Complex Biological Systems: Adaptation and Tolerance to Extreme Environments

MODELING AND APPROACHES

Chapter 1

Critical Impacts on Complex Biological and Ecological Systems: Basic Principles of Modeling

R.G. Khlebopros1, V.G. Soukhovolsky1,2,*

1International Scientific Centre for Organism Extreme States Research, Krasnoyarsk Scientific Centre, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk 660036, Russia

2Institute of Forest, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk 660036, Russia

*Corresponding author: [email protected]

Abstract

In the chapter, methods of description and modeling of complex biological systems, as well as the influences of various environmental factors are considered. A complex system consists of a greater number of components cooperating with each other. Any complex biological system is stable to external impacts below a certain intensity and frequency. This property is a condition for the duration of the existence of such a complex system. This chapter discusses problems that are associated with modeling of such complex systems. Existing methods for macro- and microscopic description of complex biological systems are analyzed. In particular, methods of optimization modeling of complex biological systems are considered. In optimization models it is offered, that the biological system is characterized by the presence of a certain set of goals, which it aspires to achieve. A set of criterion functions, which describe the purposes facing to the system, aspiring to survive in a changing environment is entered into the models. It is supposed that the value of the criterion functions depends on the parameters of the state of the biological system. The process of adaptation of the system is that the system parameters accept following values, at which sizes of the target achieve a maximum, and the probability of survival of such a system appears maximum under existing environmental conditions. Further in the book optimization approach is used for modeling processes of photosynthesis in the analysis of the stability of closed ecological systems. Here, the problems of the response of complex biological systems to external impacts are discussed. If the intensity and speed of the external impact surpass some critical level, the result of such critical impact is a system transformation. Thus, the state of the components and / or the spatial order in the system is changed. Classification of types of critical impacts on systems and classification of the properties of the systems shown at critical impacts are given. In particular, consider these types of complex biological systems, such as elastic and fragile systems. Methods of modeling of state changes of complex systems are considered at critical impacts. Using the methods of modeling of complex biological systems, the methods being by analogues used in the analysis of critical phenomena in physical systems, is considered in detail.

Keywords: Complex biological systems, Methods of modeling, Optimization models

1.1 Complex Ecological Systems: The Principle of Decomposition, Taking into Account the Characteristic Times of Components

In contemporary science, analysis and prediction of the behavior of complex systems is in great demand. Once, the use of mathematical approaches in different sciences, in natural ones in particular, was limited to the construction and subsequent analysis of model systems such as pendulum, solid-solid interaction, etc. Although these model systems were also good for presenting events, we now deal with complex systems and in every particular case we aim to reduce the number of variables describing a system to a minimum, ideally to one. Thus, the main objective now is to examine a complex system and to construct an analogous model system that would have the necessary properties for predicting its dynamics. This is possible if the complex system is defined as a system with a large number of variables. However, not every system with a large number of variables is a complex system. The complex system is the system that exists in reality. This implies that for any of its coordinates there is a domain in which negative feedback dominates. If there are no domains in which for all components there is negative feedback, the system does not exist. This is true for all systems – living, nonliving, economic, social, etc.

Let us consider the issue of reducing the number of variables of the analyzed system to the level that cannot be further reduced if we are to stay within the range of correct answers (Tikhonov, 1943; Bogolyubov and Krylov, 2005; Khlebopros et al., 2007). Imagine a complex system and arrange the equations based on characteristic relaxation times (τ) in descending order. If the system analyzed is forest, the longest characteristic times are the times of formation of forested areas – usually several thousand years. Next can be the characteristic time of the existence of tree stands; different plant species growing in the forest have different characteristic times, but these are much shorter times anyway – decades or centuries. The next in the series are forest mammals, whose lifetime is several decades; they are followed by insects with the life cycles lasting 1 or 2 years; then come equations for microorganisms, with their lifetimes of several weeks, days, hours, etc.

If we arrange the scores of existing components in this way, we can ask, for example, what changes will occur in this forest if insects reach outbreak levels and eat up the mature forest. Over the lifetime of one generation of insects, the forest will not actually undergo any changes, but microorganisms will act as a quick component. There are very slow components, which for the insect equation will act as continuous, slowly evolving, factors. It will take decades or centuries for the forest composition to change so much that insects will live in the same place but in a different forest, with trees of different sizes and species. Microorganisms, on the other hand, will multiply and vanish several times over the same period, and this component will be averaged over the characteristic times equal to several decades. Then we can only examine the components that have equal characteristic times. These can be insects that consume other insects that, in turn, consume forest trees. Their equations should be retained, while other components will either be present as continuous parameters or become averaged over this time.

Thus, the number of analyzed components will be reduced dramatically. However, this is only possible if we do not have to find out whether insect outbreaks will ruin forests. Forests may be replaced by marshes, and then the characteristic times that we need are characteristic times of the formation of marshes and forests. In this case we deal with changes in the soil under the impact of forest or marsh rather than with insects and trees. These processes last several thousand years, and the effect of all the components with much shorter characteristic times will become averaged. Thus, these equations need not be used in the model.

