Abstract. We apply to aquatic ecosystem components, formal measures of emergence, self-organization, homeostasis, autopoiesis and complexity developed by the authors. These measures are based on information theory, created by Claude Shannon. In particular, they were applied to the physiochemical component of an aquatic ecosystem located in the Arctic Polar Circle. The results show that variables with a “homogeneous” distribution of its values in all states presented obtained higher values of emergence, while variables with a more “heterogeneous” value’s distribution had a higher self-organization. It is confirmed that variables having high complexity values reflect a balance between change (emergence) and regularity/order (self-organization). In addition, homeostasis values were coinciding with the variation of winter and summer season. Also, autopoiesis values confirmed a higher degree of independence of some components (physiochemical and biological) over others. This approach showed how the ecological dynamics can be described in terms of information.
Keywords Complex Systems, Information Theory; Complexity, Self-organization, Emergence, Homeostasis, Autopoiesis.
Traditionally, science has been reductionistic. Reductionism—the most popular approach in science—is not appropriate for studying biological and ecological systems, as it attempts to simplify and separate in order to predict their future behavior and states. Due to prediction difficulty, biological and ecological systems have been considered as complex. This “complexity” is due to the relevant interactions between components. It is important to highlight that, etymologically, the term complexity comes from the Latin plexus, which means interwoven. In other words, something complex is difficult to separate.
Being complex, biological and ecological systems have properties like emergence, self-organization, and life. It means, biological and ecological systems dynamics generate novel information from the relevant interactions among components. Interactions determine the future of systems and its complex behavior. Novel information limits predictability, as it is not included in initial or boundary conditions. It can be said that this novel information is emergent since it is not in the components, but produced by their interactions. Interactions can also be used by components to self-organize, i.e. produce a global pattern from local dynamics. The balance between change (chaos) and stability (order) states has been proposed as a characteristic of complexity. Since more chaotic systems produce more information (emergence) and more stable systems are more organized, complexity can be defined as the balance between emergence and self-organization. In addition, there are two properties that support the above processes: homeostasis refers to regularity of states in the system and autopoiesis that reflects autonomy.
Due to a plethora of definitions, notions, and measures of these concepts have been proposed, the authors proposed abstract measures of emergence, selforganization, complexity, homeostasis an autopoiesis based on information theory, in focus to clarify their meaning with formal definitions (Gershenson and Fernández, 2012). Now we propose apply to aquatic ecosystem formal measures developed. From the application to the case of study, an Artic lake, we clarify the ecological meaning of these notions, and we showed how the ecological dynamics can be described in terms of information. This way the study of the complexity in biological and ecological cases, now is easier.
In the next section, we present a brief explanation of notions of self-organization, emergence, complexity, homeostasis, autopoiesis and limnology (as the field that studies the lakes9. In section 3, we present the synthetic measures of results of emergence, selforganization, complexity, and homeostasis. Section 4 describes our experiments and results with the Artic lake, which illustrate the useful of the proposed measures. This is followed by a discussion in Section 5. The article closes with proposals for future work and conclusions.
The measures applied in this paper have recently developed and compared with other previously proposed in the literature (Fernández et al., 2012; Gershenson and Fernández, 2012); more refined measures, based on axioms, have been presented in Fernández et al., (2013).
In general, Emergence refers to properties of a phenomenon that are present now and were not before. If we suppose these properties as non-trivial, we could say it is harder now than before to reproduce the phenomenon. In other words, there is emergence in a phenomenon when this phenomenon is producing information and, if we recall, Shannon proposed a quantity which measures how much information was “produced” by a process.Therefore, we can say that the emergence is the same as the Shannon’s information I. Thus =I
Self-organization has been correlated with an increase in order, i.e. a reduction of entropy (Gershenson and Heylighen, 2003). If emergence implies an increase of information, which is analogous to entropy and disorder, self-organization should be anti-correlated with emergence. We propose as the measure S = 1 − I = 1 − E.
We can define complexity C as the balance between change (chaos) and stability (order). We have just defined such measures: emergence and self-organization. Hence we propose: C = 4 · E · S.. Where the constant 4 is added to normalize the measure to [0, 1]
For homeostasis H, we are interested on how all variables of a system change or not in time. A useful function for comparing strings of equal length is the Hamming distance. The Hamming distanced measures the percentage of different symbols in two strings X and X’.
As it has been proposed, adaptive systems require a highC in order to be able to cope with changes of its environment while at the same time maintaining their integrity (Langton,1990; Kauffman, 1993). If we have X represent the trajectories of the variables of a system and Y represent the trajectories of the variables of the environment of the system, If X had a high E, then it would not be able to produce its own information. With a highS, X would not be able to adapt to changes in Y . Therefore, we propose: A = C(X )/C(Y) .
