elsarticle template

Lakes Complexity in a Latitudinal Gradient

Nelson Fernandeza,b,1,, Cristian Villatea,d, Oswaldo Terana,d, JoseAguilara,d, Carlos Gershensond

aLaboratorio de Hidroinformatica, Universidad de Pamplona, ColombiabCentro de Micro-electronica y Sistemas Distribuidos, Universidad de los Andes, Merida,

VenezuelacDepartamento de Ciencias de la Computacion Instituto de Investigaciones en

Matematicas Aplicadas y en Sistemas,Universidad Nacional Autonoma de MexicodCentro de Ciencias de la Complejidad Universidad Nacional Autonoma de Mexico

Abstract

Measuring complexity in ecological systems has demanded general formaliza-

tions, in order to compare different components and ecosystems at different

scales. We apply formal measures of emergence, self-organization, home-

ostasis, autopoiesis and complexity to four aquatic ecosystems disposed in a

latitudinal gradient. The measures are based on information theory. Vari-

ables representing more complex dynamics in the different subsystems of lakes

were: In the Physco-chemical, variables related with temperature, oxygen,

Ph and hydrology. In the Limiting Nutrients, silicates and phosphorous.

In the biomass, Piscivorous and planktivorous fishes. Lakes Homeostasis

were associated with the spatial-temporal changes according with the sea-

IThis is only an exampleCorresponding authorEmail addresses: [email protected] (Nelson Fernandez),

[email protected] (Cristian Villate), [email protected] (Oswaldo Teran),[email protected] (Jose Aguilar), [email protected] (Carlos Gershenson)

URL: http://unipamplona.academia.edu/NelsonFernandez (Nelson Fernandez),http://turing.iimas.unam.mx/cgg (Carlos Gershenson)

Preprint submitted to Ecological Complexity December 6, 2013

sons. Biomass subsystem seems follow the temporal dynamics of the physic-

chemical subsystem than limiting nutrients dynamics. Autopoiesis results

showa higher degree of independence of photosynthetic biomass over their

environment. On the latitudinal gradient from the Arctic to Tropical, North

Lowland Lake appears to represent a transition point for complexity values

in all subsystems. Our approach shows how the ecological dynamics can be

described in terms of information and can increasing our understanding of

ecosystems and complexity itself

Keywords: Ecological Dynamics,Complex Systems, Information

Theory,Self-Organization, Emergence, Complexity,Homeostasis,

Autopoiesis.

1. Introduction1

In the last years, the complexity theory and their associated properties2

like the self-organization, emergence, criticality have been increasing numbers3

of applications in ecological systems ?. The study of complexity in ecology4

has been tried to relate with ecological richness, abundance, and hierarchi-5

cal structure. As a result, different approximations have been explored to6

develop mathematical formalisms, in order to represent the ecological com-7

plexity in particular as ecological indicator (Parrot, 2005).8

The information theory has been useful for the development of several9

models of complexity and it is hasbeen used in different ways as it can see10

in Prokopenko et al (2009). In Ecology the formalizations resulting of ap-11

plication of information theory, often relate the highest degree of complexity12

to random states; others in an opposite way, relates the highest degree of13

2

complexity with regularity ?. Thus, the meaning, interpretation and appli-14

cability of the complexity notion and their associated properties in ecology15

remains a challenge in ecology ?. As result, general and simple proposals for16

modeling and measuring complexity in ecology is needed.17

According with Fernandez et al.(2014) and Fernandez & Gersheson (2014),18

the description of the complexity in terms of properties like emergence, self-19

organization, and others related with the self-regulation and autonomy such20

as homeostasis and autopoiesis is suitable. The importance of these proper-21

ties is that they comes from the relevant interactions among systems com-22

ponents and generates novel information. Novel information is useful to23

describe the complex behavior in ecological systems. It can be said that this24

novel information is emergent, since it is not in the components, but it is25

produced by interactions. Interactions can also be used by components to26

self-organize, i.e. produce a global pattern from local dynamics. Interactions27

are also key for feedback control loops, which help systems regulate their28

internal states, an essential aspect of living systems(Fernandez et al, 2014).29

Among multiple ways to describe the state of an ecosystem, the balance30

between change (chaos) and stability (order) states has been proposed as31

a characteristic of complexity Langton1990,Kauffman1993. This way, we32

can say that more chaotic systems produce more information (emergence),33

and more stable systems are more organized. Thus we propose, based on34

information theory, that complexity can be defined as the balance between35

emergence and self-organization (Gershenson & Fernandez 2012; Fernandez36

et. al. 2015). This approach has been applied to some ecological systems37

(Fernandez et al. 2013*eccs) with good results indicating that ecological38

3

dynamics can be described in terms of information.39

In this context, this papers expand the useful of the application of com-40

plexity measuring applying formal expressions of complexity, self-organization,41