This mathematical approach can conveniently be used to analyze complex systems because it dramatically reduces the number of equations, components, and variables that have to be taken into account in solving different problems.

So, analysis of a complex system should begin with composing time hierarchy for its components, followed by defining of characteristic times that will be used to examine the system and its dynamics. The system always evolves, but evolutionary processes over very short, short, and long times are different. Analysis of the behavior of the system over these characteristic times will answer the questions about steady and unsteady development, evolution of different complex systems or, vice versa, the risk of catastrophic events (Isaev and Khlebopros, 1973a, 1973b, 1974; Isaev et al., 1974, 2001).

Another opportunity to reduce the number of coordinates to be taken into account arises when the analyzed system changes from one state to another: either from the current state into a new one or from one stable state into another stable state or from the equilibrium state into evolution around the cycle or along another complex trajectory. All changes have the following property: close to the transition boundary, the number of the components that we have retained is again dramatically reduced. As in this n′-dimensional space (n′ << n), close to the boundaries the crossing of which will change the state, we reduce the number of coordinates again, the only ones left will be the movement along which is orthogonal to the interface separating the states, while all the coordinates that run almost parallel to this interface can be ignored. This is the second way to reduce the number of coordinates and, thus, to considerably increase the possibility of accurate solution of different extremal problems.

To analyze evolution of the complex system that exists in reality and, hence, has a negative feedback domain for any component, we should construct a phase portrait. The most correct approach is to construct it for the component that has the characteristic time (relaxation time) over which we examine evolution of the system and make forecasts (Volterra, 1931; Kolmogorov, 1936, 1972; Kolmogorov et al., 1937; Poincaré, 1947; Lyapunov, 1954–1959). If, for instance, we examine insect outbreaks that feed on trees and shrubs, we should take into account characteristic times of several generations of these insects.

Prior to constructing the phase portrait by experimental points, we can do the following: plot the number of insects per square meter (tree) on the abscissa and the rate of change in this number on the ordinate (in the case of insects, we should take two successive measurements of insect density and divide the difference between these densities by the characteristic time between these two measurements). Thus, we have the coordinate and its rate of change. There are two types of the phase portrait. One is the narrow phase portrait (Figure 1.1) (Isaev and Khlebopros, 1973 a, 1973b, 1974).

Fig. 1.1. The narrow phase portrait of complex system

The narrow phase portrait implies that we can actually use one equation, i.e. the rate depends upon coordinate , and all the other coordinates will either be averaged (if they change rapidly) or function as parameters (if they change slowly). Thus, of all the numerous coordinates that describe the forest ecosystem we may need just this coordinate (this insect abundance and the rate of change of this abundance) to describe evolution (behavior) of the forest area. Only one equation is left. The intersection point of this narrow phase portrait with axis Xi is the equilibrium state around which insect abundance will vary. It can certainly evolve over time and have slow trends, but these trends will correlate with other characteristic times such as the time of tree growth and the time of a change in the tree stand. If the rate of change in the insect abundance versus insect abundance is a nonmonotonic function, there may be several intersections (Figure 1.2); in the simple case there will be three intersections and two of them, X1 and X2, are stable equilibrium points and Xr is the divide point.

Fig. 1.2. The narrow phase portrait with two stable equilibrium points

If there are more variables (if we take these same insects but with their competitors and enemies that have similar characteristic times), the phase portrait will be broad (Volterra, 1931; Kolmogorov, 1936, 1972; Kolmogorov et al., 1937; Poincaré, 1947; Lyapunov, 1954-1959). That is, for the same Xi values (in the case of insects, for the same densities), the rate will be different depending on the abundance of competitors or enemies – insects or other animals. When the system is delay-free (all enemies and competitors have agreed to Xi being constant), this curve can cross the abscissa once (Figure 1.3), i.e. there is one equilibrium point, or three times, i.e. there are two stable equilibrium points. Although the phase portrait is broad, the behavior of the entire system is sometimes similar to the behavior of the system with the narrow phase portrait (Figure 1.4). The only difference is that in the narrow phase portrait all movements are along the reproduction curves, whereas in the case just described the movements are above or below rather than along the reproduction curves (Isaev and Khlebopros, 1973a, 1973b, 1974; Isaev et al., 2001).

Fig. 1.3. The broad phase portrait with one stable equilibrium point

Fig. 1.4. The broad phase portrait with two stable equilibrium points

However, when there is a negative feedback domain (Figure 1.5), there may be positive feedback domains in it. Then, if the curve intersects the abscissa, different variants of the system dynamics can be obtained, depending on the position of points X1 and X2 (Figure 1.6) (Isaev and Khlebopros, 1973a, 1973b, 1974; Isaev et al., 2001; Bazykin et al., 1997).

