Without seasonal forcing, the model predicts various kinds of asymptotic regime, including stable equilibria (a), simple limit cycles (b), complex periodic cycles (c) and chaos (d). At first sight, seasonal forcing seems to have little influence on the dynamical repertoire of the model. With seasonal forcing, the model also displays simple limit cycles (e), complex periodic cycles (f) and chaos (g). However, a closer look reveals differences between the model behaviour with and without seasonal forcing. With seasonal forcing, the periodic solutions are ‘locked’ within the seasonal cycle: the same pattern repeats each year (e) or after some years (f). In addition, the model can also produce quasi-periodic cycles, where solutions are entrained within the seasonal cycle yet never repeat themselves as they slightly shift phase every year. Chaotic communities seem to experience similar seasonal patterns. However, the fluctuations of phytoplankton and zooplankton species in chaotic communities remain irregular even when entrained in a regular seasonal environment (g).
Figure 2. Community dynamics predicted by the model. The two top panels indicate the nature of the environmental forcing. Without seasonal forcing, the model produces (a) stationary equilibria, (b) simple cycles, (c) complex periodic cycles or (d) chaotic dynamics. (more ...)
These dynamics can be illustrated by Poincaré maps sampling the model communities once per year for many consecutive years. Model communities with a periodicity of 1 year return to exactly the same species composition year after year, which appears as a single point on the Poincaré map. Communities with a periodicity of N years produce N points on the Poincaré map, quasi-periodicity produces a closed curve (a), while chaos produces a complex fractal structure (b).
Figure 3. Poincaré maps with annual snapshots of the model community collected over many years. More specifically, the maps plot the biomasses of two plankton species sampled from the model community at the 1st of January of each year for 100 000 years. (more ...)
Many of the model communities exposed to seasonal forcing displayed chaos with remarkable synchronization patterns at the species level (). The species fluctuations are irregular, yet these irregular fluctuations are squeezed within the seasonal cycle. As a consequence, species enter the winter season in different proportions, and this affects the species composition of the next spring bloom. For instance, a shows a typical phytoplankton spring species. It reaches peak abundance in March, although its peak abundance varies from year to year, and some years it does not peak in spring at all. b shows another phytoplankton species from the same plankton community. This species could be called a typical summer species. It is present every summer. However, some years it peaks twice, with a first peak in May–June and a second smaller peak in September. In other years, it peaks in September only. The zooplankton species show similar seasonal patterning. For instance, some zooplankton species are mainly present in winter (c), while others dominate during the summer period (d). The example in d is particularly interesting. In some years, this zooplankton species shows little variability from March to September, while in other years, it fluctuates wildly during the same period. Accordingly, the species composition in our model communities shows distinct patterns of seasonal organization, but with strong year-to-year variability.
Figure 4. Year-to-year variability in the population dynamics of (a) phytoplankton species 9, (b) phytoplankton species 6, (c) zooplankton species 5 and (d) zooplankton species 3. All four species are from the same model community, simulated for a total period (more ...)
Which species traits and environmental conditions are responsible for the widespread chaotic dynamics in our model communities? A complete answer to this question is beyond the scope of this paper. However, some insight can be obtained by modifying the model assumptions systematically. This shows that more than 50 per cent of the model simulations produced chaos when using our default parameter settings (, first row). The occurrence of chaos was not very sensitive to the relative magnitude of intraspecific versus interspecific phytoplankton competition (). In contrast, modifying zooplankton predation had a striking effect on the occurrence of chaos. When zooplankton was removed from the model, very few simulations showed chaotic dynamics and they did so only under seasonal forcing (). Similarly, inefficient zooplankton grazing and specialist zooplankton reduced the occurrence of chaos. This shows that predator–prey oscillations, and the nature of predation, played a key role in the generation of complex dynamics in our model communities.
Table 2. Occurrence of chaos in our simulated communities under different model assumptions. The first row (in italics) shows the percentage of chaotic communities predicted by the reference model used in our study, both without seasonal forcing (a = 0) and with (more ...)
Productivity also had a clear effect on the occurrence of chaotic dynamics. At low productivity (K = 2 mg l−1), stationary dynamics prevailed in constant environments, simple periodic dynamics prevailed in seasonal environments and chaos occurred only in a few model communities with strong seasonal forcing (; see also appendix S1 in the electronic supplementary material). Chaos was widespread at intermediate productivity (K = 5 and K = 10 mg l−1). At high productivity (K = 20 and K = 50 mg l−1), the occurrence of chaos declined and the population dynamics often shifted to periodic cycles in both constant and seasonal environments.
In all cases summarized in , seasonal forcing increased the occurrence of chaos. To investigate this aspect in further detail, we estimated whether the amplitude of seasonal forcing affected the occurrence of chaos in our model communities (). We focused on the intermediate productivities (K
= 5 and K
= 10 mg l−1
). At K
= 5 mg l−1
, the amplitude of seasonal forcing increased the occurrence of chaos (a
; linear regression: R2
= 0.45, N
= 11, p
= 0.024). At K
= 10 mg l−1
, mild forcing (0.1 < a
< 0.4) caused a slight increase in the probability of chaos, but when the amplitude of seasonal forcing was further increased (a
> 0.6), the probability of chaos declined (b
; quadratic regression: R2
= 0.66, N
= 11, p
= 0.013). We further explored the predictability of these communities by calculating their Lyapunov exponents. A positive Lyapunov exponent indicates chaos. The inverse value of the Lyapunov exponent is often used as a simple metric of the predictability of chaotic systems (Strogatz 1994
). In those simulations that displayed chaotic dynamics, the magnitude of the Lyapunov exponent was not affected by the amplitude of seasonal forcing (c
). This indicates that the predictability of the chaotic plankton communities was neither enhanced nor reduced by a stronger seasonality. However, the median values of the Lyapunov exponents were significantly higher at K
= 10 mg l−1
than at K
= 5 mg l−1
= −3.77, d.f. = 20, p
< 0.002), which indicates that the predictability of the model communities was affected by productivity.
Figure 5. (a,b) Relative frequency at which randomly generated model communities display chaos, plotted as a function of the amplitude of seasonal forcing. Results are shown for model communities grown at two productivities: (a) K = 5 mg l−1 and (b) K = (more ...)