emergence, homeostasis and autopoiesis to the Physico-chemical, nutrients42

and biomass subsystems to four types of lakes located in a latitudinal gradi-43

ent (Arctic, North Lowland, North Highland to Tropical), in focus to evaluate44

the usefulness and benefits in ecological systems.45

2. Measures46

The measures applied in this paper have recently developed and compared47

with other previously proposed in the literature (Fernandez et al., 2012;48

Gershenson and Fernandez, 2012); more refined measures, based on axioms,49

have been presented in Fernandez et al., (2013).50

In general, Emergence refers to properties of a phenomenon that are51

present now and were not before. If we suppose these properties as non-52

trivial, we could say it is harder now than before to reproduce the phe-53

nomenon. In other words, there is emergence in a phenomenon when this54

phenomenon is producing information and, if we recall, Shannon proposed55

a quantity which measures how much information was produced by a pro-56

cess.Therefore, we can say that the emergence is the same as the Shannons57

information I. Thus E=I58

Self-organization has been correlated with an increase in order, i.e. a59

reduction of entropy (Gershenson and Heylighen, 2003). If emergence implies60

an increase of information, which is analogous to entropy and disorder, self-61

organization should be anti-correlated with emergence. We propose as the62

4

measure S = 1 I = 1 E.63

We can define complexity C as the balance between change (chaos) and64

stability (order). We have just defined such measures: emergence and self-65

organization. Hence we propose: C = 4 E S.. Where the constant 4 is66

added to normalize the measure to [0, 1]67

For homeostasis H, we are interested on how all variables of a system68

change or not in time. A useful function for comparing strings of equal length69

is the Hamming distance. The Hamming distanced measures the percentage70

of different symbols in two strings X and X.71

As it has been proposed, adaptive systems require a highC in order to be72

able to cope with changes of its environment while at the same time maintain-73

ing their integrity (Langton,1990; Kauffman, 1993). If we have X represent74

the trajectories of the variables of a system and Y represent the trajectories75

of the variables of the environment of the system, If X had a high E, then it76

would not be able to produce its own information. With a highS, X would not77

be able to adapt to changes in Y . Therefore, we propose: A = C(X )/C(Y) .78

3. Case Studies79

The data of different lakes models used in this section was obtained us-80

ing The Aquatic Ecosystem Simulator (Randerson and Bowker, 2008). The81

model used is deterministic, so there is no variation in different simulation82

runs. All variables and daily data we obtained from Arctic, North Highland,83

North Lowland and Tropical are shown in Annex A.84

The criteria for lakes choosing was the location of each lake in a latitu-85

dinal gradient from the Polar to the Tropical Zone (Ar-T), where light and86

5

temperature conditions have a high amplitud and variation.87

3.1. Arctic Lake (Ar)88

Arctic lakes are located at the Arctic Polar Circle. Their mean surface89

temperature is around 3C and their maximum is near to 9C and their mini-90

mum is 0C.91

In general, Arctic lake systems are classified as oligotrophic due to their92

low primary production, represented in chlorophyll values of 0.8-2.1 mg/m3.93

The lakes water column, or limnetic zone, is well-mixed; this means, there94

are no stratification (layers with different temperatures). During winter (Oc-95

tober to March), the surface of the lake is ice covered. During summer (April96

to September), ice melts and the water flow and evaporation increase. Con-97

sequently, the two climatic periods (winter and summer) in the Arctic region98

cause a typical hydrological behavior in lakes. This hydrological behavior in-99

fluences the Physico-chemical subsystem of the lake.100

Limiting Nutrients in the form of nitrates, silicates and carbon dioxide101

are between 90 and 100% available for phytoplankton in all year. Thus, phy-102

toplankton and periphyton biomass is dominated by planktonic (38.6%) and103

Periphytic diatoms (45%). For zooplankton, the 91.7% is dominated by her-104

bivorous. At the Benthic Zone, detritivorous invertebrates with a 86.8% of105

total abundance and piscivorous fishes with the 85.8% are the two groups with106

high dominance in their respectively group.107

3.2. North Highland Lake (NH)108

.109

6

North highland lake corresponds with a mesotrophic ecosystem in a cool110

North-Temperate climate (Mean=5.3C). Levels of chlorophyll are between111

2.2.-6.2 mg/sec. The surface is covered with ice in winter (end of November,112

December, January and early February). Ice covering forms a barrier to the113

wind which minimizes losses of water evaporation while the bottom of the114

lake remains unfrozen. The water column does not thermostratified and is115

permanently well mixed whit levels of 50 percent in summer and 90 percent116

in winter. The maximum flows are in spring and autumn (9.6 m3/sec) with117

minimum flow in summer(0.6 m/sec). Evaporation is reduced because their118