Fig. 1.5. The broad phase portrait with one stable equilibrium point and one unstable equilibrium point

Fig. 1.6. The broad phase portrait with two stable equilibrium points – fixed outbreak (a), with one stable equilibrium point and one unstable equilibrium point – anti-outbreak (b), with two unstable equilibrium points – sustained outbreak (c)

One of the possible variants is when X1 and X2 are in the negative feedback domain, i.e. are stable points. Again, we have the situation that has already been examined, one described by the narrow phase portrait, i.e. the broad phase portrait whose phase space is entirely the negative feedback domain. If, however, either X1 or X2 is out of the stable domain, in the positive feedback domain, either outbreak or anti-outbreak occurs, i.e. there is one stable point from which the system is ejected from time to time either towards a dramatic increase or towards a dramatic decrease in insect abundance. An interesting situation occurs when both X1 and X2 are in the positive feedback domain. Then we have a very rare event, which can be termed a sustained outbreak, if we deal with insects, or attractor formation, if we analyze any other complex system. The existence of the attractor indicates that the system all the time moves around points X1 and X2, without going along the same trajectory twice.

Thus, the detailed analysis of the phytophagous insects – forest area system yielded very interesting trends that can be observed in any complex system. The various behaviors of insects in the forest can be reduced to just six variants:

– one stable equilibrium point;

– one unstable equilibrium point, when the system is far from the equilibrium, constantly moving along an intricate and always changing trajectory of the phase space;

– fixed outbreak, when there are two stable states: one with high abundance and the other with low;

– outbreak;

– anti-outbreak;

– sustained outbreak.

Interestingly, we need not always keep the complex system really complex, i.e. we can choose a characteristic time that we are interested in to examine the evolution of the system. Then we can choose the coordinates, the variables whose characteristic times are the same as or close to the characteristic time we are interested in. The next step is to construct a phase portrait for one of these coordinates. A narrow or a broad phase portrait can result. If it is narrow, all depends on how many times the characteristic curve crosses the abscissa: if there is one intersection, there is one stable point; if there are several intersections, there are two, three, four … equilibrium points. If the resulting phase portrait is broad, we cannot use just one coordinate; the number of coordinates is actually larger than one but much smaller than in the initial system. Then, plotting experimental points in the phase portrait, we can.

First: determine that it is broad;

Second: determine that it is singly connected, that there are no positive feedback domains in it. If this is so, the results are actually the same as in the narrow phase portrait.

If in the phase portrait, within the negative feedback domain there is a positive feedback domain, there may be six variants of the behavior of the system.

The approach that we have demonstrated can be used to examine a complex system step by step. Let us imagine an artist painting a landscape he sees in front of him. This landscape contains mountains, hills, trees, meadows, and, probably, animals. If we scrutinize the landscape closely, we can ask: What may change in a day? The position of the goats may change, but not the position of the trees. The position of the trees will remain unchanged for decades and the positions of the mountains and the hills for several thousand years. One can see quick movements in this system and analyze the dynamics based on this; the same system can be analyzed in terms of slower movements; and, finally, it can be analyzed in terms of very-very slow movements. Anyway, each time the phase portrait is only constructed for the characteristic time that the researcher is interested in to examine the evolution of the entire system.

1.2 Analysis of Critical Impacts on Complex Systems and Extreme Principles of Modeling

In physics, complex systems are described using not only kinetic equations, which account for the dynamics of microscopic variables of the system, but also the so-called extremal principles. In this case, n equations of dynamics are replaced by a certain function relative to which it is assumed that the system tends to reach the state in which the value of this state function is the lowest. Then there are not more than 3–4 independent variables in the model of a physical system, whatever the number of the system components is. This reduction is convenient for finding stable states of the system but not for describing the dynamics of the processes occurring in it (Landau and Lifshits, 1964; Dmitriev, 2004).

Yet, simplification of the models of ecosystems using this metamodel approach seems a promising tactic. There are various approaches to the construction of ecological models, based on using extremal principles (Fursova et al., 2003; Soukhovolsky et al., 2008). In this study we use extremal principles to describe such critical events in forest ecosystems as insect outbreaks, forest fires, etc.

1.2.1 Meta-Models of Phase Transitions for Describing Critical Events in Complex Systems

To describe the state of a complex system, physics introduces the notion of phase as an integral state of a certain substance of a physical system (Bruce and Cowley, 1981). A very simple example is three phases of water (solid, liquid, and gaseous). A change in the state of the object or the system is termed phase transition (e.g., ice melting or water evaporation). Phase transitions occur under an external influence (such as a temperature change). It is assumed that the state of the system and phase transitions can be described using a certain thermodynamic potential, G, and condition G → min is an optimization principle, regulating the process of phase transition (Landau and Lifshits, 1964). The thermodynamic potential of a complex system depends on a large number of factors, and in most cases, the general form of function G is unknown. However, in the case of second-order phase transitions, such as superconductivity, super fluidity, and magnetization, the number of factors that influence the behavior of the system is usually dramatically reduced, suggesting that function G depends on just two variables: the so-called order parameter, q, characterizing certain general properties of the examined object, and an external variable such as temperature T (Landau and Lifshits, 1964). In the theory of phase transitions, the order parameter is regarded as an independent variable. The phase transition begins when the value of the external variable becomes less (or more – depending on the particular object) than a certain critical value.

Using this simplified approach to describing the process of phase transition, the G = f(T,q) relation can be sufficiently accurately represented as an even power series of the order parameter (Landau and Lifshits, 1964).

