In this paper, a five-dimensional energy demand-supply system has been considered. On the one hand, we analyze the stability for all of the equilibrium points of the system. For each of equilibrium point, by analyzing the characteristic equation, we show the conditions for the stability or instability using Routh-Hurwitz criterion. Then numerical simulations have been given to illustrate all of cases for the theoretical results. On the other hand, by introducing the phenomenon of time delay, we establish the five-dimensional energy demand-supply model with time delay. Then we analyze the stability of the equilibrium points for the delayed system by the stability switching theory. Especially, Hopf bifurcation has been considered by showing the explicit formulae using the central manifold theorem and Poincare normalization method. For each cases of the theorems including the Hopf bifurcation, numerical simulations have been given to illustrate the effectiveness of the main results.
1. The Heteroptera, principally mirids, collected in a light-trap run on a field margin at Rothamsted Experimental Station for various periods between 1933 and 2000, have been identified, and the catches analysed to show the extent of change and stability in the community.
2. Trap catch, both in terms of individuals and species, was correlated with maximum daily temperature.
3. α-diversity showed a U-shaped curve over the period. The dip may have been associated with pesticide use, although a lack of days with high maximum temperatures cannot be ruled out.
4. By the late 1990s, α-diversity had again reached a peak (Fisher's = 11), comparable to that in the 1930s.
5. However, the change in the composition of the community over the whole period (β-diversity) was significant, the index of difference being 0.66 on a scale where 0 is no change in composition or relative abundance and 1 no species in common.
6. The value of β-diversity was highest in the water bugs, which disappeared altogether. Categorising the others by host plant type, the greatest change over time was in those associated with perennial herbs. There were decreasing differences in tree-dwellers and grassland species respectively, and the least change was in the community associated with annual plants (arable weeds).
7. Changes in the abundance of Heteroptera since 1933 follow closely those of the macrolepidoptera from the same samples. However changes in diversity show very different patterns, with moth diversity continuing to decline since 1960 in contrast to the increases apparent from the Heteroptera data.
A one-dimensional version of a gradient system, known as ''Kobayashi-Warren-Carter system'', is considered. In view of the difficulty of the uniqueness, we here set our goal to ensure a ''stability'' which comes out in the approximation approaches to the solutions. Based on this, the Main Theorem concludes that there is an admissible range of approximation differences, and in the scope of this range, any approximation method leads to a uniform type of solutions having a certain common features. Further, this is specified by using the notion of ''energy-dissipative solution'', proposed in a relevant previous work.
In this paper we use the fixed point method to prove asymptotic stability results of the zero solution of a generalized linear neutral difference equation with variable delays. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Y. N. Raffoul (2006), E. Yankson (2009), M. Islam and E. Yankson (2005).
In the paper we study the subject of stability of systems with h-differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with n fractional orders. The equivalent descriptions of fractional h-difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with n-orders.
Optimization problems with stochastic dominance constraints are helpful to many real-life applications. We can recall e. g., problems of portfolio selection or problems connected with energy production. The above mentioned constraints are very suitable because they guarantee a solution fulfilling partial order between utility functions in a given subsystem U of the utility functions. Especially, considering U:=U1 (where U1 is a system of non decreasing concave nonnegative utility functions) we obtain second order stochastic dominance constraints. Unfortunately it is also well known that these problems are rather complicated from the theoretical and the numerical point of view. Moreover, these problems goes to semi-infinite optimization problems for which Slater's condition is not necessary fulfilled. Consequently it is suitable to modify the constraints. A question arises how to do it. The aim of the paper is to suggest one of the possibilities how to modify the original problem with an "estimation" of a gap between the original and a modified problem. To this end the stability results obtained on the base of the Wasserstein metric corresponding to L1 norm are employed. Moreover, we mention a scenario generation and an investigation of empirical estimates. At the end attention will be paid to heavy tailed distributions.
In this paper, we propose a new impulsive predator prey model with impulsive control at different fixed moments and analyze its interesting dynamic behaviors. Sufficient conditions for the globally asymptotical stability of the semi-trivial periodic solution and the permanence of the present model are obtained by Floquet theory of impulsive differential equation and small amplitude perturbation skills. Existences of the "infection-free" periodic solution and the "predator-free" solution are analyzed by bifurcation theory of impulsive differential equation. Finally, the analytical results presented in the work are validated by numerical simulation figures for this proposed model.
We present the Rothe method for the McKendrick-von Foerster equation with initial and boundary conditions. This method is well known as an abstract Euler scheme in extensive literature, e.g. K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. Various Banach spaces are exploited, the most popular being the space of bounded and continuous functions. We prove the boundedness of approximate solutions and stability of the Rothe method in $L^\infty $ and $L^1$ norms. Proofs of these results are based on comparison inequalities. Our theory is illustrated by numerical experiments. Our research is motivated by certain models of mathematical biology.
The paper deals with a difference equation arising from the scalar pantograph equation via the backward Euler discretization. A case when the solution tends to zero but after reaching a certain index it loses this tendency is discussed. We analyse this problem and estimate the value of such an index. Furthermore, we show that the utilized proof technique enables us to investigate some other numerical formulae, too.
Optimization problems depending on a probability measure correspond to many applications. These problems can be static (single-stage), dynamic with finite (multi-stage) or infinite horizon, single- or multi-objective. It is necessary to have complete knowledge of the "underlying" probability measure if we are to solve the above-mentioned problems with precision. However this assumption is very rarely fulfilled (in applications) and consequently, problems have to be solved mostly on the basis of data. Stochastic estimates of an optimal value and an optimal solution can only be obtained using this approach. Properties of these estimates have been investigated many times. In this paper we intend to study one-stage problems under unusual (corresponding to reality, however) assumptions. In particular, we try to compare the achieved results under the assumptions of thin and heavy tails in the case of problems with linear and nonlinear dependence on the probability measure, problems with probability and risk measure constraints, and the case of stochastic dominance constraints. Heavy-tailed distributions quite often appear in financial problems \cite {Meer 2003)} while nonlinear dependence frequently appears in problems with risk measures \cite {Kan (2012a),Pflu (2007)}. The results we introduce follow mostly from the stability results based on the Wasserstein metric with the "underlying" L1 norm. Theoretical results are completed by a simulation investigation.