water is more or less cold and vapour-pressure gradients are no large (mean119

of 9,262 m3/day). Retention Time is maximum in summer with 100 days.120

Oxygen concentration is upper to 10 mg/lt in the three layers. pH mean121

values are around 7 to 7.3 units, but it moves in a range of 6.7 to 7.8 units122

from the surface to bottom.123

The correlation among variables are more seasonal in NH than NL. This124

means, the period of summer is related with high retention time, higher pH.125

Winter season is related with the higher levels of oxygen, inflow and out-126

flow and oxygen. However, there is a more strong correlation of benthic and127

sediment Oxygen.128

Limiting Nutrients like nitrates and carbon dioxide are around of 95%129

available for phytoplankton. Phosphates and Silicates shown variations and130

less percentage of availability. The former around of 80% and the second131

one around 95% all year. Biomass composition is dominated by planktonic132

(46.7%) and benthic (41%) diatoms. Zooplankton composition is almost of133

herbivorous zooplankton (91.4%), but carnivorous zooplankton reaches a low134

7

percentage of 8.6. In the group of benthic invertebrates detritivorous domi-135

nates with the 87.5%. Fish community is dominated by benthic fish again,136

but in a high proportion (88.9%).137

3.3. North Lowland Lake (NL)138

.139

North lowland is an eutrophic lake, located in a warm North-Temperate140

climate (mean T of 14C). Their primary production expressed in mg/m3 of141

chlorophyll is around 6.3-19.2. There are four seasons in a year, winter,142

spring, summer and autumn. In summer, the flow variations between inflow143

and outflow fall to 3.5 from 25.2 m3/sec. Retention time increases to 100144

days. The lack of wind and high temperatures (24C), causes the water column145

thermostratification. Stratification is expressed in generation of two layers.146

At the border of these layers, temperature changes dramatically (24C Surface147

to 20.6C in Planktonic layer, to 17.3C in Benthic layer). Water above and148

below of thermocline do not mix. The warmer water is near the surface and149

denser water is near the bottom. In winter, there is no ice covering in the150

surface. Opposite to the summer when the flow is minimum, in spring and151

autumn the water column overturns (Retention Time of 14 days and Zone152

Mixing of 100%), causing increases in conductivity. In summer, depletions153

of oxygen at the three layers are more drastic than Artic lakes (below 8.7154

mg/lt). Oxygen is directly correlated with the zone mixing, inflow and out-155

flow, and inverse correlated with the others parameters, especially with pH156

and retention time.157

All limiting nutrients are above of 90% available for phytoplankton in158

all seasons. According with this availability, phytoplankton and periphyton159

8

biomass composition is dominated by planktonic (47%) and benthic (34.3%)160

diatoms. This way, 100% of zooplankton composition is reached by herbivo-161

rous zooplankton and fish community is dominated by benthic fish (67.6%).162

3.4. Tropical Lake (T)163

The Tropical Lake is a hyper-eutrophic ecosystem (Chlorophyll 19.2164

mg/lt) located in a moist Tropical climate, at north of the equator, near165

to the tropic of cancer with a mean temperature of 25C in the surface layer.166

Tropical lake has a wet season and a dry season. Higher irradiance conducts167

to higher temperatures and smaller thermal differences between layers. For168

that reason, the water column is permanently warm and stratified. Stratifica-169

tion is by the heat exchange, but it is less stable than stratification in lakes at170

higher latitudes. Specially, because the wind could have great incidence in the171

mixing of the water column. Thus, intra-seasonal variations have an effect172

in thickness of the mixed layer than other morphometrically similar temper-173

ate lakes (AES, Lewis**). The maximum flow of water is in the wet season,174

and minimum flow is in the dry season. Episodes of heat and mixing, affects175

the nutrient cycling and plankton dynamics. It is highlighted that primary176

production in tropical lakes is about twice than higher latitudes. Also, it is177

known that Nitrogen is the more limiting nutrient.178

The equilibrium among species inside phyto and periphyton communities179

(around 33% for diatoms, green algae and cyanobacteria) is higher. Zoo-180

plankton populations are dominated by herbivorous (90%), benthic by detri-181

tivorous invertebrates (84.4%) and fishes (87%).182

9

4. Results183

4.1. Complexity as the Balance Between Emergence and Self-organization184

4.1.1. Complexity in the Physico-chemical Subsystem185

At Ar variables related with light in surface, planktonic and benthic zones186

(SL, PL and BL) have high value of emergence. Variation of light was 10.28187

10.44. Meantime, benthic conductivity (BCd, 600.3 40.8 S) and percent-188age of water mixing between planktonic and benthic zones (ZM, 50 3.53%)189have very high self-organization. Remaining variables were classified in very190

high complexity category, with the exception of two variables associated con-191

ductivity (ICd=3896.96 17.29 S and BCd= 600,32 40.53 S) which were192ranking in very low complexity category.193

