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Real-time Vulnerability of Synchronization in Ecological Networks

By Everton Medeiros

In a world in which ecosystems are constantly losing spatial extensions due to human activity (like land use), some species’ capability to move across different territorial patches is a crucial survival strategy. This remarkable ability to search for better resource conditions can increase the stability of populations in ecosystems.

However, such spatial dispersal of species constitutes interconnections between patches that facilitate synchrony among their population dynamics. The impact of this synchronized behavior on ecosystem functioning is controversial. On one hand, synchrony elevates the risk of localized extinctions becoming global and consequently causing the whole system to collapse. On the other hand, the occurrence of a soft manifestation of synchronization—called phase synchronization—generates traveling waves of population abundances that rescue local patches from extinction due to the wavefront’s arrival. In light of these facts, it is clear that synchronized behavior should be avoided or even suppressed in certain circumstances, and preserved in others. To address these demands, we provide a mechanism that numerically observes synchronized behavior to be vulnerable to external perturbations and therefore prone to suppression in certain time windows in order to preserve ecosystem functioning.

Despite the reported phenomenon’s generality, we demonstrate our findings and explain the underlying mechanism in a network of patches containing the dynamics of a simple food web model consisting of only three trophic levels: a producer, a primary consumer, and a top predator. To represent the species’ mobility throughout patches across the network, we consider a network coupling in which the mobile species—i.e., the primary consumers as well as the top predators—are able to migrate from an individual patch to the connected patches according to a diffusion ansatz. The topology of connections is random. Considering the population dynamics of each identical isolated patch, it ultimately converges to an oscillating regime — a limit cycle. However, close to these well-behaved oscillations is an unstable chaotic set covered by infinitely many unstable orbits offering transient chaotic trajectories for the population dynamics, depending on the initial conditions. Next, the coupled system tends to end up in a completely synchronized state around the existing limit cycle for each patch; this is due to the coupling dynamics.

Figure 1. Time evolution of the perturbed patch’s rabbit population. The time intervals at which the applied perturbations desynchronize the network of patches appear in red. The time intervals during which the system is safe under the same perturbation are in blue.
After synchronization is established, we apply a perturbation to one individual patch; this patch desynchronizes and visits the aforementioned unstable chaotic set. The perturbed patch would typically come close to this set only for some time before returning to the synchronized state. Yet we found that the perturbed patch may approach the chaotic set very closely depending on the timing of the mentioned intervention, leading to long-lasting chaotic trajectories. As a result, the patch may stay in the chaotic set long enough to pull extra patches out of the synchronized state. After the dynamics of a critical number of patches reach the chaotic set, they mutually perturb each other; this makes escape from the set highly unlikely, causing the breakdown of synchronization in the ecological system. Figure 1 shows the time evolution of a rabbit population in the perturbed patch, with the time instants at which the applied perturbation can desynchronize the patch network appearing in red. The time intervals at which the system is safe under the same perturbation are in blue.

Ultimately, this finding allows one to manage the timing of interventions to increase the probability of success in ecosystem conservation strategies. For example, when synchrony threatens the ecosystem health due to higher extinction rates, the time intervals of vulnerability become “windows of opportunity” for interventions in order to suppress the synchronized behavior. In contrast, for cases in which synchrony is favorable, the ecosystem must be free from disturbances of any nature during the time intervals of vulnerability. Additionally, the reported methods are not restricted to ecosystems; the timing of perturbations may also be relevant for heart and brain interventions. We are preparing a manuscript with the reported ecological implications, and general results for a network composed of electronic circuits are available in [1].

Everton Medeiros presented this work during a minisymposium at the 2019 SIAM Conference on Applications of Dynamical Systems, which took place in May in Snowbird, Utah.

[1] Medeiros, E.S., Medrano-T, R.O., Caldas, I.L., Tél, T., & Feudel, U. (2019). Real-time vulnerability of synchronized states. Preprint, arxiv:1904.11420.

Everton S. Medeiros is postdoctoral fellow at the Institute of Physics at the University of São Paulo in Brazil.

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