Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks. DOI: 10.1103/PhysRevLett.111.024101 PACS numbers: 05.45.Xt, 05.65.+b, 05.90.+m, 89.75.Fb Dynamical systems are generally seen as being either deterministic or stochastic. The advent of dynamical chaos several decades ago attracted much attention and illustrated that even complex dynamics can be deterministic. Dynamical systems can also be classified as autonomous (self-contained) or nonautonomous (subject to external influences). In reality, nonautonomous systems are the commoner, but they are far harder to treat. Until now they were mostly treated as stochastic or, alternatively, attempts were made to reformulate them as autonomous. Neither approach captures the characteristic properties of these systems.In this Letter we propose a new class of nonautonomous systems and name them chronotaxic to characterize oscillatory systems with time-varying, but stable, amplitudes and frequencies. Nonautonomous oscillatory systems appear in various fields of research including neuroscience [1,2], cardiovascular dynamics [3], climate [4,5] and evolutionary science [6], as well as in complex systems and networks [7][8][9]. Although we are witnessing a rapid development of the theory of nonautonomous [10,11] and random dynamical systems [12,13], nonautonomous oscillatory systems with stable but time-varying characteristic frequencies have to date not been addressed. When treated in an inverse approach such systems are usually considered as stochastic. In an attempt to cope with the problem, several methods for the inverse approach were introduced, including wavelet-based decomposition [14] Common to all these systems is that they are oscillatory, have stable amplitudes, and frequencies that are resistant to external perturbations. The variety of systems with these characteristics suggests that their dynamics are generated from a universal basis. To date, the description of stable oscillatory dynamics has been based on the model of autonomous self-sustained limit cycle oscillators [23]. While this model provides stable amplitude dynamics, frequencies of oscillations within this model can be easily changed even by weakest external perturbations.The new class of nonautonomous oscillatory dynamical systems that we now propose account for such dynamics. The novelty of these systems is that not only are the amplitude dynamics stable but also the frequencies of the oscillations are time dependent and stable-i.e., their timedependent values cannot be easily altered by external perturbations. Their c...

Individual-based models, ‘IBMs’, describe naturally the dynamics of interacting organisms or social or financial agents. They are considered too complex for mathematical analysis, but computer simulations of them cannot give the general insights required. Here, we resolve this problem with a general mathematical framework for IBMs containing interactions of an unlimited level of complexity, and derive equations that reliably approximate the effects of space and stochasticity. We provide software, specified in an accessible and intuitive graphical way, so any researcher can obtain analytical and simulation results for any particular IBM without algebraic manipulation. We illustrate the framework with examples from movement ecology, conservation biology, and evolutionary ecology. This framework will provide unprecedented insights into a hitherto intractable panoply of complex models across many scientific fields.

Following the development of a new class of self-sustained oscillators with a time-varying but stable frequency, the inverse approach to these systems is now formulated. We show how observed data arranged in a single-variable time series can be used to recognize such systems. This approach makes use of time-frequency domain information using the wavelet transform as well as the recently developed method of Bayesian-based inference. In addition, a set of methods, named phase fluctuation analysis, is introduced to detect the defining properties of the new class of systems by directly analyzing the statistics of the observed perturbations.We apply these methods to numerical examples but also elaborate further on the cardiac system.

The competitive exclusion principle is one of the most influential concepts in ecology. The classical formulation suggests a correlation between competitor species similarity and competition severity, leading to rapid competitive exclusion where species are very similar; yet neutral models show that identical species can persist in competition for long periods. Here, we resolve the conflict by examining two components of similarity -niche overlap and competitive similarity -and modeling the effects of each on exclusion rate (defined as the inverse of time to exclusion). Studying exclusion rate, rather than the traditional focus on binary outcomes (coexistence versus exclusion), allows us to examine classical niche and neutral perspectives using the same currency. High niche overlap speeds exclusion, but high similarity in competitive ability slows it. These predictions are confirmed by a well-known model of two species competing for two resources. Under ecologically plausible scenarios of correlation between these two factors, the strongest exclusion rates may be among moderately similar species, while very similar and highly dissimilar competitors have very low exclusion rates. Adding even small amounts of demographic stochasticity to the model blurs the line between deterministic and probabilistic coexistence still further. Thus, focusing on exclusion rate, instead of on the binary outcome of coexistence versus exclusion, allows a variety of outcomes to result from competitive interactions. This approach may help explain species coexistence in diverse competitive communities and raises novel issues for future work.