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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
Complexity
0.0
0.2
0.4
0.6
0.8
1.0
0 100 200 300
0.0
0.2
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Homeostasis
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
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.
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
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
Complexity
0.0
0.2
0.4
0.6
0.8
1.0
0 100 200 300
0.0
0.2
0.4
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Homeostasis
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
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
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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
SL PL BL ST PT BT I.O RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
SL PL BL ST PT BT I.O RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
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
SL PL BL ST PT BT I.O RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
Autopoiesis
0.0
0.4
0.8
1.2
Figure 8: Complexity in Physico-chemical Subsystem for a North Lowland Lake.
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
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
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
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1.0
Homeostasis
SL PL BL ST PT BT IO RT Ev ZM ICd PCd BCd SO2 PO2 BO2 SdO2 IpH PpH BpH
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
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
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
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Autopoiesis
0.0
0.5
1.0
1.5
Figure 10: Complexity in Limiting Nutrients Subsystem for an Arctic Lake.
28
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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
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Complexity
0.0
0.2
0.4
0.6
0.8
1.0
0 100 200 300
0.0
0.2
0.4
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Homeostasis
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Autopoiesis
0.0
0.5
1.0
1.5
Figure 11: Complexity in Limiting Nutrients Subsystem for a Noth Higland 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
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Complexity
0.0
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0.6
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1.0
0 100 200 300
0.0
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Homeostasis
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Autopoiesis
0.0
0.4
0.8
1.2
Figure 12: Complexity in Limiting Nutrients Subsystem for a Noth Lowland 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
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Complexity
0.0
0.2
0.4
0.6
0.8
1.0
0 100 200 300
0.0
0.2
0.4
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1.0
Homeostasis
IS PS BS IN PN BN IP PP BP ICD PCD BCD Pde Bde
Autopoiesis
0.0
0.4
0.8
1.2
Figure 13: Complexity in Limiting Nutrients Subsystem for a Tropical Lake
29
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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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Autopoiesis
0.0
1.0
2.0
3.0
Figure 14: Complexity in Biomass Subsystem for an Arctic Lake.
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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Autopoiesis
0.0
1.0
2.0
3.0
Figure 15: Complexity in Biomass Subsystem for a North Higland Lake
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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Autopoiesis
0.0
0.4
0.8
1.2
Figure 16: Complexity in Biomass Subsystem for a North Lowland Lake.
30
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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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Selforganization
0.0
0.2
0.4
0.6
0.8
1.0
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
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
PD PCy PGA PCh BD BCy BGA SurM SubM HZ CZ BH BDt PlF BF PiF
Autopoiesis
05
1015
Figure 17: Complexity in Biomass Subsystem for a Tropical Lake.
31