At NH, light variables increase their ranking to very high emergence cat-194

egory, in consequence its complexity was reduced to low and very low cate-195

gories. The light variation for all zones was between 12.31 9.31. Meantime196BCd (598.2 95.91 S) and ZM (62.05 19.36) variables increase its com-197plexity to high category.198

At NL, temperature variables increases its level of change to very high.199

The variation for all zones was between 11.3 6.5. Also light variables200(14.6 7.4). Remaining variables had fair to very high emergence. Thus,201self-organization, in general is low. As the result of the above facts, variables202

with very high complexity were relating with hydrology (IO= 10.7 6.5, RT=20361.9 40.4) conductivity (ICd=870 93.5, PCd=1132.6 182.2) and pH204in all zones of lake (7.0 14,67 0.2)205

At T, all Physico-chemical variables have similar levels of regularity (S)206

and change (E); consequently most variables have high or very high complex-207

10

Figure 1: Principal Component Analysis for Physico-chemical Subsystem.

ity with the exception of ZM (16.48 2.29) and Sediment Oxygen (SdO2 =2082 0).209

Based on Principal Component Analysis (PCA) Figure 1 the ordination210

of emergence, self-organization, complexity and autopoiesis properties, we211

can summarize that variables of Physico-chemical subsystem can conform 3212

groups. Group 1 including variables related with high changes or emergence213

as light. Group 2 conformed by variables associated with high regularity like214

conductivity and zone mixing, and Group 3 Variables expressing high com-215

plexity like temperatures, oxygen,pHs, retention time and inflow and outflow.216

11

4.1.2. Complexity in the Limiting Nutrient Subsystem217

Limiting nutrients shown at the Arctic high changes in inflow silicates218

(IS= 25014.31 3545,08), carbon dioxide in the inflow and planktonic Zones219(ICD= 12006.9 1701.8, PCD= 10105.7 979.4). Very high regularity in220nitrogen at the 3 layers of lakes (IN,PN and BN= 67.87 13.36); high221regularity in silicates (PS= 40550.7 7272.34, BS= 41160.69 7453.5),222phosphorous (PP=9.36 1.25, BP = 9.52 1.28) in planktonic and benthic223zones. Also, there is high self-organization for planktonic detritus (PDt=224

21.26 2.55). In complexity terms very high category was for IS, PP,BP,225BCD, PDt and BDt.226

For the NH, planktonic carbon dioxide (PCD= 9788.5 2119.1) had227very high emergence. Inflow and benthic carbon dioxide (ICD= 10005.7 2281418.2, BCD= 7571.1 3150.3) were in high emergence category. In con-229trast, variables with very high self-organization were silicates in planktonic230

and benthic zone (PS= 25257.32 7025.4 BS= 25703.99 7216.8), phos-231phorous in 3 layers (IP,PP and BP= 7.69 2.26) and nitrogen in inflow232and planktonic zone (IN and PN= 62.91 15.1); nitrogen in benthic was233high self-organization (79.21 9.34). Variables in the very high complex-234ity category were inflow silicates (IS), carbon dioxide in inflow and Benthos235

(ICD,BCD), and detritus (PDt,BDt).236

At the NL, due to an increasing in the emergence of nitrogen (143.4237

39.08) and decreasing in the self-organization of detritus (228989.6 238245332.9), 13 of the 16 variables of the limiting nutrient components was239

classified in very high and high complexity categories. Carbon dioxide in240

planktonic and benthic zones (PCD= 9746.7 1477.7 and BCD= 10888.03241

12

2105), and benthic detritus (BDt= 457320.9 126432.4) were catego-242rized as low complexity variables because of their very high emergence. At243

this point of the gradient Ar-T, complexity of the limiting nutrients subsys-244

tem has an important variation in terms of its increasing with respect of Ar245

and NH levels. This levels continuous its increment due the balance between246

emergence and self-organization values at the end of the gradient. This way,247

at the tropic lake a very high levels of complexity for the majority of variables248

