Toward Analytical Chaos in Nonlinear Systems
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Downloaded: More Print chapter. How to cite and reference Link to this chapter Copy to clipboard. Cite this chapter Copy to clipboard Nikolai A. Magnitskii October 24th Available from:. These 8 individuals were assigned random but fixed values for maximum lifetime. As the chaotic parts of the population dynamics would be lost in non-linear analysis when based on mean values from these simulation runs, non-linear dynamical characteristics were calculated for each simulation run separately.
Statistics were subsequently calculated from the resulting distributions of non-linear characteristics. In this way the data structure allowed for statistical analysis of population age class dynamics. Data analysis was based on the time series data described above.
To exclude the effects from transient dynamics, the first 50 data points were omitted from all analyses. First, 3-dimensional phase space was plotted from the raw data of neonate, juvenile and adult daphnids. Secondly, exploratory analysis was performed on the time series data by use of linear autocorrelation functions. In a 3-dimensional plot neonate, juvenile, and adult Daphnia abundances were plotted for each DCA-concentration. Trajectories were assessed in terms of compactness and orientation in phase space. Autocorrelation analysis was performed with function corr from the Tisean package to assess the amount of linear structures within the data.
To find the best unfolding of the time series in the reconstructed phase space a series of time-delay embeddings was performed by means of the function delay from the Tisean package and the optimal parameters were chosen. Recurrence analysis  was performed to quantify the number and duration of recurrences of the system in its state space.
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Quantification of attractor characteristics was done by calculation of global Lyapunov exponents. Values larger than 1 were assumed to be a strong evidence for chaotic dynamics within the time series. From the distributions of Lyapunov coefficients mean values and standard deviations per age class and treatment level were calculated and tested for significant differences by means of a Wilcoxon signed-rank test function wilcoxon.
All results are described for the overall population as well as the age classes neonates, juveniles and adults. Figures are only shown for the overall population. The according figures for the age classes can be found in the supporting information. For the highest DCA concentration level the dynamics were obviously different. In general the autocorrelation functions Fig. This effect was especially strong for juveniles and adults, while in the neonate age class the effect was clearly less pronounced see Fig. S2 in supporting information.
From time-delay embedding Fig.
Especially for neonates and juveniles see Fig. Increased pattern formation was observed with increasing DCA concentrations in the recurrence plots Fig. This was also observed for the age classes see Fig. S4 in supporting information. For neonates a very strong lineal pattern occurred in the highest concentration level, while for juveniles and adults this pattern formation was less pronounced.
Especially for juveniles additional circular structures were observed in the highest concentration level. This effect was especially pronounced for juveniles and adults see Fig. S5 in supporting information. For neonates the attractor was always located near the phase space origin. Table 1 shows mean values and standard deviations for maximum Lyapunov coefficients calculated from the Lyapunov distribution for the overall population in the treatment levels.
All maximum Lyapunov coefficients for all treatment levels were larger than zero Fig.
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Table 2 shows the results from the Wilcoxon signed-rank tests on statistical differences between mean maximum Lyapunov coefficients for all treatment levels. Also, all mean maximum coefficients for the treatment levels were significantly different from the control.
From the time series data in Fig. The existence of equilibrium is prerequisite in the attempt to define recovery because only if population structure is relatively stable, the population can return to defined pre-disturbance conditions if given sufficient time following the disturbance . Generally the system seemed to converge to a quasi-stable state in the long run see Fig. This region occurred well bounded a prerequisite for chaos , but the trajectory seemed to follow a complex attractor, exhibiting considerable population fluctuations. The trajectory settled within a much smaller phase space region, nearly approaching the zero abundance level, especially for the adult age class.
These dynamics clearly exhibited an increased extinction risk for the overall population if this effect would have lasted too long although this does not pose a problem to the age-class itself in the first place, as the classes are refilled by aging individuals. For the overall population as well as for all age classes in the control and all treatment levels a sinusoid like oscillation was observed see Fig.
Toward Analytical Chaos in Nonlinear Systems
Whether unpredictability results from a random process or is based on non-linear effects deterministic chaos can not be assessed by linear autocorrelation analysis as it is unable to distinguish between these two sources of fluctuation. However, since individual properties, except random individual maximum lifetime, were switched of to reduce the stochasticity of the model it is fare assumption that the unpredictability resulted exclusively from the non-linear effects.
