detailed trophic network (something very difﬁcult to do 457 in general), one could use standard network analysis tools to understand, for example, the topology of the 458 network, such as connectance. In that sense, the authors summarize evidence from the literature to show 459 that an increase in links dissipates the impact of variability in species distribution and increases stability 460 [65]. Higher connectivity thus improves the strength of the ecosystem as well as resilience [20] . If so, 461 connectivity seems to be a useful measure of the robustness of food webs and indirectly of ecosystem 462 stability.

463 In addition to connectivity, they show the importance of interaction strength diversity. Following 464 ideas of [104], They claim that ecosystem stability requires a balanced presence of weak and strong 465 interactions. Suppression of weak interactions destabilizes the system for a certain amount of species.

16/25 466 Moreover, the food web would be stable if and only if main predator-prey interactions are combined 467 with weak interactions in the context of high diversity. Thus, due to their ability to ﬂuctuate and adapt 468 within ecosystems, weak interactions function as a stabilizing force in food webs and consequently the 469 ecosystem.

470 To gain a deeper understanding of stability, Saint-Beat´ and colleagues examine the effect of cycling, 471 the presence of omnivorous and ascending.

472 For instance, the presence of omnivory gives an ecosystem trophic flexibility, a clear beneficial feature 473 that reflects integrity and resilience of the ecosystem. The researchers claim that omnivory provides to 474 the ecosystem a superior buffer to deal with environmental disturbances Because omnivorous species 475 enable faster ecosystem reaction by rapidly moving trophic routes following disturbance.For example, if a 476 disturbance impacts low trophic levels, omnivorous species that are directly linked to it, would respond 477 rapidly. In comparison, a particular predator must wait until the disturbance reaches its own level ; 478 therefore, the response time will be longer.

479 As in the reviewed paper by Gustavson and co-workers [39], Saint-Beat´ and co-workers show how 480 ascendency could be used as a key indicator to evaluate ecosystems functioning. The authors indicate that 481 to understand the function of ”ascendancy” two kinds of stability must be differentiated. A system with 482 elevated inner stability is a system with adequate inner limitations to enable a strongly organized structure, 483 corresponding to a high ascendency (high mutual information). Typically under this condition, ecosystems 484 are some how protected against internal perturbations but leave them vulnerable to external ones. On the 485 other hand, since low ascendancy is linked to redundancy, ecosystems become more resilient to external 486 disturbances. Interestingly enough, too high level of ascendancy is recognized as a a characteristic of 487 stress and may indicate a decrease ecosystem resilience.

488 Summarizing, in the dynamic response of ecosystems under the criticality framework a healthy 489 ecosystem is found where a balance between robustness and adaptation develops. In the case of network 490 topology, the ecosystems need to develop a good balance between strong and weak interactions in order 491 to be stable. In the case of the ascendancy narrative, a stable ecosystem should develop a good balance 492 between ascendancy and overhead, which seems to give resistance and resilience to ecosystems. This 493 leads us to think what we develop in the next section, in which we will ascertain whether all these three 494 kinds of balance could be particular cases of a more general evolutionary strategy of living systems: the 495 antifragility.

496 BEYOND RESILIENCE, ANTIFRAGILITY

497 Living systems can and must do much more than merely react to the environment’s variability through 498 random mutations followed by selection; they must certainty have built-in characteristics that enable them 499 to discover alternatives to cope with adversity, variability and uncertainty. Anti-fragility is one of these 500 characteristics [17, 101].

501 If one considers what does really mean that something is fragile, the key property is that it gets 502 damaged by environmental variability. Now if we ask our nearest colleague at random, about the exact 503 opposite of fragile, most likely we would get concepts such as robustness or resilience. But at close 504 inspection it is clear that none of them are the exact opposite of fragile. Both represent systems that are 505 insensitive to environmental variability or get affected only momentarily, quickly returning to its initial 506 state.

507 The exact opposite of fragility as defined by Taleb is antifragility, which is a property that enhances 508 the system’s functional capacity to response to external perturbations[102]. In other words, a system 509 is antifragile if it benefits from environmental variability, works better after being disturbed. Then 510 antifragility is beyond robustness or resilience. While the robust / resilient systems tolerate stress and 511 remain the same, antifragile structures not only withstand stress but also gain from it. The immune system 512 provide significant illustration of antifragile systems. When subjected to various germs at a young age, our 513 immune system will improve and gain different capabilities to overcome new illnesses in the future[81]. 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 2 convexity will be defined by the relation ∂ f 0 which can be simplified under the right conditions ∂x2 ≥ 1 ∆ ∆ to 2 [ 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:

E E ∀σ2 > σ1, σ2 [ f (x)] ≥ σ1 [ f (x)]. (16)

