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We have defined the environmental interface through the exchange processes between media forming this interface. Considering the environmental interface as a complex system we elaborated the advanced mathematical tools for its modelling. We have suggested two coupled maps serving the exchange processes on the environmental interfaces spatially ranged from cellular to planetary level, i.e. 1) the map with diffusive coupling for energy exchange simulation and 2) the map with affinity, which is suitable for matter exchange processes at the cellular level. We have performed the dynamical analysis of the coupled maps using the Lyapunov exponent, cross sample as well as the permutation entropy in dependence on different map parameters. Finally, we discussed the map with affinity, which shows some features making it a promising toll in simulation of exchange processes on the environmental interface at the cellular level.

Complex systems science has contributed to our understanding of environmental issues in many areas from small to large temporal and spatial scales (from the cell behavior to global climate and its change). Environmental systems by themselves are both complicated and complex. Complicated, in that many agents act upon them; complex, in that there are feedback loops connecting the state of the system back to the agents, and connecting the actions of the agents to one another. Complex systems have complex dynamics usually characterized by so-called tipping points, abrupt changes in the state of the system caused by seemingly gradual change in its drivers [

There are many existing researchers that deal with specific aspects of the environmental interface (for example, see references listed above). However, in this paper we rather made an introductory step in establishing the strategy for: 1) modeling of the processes in whole and 2) understanding the functional time of the exchange processes on the environmental interfaces, using new mathematical tools.

Technically speaking the interface is a point at which independent systems or components meet and act or communicate with each other. Interfaces can exist between system elements and they can also exist between a system element and the system environment when we speak about environmental interface. It can be specifically defined depending on the science where it is used (ecology [

The intention of this paper is to be an introductory step in creation of the strategy in modeling the processes on environmental interfaces. It is done trough the following steps: 1) Definition of environmental interface (Subsection 1.1); 2) A concise elaboration of the fundamental tools in environmental interface systems modeling (Section 2) through description of the modeling architecture, use of Category Theory, Mathematical Theory of General Systems and Formal Concept Analysis in Subsections 2.1, 2.2 and 2.3 respectively and 3) description of the two coupled maps, which serve exchange processes on the environmental interface (Subsection 3.1) and numerical simulations with maps of exchange processes on the environmental interface (Subsection 3.2). In papers that would follow we will continue with elaboration of the 1) model of evolvable environmental interfaces and 2) model of forming the functional time of the exchange

processes on the environmental interfaces.

The environmental interfaces are formed in a space that is rich with complex systems. Each system, as an open one, interacts in a coherent way, producing new structures and building cohesion and new structural boundaries. It undergoes emergence and self-organization. In modeling the complex environmental interface systems, except the traditional mathematical, are often used the new mathematical tools like Category Theory (originally proposed by Rosen [

Modelers of environmental interface systems in numerically oriented studies base their calculations on mathematical models for the simulation and prediction of different processes, which are exclusively nonlinear in describing relevant environmental quantities [

In modeling community dealing with complex systems, Rosen’s diagram ([

and another for decoding the propositions of F back to the phenomena in N. As mentioned above, there are two paths in diagram (1) and (2) + (3) + (4). According to [

Category Theory recommended by Rosen [

Following Mesarovic’s Mathematical Theory of General Systems [

Thus, it is obvious that changes in an environment induce appropriate responses in agents through the model of coupled input/output pairs. In real systems, the reverse situation is also possible such that some external changes can be influenced by the activity of organisms, but for the sake of simplicity and because it is not directly related to the main topic of this presentation, we will not consider that problem. It is clear that a critical factor in building an evolvable model as described above is choosing the appropriate structure for the mapping. When dealing with models usually developed as prediction tools, it is sufficient to assume the attitude of analyzing a “black box”. Therefore, we can propose a function that should summarize all available experimental data and obtain a set of more or less accurate predictions for various initial conditions. However, in such a case we will neglect the real meaning of the nature of mappings within E and P. Taking a slightly closer look at these relations we can see that a somewhat hidden problem is that of how I is generated from the

wholeness of external changes and what is the connection between generating I with a constitution of corresponding R. Although this connection can be efficiently represented using the FCA [

