The Spatio-Temporal Dynamics of Synchronization in Theory, Models, and Natural Systems: A Multidisciplinary Investigation

This essay outlines my interest in one of Nature's most spectacular and widespread phenomena - the synchronization of ecological and biological systems. The very presence of synchronization almost always has an element of mysterious beauty. In Australia's Great Barrier Reef for example, many coral species have the remarkable ability to tune their reproductive cycle with astonishing precision. On a single day each year, just after the November full moon, corals synchronize by collectively releasing millions of gametes in an extraordinary large-scale mass-spawning event thus enhancing the survival of the next generation. Similar large-scale synchronization phenomena may be found in the insect world. Certain Asian firefly species are famous for their ability to gather in dense swarms where they rhythmically flash on and off in perfect unison, in a dizzying synchronized display of courtship. Equally curious is the manner in which many animal populations are able to collectively synchronize over enormous spatial scales, sometimes over continents. For the classic ten-year Canadian hare-lynx cycle, populations from different regions have been tightly synchronized for hundreds of years now, creating one of Ecology's greatest enigmas.

Mathematically, the principles driving synchronization are reasonably well understood for systems that are periodic; far less is known for systems that are erratic or chaotic. Complex systems theory has helped to explain why the presence of nonlinearities in the simplest of oscillating systems can generate irregular but bounded and persistent chaotic oscillations. Despite the manner in which tiny differences can amplify in chaotic systems, it is possible for two such oscillators to quickly synchronize to one another if, like Huygens's clocks, they are weakly coupled. Yet the synchronization is not always easy to identify. Sometimes it is subtle, and can only be detected after applying special mathematical techniques. Sometimes the synchronization appears and disappears erratically. On other occasions, there may be a "phase-synchronization" which holds only between the phase or rhythm of the signals, while the chaotic amplitudes (magnitudes of the oscillations) curiously remain largely uncorrelated. As discussed below, these complex forms of synchronization may be the key to understanding and predicting events in contexts that vary from flu epidemics to the extinction of endangered species, yet the basic notions have only just begun to be explored.

In the current ecological paradigm, which stresses the study of "metapopulations" (networks of spatially interconnected subpopulations), the variability and synchronicity of subpopulations can have decisive consequences for the conservation of species. When subpopulations are asynchronous, extinction risks are spread and "diluted" through the metapopulation. Should any external disturbances endanger or even extinguish a local population, asynchrony ensures that there will be other abundant local populations that can potentially recolonize and speed-up the recovery process. In contrast, when subpopulations are strongly synchronized, disturbances can often be hazardous. For example, if subpopulations are at relatively low levels to begin with, a disturbance might result in all subpopulations synchronizing to extinction. The ideas are just as relevant for understanding disease dynamics, where extinction rather than persistence of a disease is the preferred scenario, and synchronization may act to enhance this.

Many of the existing theories concerning metapopulation dynamics and synchronization are derived from the analysis of discrete mathematical "maps", often referred to as "toy" models because of their simplicity. My aim is to advance the understanding of synchronization through the more difficult study of continuous-time population models, which in many contexts are more realistic. Only a few ecological models of this type have been described in the literature, and in some sense there is a need to develop the field. My recent work has led to the design of an important new class of continuous-time foodweb models in which population abundances cycle with precise regularity (uniform from cycle to cycle) yet the actual population abundances are themselves highly variable or chaotic - a behavior termed UPCA (Uniform Phase but Chaotic Amplitudes). Although UPCA is characteristic of numerous ecological and biological systems, such as the Canadian hare-lynx cycle mentioned above, the equations I developed with my colleagues constitute the first ecological model able to generate this fundamental behavior. Furthermore, the introduction of small levels of migration between such models can lead to a whole range of synchronization behaviors and points to a rich unexplored area of research awaiting study.

For spatially structured ecosystems, population "waves" bounce around the spatial landscape and the effects of pattern-formation enter the picture. Spatial traveling wave patterns have been noted in a wide range of dispersing species, and also emerge naturally in lattice structures of spatially connected foodweb models coupled via migration. The foodweb models are intrinsically chaotic, yet the traveling waves they generate through their mutual synchronization are often highly ordered with periodic phase structure. Little is known about the spatio-temporal structures associated with complex synchronization, but they have important implications for conservation ecology. If, for example, a disturbance perturbs a local patch-population to the brink of extinction, the periodicity of spatial phase-synchronization guarantees the recurring arrival of wave fronts in which new colonizers can buffer endangered populations. This stands in contrast to the common view mentioned above, that population synchronization accelerates global extinctions.

Another important aspect of synchronization is the manner in which ecological communities entrain to periodic environmental or climatic driving variables. Many organisms have no choice but to synchronize their life cycle to the seasons by thriving only in a small "window" of the larger annual cycle, when resources are most favorable. In Lake Kinneret (Sea of Galilee), Israel, for example, seasonal stratification and nutrient availability patterns ensure that almost every March a large-scale algae bloom appears. As in many other freshwater lakes, the presence of a phytoplankton bloom signals dangerous nutrient loading (eutrophication) and water quality problems, making the study of phytoplankton blooms absolutely vital. A deeper understanding of these blooms is also vital in marine settings, where they are a major source of primary production and are intimately connected with the global carbon cycle. The seasonal forcing adds a complicating factor to the study of bloom dynamics since it can induce irregular chaotic oscillations synchronized to the seasonal cycle. The explosive nonlinear growth of the Kinneret bloom has all the hallmarks one might expect for a synchronized chaotic process, with some erratic years when a threshold effect prevents the bloom from appearing (analogous to the dynamic features of epidemics). As yet there is little theoretical understanding of the nutrient-driven dynamics of phytoplankton blooms, their temporal synchronization and their spatial pattern-formation mechanisms. Yet such an understanding should prove helpful for dealing with the water resource problems we face at the dawn of this new millennium.

In the same spirit but in the context of epidemiology, synchronization has great control over the (often chaotic) dynamics of diseases as they spread through complex networks of suburbs and cities. With current patterns of globalization, major diseases are transmitted from airport to airport across the globe, passed along via traveling human cargo. This increasing "connectance" between distantly related cities dramatically changes the transmission dynamics of infections, and the methods for studying spatio-temporal synchronization should provide powerful tools for investigating such scenarios. Many epidemic oscillations are characterized by inherent UPCA-like dynamics, yet conventional models are incapable of reproducing this behavior. The UPCA models I am developing should provide a technical-fix and form the basis of a theoretical study of complex spatio-temporal synchronization under different realistic scenarios. Existing long-term epidemiological data sets are characterized by a variety of complex synchronizations and will provide an unexplored test-bed for complementing this study. Finally, theoretical ideas of synchronization may be harnessed to design optimal vaccination schedules. Both "pulse vaccination" and modern chaos-control are techniques that have the potential to induce spatio-temporal synchronization in regional networks, and thereby increase the extinction probabilities of infections.

Complex synchronization is the common thread that ties this suite of multidisciplinary projects together. While this essay focuses mainly on the ecological applications, these are in fact only the tip of the iceberg: Synchronization is widespread and pervades all areas of biology. Many of the models, tools and data analysis techniques proposed here should be of interest for the way they grope at deducing fundamental biological principles, rather than being restricted to any particular type of system. The interplay between theoretical investigations and the observation of natural systems is thus expected to lead to new advances and insights in our understanding of synchronization processes, in general, and the way they contribute to the self-organization of large complex systems.