(1)

where b = const, and A is a linear function of external variable T:

(2)

where Tc is the critical value of T marking the onset of phase transition.

A characteristic feature of static models of phase transitions is that they are time independent. This assumption significantly simplifies the model: the description of the system only needs stable minimums of function G, which can be derived from the conditions . The value of the order parameter, q, at which the G minimum is reached, will depend on the values of parameters A and b of equations (1) and (2).

The models of phase transitions incorporate the universality principle, in accordance with which processes of phase transitions depend on just a few basic properties of the system, such as dimensionality, the number of components of the order parameter, and distance dependence of interactions in the system (Bruce and Cowley, 1981). The idea of universality of phase transitions in systems with one order parameter and homogeneous spatial structure simplifies and unifies the construction of models of phase transitions. Even without knowing the exact form of function G, one can describe the process of phase transition and find critical values of the external factor and values of the order parameter in stable states of the system.

Below are models describing some critical events in forest ecosystems as second-order phase transitions, based on the universality principle in theory of phase transitions.

1.2.2 A Model of Outbreak as Second-Order Phase Transition

Forest insect populations are characterized by both the density of X individuals (absolute colonization) and the distribution of individuals over the area, specifically, the distribution of individuals on sample units: trees, model branches, i.e. In entomological monitoring, the simplest and most frequently used indicator of spatial distribution of insects in sample units is the relative colonization A – the fraction of sample units with individuals of the studied insect species (Isaev et al., 2001).

In accordance with the phenomenological model of population dynamics, an outbreak occurs when the pest population density exceeds a certain critical value, Xr, and the population has colonized all hosts available in the area (Isaev and Khlebopros, 1973a, 1973b; Isaev et al., 2001).

The relative colonization of the trees in the stand is then close to 1. During the sparse stable population phase the population density is the lowest possible and the insects do not colonize all of the hosts. During the outbreak phase, the abundance of the insects in the population is the highest, and they colonize all suitable host trees. Thus, insect population can be either in the sparse stable phase or in the outbreak phase. Let us hypothesize that insect survival in the population is described by a certain function, G (the function of ecological risk). The function of ecological risk can be represented as the probability of insect death in the population, depending on the effects of various regulating and modifying factors. For every particular insect population, the general form of this dependence is certainly unknown, but one can assume that stable states of the population are characterized by the minimum values of the function of ecological risk, G (Soukhovolsky et al., 2005). Then, to describe the insect outbreak as an ecological analog of phase transitions in physical systems, we should introduce the order parameter, q, to determine the external factor leading to the phase transition, and introduce an equation that would relate these variables.

Let us define the order parameter, q, as the value related to the relative insect colonization, A, of sample units of the tree stand:

(3)

where is the relative insect colonization of sample units (trees, plots, etc.), n is the number of sample units, and k is the number of sample units colonized by insects. It follows from (3) that 0 ≤ q ≤ 1.

It follows from (3) that the value q = 0 of the order parameter corresponds to the outbreak phase, when insects occur in all sample units (trees, plots, etc.). Characteristic values of the order parameter do not populate all sample units. As the external parameter, corresponding to the temperature in physical systems, we have chosen population density X. In accordance with the phenomenological theory of forest insect population dynamics, an outbreak (i.e., phase transition) occurs when the population density exceeds a certain critical value, Xr. When the sparse stable population phase is replaced by the outbreak phase, the value of the order parameter is reduced to zero, i.e. all trees in the outbreak site are colonized by the insects.

To describe the outbreak as phase transition, we use equation (1). The order parameter (3) will describe the state of the tree stand. The external factor will be population density, X. Coefficient A in (1) will be written as one linearly dependent on population density:

(4)

It follows from (4) that if the population density is higher than the critical value X = Xr, A > 0.

The value of the order parameter for the outbreak phase and the sparse density phase will be found from equation characterizing the minimum value of function G:

(5)

Equation (5) has two solutions:

(6)

Solution q = 0 describes the outbreak phase. The second solution is true for the population density below the critical value, Xr. This solution characterizes a sparse population.

At q = 0, the value of ecological risk G1 = G0, and at

. Hence, G1 > G2, and the ecological risk for populations of non-outbreak insect species that have sparse stable densities is lower than the ecological risk for outbreak species. The number of non-outbreak species should be significantly greater than the number of outbreak ones. This conclusion is in good agreement with the data of forest insect surveys. Our investigations in boreal forests of Eastern Siberia showed that of 315 economically important species of forest insects, just 74 were able to reach outbreak levels (23.5%) (Isaev et al., 2001).

Knowing insect population densities provided by insect surveys and relative estimates of insect colonization of tree stands in different phases of insect outbreaks, we can verify the proposed model by determining the relationship between X and q² in the {X, q²} plane. Based on model (1), at q = 0, points characterizing population density must be located on the abscissa on the right of point Xr. For X < Xr, in accordance with (6), between q² and density X there must be a negative linear relationship. Point X = a/2b of the intersection of line and the abscissa gives the value of Xr.

