Oscillations in I/O monotone systems under negative feedback

Oscillatory behavior is a key property of many biological systems. The Small-Gain Theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions. One advantage of the use of the monotone SGT technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low-dimensional (in many interesting examples, just one-dimensional) model reduction. This robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters. However, verifying the conditions needed in order to apply the SGT is not always easy. This paper provides an approach to the verification of the needed properties, and illustrates the approach through an application to a classical model of circadian oscillations, as a nontrivial ``case study,'' and also provides a theorem in the converse direction of predicting oscillations when the SGT conditions fail.Comment: Related work can be retrieved from second author's websit

Asymptotic behaviour for a class of non-monotone delay differential systems with applications

The paper concerns a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.Comment: 26 pages, J Dyn Diff Equat (2017

Dichotomy results for delay differential equations with negative Schwarzian

We gain further insight into the use of the Schwarzian derivative to obtain new results for a family of functional differential equations including the famous Wright's equation and the Mackey-Glass type delay differential equations. We present some dichotomy results, which allow us to get easily computable bounds of the global attractor. We also discuss related conjectures, and formulate new open problems.Comment: 16 pages, submitted to Chaos,Solitons,Fractal

Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays

Copyright [2006] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this letter, the global asymptotic stability analysis problem is considered for a class of stochastic Cohen-Grossberg neural networks with mixed time delays, which consist of both the discrete and distributed time delays. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, a linear matrix inequality (LMI) approach is developed to derive several sufficient conditions guaranteeing the global asymptotic convergence of the equilibrium point in the mean square. It is shown that the addressed stochastic Cohen-Grossberg neural networks with mixed delays are globally asymptotically stable in the mean square if two LMIs are feasible, where the feasibility of LMIs can be readily checked by the Matlab LMI toolbox. It is also pointed out that the main results comprise some existing results as special cases. A numerical example is given to demonstrate the usefulness of the proposed global stability criteria

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

State estimation for delayed neural networks

Copyright [2005] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this letter, the state estimation problem is studied for neural networks with time-varying delays. The interconnection matrix and the activation functions are assumed to be norm-bounded. The problem addressed is to estimate the neuron states, through available output measurements, such that for all admissible time-delays, the dynamics of the estimation error is globally exponentially stable. An effective linear matrix inequality approach is developed to solve the neuron state estimation problem. In particular, we derive the conditions for the existence of the desired estimators for the delayed neural networks. We also parameterize the explicit expression of the set of desired estimators in terms of linear matrix inequalities (LMIs). Finally, it is shown that the main results can be easily extended to cope with the traditional stability analysis problem for delayed neural networks. Numerical examples are included to illustrate the applicability of the proposed design method