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Nevertheless, it is useful to consider briefly the possible implications of the theory of deterministic chaos in ecology. In random motion initially adjacent points are distributed with equal probability over all new allowed intervals (Fig. 2.15A). In regular motion initially adjacent points stay adjacent (Fig. 2.15B). In chaotic motion initially adjacent points become exponentially separated (Fig. 2.15C). Thus, regular motion and random motion show complete predictability and unpredictability, respectively. Chaotic motion offers some short term predictability (Schuster 1995). Although chaotic systems totally lack long term predictability their short term predictability is better than that of stochastic random processes. One can determine the error of such predictions, and with the appropriate algorithms one can use them to regulate a system via a transputer so that the system is forced to stay in one of the chaotic paths (or trajectories) in time and space. Such fine regulation of chaotic systems, which offers opportunities for much more delicate manipulation than regulation of deterministic systems, is currently assessed in physics (Hubinger et al. 1993; Schuster 1995) and even explored for practical applications in engineering. Biological systems must obey physical laws. Conversely, we may also say that biological systems use physical laws to develop the diversity of life. It would be most surprising if the wide scope of possibilities inherent in deterministic chaos had not been used by life during evolution. It may be noted that with pure stochastic randomness life would be deprived of any significance and would be cast into meaninglessness. Conversely, pure deterministic regularity, allowing one to retrace all things accurately in the past and predict them precisely in the future for all (mathematical) infinity would cast life into tedious monotony. Only deterministic chaos, with its strict mathematical rules and yet high variability, with its unpredictability and yet delicate means of fine regulation, provides an opportunity for the adaptability, plasticity, diversity and beauty of life to unfold (W. Martienssen, quoted after a public lecture; Lloyd and Lloyd 1995).

Due to the pioneering contributions of R. May (May 1976) population dynamics has become one of the roots of the development of the chaos theory. Possibly the apparently simple logistic equation discussed by May (1976) may allow biologists a ready access to an intuitive comprehension of the theory of deterministic chaos. Let xt be the size of a population at a certain state at time t. We then want to know the size of the population at the next possible state in time, i.e. xt+1. It is obvious that the development of the population depends on its resources. These may be given by a growth factor r or more generally an external control parameter, such that xt+1 is proportional to r ■ xt. However, it is not only evident from the exorbitant increase of the human population on the globe, but a general experience of population ecology, that increasing population densities also bear inhibitory mechanisms in themselves, i.e. xt+1 is also proportional to xt ■ (1 - xt). Hence, the logistic equation for the development of the population x is xt+1 = r ■ xt ■ (1 - xt). (2.10)

Subsequent population sizes can be calculated by recursions, where xt+1 is used in the place of xt to obtain xt+2 and so forth.

However, the equation is only apparently simple. It describes one of several possible routes from order or regularity into chaos (Schuster 1995; Fig. 2.16). It shows that ordered predictability only occurs for a narrow range of the value of the control parameter r. At low values of r there is a steady state, while at larger values of r a bifurcation (i.e. a branching or dichotomy) leads first to phase doubling and ordered oscillations between two states and then further bifurcations give four states. However, very tiny additional changes of r lead the way into the non-predictability of deterministic chaos. This is seen in computer simulations of (2.10), shown in Fig. 2.16. The lower diagram shows the initial steady state, effective for a large range of values of r, then the first and second bifurcation leading to periods of 2 and 4 states, and with increasingly smaller increments of r there are then chaotic responses to tiny

Fig. 2.16 The route from order to chaos via increasing periods by augmenting r in the logistic recursion equation (2.10). The lower diagram shows the route from steady state via bifurcations (period 2, period 4) into chaos given by increasing r. The upper four diagrams give the calculated population sizes by iteration of (2.10) for the steady state, period 2 (two states), period 4 (four states) and chaos. (May 1976; Hastings et al. 1993)

Fig. 2.16 The route from order to chaos via increasing periods by augmenting r in the logistic recursion equation (2.10). The lower diagram shows the route from steady state via bifurcations (period 2, period 4) into chaos given by increasing r. The upper four diagrams give the calculated population sizes by iteration of (2.10) for the steady state, period 2 (two states), period 4 (four states) and chaos. (May 1976; Hastings et al. 1993)

changes of r. The top four graphs show the results of iterative calculations of population sizes xt. At low r in the steady state the population is stable, at period 2 there are two and at period 4 there are 4 predictable states, while in chaos, prediction of subsequent population sizes from existing ones has become impossible.

This little excursion to population theory appeared useful to explain some basic implications of the chaos theory. Chaos is a property of non-linear dynamic systems and these are the rule and not the exception both in the living and non-living world. However, while the theory of deterministic chaos has already had some impact in population biology (May 1976; Hastings et al. 1993) it is intriguing that in ecology in general it has only been accepted very reluctantly (Linsenmair 1995; Stone and Ezrati 1996), or even been rejected in the exclusive distinction of deterministic and stochastic development of diversity. It is intriguing because population biology is a field so close to or even part of ecology. For example, we may consider the control parameter r in (2.10) as an indicator of general resources or even as stress. If then we envisage that a high state of order may be given by complexity, which integrates functioning of diversity (Cramer 1993), we realize that this must occur only within a rather narrow window of stress conditions, as it is in fact borne out in experiments like those of Grime et al. (1987) and Tilman (1982) described in Sect. 3.3.2. Likewise, we are right back in the realm of deterministic chaos when we reject the climax theory of formation of stable steady state vegetation types and adopt instead the oscillating mosaic model with continuous dieback and renewal of "unpredictable irregularity" for tropical forests (Sect. 3.3.3). It is not unlikely that deterministic chaos, which certainly governs the ecology of populations, also determines the structure of tropical forests (as of other environments) with their oscillatory and non-linear behaviour.

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Renewable Energy 101

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