are shown. Only a very high emergence of detritus were the exception.249

From PCA ordination Figure 2 for limiting nutrients at all lakes, the250

groups that can be identified were a first group representing emergence with251

detritus and carbon dioxide variables. A second group representing self-252

organization in nitrogen and inflow phosphorous. A third group representing253

complexity variables with silicates and phosphorous in planktonic and benthic254

zone.255

4.1.3. Complexity in Biomass Subsystem256

At the Ar, self-organization for all groups of phyto and zooplankton species257

in all zones, were high or very high. Only the low values of emergence of di-258

atoms (PD and BD= 185.4 191.3), cyanobacteria (PCy and BCy= 118.9259 169.3) and green algae (PGA and BGA= 164.2 160.6) permits that260these photosynthetic organisms reach very high levels of complexity and au-261

topoiesis. This situation continuing in NH in spite of planktonic diatoms262

(PD= 281.12 209.35) and cyanobacteria (PCy= 162.9 169.5) reached263the fair category of emergence. It means, the feature of this two types of lakes264

is their regularity.265

In a similar way with limiting nutrient subsystem, when gradient Ar-T266

13

Figure 2: Principal Component Analysis for Limiting Nutrients Subsystem

reaches the NL point, the dynamics of emergence and self-organization varies267

in considerable level. Here, the complexity of almost all variables were maxi-268

mum due the balance in self-organization and emergence. Only chlorophyceas269

(PCh= 6.2 5.1), benthic detritivorous (BDt= 3.84 71.71) and fishes in270planktonic and benthic zones (PiF, BF and PF= 0.2 4.51) have very high271regularity for all annual cycle.272

In contrast to the NL, the biomass subsystem in the tropic reflects very273

low complexity due the very high self-organization of the living taxa. Only274

planktonic and piscivorous fishes (PlF= 0.099 0.005; PiF= 0.13 0.67)275have very high and high complexity, respectively.276

From PCA ordination Figure 3, it can be seen that photosynthetic taxa of277

14

planktonic and benthic zone are more emergent. In addition with planktonic278

primaries and secondaries consumers, clorophyceas and benthic detritivorous279

are more self-organized than other taxa.On the other hand, piscivorous and280

planktivorous fishes are more complex.281

Figure 3: Principal Component Analysis for Biomass Subsystem

4.1.4. Complexity in Latitudinal Gradient282

Comparing the average of complexity for an annual cycle in Ar-Ttransect283

as latitudinal gradient, we can see thatNLappears to represent a transition284

point for complexity values Figure 4. At this point for Physico-chemical285

subsystem decreasing complexity goes from very high to highcategory. This286

isby reason of emergence increasing (0.75). Then, at the tropical lake the287

category of complexity returns to very high level due to the increasing of288

15

self-organization (regularity in variables) and their consequently emergence289

reduction.290

0.0

0.5

1.0

1 2 3 4date

pop

group

1

2

3

Figure 4: Complexity in The Latitudinal Gradient Ar-T

For limiting nutrients subsystem, complexity goes from high at the Ar291

to fair in NH; high category is maintaining in NL and T. The transition292

pointinNL is more evident fromtheiremergence values. Emergence is almost293

0.62 (fair category).Also, emergence starts in the low category in Ar,and294

finish in it Tin the same category. This means, limiting nutrient change to295

a greater proportion at NL latitudes.296

For Biomass, the transition at NL point is more evident than other sub-297

systems because complexity values reach the higher category of Ar-T transect298

(0,74; very high category). For Ar and NH biomass, complexity value was299

classified in the low category and for T in very low.300

In terms of mean complexity by subsystem, it is seen that Physico-chemical301

Limiting Nutrients Biomass. This order corresponding with the auton-302

omy, which means in general biomass is affected in an important way by303

changes in their environment. However, the dispersion of Biomass is more304

than the other two (Figure 4) which means that biomass can respond accord-305

16

ing with the law of required variety to the environmental changes between its306

complexity range (0.382 0.22).307

By lake, we can observe that Ar and NL were in the high category and308

NH and T were in the fair category. In terms of dispersion T ArNHNL309

(Fig.**).310

Parametric multiple comparison by means of the test of Tukey shows that,311

in terms of average complexity, physic-chemical and limiting nutrients did312

not have significance differences (p= 0.85; p0.05) while biomass has sig-313

nificance differences with the other two subsystems (*p0.05). On the other314

hand, ANOVA test shows that there are not significance differences among315

complexity of lakes in the Ar-T transect (p0.05).316

In ecological terms, the dynamics observed at NL point in the transect Ar-317

T could be estimates as a complexity ecotone or complextone (tone, from the318

Greek tonos or tension). That means that NL point could be considered as319

a physical transition zone for complexity values among lakes in a latitudinal320

gradient. Consequently for some variables therein subsystems it is estimates321

that it could be represents diverse complexity ecoclines or complexcline (cline322

from Greek: to possess or exhibit gradient, to lean), due their complex-323

ity variation. For example for biomass, we can that there is a biocline in324

the transect Ar-T and in particular for cyanobacteria at the planktonic zone325

(PCy) we can name a complex cyanocline.326

From PCA ordination from variables of all subsystems conducts to the327

conformation of groups of lakes based on emergence, self-organization, com-328

plexity and autopoiesis properties. However, we chose the complexity criteria329

for definition of groups due to complexity relates the regularity and changing330

