Ecology: the biology of interaction. 1.09. Stability of biosystems

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1.08. Properties of Complex Systems

D. Shabanov, M. Kravchenko. Ecology: Biology of Interaction Chapter 1. Ecology and the Biosystems it Studies

1.10. (supplement) Dynamic Typology of Biosystems

1.09. Stability of biosystems A characteristic feature of regulation by the principle of negative feedback is that it leads to oscillations of the regulated variable (Fig. 1.9.1). It is interesting to compare, for example, phenomena occurring in a standing table and in the body of a person standing nearby. The table undergoes an elastic deformation in which the action of gravity is balanced by the elastic force, and it remains stationary. The person continuously controls his posture (using a substantial part of the “power” of his brain). A slight deviation of the body to one side triggers receptors that activate effectors (muscles) that return the system to the required state. The action of these effectors compensates the deviation with a certain excess. Because of this, the system passes through an optimal state and requires regulatory influence in the opposite direction. The standing person’s body becomes involved simultaneously in several oscillatory processes with different frequencies and amplitudes! This is why a standing person gets tired and expends energy, whereas the table does not expend energy and does not wear out. What will happen if an external influence that changes its parameters is applied to a biosystem whose state is regulated by the principle of negative feedback? Let us consider this using the example of population size regulation. A population in a stable state keeps its size relatively constant while experiencing continuous fluctuations. Suppose catastrophic events or human impacts have destroyed a significant portion of the individuals of such a population. How will it react to this impact? If the impact pushes the population beyond the regulation of its size by negative feedbacks, it will transition to another state (for example, it will die; see Fig. 1.9.1). Thus, if the number of individuals becomes very low, their reproduction may be disrupted (for example, because partners cannot find each other). Positive feedbacks will be triggered: a decline in population size will cause a reduction in the birth of new individuals. As a result, the biosystem will move to a different state (the population will perish). If, however, the impact is not critical, negative feedbacks can restore the population size to normal. This means that a decrease in size will weaken intraspecific competition, enhance reproduction, and improve juvenile survival. [IMG_1] Fig. 1.9.1. Trajectory of a system governed by feedbacks. The first external impact removed the system from the normal range, but stronger negative feedbacks returned it to the normal range. The second external impact shifted the system into the range of positive feedbacks, which amplified deviations and led the system to collapse. To describe processes analogous to those mentioned above (and those shown in Fig. 1.9.1), the concept of stability is essential. Let us analyze Fig. 1.9.1. Is the state of a biosystem that lies within the normal range stable? Yes, and this state is characterized by the fact that a deviation from equilibrium triggers mechanisms that bring the biosystem back. This stability is not impeded by the fact that the considered biosystem is in a state of continuous change. As a result of external influences, the biosystem whose dynamics are shown in Fig. 1.9.1 died. Is this state stable? Yes, but it is a different kind of stability than in the normal case. The state of a system located on the “ridge” separating the ranges governed by negative and positive feedbacks (as well as a system “sliding down” under the influence of positive feedbacks) can be called unstable. This means that, from the viewpoint of stability, these states belong to special types. Thus, for further discussion it is necessary to define the notion of “stability”. It became one of the most important ecological concepts from the early stages of ecological development. For example, it was key to F. Clemens’ theory of climax plant communities, which emerged at the beginning of the 20th century. Stability of a plant community is its ability “to preserve its composition and structure over a prolonged period. This stability is due to the community’s capacity for self‑renewal.” (V. N. Sukachev) The use of the term “stability” in ecology is a particular case of its general scientific usage. The classic definition of “stability” is the one given in 1892 by the founder of stability theory, Alexander Mikhailovich Lyapunov (associate professor, later professor at Kharkiv University, now named after V. N. Karazin). Lyapunov defined stability in terms of the effect of perturbations on motion with given initial conditions. Applying this approach to describe biosystem dynamics encounters significant difficulties and does not fully correspond to the tradition established in ecology. In our opinion, among mathematical approaches for ecology the most useful is the notion of stability according to J. L. Lagrange, which is a particular case of Lyapunov stability. From this perspective, stability is the ability of a system to remain within a bounded region of its phase space. The phase space in this case is the set of all possible states of the system, described by a set of its state variables (for example, the movement of the system in phase space is shown in Fig. 1.9.1). Choosing the variables that define the phase space allows, for instance, to consider community stability both in terms of species number and in terms of structural constancy. The essence of the stability phenomenon remains the same; only the coordinate set in which the trajectory of changes of the studied system is examined changes. What limits a particular region of phase space? The trajectories of systems that lie within that region. In Fig. 1.9.1 one can see certain bounded regions of phase space: a zone of stable equilibrium, where the system’s state is governed by negative feedbacks, and two zones (to the right and left of the stable equilibrium zone) where the system’s dynamics are determined by positive feedbacks. A special state is probably the position on the “ridge” separating the “+” and “–” control zones, as well as the state of a dead system. These regions of phase space can be called basins of stability. Each basin of stability corresponds to its own dynamic type of the considered biosystems (the concept of dynamic type is analyzed in more detail in section 1.10). Based on the study of changes in the composition of hemiclonal population systems of the hybrid complex of green frogs, the authors of this manual propose a classification of biosystem stability types, shown (with physical analogies) in Fig. 1.9.2. It can be assumed that this classification is applicable not only to the studied category of systems but also to any others. [IMG_2] Fig. 1.9.2. Classification of system stability types The difference between the physical analogies used in Fig. 1.9.1 and Fig. 1.9.2 lies in the fact that in the first figure the profile of the surface (determining the characteristic dynamics of a given stability type) is shown in a single plane, whereas in the second it is shown in two perpendicular planes. Analyzing the system dynamics in Fig. 1.9.1, one can find all the stability types shown in Fig. 1.9.2 except type II (wandering). This case corresponds to a situation where a change in the system’s state is indifferent to the mechanisms governing its dynamics. Type I (fragile stability) corresponds to the position on the “ridge” separating the ranges of regulation by negative and positive feedbacks. A deviation from the fragile equilibrium will lead the system to transition to one of the alternative states. It should be emphasized that in most cases authors writing about system stability refer to the states of their stable stability (type IV in the presented classification). Naturally, in many cases such states are of greatest interest. For example, the task of protecting a particular biosystem (e.g., a valuable ecosystem in a nature reserve) can be formulated as the task of maintaining it in type IV stability. The dynamics of indicator changes in a system that is in type IV stability are of great interest for its diagnosis (and forecasting of future changes). The speed at which a perturbed system returns to its normal state is an important characteristic. Analyzing its dynamics, two parameters can be extracted: resistance to impact and the ability to return to normal after change. These parameters reflect different properties of the system (they can be compared to hardness and elasticity in mechanics: diamond is hard but not elastic, rubber is not hard but is elastic). These parameters are negatively correlated (as strength and elasticity are in classical mechanics). They constitute two components of overall stability (illustrated in Fig. 1.9.3 as the area between the curve describing the system’s state dynamics and the “norm corridor”). [IMG_3] Fig. 1.9.3. Main measures of stability applicable to biosystems in type IV stability (see Fig. 1.9.2)

1.08. Properties of Complex Systems

D. Shabanov, M. Kravchenko. Ecology: Biology of Interaction Chapter 1. Ecology and the Biosystems it Studies

1.10. (supplement) Dynamic Typology of Biosystems