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Noise is ubiquitous in nature and usually results in rich dynamics in stochastic systems such as oscillatory systems, which exist in such various fields as physics, biology and complex networks. The correlation and synchronization of two or many oscillators are widely studied topics in recent years.
In this thesis, we mainly investigate two problems, i.e., the stochastic bursting phenomenon in noisy excitable systems and synchronization in a three-dimensional Kuramoto model with noise. Stochastic bursting here refers to a sequence of coherent spike train, where each spike has random number of followers due to the combined effects of both time delay and noise. Synchronization, as a universal phenomenon in nonlinear dynamical systems, is well illustrated in the Kuramoto model, a prominent model in the description of collective motion.
In the first part of this thesis, an idealized point process, valid if the characteristic timescales in the problem are well separated, is used to describe statistical properties such as the power spectral density and the interspike interval distribution. We show how the main parameters of the point process, the spontaneous excitation rate, and the probability to induce a spike during the delay action can be calculated from the solutions of a stationary and a forced Fokker-Planck equation. We extend it to the delay-coupled case and derive analytically the statistics of the spikes in each neuron, the pairwise correlations between any two neurons, and the spectrum of the total output from the network.
In the second part, we investigate the three-dimensional noisy Kuramoto model, which can be used to describe the synchronization in a swarming model with helical trajectory. In the case without natural frequency, the Kuramoto model can be connected with the Vicsek model, which is widely studied in collective motion and swarming of active matter. We analyze the linear stability of the incoherent state and derive the critical coupling strength above which the incoherent state loses stability. In the limit of no natural frequency, an exact self-consistent equation of the mean field is derived and extended straightforward to any high-dimensional case.
Partial synchronous states exist in systems of coupled oscillators between full synchrony and asynchrony. They are an important research topic because of their variety of different dynamical states. Frequently, they are studied using phase dynamics. This is a caveat, as phase dynamics are generally obtained in the weak coupling limit of a first-order approximation in the coupling strength. The generalization to higher orders in the coupling strength is an open problem. Of particular interest in the research of partial synchrony are systems containing both attractive and repulsive coupling between the units. Such a mix of coupling yields very specific dynamical states that may help understand the transition between full synchrony and asynchrony. This thesis investigates partial synchronous states in mixed-coupling systems. First, a method for higher-order phase reduction is introduced to observe interactions beyond the pairwise one in the first-order phase description, hoping that these may apply to mixed-coupling systems. This new method for coupled systems with known phase dynamics of the units gives correct results but, like most comparable methods, is computationally expensive. It is applied to three Stuart-Landau oscillators coupled in a line with a uniform coupling strength. A numerical method is derived to verify the analytical results. These results are interesting but give importance to simpler phase models that still exhibit exotic states. Such simple models that are rarely considered are Kuramoto oscillators with attractive and repulsive interactions. Depending on how the units are coupled and the frequency difference between the units, it is possible to achieve many different states. Rich synchronization dynamics, such as a Bellerophon state, are observed when considering a Kuramoto model with attractive interaction in two subpopulations (groups) and repulsive interactions between groups. In two groups, one attractive and one repulsive, of identical oscillators with a frequency difference, an interesting solitary state appears directly between full and partial synchrony. This system can be described very well analytically.
While habitat loss is a known key driver of biodiversity decline, the impact of other landscape properties, such as patch isolation, is far less clear. When patch isolation is low, species may benefit from a broader range of foraging opportunities, but are at the same time adversely affected by higher predation pressure from mobile predators. Although previous approaches have successfully linked such effects to biodiversity, their impact on local and metapopulation dynamics has largely been ignored. Since population dynamics may also be affected by environmental disturbances that temporally change the degree of patch isolation, such as periodic changes in habitat availability, accurate assessment of its link with isolation is highly challenging. To analyze the effect of patch isolation on the population dynamics on different spatial scales, we simulate a three-species meta-food chain on complex networks of habitat patches and assess the average variability of local populations and metapopulations, as well as the level of synchronization among patches. To evaluate the impact of periodic environmental disturbances, we contrast simulations of static landscapes with simulations of dynamic landscapes in which 30 percent of the patches periodically become unavailable as habitat. We find that increasing mean patch isolation often leads to more asynchronous population dynamics, depending on the parameterization of the food chain. However, local population variability also increases due to indirect effects of increased dispersal mortality at high mean patch isolation, consequently destabilizing metapopulation dynamics and increasing extinction risk. In dynamic landscapes, periodic changes of patch availability on a timescale much slower than ecological interactions often fully synchronize the dynamics. Further, these changes not only increase the variability of local populations and metapopulations, but also mostly overrule the effects of mean patch isolation. This may explain the often small and inconclusive impact of mean patch isolation in natural ecosystems.