For the highest concentration level the amplitude of the periodic oscillations clearly increased while the frequency decreased. This argues for an increased formation of long-range correlations within the data  for high concentration levels, enhancing the long-term predictability. It was thus concluded that the linear predictability of system dynamics increased with increasing disturbance. As already observed for the raw data phase space, the system was forced towards an asymptotically stable equilibrium point for all DCA concentrations in embedded phase space see Fig.
The fluctuations within the trajectory's settlement area decreased for the overall population and especially for the juvenile daphnids for the highest concentration level, reflecting the population's decreasing ability to respond to changes in environmental conditions decreasing resilience.
Additionally, as the equilibrium point moved towards the phase space origin with increasing DCA concentration, the risk of extinction also increased. The generally increasing pattern formation in the recurrence plots revealed, that the system dynamics increasingly tended to return to formerly engaged states when the disturbance increased see Fig.
While the more uniform spreading of points in the lower concentration levels suggested a high portion of chaotic dynamics especially for the neonate daphnids , the lineal pattern found for in the highest concentration level suggested a high amount of periodic dynamics within the time series . In contrast, the circular structures observed for juveniles in the highest concentration level suggested the occurrence of quasi-periodic dynamics, like superimposed harmonic oscillations.
These structures can be suspected to originate from the realistic and model-built in time delay between juvenile response, their maturation and the according effects on reproduction.
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For the juvenile daphnids a mixture-like pattern was observed in the highest concentration level. It seems like the system settled at a transition between quasi-periodic and periodic dynamics. Such a transition has already been observed in a population model for Tribolium  , in which the varied parameter of disturbance was the rate of larvae developing to adults, resembling the effect on age class dependent mortality of juveniles within the Daphnia model. As already deduced from the autocorrelation function, from our findings it was concluded, that the predictability of the system dynamics increased with increasing disturbance.
The abruptly changing location of the attractor esp.
Consulting the Lyapunov coefficient as a measure of rate of chaotic dynamics within the population development, this rate decreased for DCA treatments compared to control see Fig. From the biological point of view this seems reasonable as positive feedback influence of exponential growth is likely to be smaller in the presence of a disturbance affecting reproduction.
The decrease of the coefficients indicated that higher amounts of disturbance decreased the potential for exponential increase of trajectory distances. This was already obvious from the former analyses, but the statistics on the coefficients quantified and validated this. It suggested some sort of concentration-response relationship, thereby approving the Lyapunov coefficient as an indicator to detect non-linear effects of population dynamic disturbance due to environmental stress.
Additionally, the significant decrease of the Lyapunov coefficient revealed a true and important emergent effect which could not have been expected from the raw population abundance pattern. The decreasing Lyapunov coefficients with increasing DCA concentration levels resembled a concentration-response relationship.
This is an important finding as it is directly connected to the important species trait of reproductive capacity that strongly triggers the daphnids' population resilience see discussion below. Unfortunately it is not straightforward to quantify the effect as the Lyapunov coefficient and the treatment level exhibit a reciprocal relationship.
Furthermore, the upper effect bound is difficult to define, because Lyapunov coefficients approaching zero or values below zero mean a discrete change in system dynamics phase transition towards a limit cycle or a stable equilibrium point respectively , not a steady change as required for a continuous concentration-response relationship. Thus the concentration-response relationship would exhibit discontinuities, difficult to interpret. This would mean an extra discontinuity within the system's reaction to the increased treatment, which is a further clear contradiction to the concept of concentration-response relationships.
Nevertheless, a no-effect threshold value can easily be inferred from the distribution of Lyapunov coefficients. A simple test on significant differences between the mean values of controls and treatment levels see Tab. In the present study this threshold turned out to be smaller than 2. Such a threshold could serve as a relevant indicator to assess non-linear effects in population time-series data. The population dynamics can generally be decomposed into a deterministic periodic plus chaotic and a stochastic part, the latter not being considered in this study. The authors are well aware about the diverse discussion about the problem how to distinguish stochastic fluctuations from chaotic dynamics e.
The control level was found to exhibit a larger amount of chaotic dynamics than the treatments. Nevertheless, this does not mean that the controls were not predictable and thus not suitable for assessment.