514

515 This way, Taleb defines local antifragility as ”a situation in which, over a specific interval [a,b], either 516 the expectation increases with the scale of the distribution as in Eq 2.5, or the dose-response is convex 517 over the same interval”. 518 Although Antifragility framework was developed by Taleb in the context of financial risk analysis, duo 519 to its universal mathematical formalism it has track attention and has been applied far away its original 520 scope. There are applications of the Antifraglity concept from molecular biology to urban planning 521 (see [81] and references inside). In their work, Pineda and co-workers [81] proposed a straightforward 522 implementation of antifragility by defining as payoff function the complexity of the system. which makes 523 a lot of sense in the context of our review because complexity is highly related with critically and hence 524 with these these trade-off balance between robustness and adaptability. 525 The authors defined fragility as

= −∆C |∆x|, (17) I

526 where ∆C is the change in system complexity due to a perturbation of degree |∆x|. As complexity can 527 always be normalized to [0,1], then positive values of deﬁne fragile systems; when is zero the system 528 is robust/resilient; and for negative values of the system is antifragile. H H 529 Then, Pineda and co-workers [81] apply it to random Boolean networks (RBNs) of a model of genetic H 530 regulatory works. They found that ordered RBNs are the most antifragile and demonstrated that, as 531 expected, seven biological well studied networks such as CD4+ T cell differentiation and plasticity or 532 Arabidopsis thaliana cell-cycle, are antifragile. 533 We know, from Central Limit Theory, that normal distributions can only emerge from (simple) 534 systems without interactions (probabilistic independence). When we take into account interactions (no 535 probabilistic independence) then the corresponding probability distribution will have fat-tails. In that 536 sense complexity is related with fat-tails and fat-tails with fragility/antifragility [101]. In Taleb’s narrative, 537 normal distribution in the response function characterize robust systems; whereas left fat-tailed are fragile, 538 and right fat-tailed are antifragile systems. Most interestingly, Fossion and co-workers [29] have related 539 homeostasis (physiological resilience?) to pairs of physiological variables, one to be controlled (the 540 one that remains in homeostasis) and another one that controls the former. The main idea is that in 541 order to have a homeostatic physiological variable (normal), the body must use other variables (right 542 fat-tailed) to absorb a random injection of matter, energy, information or any combination of them from 543 the environment. In Figure 4 of their paper, they present results for variability analysis of heart rate 544 HR and blood pressure BP for (a) healthy control(s), (b) recently diagnosed diabetic patient(s) and (c) 545 long-standing diabetic patient(s). They show that for healthy patients BP is normal and HR is right 546 fat-tailed. In the case of recently diagnosed diabetic patients, BP start to lose normality and develop a left 547 tail and HR tends to normality. Eventually, long-standing diabetic patients, BP has a clearly left fat-tailed 548 behavior and HR has become normal. This is very compelling evidence of the role of antifragility in 549 human health. 550 In a general manner, Taleb [101] Suggests the so-called barbell (or bimodal) approach as an archetypal 551 strategy for achieving antifragility. The ﬁrst step towards antifragility is to reduce downsides instead 552 of increasing upsides. In other words, by reducing exposure to adverse low probability but elevated

18/25 553 adverse payoff occurrences (i.e. ’ ’ black swans ” events) and allowing natural antifragility to function on 554 its own. We follow Taleb with a vulgar finance instance, where the idea is easiest to explain, although 555 most of them are misunderstood. The barbel approach in finances comprises of placing 90% of your 556 resources in safe instruments (provided that you are protected against inflation) or what is referred to as the 557 ”value repository number,” and 10% in very risky, maximal risky bonds, exposing yourself to unpredicted 558 massive gains in a convex way. In this way, one ends up with some sort of bimodal optimization taking 559 advantage of at the same time of the robustness of safe inversion and on the other hand the adaptability of 560 high risky ones. Anyone who has a 100% stake in so-called ”medium” securities (unimodal optimization) 561 is at danger of ”complete risk ruin”. This Barbel Strategy addresses the issue of incomputability and 562 fragility in the assessment of the hazards of unusual occurrences. 563 As in the barbell strategy, a basic mechanism to achieve antifragility, is a thorough strategy to risk 564 management under fat-tailed distributions, and those are widely present in nature, then it should be very 565 ubiquitous in natural systems. We identify this barbell risk strategy as the ”good balance” property in 566 network topology by means of the relation of strong and weak interactions or between ascendancy and 567 overhead; and balance robustness and adaptability, identified as fingerprint of critically (scale invariant 568 and i/ f type of noise), in the dynamic of system’s fluctuations. All these three ”good balances” are 569 related, as we showed with ecosystem integrity and resilience.

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.

570 DISCUSSION AND CONCLUSIONS

571 From the analysis of the literature, we found that the citation network from reviewed network is not 572 percolated, what we interpret as a lack of uniﬁcation in the research ﬁeld and an opportunity for inter-

19/25 573 disciplinary work. We found (see Figure 9) three main narratives (a) Ecosystem properties that enable 574 them to be more resilient; (b) Ecosystem response to perturbations; and (c) Complexity. From this and 575 complementary literature consulted we have identiﬁed 11 possible indicators for ecosystem resilience 576 (See Table 5). In particular we show how to apply Fisher information in a study case which we consider a 577 very promising proxy of resilience, since it has a solid formal framework, it is easy to implement and it 578 can be applied to any kind of system.