Many physical, biological as well as the environmental interface issues, can be described by the dynamics of coupled maps. In environmental models various non-linear dynamics methods are used (for example, [8,12,31-33]). However, in modeling of exchange energy and matter in environmental interface it is useful to use 1) the diffusive coupling, which describes the energy exchange [8,13] and 2) the mapping with internal affinity, that describes the matter exchange [

where parameter is the so-called logistic parameter, while is the coupling parameter. Since the first map is described in [

The map with internal affinity (in further text-map with affinity) can be used for describing the matter exchange in various biological as well as biophysical environmental interfaces. However, here we will consider it through intercellular exchange with the cell membrane as an environmental interface. As it is obvious from the empirical description, we can infer the successfulness of the communication process by monitoring: 1) the number of signaling molecules, both inside and outside of the cell and 2) their mutual influence. The concentration of signaling molecules in an intercellular environment is subject to various environmental influences, and taken alone often can indicate more about state of the environment than about the communication itself. Therefore, we choose to follow the concentration of signaling molecules inside of the cell as the main indicator of the process. In this case, the parameters of the system are 1) the affinity by which cells perform uptake of signaling molecules (a2), which depends on the number and the state of the appropriate receptors, 2) the concentration c of the signaling molecules in the intercellular environment within the radius of interaction, 3) the intensity of the cellular response (a1) and and 4) the influence of other environmental factors, which can interfere with the process of communication. In this case we estimate parameter which can be taken collectively for intraand intercellular environments inside of the one variable, indicating the overall disposition of the environment to the communication process.

The time development (is the number of time steps) of the concentration in cells can be expressed as

The map, represents the flow of materials from cell to cell, and and are defined by a map that can be approximated by a power map, and. If and, the interaction is expressed as a nonlinear coupling between two cells. The dynamics of intracellular behavior is expressed as a logistic map (e.g. [34,35]),

Because the concentration of the signaling molecules can be regarded as their population for a fixed volume, and because we are focused on the mutual influence of these populations, it is clear that we should use the coupled logistic equations. Instead of considering cell-to-cell coupling of two explicit n-gene oscillators [

with notation

where is a vector representing the concentration of the signaling molecules inside of the cell, while denotes the stimulative coupling influence of members of the system, which is here restricted only to positive numbers in the interval (0, 1). The starting point is determined such that. Parameter in logistic difference equations determines an overall disposition of the environment to the given population of signaling molecules and exchange processes. The affinity to uptake signaling molecules is indicated by. Let us note that we require that the sum of all affinities of cells exchanging substances has to satisfy the condition or in the case of two cells,

, i.e.,. Because is a fixed point (4), in order to ensure that zero is not simultaneously the point of attraction, we define as an exponent. Finally, represents coupling of two factors: the concentration of the signaling molecules in the intracellular environment and the intensity of response they can provoke. This form is taken because the effect of the same intracellular concentration of signaling molecules can vary greatly with variation of affinity of genetic regulators for that signal, which is further reflected in the ability to synchronize with other cells. Therefore, influences both the rate of intracellular synthesis of signaling molecules and the synchronization of the signaling processes between two cells, so the parameter (the coupling parameter) is taken to be a part of both and. However, the relative ratio of these two influences depends on the current model setting. For example, if for both cells, is strongly influenced by the intracellular concentration of signals that can provoke relatively smaller responses, then the form of equation will be

We now analyze our coupled system, given by (6a) and (6b). For and, we have . So, for small, the dynamic of our investigated system is similar to the dynamic of the following systems obtained by minorization, as follows

and

If we apply a majorization, the considered system becomes

For all of these systems, it is obvious that they do not depend on the parameter. Because, where are the components of in (8), their dynamics are symmetric to the diagonal, . This symmetry also exists for system (6a)-(6b), if.

Recalling the aforementioned conditions for, and, we consider only systems (7) and (9) because the behavior of the system (6a)-(6b) comes from the properties of the mentioned ones. It is seen that the systems (7) and (9) consist of uncoupled logistic maps on the interval (in (7)) or on the interval (in (9)), where is the smaller solution of the equation i.e.

.(10)

Finally, let us note that the map, which describes exchange processes, can be generalized in the form

This system we will call map of exchange processes. Specially, for and we get the diffusive map, while for and, we get the map with affinity.