17

aspects, and it is the base for autopoiesis calculation. This way, the general331

ordination shows theNL disjoins of the group conforming by NH-Ar with T.332

The separation of NL from other lakes is by cause of their variability and load333

of some variables of biomass and Physico-chemical subsystem. These vari-334

ables were macrophytes, clorophyceas and planktonic phosphorous and by the335

low level in complexity of the fishes and light variables. NL separation also336

supports the consideration that NL represents a transition in the values for337

all properties, marking this location as differential on the latitudinal gradient338

from the arctic to tropic.339

4.2. Autopoiesis340

There are two ways for observing autopoiesis (A). The first one is the A of341

the each variable. Variables with more complexity than other have a positive342

A reflecting more autonomy. Variables with low complexity than other have343

negative A, reflecting less autonomy. Results of A by variable in each subsys-344

tem can be seen in Annex A. In general, variables in categories of very high345

and high complexity, have more A and they resulting as more autonomous, as346

well.That means, they have more capacity of adaptation in front the changes347

of their environment which is constituted by the other subsystem variables.348

The second one form of determine autopoiesis (A) is among variables of dif-349

ferent subsystems, according with the matter-energy flux in ecosystems. It is350

well-know that photosynthetic living beings depending of solar radiation and351

nutrients availability as the base for its metabolism process. Also, zooplankton352

depending of grazing phytoplankton communities.353

Starting from complexity values of selected variables of Physico-chemical,354

Limiting Nutrients and Biomass are shown in the Table 1. We compare A355

18

of Biomass related with the their Physico-chemical and Limiting Nutrients356

environment. A Biomass values for planktonic and benthic zone are depicted357

in Figure 5.358

Table 1: Selected Variables for Phytoplanktonic Biomass Autopoiesis Calculation

Component Planktonic zone Benthic zone

Physiochemical Light, Temperature, Conductivity, Oxygen, pH. Light, Temperature, Conductivity, Oxygen, Sediment Oxygen, pH.

Limiting Nutrients Silicates, Nitrates, Phosphates, Carbon Dioxide. Silicates, Nitrates, Phosphates, Carbon Dioxide.

Biomass Diatoms, Cyanobacteria, Green Algae, Chlorophyta. Diatoms, Cyanobacteria, Green Algae.

From the Table 1, we notice that Biomass in T is near to zero in the359

planktonic zone and zero in the benthic zone. It means that tropical biomass360

is almost static. We can verify this with their very-high category of self-361

organization obtained. This implies that any pattern in complexity can be362

observed in biomass as the result of the influence of its environment which is363

represented by its Physico-chemical and Limiting Nutrients subsystem. This364

case gives a minimal A for all comparisons carried out with the trajectories365

of Biomass in tropical lake.366

Values of A1 were reached by photosynthetic living beings located at the367

benthic zone of Ar,NH and NL in front of Physico-chemical and Limiting nu-368

trients. Also, A1 was reached by Biomass/Limiting Nutrients of planktonic369

zone at Ar and NH and Biomass/Physico-Chemical of planktonic zone of NL370

Figure ??. In terms of the Ashbys Law of Requisite Variety(Ashby, 1956),371

photosynthetic biomass in Polar and Template latitudes have more variety372

than its environment. More variety is related with more number of states.373

19

More states permit to face on environmental changes. Variety is the result374

of very high complexity and could be reflecting as more autonomy of the taxa375

therein.376

Ar

NH

NL

T

B.P.P B.P.B B.LM.P B.LM.B

Figure 5:

The remaining cases of biomass obtain values less than one (A1) and377

were related with their response in front of Physico-chemical in the planktonic378

zone of Ar and NH. Also NL biomass response in front of limiting nutrient379

at the same zone obtain A1. This values between 0.72 and 0.98 shows that380

their environment changes more than the populations of photosynthetic living381

beings. As we can see, the weak or almost fair lake-specific response of the382

biomass, suggest that species involved in this subsystem could be affected in383

a high proportion in case of strong change events.384

On the base of above findings, we thought that relationships evaluated in385

20

biomass of different lakes might be evaluated in the context of environmental386

variability.387

4.3. Homeostasis388

The homeostasis h calculation by comparing the daily values of all vari-389

ables is a useful indicator for periodic or seasonal dynamics characterization390

and determination. h can represents the temporal variation of the state of391

each subsystem in each lake. This situation is more evident in the Physico-392

chemical subsystem of lakes which responds proportionally with the seasonal393

changes affecting variables like temperature, light and others related with the394

hydrological cycle. Limiting Nutrient subsystems works in a shorter scale than395