Key Indicator Measure/proxy Requires Resilience First, more stable ecosystem are more resilient and according to Cabezas et al. for a system to be FI Fisher Information Stability Time series resilient, after a disturbance the FI values prior to it must be recovered In general to greater diversity, greater resilience Optional / use of Div Diversity Presence field data But there are exceptions related to changes of resource space composition and use of resources Knowing the networks and being able to quantify the intensity of the Increase in the number of connections dissipates the effect of Co Network Conectance Stability connections, Gustavson proposes ways variation in distribution of species and enhances stability species to deal with the lack of information about it Omn Presence of omnivore species Communication between different scales Presence of omnivore species Presence of omnivore species enhance stability and resilience Knowing the networks and being Balance between Observations show that ecosystems are more resilient when able to quantify the intensity of the NC Network Criticality robustness (strong Interactions) and there is a good balance between the number of strong and weak connections, Gustavson proposes ways adaptability (Weak Interactions) connections to deal with the lack of information about it Given a community matrix, Lotka-Volterra if all the real parts of its L-VC Community matrix More stable ecosystem are more resilient Coefficients eigenvalues are negative the ecosystem is stable Given a network of interactions (i.e trophic Capture in a single index the ability of an ecosystem network) it measures how well, on average, As Ascendency Mean mutual information to prevail against disturbance by virtue of its the network articulates a flow event between combined organization and size. any two nodes. Scaling coefficient of foraging It has been shown that Levy flights foraging patterns Levy Levy Flights patterns for key species such It is a proxy of resources spatial complexity are related and enhance ecosystems resilience as puma or jaguar Frac Fractality Spatial complexity High resolution satellite images More complex ecosystems should be more resilient Network of interactions, can be a Change in the complexity of a biotic (i.e trophic) Boolean network of co-occurrences Resilience would be an intermediate state between AF Antifragility network, in the face of disturbances of a key species such as puma or jaguar fragility and antifragility with its prey for example H Homeostasis System Homeostasis Time Series Equivalent of resilience Table 5. Resilience measure found in the literature review and complementary papers

579 Nevertheless a new way to reinterpret resilience emerged from this critical literature review: an- 580 tifragility. This novel framework developed by N.N. Taleb [101, 103, 102] is based on fat-tailed, non- 581 linear responses of the system to variability (see Figure 11). In a simple way, if a system has a concave 582 (non-linear) payoff function dependent of certain variable, then the system is fragile to it. On the contrary, 583 if the payoff in convex then it is antifragile and if the system is essentially insensible to variability, then 584 is robust /resilient. In Taleb’s work, antifragility is associated with bimodal risk strategy called “The 585 Barbell” which we believe manifest itself in the narratives as “a good balance” between (i) strong and 586 weak interactions in network topology; (ii) adaptability and robustness (criticality); and (iii) ascendancy 587 and overhead.

588 In the long term, considering the coupling of ecosystem with human systems (i.e. via climate change) 589 we consider that antifragility is a more desirable feature than resilience. Thinking in socioecosystems, we 590 can see that they usually not only keep on living, but they do ﬂourish and evolve, even in the presence 591 of great stressors such as climate crisis or land change. In fact, in a recent work [19], It has been shown 592 that the outcome of using antifragility as a design criterion is that the scheme being studied demonstrates 593 a more favorable behavior than a ”simply” robust model in a setting that is susceptible to black swans 594 (unpredictable, very low frequency of ocurrancebut vey high impact events) . Then, for socioecosystem 595 governance, planning or in general, any decision making perspective, antifragility might be a valuable and 596 more desirable goal to achieve than a resilience aspiration [7].

597 REFERENCES [1] 598 Abe, S., Sarlis, N., Skordas, E., Tanaka, H., and Varotsos, P. (2005). Origin of the usefulness of the 599 natural-time representation of complex time series. Physical review letters, 94(17):170601. [2] 600 Ahmad, N., Derrible, S., Eason, T., and Cabezas, H. (2016). Using ﬁsher information to track stability 601 in multivariate systems. Royal Society open science, 3(11):160582. [3] 602 Aronson, J., Floret, C., Le Floc’h, E., Ovalle, C., and Pontanier, R. (1993). Restoration and rehabilita- 603 tion of degraded ecosystems in arid and semi-arid lands. ii. case studies in southern tunisia, central 604 chile and northern cameroon. Restoration ecology, 1(3):168–187.

