## At the Edge of Chaos

The prevailing conditions including the abundance of other species determine which growth rate is optimal for any organism. If the growth rate is too high, then the resources (food) will be depleted and the growth will cease. If the growth rate is too low, then the species does not utilize the resources (food) to the extent that is possible. The optimal growth rate also yields the highest system eco-exergy. If, in a well-calibrated and validated eutrophication model (state variables include phyto-plankton, nitrogen, phosphorus, zooplankton, fish, sediment nitrogen, and sediment phosphorus), we vary the zooplankton growth rate, then eco-exergy will show a maximum at a certain growth rate (which is frequently close to the value found by the calibration and approved by the validation). At both lower and higher growth rates, the 'average' eco-exergy is lower because the available phytoplankton is either not utilized completely or is overexploited. When overexploitation occurs, the phyto-plankton and zooplankton show violent fluctuations. When the resources are available, the growth rate is very high but the growth stops and the mortality increases as soon as the resources are depleted, which gives the resources a chance to recover and so on. At a growth rate slightly higher than the value giving maximum exergy, the model starts to show deterministic chaos: a minor difference in the initial value causes exponentially increasing changes as the time increases. Figure 1

Figure 1 Exergy is plotted versus maximum growth rate for zooplankton in a well-calibrated and validated eutrophication model. The shaded line corresponds to chaotic behavior of the model, i.e., violent fluctuations of the state variables and the exergy. The shown values of the exergy above a maximum growth rate of about 0.65-0.7 d~1 are therefore average values. By a minor change of the initial value of phytoplankton or zooplankton in the model, significant changes are obtained after 2 months of simulations as an indication of deterministic chaos.

Growth rate of zooplankton (1 per 24 h)

Figure 1 Exergy is plotted versus maximum growth rate for zooplankton in a well-calibrated and validated eutrophication model. The shaded line corresponds to chaotic behavior of the model, i.e., violent fluctuations of the state variables and the exergy. The shown values of the exergy above a maximum growth rate of about 0.65-0.7 d~1 are therefore average values. By a minor change of the initial value of phytoplankton or zooplankton in the model, significant changes are obtained after 2 months of simulations as an indication of deterministic chaos.

illustrates the exergy as function of the zooplankton growth rate in the model referred to above, focusing on the time when the model starts to show deterministic chaos. These results are consistent with Kaufmann's statement: biological systems tend to operate at the edge of chaos to be able to utilize the resources at the optimum. In response to constraints, systems move as far away from thermodynamic equilibrium as possible under the prevailing conditions, but this will imply that the system has a high probability to avoid chaos, although the system is operating close to chaos. Considering the enormous complexity of natural ecosystems, and the many interacting processes, it is surprising that chaos is not frequently observed in nature, but it can be explained by an operation at 'the edge' of chaos to ensure a high utilization of the resources - to move as far away from thermodynamic equilibrium as possible at the prevailing conditions.

See also-. Catastrophe Theory.

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