关键词：
*Composite farming paddy ecosystem;*Equilibrium;*Hopf bifurcation;*Maximum yield of rice;*Stability
摘要：
As one of the Globally Important Agricultural Heritage Systems (GIAHS), rice field composite farming is an ecological measure in rice production, which can reduce the amount of chemical fertilizers, pesticides and herbicides. This research studies the interaction among rice, weed, inorganic fertilizer and herbivore in a composite farming paddy ecosystem. We develop a differential equation model to analyze the relations and interactions among those components. Results show the existence of an equilibrium for paddy and weed extinction, one or two equilibria for rice extinction, an equilibrium for weed extinction, and an equilibrium for rice and weed coexistence. Based on the obtained stability conditions of these equilibria, measures are proposed to avoid the existence or the stability of equilibria for rice extinction. Other measures are proposed to lead to a stable equilibrium for weed extinction, which is the most desirable result in rice production. Conditions for maximizing the yield of rice are also obtained by taking the relative mortality of rice as variable. In addition, we discover the existence of Hopf bifurcation phenomenon in the system, and develop the critical value of Hopf bifurcation by taking the artificial fertilizer rate as the bifurcation parameter. Our findings provide effective guidance and insights for rice production in a composite farming paddy ecosystem.
摘要：
By introducing a delayed fractional-order differential equation model, we deal with the dynamics of the stability and Hopf bifurcation of a paddy ecosystem with three main components: rice, weeds, and inorganic fertilizer. In the system, there exists an equilibrium for rice and weeds extinction and an equilibrium for rice extinction or weeds extinction. We obtain sufficient conditions for the stability and Hopf bifurcation by analyzing their characteristic equation. Some numerical simulations validate our theoretical results.
摘要：
Abstract: The sufficient conditions are obtained that the f(x) is a periodic function with period 2T, 3T or 4T for the functional equation f(x+T)f(x) =af(x+T)+bf(x)+c. The results generalize the existing conclusions.#@#@#摘要: 本文针对函数方程f(x+T)f(x) =af(x+T)+bf(x)+c分别获得f(x)是周期为2T、3T、4T函数的充分条件，推广了已有结论。
关键词：
periodic solution;biological rhythms;hepatitis B virus infection;coincidence degree theory
摘要：
In this paper, a periodic model for hepatitis B virus infection is proposed. The model describes the breeding of the infected cells and the periodic variation of the environment. On the basis of the continuation theorem of coincidence degree theory, a condition for the existence of a positive periodic solution to the model is established. The result can be used to explain the wave phenomenon on the density of the pathogens in patients blood and the occurrence of superinfection in hepatitis B virus infections. Copyright (c) 2014 John Wiley & Sons, Ltd.
关键词：
This paper focuses on the multidirectional associative memory (MAM) neural networks with m fields which is more advanced to realize associative memory. Based on the Brouwer fixed point theorem and Dini upper right derivative;it is confirmed that the multidirectional associative memory neural network can have 3l equilibria and 2l equilibria of them are stable;where l is a parameter associated with the number of neurons. Furthermore;an example is given to illustrate the effectiveness of the results. Published: 2013 First available in Project Euclid: 14 March 2014 zbMATH: 06950768 MathSciNet: MR3138931 Digital Object Identifier: 10.1155/2013/592056
摘要：
This paper focuses on the multidirectional associative memory (MAM) neural networks with m fields which is more advanced to realize associative memory. Based on the Brouwer fixed point theorem and Dini upper right derivative, it is confirmed that the multidirectional associative memory neural network can have 3(l). equilibria and 2(l). equilibria of them are stable, where l is a parameter associated with the number of neurons. Furthermore, an example is given to illustrate the effectiveness of the results.
摘要：
The paper proposes several mathematical models of the multidirectional associative memory (MAM) neural network by analyzing its structure. A model of MAM with distributed delays is studied. Under some new assumptions on activation functions, $${2^{n_0[m/2]}}$$ invariant subsets of MAM are constructed. Then the existence and the exponential stability of $${2^{n_0[m/2]}}$$ periodic solutions located on invariant subsets are obtained by constructing a suitable Liapunov function and a Poincaré mapping. An estimating method of the exponential convergence rate is given. The obtained results are new to MAM neural networks. An example is given to illustrate the effectiveness of the results. The paper proposes several mathematical models of the multidirectional associative memory (MAM) neural network by analyzing its structure. A model of MAM with distributed delays is studied. Under some new assumptions on activation functions, $${2^{n_0[m/2]}}$$ invariant subsets of MAM are constructed. Then the existence and the exponential stability of $${2^{n_0[m/2]}}$$ periodic solutions located on invariant subsets are obtained by constructing a suitable Liapunov function and a Poincaré mapping. An estimating method of the exponential convergence rate is given. The obtained results are new to MAM neural networks. An example is given to illustrate the effectiveness of the results.
