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We consider a nonlocal delayed reaction–diffusion equation in an unbounded domain that includes some special cases arising from population dynamics. Due to the non-compactness of the spatial domain, the solution semiflow is not compact. We first show that, with respect to the compact open topology for the natural phase space, the solutions induce a compact and continuous semiflow ${\Phi}$ on a bounded and positively invariant set Y in C +?=?C([?1, 0], X +) that attracts every solution of the equation, where X + is the set of all bounded and uniformly continuous functions from ${\mathbb{R}}$ to [0, ∞). Then, to overcome the difficulty in describing the global dynamics, we establish a priori estimate for nontrivial solutions after describing the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. The estimate enables us to show the permanence of the equation with respect to the compact open topology. With the help of the permanence, we can employ standard dynamical system theoretical arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated with the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.  相似文献
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In this paper, we study the global dynamics of a class of differential equations with temporal delay and spatial non-locality in an unbounded domain. Adopting the compact open topology, we describe the delicate asymptotic properties of the nonlocal delayed effect and establish some a priori estimate for nontrivial solutions which enables us to show the permanence of the equation. Combining these results with a dynamical systems approach, we determine the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker?s birth function and Mackey-Glass?s hematopoiesis function, we obtain threshold results for the global dynamics of these two models. We explain why our results on the global attractivity of the positive equilibrium in C+?{0} under the compact open topology becomes invalid in C+?{0} with respect to the usual supremum norm, and we identify a subset of C+?{0} in which the positive equilibrium remains attractive with respect to the supremum norm.  相似文献
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In this paper, essentially strongly order-preserving and conditionally set-condensing semiflows are considered. Obtained is a new type of generic quasi-convergence principles implying the existence of an open and dense set of stable quasi-convergent points when the state space is order bounded. The generic quasi-convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations, giving some results of theoretical and practical significance.  相似文献
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In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson's equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities.  相似文献
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In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein-Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known.  相似文献
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In this paper, we introduce a class of pseudo-monotone maps on ordered topological spaces. By exploiting monotonicity methods and the invariance of the omega limit set, we establish a convergence principle for discrete-time semiflows generated by the maps introduced. The convergence principle is then applied to a class of periodic neutral delay differential equations, which leads to some novel and sharper results.  相似文献
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This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction-diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption.  相似文献
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This paper formulates and studies a model of planar systems. The model can well describe many practical architectures of delayed neural networks, which is generalization of some existing neural networks under a time-varying environment. Without assuming the smoothness, monotonicity and boundedness of the activation functions, the existence and global exponential stability of its periodic solutions are investigated. Some explicit and conclusive results are established. Our approach is based on the continuation theorem of the coincidence degree, a priori estimates, and differential inequalities.  相似文献
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