| The theory of time scales has been developed in the1980s. More and moreresearchers have paid attention to studying time scales due to its tremendous poten-tial for applications in real life such as population dynamics, economics and physics.In recent years, researchers have obtained many useful results by researching thetheory of time scales, which promote the development of the theory on time scales.Since time scales unifies continuous mathematics and discrete mathematics, andextends the classical theory, the theoretical study of time scales has important appli-cation value. This paper extends function series theory in the classical mathematicalanalysis to time scales to further improve the theory of time scales and studies thestability of a class of neural networks and patch single-species model in the sense oftime scales.Firstly, this paper extends some related concepts of function series in the cla-ssical mathematical analysis to time scales, and then discusses the necessary and su-fficient conditions and several criteria which ensure that the function series is uni-formly convergent. Furthermore, the paper studies the fundamental problems of fun-ction series on general time scales and obtains several important analytical proper-ties of function series, namely rd-continuity, differentiability, integrability and so on.At the same time, we use some specific examples to explain the conclusions obtain-ed in this paper.Secondly, sufficient conditions to determine the stability of systems on specialtime scales are obtained by applying the extended Kirchhoff’s Matrix Tree Theoremand Lyapunov method. A simple criterion which ensures that the fixed point of theneural network satisfies the global asymptotical stable is established by demonstra-tion. Meanwhile, an example shows the effectiveness of the provided criterion is gi-ven. Regarding the patch single-species model, the stability condition of the positiveequilibrium point is given by applying Lyapunov method and graph theory. |
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