Figure 1.7 shows the data of surveys that demonstrate the relation between q² and X for populations of the pine looper Bupalus piniarius L. in Altai, Minusinsk, and North-Kazakhstan pine forests during 1962–1988.

Fig. 1.7. Relationship between parameter q² and the density of pine looper Bupalus piniarius L. populations in different outbreak phases in Altai, Minusinsk, and North-Kazakhstan pine forests during 1962–1988

Bupalus piniarius L. surveys were performed in spring, during the pupal stage, on 1 × 1 m plots (Palnikova et al., 2002). The survey data are clearly in good agreement with the theoretical model (Figure 1.7). For population densities below the critical value, there is a significant linear relationship between population density and the squared order parameter. For population densities higher than the critical value, the order parameter is close to zero. The critical density Xr for the pine looper is 5.65 pupae per plot, and this is in good agreement with the data reported in the literature – 6 pupae per plot (Isaev et al., 2001).

Figure 1.8 shows the relation between parameter q² and population densities of the Siberian silk moth Dendrolimus superans sibiricus Tschetv. in different phases of the outbreak cycle.

Fig. 1.8. Relationship between parameter q² and the density of Dendrolimus superans sibiricus Tschetv. populations in different phases of the outbreak cycle

Calculations were done using Y.P. Kondakov’s data (Kondakov, 1974). Here we also have a good agreement between survey data and theoretical equation (1). The critical value Xr calculated from equation (1) is equal to 657 larvae per tree, and the number given in the handbook on monitoring of forest pests for the critical density of the Siberian silk moth population is 600 larvae per tree (Isaev et al., 2001).

Compare model (1) with the Poisson model describing the relation between the relative and absolute estimates of insect colonization and based on the assumption that all sample units are similar in their properties, individual insects choose their food randomly, and the interaction between individuals is weak if any. In this case, the distribution of insects among sample units can be described by the Poisson model. According to this model, the portion q = 1 – A of sample units without insects (i.e. the analog of the order parameter in model (1)) is expressed with the following formula:

(7)

It follows from (7) that at X → 0, i.e. under low population density, q → 1, that is, there are very few sample units with insects in them. At X → ∞, q → 0, i.e. almost all sample units are colonized by the insects. It also follows from (7) that the type of relation between relative and absolute colonization must be similar for all insect species, as in (7) there are no species-specific constants.

According to (7), as the population density grows relative colonization of tree stands increases rapidly. At X = 3 individuals / sample unit, q ≈ 0.05 and A ≈ 0.95. Obviously, if the distribution of insects among sample units is random, at a population density above 3 insects per sample unit there will be very few sample units without insects on the survey plot.

Thus, analysis of survey data on the relation between the relative and absolute colonization by insect populations can provide a basis for choosing between model (1) and model (7). If the results of the analysis suggest the equation of the relation between the relative and absolute colonization is significantly different from the Poisson model (7), this will be indicative of interactions between insects and their host plants as well as between phytophagous insects. One manifestation of these interactions is cooperative colonization of trees. Parameters of these interactions can be estimated from the values of constants b and Xr of model (1). Statistical analysis showed that the survey data disagree with calculations based on model (7). Both survey data and results of calculations based on model (1) suggest that for the Siberian silk moth, population density of 3 insects per tree corresponds to relative colonization of about 4% of the trees and for the pine looper population – about 45% of the trees. Calculations based on model (7) suggest that insects must be present on 95% of the trees.

Figure 1.9 shows survey data and model results of the squared order parameter calculated from models (1) and (7) to estimate the distribution of Homoptera insects (Lepyronta coleopterata L., Philaenus spumarius L., Cinara nuda Mordv., C. hyperophila Koch.) on trees of middle-taiga plain forests of the Krasnoyarsk Territory (Bulanova et al., 2008).

Fig. 1.9. Order parameters determined from the survey data and calculated from models (1) and (7) for insects of the order Homoptera in the forests of the Yemelyanovskii District of the Krasnoyarsk Territory.

Note: 1, calculations from model (7); 2, calculations from model (1); 3, survey data

Results of the model of phase transitions (1) are in good agreement with the survey data, whereas results of the Poisson model (7) significantly disagree with field measurements. Table 1.1 compares values of relative colonization of the trees found using survey data and calculated from models (1) and (7), based on the highest values, X(max), of absolute colonization by insects observed during surveys.

Table 1.1. elative colonization of trees estimated using survey data and models (1) and (7)

Note: * the value is significantly (p = 0.95) different from the survey data

The survey data and results of calculations using model (1) are very similar. On the other hand, the calculation of relative colonization based on the Poisson model (7) is in some cases significantly different from survey data, and the higher the values of absolute colonization the greater the differences between the calculations of relative colonization based on model (7) and the survey data. At low population densities, calculations from both model (1) and model (7) are in good agreement with the survey data. This good agreement may be accounted for by the fact that at low population densities (X < 0.5), the exponent in the right-hand side of (1) can be expanded into Taylor series to within a linear term. Then from (7) we obtain

(8)

This may suggest that at low population densities, both models predict a linear relationship between relative colonization and population density. As the population density grows, expansion (8) becomes incorrect, and the Poisson model gives too high calculated values of relative colonization, while model (1) remains more correct even at high densities.