20/25 [4] 605 Aronson, J. and Le Floc’h, E. (1996). Vital landscape attributes: missing tools for restoration ecology. 606 Restoration Ecology, 4(4):377–387. [5] 607 Bak, P., Tang, C., and Wiesenfeld, K. (1988). Self-organized criticality. Physical review A, 38(1):364. [6] 608 Bartumeus, F. (2007). Levy´ processes in animal movement: an evolutionary hypothesis. Fractals, 609 15(02):151–162. [7] 610 Bleciˇ c,´ I. and Cecchini, A. (2018). Planning for antifragility and antifragility for planning. In 611 International Symposium on New Metropolitan Perspectives, pages 489–498. Springer. [8] 612 Boyer, D. and Lopez-Corona,´ O. (2009). Self-organization, scaling and collapse in a coupled automaton 613 model of foragers and vegetation resources with seed dispersal. Journal of Physics A: Mathematical 614 and Theoretical, 42(43):434014. [9] 615 Brose, U. (2008). Complex food webs prevent competitive exclusion among producer species. 616 Proceedings of the Royal Society B: Biological Sciences, 275(1650):2507–2514. [10] 617 Brown, C. T., Liebovitch, L. S., and Glendon, R. (2007). Levy´ flights in dobe ju/’hoansi foraging 618 patterns. Human Ecology, 35(1):129–138. [11] 619 Cabezas, H. and Fath, B. D. (2002). Towards a theory of sustainable systems. Fluid phase equilibria, 620 194:3–14. [12] 621 Cabezas, H., Pawlowski, C. W., Mayer, A. L., and Hoagland, N. T. (2005). Sustainable systems 622 theory: ecological and other aspects. Journal of Cleaner Production, 13(5):455–467. [13] 623 Chapin Iii, F. S., Zavaleta, E. S., Eviner, V. T., Naylor, R. L., Vitousek, P. M., Reynolds, H. L., Hooper, 624 D. U., Lavorel, S., Sala, O. E., Hobbie, S. E., et al. (2000). Consequences of changing biodiversity. 625 Nature, 405(6783):234. [14] 626 Courchamp, F., Chapuis, J.-L., and Pascal, M. (2003). Mammal invaders on islands: impact, control 627 and control impact. Biological Reviews, 78(3):347–383. [15] 628 Crabbe,´ P., Holland, A. J., Ryszkowski, L., and Westra, L. (2000). Implementing Ecological 629 Integrity: restoring regional and global enivronmental and human health:[proceedings of the NATO 630 Advanced Research Workshop on Implementing Ecological Integrity: Restoring Regional and Global 631 Environmental and Human Health, Budapest, Hungary, June 26-July 1, 1999], volume 1. Springer 632 Science & Business Media. [16] 633 Dakos, V., Matthews, B., Hendry, A., Levine, J., Loeuille, N., Norberg, J., Nosil, P., Scheffer, M., and 634 De Meester, L. (2018). Ecosystem tipping points in an evolving world. bioRxiv, page 447227. [17] 635 Danchin, A., Binder, P. M., and Noria, S. (2011). Antifragility and tinkering in biology (and in 636 business) flexibility provides an efficient epigenetic way to manage risk. Genes, 2(4):998–1016. [18] 637 Dannemann, T., Boyer, D., and Miramontes, O. (2018). Levy´ flight movements prevent extinctions 638 and maximize population abundances in fragile lotka–volterra systems. Proceedings of the National 639 Academy of Sciences, 115(15):3794–3799. [19] 640 de Bruijn, H., Großler,¨ A., and Videira, N. (2019). Antifragility as a design criterion for modelling 641 dynamic systems. Systems Research and Behavioral Science. [20] 642 DeAngelis, D. (1980). Energy flow, nutrient cycling, and ecosystem resilience. Ecology, 61(4):764– 643 771. [21] 644 Doherty, T. S., Davis, R. A., van Etten, E. J., Algar, D., Collier, N., Dickman, C. R., Edwards, G., 645 Masters, P., Palmer, R., and Robinson, S. (2015). A continental-scale analysis of feral cat diet in 646 australia. Journal of Biogeography, 42(5):964–975. [22] 647 Doherty, T. S., Glen, A. S., Nimmo, D. G., Ritchie, E. G., and Dickman, C. R. (2016). Invasive preda- 648 tors and global biodiversity loss. Proceedings of the National Academy of Sciences, 113(40):11261– 649 11265. [23] 650 Dutrieux, L. P. (2016). Multidimensional remote sensing based mapping of tropical forests and their 651 dynamics. PhD thesis, Wageningen UR. [24] 652 Eason, T. and Cabezas, H. (2012). Evaluating the sustainability of a regional system using fisher 653 information in the san luis basin, colorado. Journal of environmental management, 94(1):41–49. [25] 654 Equihua Zamora, M., Garc´ıa Alaniz, N., Perez-Maqueo,´ O., Ben´ıtez Badillo, G., Kolb, M., Schmidt, 655 M., Equihua Ben´ıtez, J., Maeda, P., and Alvarez´ Palacios, J. (2014). Integridad ecologica´ como 656 indicador de la calidad ambiental. Bioindicadores: Guardianes de nuestro futuro ambiental, pages 657 695–718. [26] 658 Fernandez,´ N., Maldonado, C., and Gershenson, C. (2014). Information measures of complexity, 659 emergence, self-organization, homeostasis, and autopoiesis. In Guided self-organization: Inception,