Physico-chemical subsystem; it seems that nutrients could vary more at week396

scales than month scales. For all three subsystems, biomass demonstrated397

has similar scale variation with Physico-chemical subsystem; in special at398

the T which their biomass variation takes place in several months intervals.399

The above homeostasis results demonstrates the importance of the temporal400

timescale because h can vary considerably if we compare states every minute401

or every month (see homeostasis figures in Annex A).402

For H, the Table 2 shows average values and standard deviation for all403

lakes and subsystems. Considering the lakes studied maintains periods with-404

out changes, values of H are all in the category of very-high homeostasis. We405

observe that Biomass and Physico-chemical were more stable in a year than406

Limiting Nutrients subsystem. In detail, the Ar and NL Biomass were more407

regulars than NL and T. Physico-chemical subsystem has a similar regularity408

for all lakes being the lower NL. For Limiting Nutrients, the lower regularity409

is also for NL and the high for Ar.410

21

Table 2: Homeostasis averages H for Lakes

Lake Biomass Sd Phy Chem Sd LimNutr Sd

Ar 0.9803620 0.04471970 0.9594521 0.06440536 0.9574551 0.06520360

NL 0.97643440.05434534 0.94301370.0922801 0.91471510.10648702

NH 0.91729450.10604234 0.95739730.07495470 0.94516000.08047069

T 0.95799180.11550661 0.96493150.05803853 0.94828260.07298964

Global 0.95802070.03578496 0.956198630.01496563 0.94140320.01791861

22

5. Discussion411

5.1. The ecological sense of the proposed measures.412

The proposed measures characterize the different ecological configurations413

and dynamics that elements of lakes acquire through their interactions. From414

simple mathematical expressions, based on probabilistic features, we can cap-415

ture the properties and tendencies of the ecological systems, considering the416

scale at which they are described. In contrast with other complex computa-417

tional methods, our approach permits analyze the properties and tendencies418

of any variables, in different subsystems, for one or more ecosystems. Also,419

we can change scale and apply the measures there, in order to determine at420

which scale the richness dynamics is representative. In instance, for Physico-421

chemical subsystem in Ar, previous analysis shows that base 10 is very infor-422

mative and represents the dynamics as base 8 or 16,34 and 64 (Fernandez et423

al., 2014).424

The integration of self-organization and emergence aspects into our C425

measure has advantages asthe complex dynamics of an ecological system, can426

be observed as the balance between regularity and change or variability. In this427

context, it can define which variable, process or ecosystem is more complex428

than other as we can observe in429

The characterization of the behavior of the biotic component, in front430

of environmental disturbances or environmental variability, can be comple-431

mented with the autopoiesis and homeostasis measures. It is an important432

feature that will be useful for studies of global ecological change. In general433

we suggest that systems with a higher complexity are more robust while those434

with a lower complexity are less.435

23

Clarifying that chaos should not be confused with complexity (Gershen-436

son, 2013), we highlight that our measures can distinguish between random437

and non-random ecological processes or variables. The former is related with438

very high emergence (high entropy), it implies too many changes and pat-439

terns destruction. The second implies very high self-organization (very low440

entropy); it prevents that complex patterns emerging. For further details, this441

randomness can be examined in the probability distribution for any process442

or variables at different scales, as well.443

Besides our measures can be related with the temporal and organizational444

scales and fluctuating environments, they can be related with other ecolog-445

ical aspects like occupancy, movement patterns and numbers of species as446

show in Fernandez et al (2013). This way, proposed measures can be good447

complements in status and trends study, in ecological communities.448

Based on the complexity average values for the 3 subsystem, we can449

observe that before the NL point, the complexity of limiting nutrients and450

biomass have an increasing trend; then the values trend is decreasing. Mean-451

time complexity in the Physico-chemical shows a decreasing trend before NL,452

then at T ischangein an increasing way. This results suggest that there is a453

the differential trend of complexity according with the subsystems. Thus, we454

can not suggest that there is a clear global pattern in the trend of complexity455

for lakes according with the latitude.456

5.2. Complexity and others Measures of Information.457

Complexity has been correlated with other measures of information like458

Fisher Information (Prokopenko et al., 2011) and Tsallis Information (Tsal-459

lis, 2002; Gell-Mann and Tsallis, 2004). Contrasting C with Fisher Infor-460

24

mation we observed that C is smoother, so it can represent dynamical change461

in a more gradual fashion. Fisher information has a much higher steepness462

in comparison with C. On the other hand, test of Tsallis information for Ar463

lakes shows that it follows self-organization patterns in one cases and others464

