Narrow Results

This paper is concerned with new generalizations and refinements of the diamond-$\alpha$ integral Cauchy–Schwartz inequality on time scales. The new generalization and refinement of the Cauchy–Schwartz inequality related to diamond integral on time scales are also considered. These obtained inequalities unify continuous inequalities and their corresponding discrete forms associated with Cauchy–Schwartz inequality.

We study the extreme long-time behavior of the metastable phase of the three-dimensional Ising model with Glauber dynamics in an applied magnetic field and at a temperature below the critical temperature. For these simulations, we use the advanced simulation method of projective dynamics. The algorithm is described in detail, together with its application to the escape from the metastable state. Our results for the field dependence of the metastable lifetime are in good agreement with theoretical expectations and span more than 50 decades in time.

An overview of advanced dynamical algorithms capable of spanning the widely disparate time scales that govern the decay of metastable phases in discrete spin models is presented. The algorithms discussed include constrained transfer-matrix, Monte Carlo with Absorbing Markov Chains (MCAMC), and projective dynamics (PD) methods. The strengths and weaknesses of each of these algorithms are discussed, with particular emphasis on identifying the parameter regimes (system size, temperature, and field) in which each algorithm works best.

We present a classical interspecific competition model. Individuals compete for a resource on a common patch and can go to a refuge. It is assumed that if species would remain on the competition patch, species 1 survives and species 2 would go extinct. Therefore, species 1 is Locally Superior Competitor (LSC) and species 2 Locally Inferior Competitor (LIC). We study the effects of density dependent dispersal from the competition patch to the refuge on the global outcome of competition. We study two cases. The first case considers LSC density dependent dispersal of the LIC trying to escape competition and going to its refuge when the LSC density is large. The second case considers aggressiveness of LIC leading to LIC density dependent dispersal of the LSC. We show that under some conditions, tactic 2 can allow the LIC to survive and even provoke global extinction of the LSC.

In this work we extend approximate aggregation methods in time discrete linear models to the case of time varying environments. Approximate aggregation consists of describing some features of the dynamics of a general system involving many coupled variables in terms of the dynamics of a reduced system with a few "global" variables.

We present a time varying discrete model in which we distinguish two processes with different time scales. By defining the global variables as appropriate linear combinations of the state variables, we transform the system into a reduced one. The variables corresponding to the original and reduced systems can be related, therefore allowing one the study of the former in terms of the latter. The property of weak ergodicity, which has to do with the capacity of a system to become asymptotically independent of initial conditions, is explored for the original and reduced systems.

The general method is also applied to aggregate a time-dependent multiregional model which appears in the field of population dynamics in two different cases: Fast migration with respect to demography and fast demography with respect to migration.

Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler "aggregated system" governed by a few "macrovariables". Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence of different time scales.

In this work we deal with the approximate aggregation of a model for a population subjected to demographic stochasticity and whose dynamics is controlled by two processes with different time scales. There are no restrictions on the slow process while the fast process is supposed to be conservative of the total number of individuals. The incorporation of the effects of demographic stochasticity in the dynamics of the population makes both the fast and the slow processes being modelled by two multi-type Galton–Watson branching processes. We present a multi-type global model that incorporates the combined effect of the fast and slow processes and develop a method that takes advantage of the difference of time scales to reduce the model obtaining an "aggregated" simpler system. We show that, given the separation of time scales between the two processes is high enough, we can obtain relevant information about the behavior of the multi-type global model through the study of this simple aggregated system.

This paper addresses the problem of the optimal size and number of marine reserves to achieve maximum value in commercial fisheries. A simplified network planning situation is analyzed to optimize the size and number of marine reserves. We consider a general 2L-patch model of harvesting population dynamics with continuous time. Fish movements between the sites, as well as vessel displacements between the fishing sites, are assumed to take place at a faster time scale than the variation of the stock and the change of the fleet size. We take advantage of these two time scales to derive a reduced model governing the dynamics of the total fish stock and the total fishing effort. This reduced model is used to determine the optimal size and number of a marine reserves in order to maximize the catch at equilibrium. We show that the optimal number that maximizes the total fish catch at equilibrium depend intricately on the size of the reserve: A small number of reserves is optimal when the size of the reserve is higher and inverse.