It is evident from the data just presented (Figures 1.7–1.9, Table 1.1) that parameters characterizing the relation between the relative and absolute colonization by insects (model (1)) are species specific and vary both between different groups of insects and within one systematic group in different regions. Model (7) is less sophisticated. Our analysis showed that calculations based on interpretation of the insect outbreak as a second-order phase transition suggest a good agreement between the model of phase transitions and field data.

1.2.3 The Effect of Modifying Factors on the Development of an Outbreak

A modifying, i.e. density-independent, factor, in terms of phase transitions, can be treated as superposition of external field onto the system. For population dynamics and development of outbreaks, the major modifying factor, which significantly affects population increase and development of an outbreak, is ambient temperature. If temperature is regarded as constant time-independent field T, we assume that elevation of the environmental temperature will decrease the ecological risk potential. Then, equation (1) can be rewritten as follows (Dmitriev, 2004)

(9)

where V is a certain constant.

It is clear from (9) that in this case, the transition of the population to the outbreak state will not occur at q = 0, i.e. during the peak phase not all the trees will be infested by pests.

In this case, no discrete phase transition will occur, and in the external temperature field the transition is diffuse and can take place within a certain interval of pest population densities.

The parameter characterizing the effect of the external field on the order parameter that we use is the sensitivity value . It follows from (9) that

(10)

Substituting equilibrium values of the order parameter in the absence of the field into (10), we obtain the expressions for sensitivity to the effect of the external field in the ranges above and below the critical population density Xr:

(11)

It is clear from (11) that on approaching the critical density Xr, sensitivity tends to infinity. At both very low and very high population densities, the effect of the external field on changes in population density is weak; sensitivity increases dramatically close to the critical density Xr. However, according to (11), the sensitivity to the effect of external modifying factors on the sparse stable population is twice lower compared with the population in the outbreak phase.

We assume that the effect of the external temperature field is not significant, i.e. the condition Aq² » VqT is fulfilled. Let us find the value of the order parameter q at which the minimum of function (9) is attained:

(12)

or

(13)

At X > Xr and T → 0, the order parameter , when all trees are colonized by pests and an outbreak occurs, is equal to 0. At T ≠ 0, the value of is found from cubic equation (13). The solution of (13) is graphically shown in Figure 1.10.

Fig. 1.10. The graphic solution of equation (13)

Note: 1, y1 = VT; 2, y2 = a(X – Xr)q + 2Bq³

Figure 1.10 shows that the effect of the temperature causes the value of the order parameter q to increase when the outbreak takes place.

At X < Xr, we obtain from (13):

(14)

The graphic solution of (14) is shown in Figure 1.11.

Fig. 1.11. The graphic solution of equation (14)

Note: 1, y1 = VT; 2, y2 = – a(X – Xr)q + 2Bq³

Figure 1.11 shows that there are three solutions to equation (14). However, the only solution that corresponds to the maximum of function G is II. The positive root of III characterizes colonization of trees by sparse stable population. Note that the condition of the slightness of the effect exerted by the field is necessary for the condition q ≤ 1 to be fulfilled.

Thus, the effect of the temperature on the insect population in the examined model corresponds to the situation when the transition to the outbreak phase does not result in colonization of all sample units (plots, trees) by pests.

1.2.4 The Impact of Chemical Compounds on Biological Objects

A model of a second-order phase transition can be used to evaluate the effect of chemical compounds on biological objects. Typically, these exposures are characterized by a dose of LD50 exposure to a chemical substance that kills half the objects, or a dose of LD100, which kills all objects under the influence of LD50.

If we consider the effect of an external factor (chemical substance) from the point of view of the second-order phase transition model, then the relationship between the order parameter q characterizing the state of the biological object and the concentration C of the chemical substance will be described by the following equation:

(15)

where q is the proportion of biological objects (for example, tumor cells) on which the chemical compound is exposed, Cr = LD100 is the dose of exposure at which all biological objects affected by the drug die.

The effect of impact of the chemicals on tumor cells in tissue culture is shown in Figures 1.12 and 1.13 based on the data presented in (Severin et al., 2013):

Fig. 1.12. The effect of the chemical preparation paclitaxel included into biodegradable nanoparticles (NP-paclitaxel) on the survival of tumor cells in tissue culture (according to Severin et al., 2013)

Note: NP, nanoparticles; 1, the phase of partial death of tumor cells; 2, the phase of complete death of tumor cells

Fig. 1.13. The effect of the chemical preparation paclitaxel included in the biodegradable nanoparticles associated with recombinant receptor-binding fragment of alpha-fetoprotein (NP-paclitaxel-rAFP-3D) on the survival of tumor cells in tissue culture (according to Severin et al., 2013)

Note: NP, nanoparticles; rAFP-3D, recombinant receptor-binding fragment of alpha-fetoprotein; 1, the phase of partial death of tumor cells; 2, the phase of complete death of tumor cells

As can be seen from Figure 1.12, the death of tumor cells under the influence of paclitaxel included into biodegradable nanoparticles (NP-paclitaxel) is very well described by equation (15). The dose above Cr = LD100 = 82.9 μg ml–1 = 1.028 / 0.0124 leads to overkilling – an overabundance of tumor cells. The traditional dose of LD50 at q² = 0.25 is easily calculated from equation (15):

(16)

Hence LD50 = 62.74 μg ml–1. Since the parameters of equation (15) can be found from the data of experiments on the effect of small doses, in assessing the risk of exposure of chemicals to biological objects, it is possible to perform experiments on the effect of small doses on the survival of the studied objects.