21/25 660 pages 19–51. Springer. [27] 661 Filotas, E., Parrott, L., Burton, P. J., Chazdon, R. L., Coates, K. D., Coll, L., Haeussler, S., Martin, 662 K., Nocentini, S., Puettmann, K. J., et al. (2014). Viewing forests through the lens of complex systems 663 science. Ecosphere, 5(1):1–23. [28] 664 Fleming, P. A., Anderson, H., Prendergast, A. S., Bretz, M. R., Valentine, L. E., and Hardy, G. 665 E. S. (2014). Is the loss of a ustralian digging mammals contributing to a deterioration in ecosystem 666 function? Mammal Review, 44(2):94–108. [29] 667 Fossion, R., Rivera, A. L., and Estanol, B. (2018). A physicist’s view of homeostasis: how time series 668 of continuous monitoring reflect the function of physiological variables in regulatory mechanisms. 669 Physiological measurement, 39(8):084007. [30] 670 Frieden, B. R. (2000). Physics from fisher information: a unification. [31] 671 Gershenson, C. and Heylighen, F. (2003). When can we call a system self-organizing? In European 672 Conference on Artificial Life, pages 606–614. Springer. [32] 673 Gisiger, T. (2001). Scale invariance in biology: coincidence or footprint of a universal mechanism? 674 Biological Reviews, 76(2):161–209. [33] 675 Goldberger, A. L. (1992). Fractal mechanisms in the electrophysiology of the heart. IEEE Engineering 676 in Medicine and Biology Magazine, 11(2):47–52. [34] 677 Goldberger, A. L., Peng, C.-K., and Lipsitz, L. A. (2002). What is physiologic complexity and how 678 does it change with aging and disease? Neurobiology of aging, 23(1):23–26. [35] 679 Gonzalez-Mejia, A. M., Eason, T., Cabezas, H., and Suidan, M. T. (2012). Computing and interpreting 680 fisher information as a metric of sustainability: regime changes in the united states air quality. Clean 681 Technologies and Environmental Policy, 14(5):775–788. [36] 682 Gough, C. M., Bond-Lamberty, B., Stuart-Haentjens, E., Atkins, J., Haber, L., and Fahey, R. (2017). 683 Carbon cycling at the tipping point: Does ecosystem structure predict resistance to disturbance? In 684 AGU Fall Meeting Abstracts. [37] 685 Grimm, V. and Calabrese, J. M. (2011). What is resilience? a short introduction. In Viability and 686 Resilience of Complex Systems, pages 3–13. Springer. [38] 687 Gunderson, L. H. (2000). Ecological resilience—in theory and application. Annual review of ecology 688 and systematics, 31(1):425–439. [39] 689 Gustavson, K., Lonergan, S. C., and Ruitenbeek, J. (2002). Measuring contributions to economic 690 production—use of an index of captured ecosystem value. Ecological Economics, 41(3):479–490. [40] 691 Guy, G. A., Kosugi, T., and Sulzman, E. (2007). Ameriflux network aids global synthesis. Eos, 692 88(28). [41] 693 Haraway, D. (2015). Anthropocene, Capitalocene, Plantationocene, Chthulucene: Making Kin. 694 Environmental Humanities, 6(1):159–165. [42] 695 Harris, D. B. and Macdonald, D. W. (2007). Interference competition between introduced black rats 696 and endemic galapagos rice rats. Ecology, 88(9):2330–2344. [43] 697 Harrison, G. W. (1979). Stability under environmental stress: resistance, resilience, persistence, and 698 variability. The American Naturalist, 113(5):659–669. [44] 699 Hector, A., Dobson, K., Minns, A., Bazeley-White, E., and Lawton, J. H. (2001). Community 700 diversity and invasion resistance: an experimental test in a grassland ecosystem and a review of 701 comparable studies. Ecological Research, 16(5):819–831. [45] 702 Ho, M.-W. and Ulanowicz, R. (2005). Sustainable systems as organisms? BioSystems, 82(1):39–51. [46] 703 Holling, C. S. (1973). Resilience and stability of ecological systems. Annual review of ecology and 704 systematics, 4(1):1–23. [47] 705 Holling, C. S. (1996). Engineering resilience versus ecological resilience. Engineering within 706 ecological constraints, 31(1996):32. [48] 707 Hughes, J. and Macdonald, D. W. (2013). A review of the interactions between free-roaming domestic 708 dogs and wildlife. Biological Conservation, 157:341–351. [49] 709 Ivanov, P. C., Rosenblum, M. G., Peng, C.-K., Mietus, J., Havlin, S., Stanley, H. E., and Goldberger, 710 A. L. (1996). Scaling behaviour of heartbeat intervals obtained by wavelet-based time-series analysis. 711 Nature, 383(6598):323. [50] 712 Karunanithi, A. T., Cabezas, H., Frieden, B. R., and Pawlowski, C. W. (2008). Detection and 713 assessment of ecosystem regime shifts from fisher information. Ecology and society, 13(1). [51] 714 Karunanithi, A. T., Garmestani, A. S., Eason, T., and Cabezas, H. (2011). The characterization of