Limnology is related with formal study of lakes. In particular treats with the distinctive properties of individual lakes and the nature of their interactions with their surrounding environment (Catchment basin). Lakes has distinct zones of biological communities linked to the physical structure of the lake (Fig. 1). Classical zones studied are (i) Macrophyte or littoral zone, composed mainly by aquatic plants, which are rooted, floating or submerged. (ii) The planktonic zone corresponds to the open surface waters; away from the shore in which organisms passively floating and drifting on the lakes’ currents (phyto and zooplankton). Planktonic organism are incapable of swimming against a current, however some of them are somewhat motiles. (iii) Benthic zone is the lowest level of a body of water related with the substratum, including the sediment surface and subsurface layers. (iv) Mixing zone where the interchange of water from planktonic and benthic zone can be mixed.
At different zones, one or more components or subsystems can be an assessment for the ecosystem dynamics. Our case of study considered three components: physiochemical, limiting nutrients and photosynthetic biomass for the planktonic and benthic zones. The physiochemical component refers to the chemical composition of water. It is affected by various conditions and processes such as geological nature, the water cycle, dispersion, dilution, solute and solids generation (e.g. photosynthesis), and sedimentation. Related to the physiochemical component, limiting nutrients which are basic for photosynthesis are associated with the biogeochemical cycles of nitrogen, carbon, and phosphorous. These cycles permit the adsorption of gases into the water or the dilution of some limiting nutrients. In addition, among limnetic biota, photoautotrophic biomass is the basis for the trophic web establishment. The term autotrophs is used for organisms that increase their mass hrough the accumulation of proteins which they manufacture, mainly from inorganic radicals (Stumm, 2004). This type of organisms can be found at the planktonic and benthic zones.
In general, Arctic lake systems are classified as oligotrophic due to their low primary production, represented in chlorophyll values of 0.8-2.1 mg/m3. The lake’s water column, or limnetic zone, is well-mixed; this means that there are no stratifications (layers with different temperatures). During winter (October to March), the surface of the lake is ice covered. During summer (April to September), ice melts and the water flow and evaporation increase. Consequently, the two climatic periods (winter and summer) in the Arctic region cause a typical hydrologic behavior in lakes. This hydrologic behavior influences the physiochemical subsystem of the lake.
The data from an Artic lake model used in this section was obtained using The Aquatic Ecosystem Simulator (Randerson and Bowker, 2008). Table 1 show the variables and daily data we obtained from the Arctic lake simulation. The model used is deterministic, so there is no variation in different simulation runs. There are a higher dispersion for variables such as temperature (T) and light (L) at the three zones of the Arctic lake (surface=S, planktonic=P and benthic=B); Inflow and outflow (I&O), retention time (RT) and evaporation (Ev) also have a high dispersion,Ev being the variable with the highest dispersion.
Figure 2 shows the values of emergence, self-organization, and complexity of the physiochemical subsystem. Variables with a high complexity C reflect a balance between change/chaos (emergence) and regularity/order (self-organization). This is the case of benthic and planktonic pH (BpH;PpH), I&O (Inflow and Outflow) and RT (Retention Time). For variables with high emergencies (E > 0.92), like Inflow Conductivity (ICd) and Zone Mixing (ZM), their change in time is constant; a necessary condition for exhibiting chaos. For the rest of the variables, self-organization values are low (S < 0.32), reflecting low regularity. It is interesting to notice that in this system there are no variables with a high self-organization nor low emergence. Since E,S,C , these measures can be categorized into five categories as shown in Table 2. These categories are described on the basis of the range value, the color and the adjective in a scale from very high to very low. This categorization is inspired on the categories for Colombian water pollution indices. These indices were proposed by Ramírez et al. (2003) and evaluated in Fernández et al. (2005).
From figure 2 and a principal component analysis (not shown), we can divide the values obtained in complexity categories as follows:
Very High Complexity C . The following variables balance self-organization and emergence: benthic and planktonicpH (BpH, PpH), inflow and outflow (I&O), and retention time (RT). It is remarkable that the increasing of the hydrological regime during summer is related in an inverse way with the dissolved oxygen (SO_2; BO_2). It means that an increased flow causes oxygen depletion. Benthic Oxygen (BO_2) and Inflow Ph (IpH) show the lowest levels of the category. Between both, there is a negative correlation: a doubling of IpH is associated with a decline of BO_2 in 40 percent.
High ComplexityC [0.6, 0.8). This group includes 11 of the 21 variables and involves a high E and a low S. These 11 variables that showed more chaotic than ordered states are highly influenced by the solar radiation that defines the winter and summer seasons, as well as the hydrological cycle. These variables were: Oxygen (PO_2, SO_2); surface, planktonic and benthic temperature (ST, PT, BT); conductivity (ICd, PCd, BCd); planktonic and benthic light (PL,BL); and evaporation (Ev).
Very Low Complexity C [0, 0.2). In this group, E is high, and S is very low. This category includes the inflow conductivity (ICd) and water mixing variance (ZM). Both are high and directly correlated; it means that an increase of the mixing percentage between planktonic and benthic zones is associated with an increase of inflow conductivity.