Ecological systems are large scale systems involving a large number of variables. By aggregation, it is possible to obtain at each level of organization an approximate and simple system of differential equations which can be studied more easily than the whole system involving a very large number of variables. Important differences in the orders of magnitude of the time scales in ecological systems allow us to apply methods of perturbation theory in order to replace large scale systems by reduced systems described by a few number of global variables. Perturbation theory determines the conditions required for aggregation. As examples, we present prey-predator models taking into account the activity sequences of animals at the individual level. In this way, the predation pressure depends on the individual strategies selected by the animals. We compare numerical simulations for the whole system and for the reduced system. We show that as soon as the orders of magnitude for the slow and fast dynamics are sufficiently different, then the orbits obtained for the reduced system are very close to the orbits obtained for the whole system.

The aim of this work is to show that at the population level, emerging properties may occur as a result of the coupling between the fast micro-dynamics and the slow macrodynamics. We studied a prey-predator system with different time scales in a heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to the growth and the interactions between the species. Preys go on the spatial patches on which some resources are located and can be caught by the predators on them. The efficiency of the predators to catch preys is patch-dependent. Preys can be more easily caught on some spatial patches than others. Perturbation theory is used in order to aggregate the initial system of ordinary differential equations for the patch sub-populations into a macro-system of two differential equations governing the total populations. Firstly, we study the case of a linear process of migration for which the aggregated system is formally identical to the slow part of the full system. Then, we study an example of a nonlinear process of migration. We show that under these conditions emerging properties appear at the population level.

The aim of this work is to extend approximate aggregation methods for multi-time scale systems of ordinary differential equations to time discrete models. We give general methods in order to reduce a large scale time discrete model into an aggregated model for a few number of slow macro-variables. We study the case of linear systems. We demonstrate that the elements defining the asymptotic behaviours of the initial and aggregate models are similar to first order. We apply this method to the case of an agestructured population with sub-populations in each age classes associated to different spacial patches or different individual activities. A fast time scale is assumed for patch or activity dynamics with respect to aging and reproduction processes. Our method allows us to aggregate the system into a classical Leslie model in which the fecundity and aging parameters of the aggregated model are expressed in terms of the equilibrium proportions of individuals in the different activities or patches.

The aim of this work is to extend approximate aggregation methods for multi-time scale systems of nonlinear ordinary differential equations to time discrete models. Approximate aggregation consist on describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete time models with two different time scales, the fast one considered linear and the slow one generally nonlinear. We transform the system to make the global variables appear, and use a version of center manifold theory to build up the aggregated system. Simple forms of the aggregated system are enough for the local study of the asymptotic behaviour of the general system provided that it has certain stability under perturbations. The general method is applied to aggregate a multiregional density dependent Leslie model into a density dependent Leslie model in which the demographic rates are expressed in terms of the equilibrium proportions of individuals in the different patches.

On the grounds of few perceived Gronwall inequalities, we inspect some new relevant nonlinear dynamic inequalities on time scales related to one independent variable. To guarantee the evenness of information division and with the conversation of peculiar cases, it is indicated that these inequalities catch dynamic, delay, Volterra–Fredholm, discrete, fractional, etc. The inequalities suggested right here to other specific bounds on unknown parameters may be handled as powerful equipment for determining the features of various dynamic equations on time scales. It is trusted that this research plan will open new possibilities in in-depth examination of time scale scheme structure of varying nature.

The neutral delay model has many applications in biology, ecology, man-made neural networks, mechanics and engineering. The main purpose of this paper is to establish new oscillation criteria for the two-dimensional neutral delay dynamical system

This study defines a Hadamard fractional sum by use of the time-scale theory. Then a $h$-fractional difference is given and fundamental theorems are proved. Initial value problems of fractional difference equations are presented and their equivalent fractional sum equations are provided. The discrete Mittag-Leffler function solutions of linear fractional difference equations are obtained. It can be concluded that the new discrete fractional calculus of Hadamard type is well defined.