22/25 715 socio-political instability, development and sustainability with fisher information. Global Environmental 716 Change, 21(1):77–84. [52] 717 Kay, J. J. (1991). A nonequilibrium thermodynamic framework for discussing ecosystem integrity. 718 Environmental Management, 15(4):483–495. [53] 719 Kennedy, T. A., Naeem, S., Howe, K. M., Knops, J. M., Tilman, D., and Reich, P. (2002). Biodiversity 720 as a barrier to ecological invasion. Nature, 417(6889):636. [54] 721 Kiyono, K., Struzik, Z. R., Aoyagi, N., Togo, F., and Yamamoto, Y. (2005). Phase transition in a 722 healthy human heart rate. Physical review letters, 95(5):058101. [55] 723 Kleinen, T., Held, H., and Petschel-Held, G. (2003). The potential role of spectral properties in 724 detecting thresholds in the earth system: application to the thermohaline circulation. Ocean Dynamics, 725 53(2):53–63. [56] 726 Kleinn, C. (1991). Zum waldbegriff in forstlichen großrauminventuren (on forest definitions in large 727 area forest inventories. in german with english abstract). [57] 728 Kleinn, C. (1992). On the compatibility of forest inventory results. In Conference on integreating 729 forest management over space and time, pages 278–285. [58] 730 Kleinn, C. (2001). A cautionary note on the minimum crown cover criterion in forest definitions. 731 Canadian Journal of Forest Research, 31(2):350–356. [59] 732 Landa, E., Morales, I. O., Fossion, R., Stransk´ y,` P., Velazquez,´ V., Vieyra, J. L., and Frank, A. (2011). 733 Criticality and long-range correlations in time series in classical and quantum systems. Physical Review 734 E, 84(1):016224. [60] 735 Langdon, P. G., Dearing, J., Dyke, J., and Wang, R. (2016). Identifying and anticipating tipping 736 points in lake ecosystems. Past Global Changes Magazine, 24:16–17. [61] 737 Levin, S. A. (2005). Self-organization and the emergence of complexity in ecological systems. 738 Bioscience, 55(12):1075–1079. [62] 739 Lopez-Corona,´ O., Ramırez-Carrillo, E., and Magallanes-Guijon,´ G. (2019). The rise of the techno- 740 bionts: toward a new ontology to understand current planetary crisis. Researchers One. [63] 741 Loreau, M., Downing, A., Emmerson, M., Gonzalez, A., Hughes, J., Inchausti, P., Joshi, J., Norberg, 742 J., and Sala, O. (2002). A new look at the relationship between diversity and stability. Biodiversity and 743 ecosystem functioning: Synthesis and perspectives, pages 79–91. [64] 744 Lund, H. G. (2006). Definitions of forest, deforestation, afforestation, and reforestation. Forest 745 Information Services Gainesville, VA. [65] 746 MacArthur, R. (1955). Fluctuations of animal populations and a measure of community stability. 747 Ecology, 36(3):533–536. [66] 748 Mandelbrot, B. B. (1982). The fractal geometry of nature, volume 2. WH freeman New York. [67] 749 Manuel-Navarrete, D., Kay, J., and Dolderman, D. (2007). Evolution of the ecological integrity 750 debate. Sustaining Life on Earth: Environmental and Human Health Through Global Governance, 751 pages 127–138. [68] 752 Mart´ın-Mart´ın, A., Orduna-Malea, E., and Lopez-C´ ozar,´ E. D. (2018). Coverage of highly-cited 753 documents in google scholar, web of science, and scopus: a multidisciplinary comparison. arXiv 754 preprint arXiv:1804.09479. [69] 755 Mayer, A. L., Pawlowski, C., Fath, B. D., and Cabezas, H. (2007). Applications of fisher information 756 to the management of sustainable environmental systems. In Exploratory data analysis using Fisher 757 information, pages 217–244. Springer. [70] 758 Mayer, A. L., Pawlowski, C. W., and Cabezas, H. (2006). Fisher information and dynamic regime 759 changes in ecological systems. Ecological modelling, 195(1-2):72–82. [71] 760 McNaughton, S. J. (1977). Diversity and stability of ecological communities: a comment on the role 761 of empiricism in ecology. The American Naturalist, 111(979):515–525. [72] 762 Michaelian, K. (2005). Thermodynamic stability of ecosystems. Journal of theoretical biology, 763 237(3):323–335. [73] 764 Miramontes, O., Boyer, D., and Bartumeus, F. (2012). The effects of spatially heterogeneous prey 765 distributions on detection patterns in foraging seabirds. PloS one, 7(4):e34317. [74] 766 Moore, J. C. (2018). Predicting tipping points in complex environmental systems. Proceedings of the 767 National Academy of Sciences, 115(4):635–636. [75] 768 Naeem, S., Knops, J. M., Tilman, D., Howe, K. M., Kennedy, T., and Gale, S. (2000). Plant diversity 769 increases resistance to invasion in the absence of covarying extrinsic factors. Oikos, 91(1):97–108.