The homeostasis was calculated by comparing the variation of all variables, representing the state of the Arctic subsystem every day. The timescale is very important, because H can vary considerably if we compare states every minute or every month. The h values have a mean (H) of 0.95739726 and a standard deviation of 0.064850247. The minimum h is 0.60 and the maximumh is 1.0. In an annual cycle, homeostasis shows four different patterns, as shown in Figure 2, which correspond with the seasonal variations between winter and summer. These four periods show scattered values of homeostasis as the result of transitions between winter and summer and winter back again. The winter period (first and last days of the year) has very high h levels (1 or close to 1) and starts from day 212 and goes to day 87. In this period, the winter conditions such as low light level, temperature, maximum time retention due to ice covering and low inflow and outflow, water mixing interchange between planktonic and benthic zones, low conductivities and pHs and high oxygen are present. The second, third and fourth periods correspond to summer. The second period starts with the increasing of benthic pH, zone mixing, and inflow-outflow variables. Between days 83 and 154, this period is characterized for extreme fluctuations as a result of an increase in temperature and light. Homeostasis fluctuates and reaches a minimum of 0.6 in day 116. At the end of this period, the evaporation and zone mixing increase, while the oxygen at benthic and sediment decrease. The third period (days 155 to 162) reflects the stabilization of the summer conditions; It means maximum evaporation, temperature, light, mixing zone, conductivity and pH and lowest amount of oxygen. Homeostasis is maximal again for this period. The fourth period (days 163-211), which has h fluctuations near to 0.9, corresponds to the transition of summer to winter conditions.
Autopoiesis was measured for three components (subsystems) at the planktonic and benthic zones of the Arctic lake. These were physiochemical, limiting nutrients and biomass. They include the variables and organisms related in Table 3. According to the complexity categories established in Table 2, the planktonic and benthic components have been classified in the following categories: limiting nutrient variables in the low complexity category (C[0.2, 0.4); orange color), physiochemical variables in the high complexity category (C[0.6, 0.8); green color) and biomass in the very high complexity category (C; blue color). A comparison of the complexity level for each subsystem of each zone (averaging their respective variables) is depicted in Figure 3.
In order to compare the autonomy of each group of variables, equation ** was applied to the complexity data, as shown in Figure 4. For the planktonic and benthic zones, we calculated the autopoiesis of the biomass elements in relation to limiting nutrient and physiochemical variables. All A values are greater than 1. That means that the variables related to living systems have a greater complexity than the variables related to their environment, represented by the limiting nutrient and physiochemical variables. While we can say that some physiochemical variables, including limiting nutrients have more or fewer effects on the planktonic and benthic biomass, we can also estimate that planktonic and benthic biomass are more autonomous compared to their physiochemical and nutrient environments. The very high values of complexity of biomass imply that these living systems can adapt to the changes of their environments because of the balance between emergence and self-organization that they have.
The proposed measures characterize the different configurations and dynamics that elements of complex systems acquire through their interactions. That means, these measures capture the properties and tendencies of a system, that is why the scale at which they are described is appropriate. They will not indicate which element interacted with which element, how and when. If we are interested in the properties and tendencies of the elements, we can change scale and apply the measures there. Still, we have to be aware that the measures are averaging—and thus simplifying—the phenomena they describe. Whether relevant information is lost on the averaging depends not only on the phenomenon, but on what kind of information we are.
On the above context, measuring complextiy at ecological systems is very useful due to Including interactions in ecological studies, for complexity understanding has been no easy. For example, it has been tried with global models that including the greatest number of variables, resulting also in serious deficiencies in predictability, especially for the limitation for the incorporation of all interactions ecosystem multi-elements and components (Moore et al. 2011). Alternative forms of explain de complex dynamics have been trying with the assessment of attributes like resilience and robustness (Ulanowicz et al. 2009). Also, ecological complexity has been related with stability. This way, complexity characterization has been supported in variables such as species richness (number of species), connectance (fraction of the possible interspecific interactions), interaction strength (effect of one species’ density on the growth rate of another specie) and evenness (abundance variance). Meanwhile, stability has been related with resilience (velocity to return to the equilibrium), resistance (variable’ grade of change) and variability (population density variance) (Pimm, 1984). However, these interpretations of interactions are conducts to find an explanation of functional complexity, than the evaluation of how complex is an ecosystem.
We applied measures of emergence, self-organization, complexity, homeostasis, and autopoiesis based on information theory to an aquatic ecosystem, in focus to evaluate how complex are differents variables and components. The generality and usefulness of the proposed measures will be evaluated gradually, as these are applied to different ecological systems. The potential benefits of general measures as the ones proposed here are manifold. Even if with time more appropriate measures are found, aiming at the goal of finding general measures which can characterize complexity, emergence, self-organization, homeostasis, autopoiesis, and related concepts for any observable ecosystem is a necessary step to take.
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