This paper is devoted to studying some new reverse Hölder-type inequalities associated with the dynamic integral called diamond on time scale, which is expressed as an ‘approximate’ symmetric integral on time scales. Some related inequalities are also presented. The obtained inequalities extend known results related with discrete and continuous forms.

The lifetime of a system is measured in two principal time scales, L and H. We consider a family of time scales of type T_a=(1−a) · L+a · H, for which the lifetime is T=(1−a) L+aH, a∈[0, 1], where L and H are the lifetimes in the principal time scales. The optimal time scale, by the definition, provides the minimal value of the coefficient of variation of system lifetime. We consider the age replacement model in time scale T_a and develop a nonparametric numerical procedure for finding the optimal weighting parameter a* that provides the smallest value of the corresponding cost function. This procedure is applied to fatigue test data, and it is demonstrated that the best time scale with respect to the age replacement cost function is very close to the optimal time scale. The same is true for a modified age replacement scheme in which the replacement age equals the p-quantile of system lifetime.

Supremum and L₂-norms estimates of the solutions of nonlinear vector Volterra–Stieltjes equations in the complex Euclidean space are derived using recently established estimates for the norm of vector-valued functions and the resolvents of quasi–nilpotent operators. Such equations include both continuous time Volterra integral equations and discrete time Volterra difference equations as well as mixtures of these two basic types of time scales. The estimates are formulated in terms of the coefficients of the equations, which are separated into a linear part and a nonlinear part satisfying a local Lipschitz condition. Stability conditions are then derived from these estimates.

In this paper, by using the continuation theorem of coincidence degree theory we study the existence of periodic solution for a two-species ratio-dependent predator-prey system with time-varying delays and Machaelis–Menten type functional response on time scales. Some new results are obtained.

In the light of existing physical theories, it is shown that representation in terms of functional interactions and formalism (S-Propagators) should satisfy three physical and six biological constraints. Consequences are summarized for neurohormonal field, developmental phase, aging phase, functional hierarchy, Principle of Auto-Associative stability (PAAS), self-organization and neural selection, Darwinian evolution, and the intelligence of movement. Abstraction and complexity of the proposed theories are discussed relatively to their advantages for integrative neuroscience.

Variable contributions of state and trait to the electroencephalographic (EEG) signal affect the stability over time of EEG measures, quite apart from other experimental uncertainties. The extent of intraindividual and interindividual variability is an important factor in determining the statistical, and hence possibly clinical significance of observed differences in the EEG. This study investigates the changes in classical quantitative EEG (qEEG) measures, as well as of parameters obtained by fitting frequency spectra to an existing continuum model of brain electrical activity. These parameters may have extra variability due to model selection and fitting. Besides estimating the levels of intraindividual and interindividual variability, we determined approximate time scales for change in qEEG measures and model parameters. This provides an estimate of the recording length needed to capture a given percentage of the total intraindividual variability. Also, if more precise time scales can be obtained in future, these may aid the characterization of physiological processes underlying various EEG measures. Heterogeneity of the subject group was constrained by testing only healthy males in a narrow age range (mean = 22.3 years, sd = 2.7). Eyes-closed EEGs of 32 subjects were recorded at weekly intervals over an approximately six-week period, of which 13 subjects were followed for a year. QEEG measures, computed from Cz spectra, were powers in five frequency bands, alpha peak frequency, and spectral entropy. Of these, theta, alpha, and beta band powers were most reproducible. Of the nine model parameters obtained by fitting model predictions to experiment, the most reproducible ones quantified the total power and the time delay between cortex and thalamus. About 95% of the maximum change in spectral parameters was reached within minutes of recording time, implying that repeat recordings are not necessary to capture the bulk of the variability in EEG spectra.