23/25 [76] 770 Newman, M. E. (2005). Power laws, pareto distributions and zipf’s law. Contemporary physics, 771 46(5):323–351. [77] 772 Noble, I. R. and Slatyer, R. (1980). The use of vital attributes to predict successional changes in plant 773 communities subject to recurrent disturbances. Vegetatio, 43(1-2):5–21. [78] 774 Parrondo, J. M., Horowitz, J. M., and Sagawa, T. (2015). Thermodynamics of information. Nature 775 physics, 11(2):131. [79] 776 Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E., and Goldberger, A. L. (1994). 777 Mosaic organization of dna nucleotides. Physical review e, 49(2):1685. [Pimm] 778 Pimm, S. L. The balance of nature?: ecological issues in the conservation of species and 779 communities/stuart l. pimm. [81] 780 Pineda, O. K., Kim, H., and Gershenson, C. (2018). Antifragility of random boolean networks. arXiv 781 preprint arXiv:1812.06760. [82] 782 Putz, M. V. and Chattaraj, P. K. (2013). Electrophilicity kernel and its hierarchy through softness in 783 conceptual density functional theory. International Journal of Quantum Chemistry, 113(18):2163–2171. [83] 784 Ram´ırez-Carrillo, E., Lopez-Corona,´ O., Toledo-Roy, J. C., Lovett, J. C., de Leon-Gonz´ alez,´ F., 785 Osorio-Olvera, L., Equihua, J., Robredo, E., Frank, A., Dirzo, R., et al. (2018). Assessing sustainability 786 in north america’s ecosystems using criticality and information theory. PloS one, 13(7):e0200382. [84] 787 Ramos-Fernandez,´ G., Mateos, J. L., Miramontes, O., Cocho, G., Larralde, H., and Ayala-Orozco, B. 788 (2004). Levy´ walk patterns in the foraging movements of spider monkeys (ateles geoffroyi). Behavioral 789 ecology and Sociobiology, 55(3):223–230. [85] 790 Ravindran, S. (2016). Inner workings: Coral reefs at a tipping point. Proceedings of the National 791 Academy of Sciences, 113(19):5140–5141. [86] 792 Rayner, M. J., Hauber, M. E., Imber, M. J., Stamp, R. K., and Clout, M. N. (2007). Spatial 793 heterogeneity of mesopredator release within an oceanic island system. Proceedings of the National 794 Academy of Sciences, 104(52):20862–20865. [87] 795 Regier, H. A. (1995). Ecosystem integrity in a context of ecostudies as related to the great lakes 796 region. In Perspectives on Ecological Integrity, pages 88–101. Springer. [88] 797 Reynolds, C. (2002). Resilience in aquatic ecosystems–hysteresis, homeostasis, and health. Aquatic 798 Ecosystem Health & Management, 5(1):3–17. [89] 799 Rivera, A. L., Estanol, B., Fossion, R., Toledo-Roy, J. C., Callejas-Rojas, J. A., Gien-Lopez, J. A., 800 Delgado-Garcia, G. R., and Frank, A. (2016a). Loss of breathing modulation of heart rate variability in 801 patients with recent and long standing diabetes mellitus type ii. PloS one, 11(11):e0165904. [90] 802 Rivera, A. L., Estanol,˜ B., Senties-Madrid, H., Fossion, R., Toledo-Roy, J. C., Mendoza-Temis, J., 803 Morales, I. O., Landa, E., Robles-Cabrera, A., Moreno, R., et al. (2016b). Heart rate and systolic blood 804 pressure variability in the time domain in patients with recent and long-standing diabetes mellitus. PloS 805 one, 11(2):e0148378. [91] 806 Saint-Beat,´ B., Baird, D., Asmus, H., Asmus, R., Bacher, C., Pacella, S. R., Johnson, G. A., David, 807 V., Vezina,´ A. F., and Niquil, N. (2015). Trophic networks: how do theories link ecosystem structure 808 and functioning to stability properties? a review. Ecological indicators, 52:458–471. [92] 809 Schmeller, D. S., Weatherdon, L. V., Loyau, A., Bondeau, A., Brotons, L., Brummitt, N., Geijzen- 810 dorffer, I. R., Haase, P., Kuemmerlen, M., Martin, C. S., et al. (2018). A suite of essential biodiversity 811 variables for detecting critical biodiversity change. Biological Reviews, 93(1):55–71. [93] 812 Seastedt, T. R., Hobbs, R. J., and Suding, K. N. (2008). Management of novel ecosystems: are novel 813 approaches required? Frontiers in Ecology and the Environment, 6(10):547–553. [94] 814 Sidle, R. C., Benson, W. H., Carriger, J. F., and Kamai, T. (2013). Broader perspective on ecosystem 815 sustainability: Consequences for decision making. Proceedings of the National Academy of Sciences, 816 110(23):9201–9208. [95] 817 Sievert, C. and Shirley, K. (2014). Ldavis: A method for visualizing and interpreting topics. In 818 Proceedings of the workshop on interactive language learning, visualization, and interfaces, pages 819 63–70. [96] 820 SILVA-RODRIGUEZ,´ E. A. and Sieving, K. E. (2011). Inﬂuence of care of domestic carnivores on 821 their predation on vertebrates. Conservation Biology, 25(4):808–815. [97] 822 Simberloff, D. (2011). How common are invasion-induced ecosystem impacts? Biological invasions, 823 13(5):1255–1268. [98] 824 Sornette, D. (2006). Critical phenomena in natural sciences: chaos, fractals, selforganization and