emergence patterns. This results generate a difficult to establish a significant465

correlation with C for Tsallis measure, is increased with its variable scale466

and determination for the optimal q choice. It seems that q=2 is the value467

in which some correlations can be appear.468

In the context of the physics, an interesting point is that different measures469

of entropy are used for describe different probability distribution. Shanon en-470

tropy is logarithmic, and it is appropriated for the phenomena with exponen-471

tial distribution. Tsallis entropy has a power model, and it is appropriated for472

phenomena with power distribution. It has been found that critical phenom-473

ena considered by someones as complex, usually have power law distribution474

referred as self-organized criticality (Per Bak, **). Consequently, Tsallis in-475

formation has been recommended as complexity measure (**). However, we476

consider that in itself Tsallis entropy might not be a complexity measure, in477

particular for its q parameter dependence and sensitivity. Tsallis information478

could be a more a general description applicable to several phenomenon, previ-479

ous their distribution inspection and knowing. As a sample of this situation,480

the results of application of Tsallis entropy to Physico-chemical subsystem481

can be observed in the Annex **482

25

6. Conclusions483

Based on Information Theory we define complexity as a type of balance a484

balance between change (emergence) and regularity/order (self-organization).485

The balance is reached in terms of their autonomy (autopoiesis) and (home-486

ostasis). It can be seen that variables, subsystem or system with a homo-487

geneous distribution have higher values of emergence whilevariables with a488

more heterogeneous distribution have a higher self-organization.489

For the two additional properties in lakes studied, Homeostasis values490

coincide with the variation of different seasons according with the latitudinal491

location of lakes. Autopoiesis values show a higher degree of independence of492

biological components over their environment.493

There are different ways to describe the state of an ecosystem and the dy-494

namical of species therein. Measures of emergence, self-organization, home-495

ostasis, autopoiesis and complexity can complement the description of ecosys-496

tems and species dynamics. They could be viewed as ecological indicators497

at different scales and have high potential for comparative analysis among498

ecosystems. In fact, the complexity analysis can be focused in either particu-499

lar system components, or a subsystem of the whole, oraecosystem as unity.500

For example, we can observe the complexity of the predatory-prey cycles re-501

lated with the movement decisions and foraging behaviors in contrasting with502

the vegetation patterns. In this sense, our measure can contribute with the503

interpretation of the six types of Complexity (spatial, temporal, structural,504

process, behavioral, and geometric- cloehle, 2004).505

A506

26

7. Complexity for Each Component507

SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH

Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 6: Complexity in Physico-chemical Subsystem for an Arctic Lake.


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.4

0.8

1.2

Figure 7: Complexity in Physico-chemical Subsystem for a North Higland Lake.

@ArticleEinstein, author = Albert Einstein, title = Zur Elektrody-508

namik bewegter Korper. (German) [On the electrodynamics of moving bod-509

ies], journal = Annalen der Physik, volume = 322, number = 10,510

pages = 891921, year = 1905, DOI = http://dx.doi.org/10.1002/andp.19053221004511

512

27

SL PL BL ST PT BT I.O RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH

Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.4

0.8

1.2

Figure 8: Complexity in Physico-chemical Subsystem for a North Lowland Lake.


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 9: Complexity in Physico-chemical Subsystem for a Tropical Lake.

IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde

Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.5

1.0

1.5

Figure 10: Complexity in Limiting Nutrients Subsystem for an Arctic Lake.

28


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.5

1.0

1.5

Figure 11: Complexity in Limiting Nutrients Subsystem for a Noth Higland Lake.


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.4

0.8

1.2

Figure 12: Complexity in Limiting Nutrients Subsystem for a Noth Lowland Lake.


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.4

0.8

1.2

Figure 13: Complexity in Limiting Nutrients Subsystem for a Tropical Lake

29

PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF

Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

1.0

2.0

3.0

Figure 14: Complexity in Biomass Subsystem for an Arctic Lake.


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

1.0

2.0

3.0

Figure 15: Complexity in Biomass Subsystem for a North Higland Lake


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

0.0

0.4

0.8

1.2

Figure 16: Complexity in Biomass Subsystem for a North Lowland Lake.

30


Emergence

0.0

0.2

0.4

0.6

0.8

1.0


Selforganization

0.0

0.2

0.4

0.6

0.8

1.0


Complexity

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Homeostasis


Autopoiesis

05

1015

Figure 17: Complexity in Biomass Subsystem for a Tropical Lake.

31

elsarticle template

Documents

ecological complexity

complexity theory

ecological dynamics

lakes complexity

ecological systems

complexity itselfkeywords

complexity valuesin

study of complexity