One can compare the effect of different preparations on the objects under study by the parameters of equation (15). The relationship between q² (quadrate of the proportion of living cells in tissue culture) and the impact of the drug paclitaxel included in the biodegradable nanoparticles associated with recombinant receptor-binding fragment of alpha-fetoprotein (NP-paclitaxel-rAFP-3D), is shown in Figure 1.13. For this chemical preparation LD50 = 20.10 μg ml–1, and LD100 = 26.84 μg ml–1. Comparing the doses for the two drugs, it can be concluded that the second drug NP-paclitaxel-rAFP-3D has a significantly greater efficacy of exposure than NP-paclitaxel.

Similar results can be obtained by analyzing the impact of herbicides on plants. The results of experiments on the impact of herbicide 2,4-dichlorophenoxyacetic acid (2,4-D) on plant seedlings are shown in Figure 1.14.

Fig. 1.14. Impact of herbicide 2,4-dichlorophenoxyacetic acid (2,4-D) in concentrations C on the survival rate of q plant seedlings

Apparently, in this case too, the second-order phase transition model describes the impact on biological objects very well. LD100 equals 0.70 μg ml–1 for the used herbicide.

Thus, the second-order phase transition model can be used to analyze the impact of various chemical compounds on biological objects. It seems that, when evaluating the impact of salts in small concentrations on plant growth, it is possible to obtain species-specific LD100 indicators that will characterize the impact on plants.

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Chapter 2

Criticality Concept and Some Principles for Sustainability in Closed Biological Systems and Biospheres

Nicholas P. Yensen1, Karl Y. Biel2,3,*

1NyPa International, Tucson, Arizona 85705, USA; Biosphere Systems International, Tucson, Arizona 85705, USA

2Institute of Basic Biological Problems, Russian Academy of Sciences, Pushchino, Moscow Region 142290, Russia

3Biosphere Systems International Foundation, Tucson, Arizona 85755, USA

*Corresponding author: [email protected]

…modeling represents the sole means available

to predict possible changes in the biosphere’s state resulting

from humanity’s large-scale influence on it, –

Academician Iosif Gitelson (cited by Gitelson et al.,

2003, Manmade Closed Ecological Systems)

Abstract

This survey of closed systems offers nomenclature, presents a new construct of criticality, describes the development of closed systems, and suggests future directions in the new science of ecobiospherics. Closed-system life-containing spheres are called biospheres and such systems with two or more biomes are named Biospheres (capitalized). Microbiospheres (smaller than humans) are classified as: biocappillaries, bioculturetubes, biopetris, biobottles, Folsome-Hansen flasks, Ecospheres® and terraspheres, bioboxes, Wardian cases. Macrobiospheres (larger than humans) are: Biotubes 1 to 8, CELSS, ALS, BPC, LMLSTP I, II and III, Biohome, Bio-Plex and ALSSIT, LSSIF, MELISSA, CEEF, Test Module, Bios-1, -2, and -3, Biosphere 2 Laboratory, Lunar and Martian bases, Earth [Biosphere 1]. Criticality is defined as a relatively rapid decrease in the ratio of organic material to inorganic forms within a system. Time to criticality is inversely related to initial organic content as a hyperbolic function which asymptotically approaches some minimum positive value. In general, the natural logarithms of time to criticality and volume are positively correlated, i.e. microbiospheres of one milliliter with fewer than 100 life forms reach criticality within 48 hours, whereas macrobiospheres may require months to years. Volumes of greater than 10 milliliters (bioculturetubes / biopetris, Ecospheres, etc.) may be necessary to achieve perpetual sustainability. The artificial macrobiospheres tested to date have not demonstrated sustainability for longer than a qualified duration of months. Microbiospheres can provide the principles to design sustainable human systems. These principles will provide the basis for sustainable space exploration systems and for the Earth’s Biosphere. The Bios systems (Krasnoyarsk, Russia), Biosphere 2 Laboratory (Tucson-Oracle, USA) and similar laboratories provide the best opportunity to understand and regulate sustainability in artificial human systems and to modulate the anthropogenic influenced Earth behaviour such as oxygen depletion and Global Climate Change.