24/25 825 disorder: concepts and tools (springer series in synergetics). [99] 826 Steffen, W., Crutzen, P. J., and McNeill, J. R. (2007). The anthropocene: are humans now overwhelm- 827 ing the great forces of nature. AMBIO: A Journal of the Human Environment, 36(8):614–621. [100] 828 Steffen, W., Rockstrom,¨ J., Richardson, K., Lenton, T. M., Folke, C., Liverman, D., Summerhayes, 829 C. P., Barnosky, A. D., Cornell, S. E., Crucifix, M., et al. (2018). Trajectories of the earth system in the 830 anthropocene. Proceedings of the National Academy of Sciences, 115(33):8252–8259. [101] 831 Taleb, N. N. (2012). Antifragile: how to live in a world we don’t understand, volume 3. Allen Lane 832 London. [102] 833 Taleb, N. N. (2018). (anti) fragility and convex responses in medicine. In International Conference 834 on Complex Systems, pages 299–325. Springer. [103] 835 Taleb, N. N. and Douady, R. (2013). Mathematical definition, mapping, and detection of (anti) 836 fragility. Quantitative Finance, 13(11):1677–1689. [104] 837 Ulanowicz, R. E. (1983). Identifying the structure of cycling in ecosystems. Mathematical Bio- 838 sciences, 65(2):219–237. [105] 839 Ulanowicz, R. E. (1986). A phenomenological perspective of ecological development. In Aquatic 840 Toxicology and Environmental Fate: Ninth Volume. ASTM International. [106] 841 Ulanowicz, R. E., Goerner, S. J., Lietaer, B., and Gomez, R. (2009). Quantifying sustainability: 842 resilience, efficiency and the return of information theory. Ecological complexity, 6(1):27–36. [107] 843 Vidal, C., Lanz, A., Tomppo, E., Schadauer, K., Gschwantner, T., di Cosmo, L., Robert, N., et al. 844 (2008). Establishing forest inventory reference definitions for forest and growing stock: a study towards 845 common reporting. [108] 846 Viswanathan, G. M., Buldyrev, S. V., Havlin, S., Da Luz, M., Raposo, E., and Stanley, H. E. (1999). 847 Optimizing the success of random searches. nature, 401(6756):911. [109] 848 Walker, B., Holling, C. S., Carpenter, S. R., and Kinzig, A. (2004). Resilience, adaptability and 849 transformability in social–ecological systems. Ecology and society, 9(2). [110] 850 West, B. J. (2010). Fractal physiology and the fractional calculus: a perspective. Frontiers in 851 physiology, 1:12. [111] 852 Wilson, E. H. and Sader, S. A. (2002). Detection of forest harvest type using multiple dates of 853 landsat tm imagery. Remote Sensing of Environment, 80(3):385–396. [112] 854 Woinarski, J. C., Burbidge, A. A., and Harrison, P. L. (2015). Ongoing unraveling of a continental 855 fauna: decline and extinction of australian mammals since european settlement. Proceedings of the 856 National Academy of Sciences, 112(15):4531–4540. [113] 857 Worm, B., Barbier, E. B., Beaumont, N., Duffy, J. E., Folke, C., Halpern, B. S., Jackson, J. B., 858 Lotze, H. K., Micheli, F., Palumbi, S. R., et al. (2006). Impacts of biodiversity loss on ocean ecosystem 859 services. science, 314(5800):787–790. [114] 860 Wyatt, K. B., Campos, P. F., Gilbert, M. T. P., Kolokotronis, S.-O., Hynes, W. H., DeSalle, R., 861 Daszak, P., MacPhee, R. D., and Greenwood, A. D. (2008). Historical mammal extinction on christmas 862 island (indian ocean) correlates with introduced infectious disease. PloS one, 3(11):e3602. [115] 863 Zellner, M. L., Theis, T. L., Karunanithi, A. T., Garmestani, A. S., and Cabezas, H. (2008). A 864 new framework for urban sustainability assessments: Linking complexity, information and policy. 865 Computers, environment and urban systems, 32(6):474–488. [116] 866 Zorach, A. C. and Ulanowicz, R. E. (2003). Quantifying the complexity of flow networks: how 867 many roles are there? Complexity, 8(3):68–76.

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