An official website of the United States government
We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way.
more »
« less
Ecological models: higher complexity in, higher feasibility out
Finding a compromise between tractability and realism has always been at the core of ecological modelling. The introduction of nonlinear functional responses in two-species models has reconciled part of this compromise. However, it remains unclear whether this compromise can be extended to multispecies models. Yet, answering this question is necessary in order to differentiate whether the explanatory power of a model comes from the general form of its polynomial or from a more realistic description of multispecies systems. Here, we study the probability of feasibility (the existence of at least one positive real equilibrium) in complex models by adding higher-order interactions and nonlinear functional responses to the linear Lotka–Volterra model. We characterize complexity by the number of free-equilibrium points generated by a model, which is a function of the polynomial degree and system’s dimension. We show that the probability of generating a feasible system in a model is an increasing function of its complexity, regardless of the specific mechanism invoked. Furthermore, we find that the probability of feasibility in a model will exceed that of the linear Lotka–Volterra model when a minimum level of complexity is reached. Importantly, this minimum level is modulated by parameter restrictions, but can always be exceeded via increasing the polynomial degree or system’s dimension. Our results reveal that conclusions regarding the relevance of mechanisms embedded in complex models must be evaluated in relation to the expected explanatory power of their polynomial forms.
more »
« less
- Award ID(s):
- 2024349
- PAR ID:
- 10216661
- Date Published:
- Journal Name:
- Journal of The Royal Society Interface
- Volume:
- 17
- Issue:
- 172
- ISSN:
- 1742-5689
- Page Range / eLocation ID:
- 20200607
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The dynamics of ecological communities in nature are typically characterized by probabilistic processes involving invasion dynamics. Because of technical challenges, however, the majority of theoretical and experimental studies have focused on coexistence dynamics. Therefore, it has become central to understand the extent to which coexistence outcomes can be used to predict analogous invasion outcomes relevant to systems in nature. Here, we study the limits to this predictability under a geometric and probabilistic Lotka– Volterra framework. We show that while individual survival probability in coexistence dynamics can be fairly closely translated into invader colonization probability in invasion dynamics, the translation is less precise between community persistence and community augmentation, and worse between exclusion probability and replacement probability. These results provide a guiding and testable theoretical framework regarding the translatability of outcomes between coexistence and invasion outcomes when communities are represented by Lotka–Volterra dynamics under environmental uncertainty.more » « less
-
A hybrid control architecture for nonlinear dynamical systems is described which combines the advantages of model based control with those of real-time learning. The idea is to generate input-output data from an error system involving the plant and a proposed model. A discretized Chen-Fliess functional series is then identified from this data and used in conjunction with the model for predictive control. This method builds on the authors’ previous work on model-free control of a single-input, single-output Lotka-Volterra system.The problem is revisited here, but now with the introduction of a model for the dynamics. The single-input, multiple-output version of the problem is also investigated as a way to enhance closed-loop performancemore » « less
-
null (Ed.)Artificial neural networks have traditionally been used to implement machine learning algorithms. There are, however, alternatives to these biologically inspired machine learning architectures that offer potentially lower complexity and stronger theoretical underpinnings. One such option in the context of control is based on using a generic input-output model known as a Chen-Fliess functional series. The main goal of the paper is to describe a specific architecture that can be used in the multivariable setting to combine both learning and model based control. It builds on recent work by the authors showing that a certain monoid structure underlies any recursive implementation of such a system. The method is demonstrated using a two-input, two-output Lotka-Volterra system.more » « less
-
Abstract The global dynamics of the two-species Lotka–Volterra competition patch model with asymmetric dispersal is classified under the assumptions that the competition is weak and the weighted digraph of the connection matrix is strongly connected and cycle-balanced. We show that in the long time, either the competition exclusion holds that one species becomes extinct, or the two species reach a coexistence equilibrium, and the outcome of the competition is determined by the strength of the inter-specific competition and the dispersal rates. Our main techniques in the proofs follow the theory of monotone dynamical systems and a graph-theoretic approach based on the tree-cycle identity.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rsif.2020.0607