Keywords: Ecobiospherics, Artificial closed biological systems, Biosphere 1, BIOSes, Biosphere 2 Laboratory

2.1 Introduction

Researchers, however, may be able to use small closed ecosystems to push the limits of stability to produce data upon which mathematical models may be tested and to accelerate our acquisition of knowledge in biospherics (Khlebopros et al., 2007). Remarkably, relatively little effort has been expended in this direction, especially compared to the extensive studies on open ecosystems. Even compared to mathematical modeling, small closed-ecosystem testing has received far less attention. Professor Iosif I. Gitelson and colleagues (Gitelson et al., 2003) agree that the use of small closed ecosystems can form a basis for mathematical modeling.

The typical ecosystem study is of an open system which, due to its interconnectedness with other ecosystems, has limited and possibly misleading paradigms for making global predictions. Because of the complexity of the Earth’s Biosphere, it is comprehensible that we, as individuals, are all blind to many aspects of the world. Metaphorically, the study of open global ecosystems is like three blind men describing the elephant’s trunk as a snake, the leg as a tree, and the ear as a leather curtain. But we, as a race, are still remarkably blind to the functional properties of closed ecological systems, not the least of which is the planet Earth’s closed ecological system, upon whose function depends the lives of all known species. Although one might think the evidence is clear, scientists disagree about the effects of: increased carbon dioxide, acid rain, the ozone hole, over fishing, and the cutting down of forests, to name a few of the perceived global anthropogenic phenomenon.

The study of small, closed ecosystems is an ideal method to investigate whole integrated ecosystem interactions and thus to discover general principles. At present, few principles of these interactions are known even though the concept of a closed ecological system has been around for over 150 years.

The following section considers some aspects of closed system history and descriptions of existing apparatuses. It also offers our suggested classification of closed systems. Further, it provides a snapshot in time of the evolving closed system terminology, including our concept of criticality. With respect to planet Earth, and to man-made closed systems, projections and suggestions are made herein as to how humanity may proceed to better understand our potential for sustaining the Earth’s Biosphere; and how miniature human-oriented Biospheres and closed systems may be developed for colonizing space, planets and beyond.

2.2 History of Manmade Closed Ecosystems

In 1827 a British physician, Nathaniel Ward, while studying cocoons under glass with soil, serendipitously discovered that ferns and other plants sprouted and grew vigorously under glass. Outside of the glass in the industrially polluted London air, however, these plants could not survive. Dr. N. Ward realized that he had discovered a way to grow plants sensitive to environmental pollutants. After some trials with small glass jars he found the right combinations of light, air, humidity and moisture to support various plants. He then developed closed-system cases large enough for houseplants. These became known as Wardian cases. With popularity, the cases became very ornate and elaborate, often mimicking architectural styles. In commerce, it was found that tea plants, rubber plants and exotic plants, when inside the cases, could be shipped to other parts of the world. This discovery resulted in the Indian tea and rubber industries. In the 1970’s environmental awareness, as exemplified by Earth Day, Rachel Carson’s Silent Spring, flower children, and the hippie back-to-nature movements, brought a renewed interest in Wardian cases leading to the modern terrarium. The 1800’s aquarium-sized and bottle-sized Wardian cases represented some of the first closed systems, although Wardian-case ecosystems were not generally hermetically sealed and were constructed principally for their attractiveness (see Arthurs, 1975; Michie, 2004).

Dr. R. Warington (1851) was perhaps the first to describe a sealed balanced aquatic microecosystem type microsphere. But it wasn’t until the second half of the 20th century that more serious work began on microcosms. Much of this work, however, still dealt with microcosms of differing degrees of closure for various experiments (Odum and Hoskin, 1957; Myers, 1958; Beyers, 1962a, 1962b; Golueke and Oswald, 1964; and various others).

Investigations of materially sealed ecosystem microcosms, i.e. the truly-closed microbiospheres, smaller than a human, hermetically-sealed biological systems, were started and pursued simultaneously and independently in both the USA (Folsome and Hanson, 1986) and in Russia (USSR) (Kovrov, 1992). Some of the sealed microbiospheres created in the 1970’s are still alive, including unpublished systems created in 1973 by K. Biel and by N. Yensen (per. obs.). Algal and microorganism-containing microbiospheres were studied by Drs. B.G. Kovrov and G.N. Fishtein (1980) and Drs. C. Folsome and J.A. Hanson (1986). Also, Dr. Bassett Maguire’s (1980) experimental studies of macroscopic organism in one-liter microsphere systems produced some of our most rudimentary knowledge of closed systems. In 1985, based on the NASA Folsome-Hansen flask (see below), N. Yensen, D. Harmony and L. Acker developed the synthetic Ecosphere® microcosm by combining species that would not normally occur together in nature into glass spheres to produce a sustainable small shrimp ecosystem (Yensen and Biel, 2005).

The first closed ecosystems were made as curiosities for their aesthetic qualities, but, through the transport of important commercial plants, had a significant impact on civilization. Even today, most closed systems are still constructed by individuals and scientists mostly out of curiosity and as aesthetic objects. It now appears that a myriad of formats is possible, despite the former scientific paradigm that, a sustainable closed ecosystem smaller than planet Earth’s Biosphere was not possible.

Today, there are several closed system types to choose from when planning an experiment (Table 2.1). Closed living systems in this classification may be defined according to their closure, size and shape.

Table 2.1. Classification of closed living systems