In the analysis of these coupled maps we will consider three parameters included in the archive of dynamical analysis of coupled maps: 1) the largest Lyapunov exponent, 2) the cross sample entropy (Cross-SampEn) and 3) the permutation entropy (PermEn). We calculate the largest Lyapunov exponent to see the behaviour of the coupled maps given by Eqs.1a-1b and 6a-6b, as particular cases of Eqs.11a-11b, depending on different values of the coupling parameter. Extending the approach in [38,39], we study, for two coupled maps representing biochemical substance exchange between cells, the stability of the fixed point by linearizing Eqs.11a- 11b.

where

is the Jacobian of the system (11) evaluated in and. By iterating Eq.12 one obtains

and thus we get Lyapunov exponent [

^{ }[

Cross-SampEn measure of asynchrony is a recently introduced technique for comparing two different time series to assess their degree of asynchrony or dissimilarity [43,44]. Let and

fix input parameters and

. Vector sequences: and and is the number of data points of time series,. For each set = (number of such that), whereranges from 1 to, and then

which is the average value of. Similarly we define

and as = (number of

such that) and

which is the average value of. Finally, we have

We applied Cross-SampEn with and for and time series, and the same range of parameters and for which the Lyapunov exponent is calculated (Figures 5(a), (b)).

Permutation Entropy (PermEn) of order is de- ﬁned as PermEn where the sum runs over all permutations of order. This is the information contained in comparing consecutive values

of the time series. Consider a time series. We consider all permutations of order which are considered here as possible order types of different numbers. For each we determine the relative frequency has type. This estimates the frequency of as good as possible for a finite series of values. To determine exactly, we have to assume an infinite time series and take the limit for in the above formula. This limit exists with probability 1 when the underlying stochastic process fulfills a very weak stationarity condition: for, the probability for should not depend on. Permutation entropy as a natural complexity measure for time series behaves similar as Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise [

a lot of uncertainties in time series of calculated physical quantities. Various measures of complexity were developed to compare time series and distinguish regular (e.g., periodic), chaotic, and random behaviour. The main types of complexity parameters are entropies and the Lyapunov exponent, among others. They are all defined for typical orbits of presumably ergodic dynamical systems, and there are profound relations between these quantities [

Our main results in this paper can be summarized as follows. 1) We have defined the environmental interface through exchange energy, matter and information between media forming this interface. 2) Considering environmental interface as a complex system we shortly described the advanced mathematical tools that can be

used in its modelling (Category Theory, Mathematical Theory of General Systems and Formal Concept Analysis). 3) We suggested the two coupled maps serving the exchange processes on the environmental interfaces spatially ranged from cellular to planetary level, i.e. a) the map with affinity, which is suitable for matter exchange processes at cellular level and b) the map with diffusive coupling for energy exchange simulation. 4) For maps (1a)-(1b), representing diffusive coupling and (6a)-(6b), that describes coupling with internal affinity, with controlling parameters () and (), respectively we calculated the largest Lyapunov exponent, sample as well as the permutation entropy. 5) The Lyapunov exponent for the diffusive map has 1) positive values, when the coupling parameter is in the interval and 2) negative values in intervals and, while logistic parameter takes values that lie in the chaotic region, i.e. low chaos (), high chaos () and intermittency (). For period 16 its values are either negative or very close to zero. 6) Similarly to diffusive map, in the map with affinity, for period 16, takes either negative or values close to zero. However, the intervals with positive values become broader with sporadic windows with negative values, when takes greater values in the chaotic region, i.e. (), () and (). 7) Calculated values of the Cross-SampEn are mostly equal or very close to zero, corresponding to the region where is negative, indicating a high level on synchronization between quantities and. Also, the calculated values of the PermEn, as a natural complexity measure for time series behave similarly as Lyapunov exponents. 8) The map with affinity shows some features, which make it a promising toll in simulation of exchange processes on the environmental interface at cellular level.

This paper was realized as a part of the project “Studying climate change and its influence on the environment: impacts, adaptation and mitigation” (43007) financed by the Ministry of Education and Science of the Republic of Serbia within the framework of integrated and interdisciplinary research for the period 2011-2014. We are appreciated to Miss Ana Firanj for designing the figures.