Wavefronts of a Nonlocal State-dependent Time Delay Model

This paper is concerned with the travelling wavefronts of a nonlocal dispersal cooperation model with harvesting and state-dependent delay,which is assumed to be an increasing function of the population density with lower and upper bound.Especially,state-dependent delay is introduced into a nonlocal reaction-diffusion model.The conditions of Schauder's fixed point theorem are proved by constructing a reasonable set of functionsΓ(see Section 2)and a pair of upper-lower solutions,so the existence of traveling wavefronts is established.The present study is continuation of a previous work that highlights the Laplacian diffusion.     

Partial differential equations with discrete and distributed state-dependent delays

This work is an attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with state-dependent delays). We study local existence and long-time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor.     

Minimum wave speed for a diffusive competition model with time delay

In this paper we study mono-stable traveling wave solutions for a Lotka-Volterra reaction-diffusion competition model with time delay. By constructing upper and lower solutions, we obtain the precise minimum wave speed of traveling waves under certain conditions. Our results also extend the known results on the minimum wave speed for Lotka-Volterra competition model without delay.     

Minimal-speed selection of traveling waves to the Lotka–Volterra competition model

In this paper the minimal-speed determinacy of traveling wave fronts of a two-species competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upper-lower solution technique on the cooperative system to study the traveling waves as well as its minimal-speed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.     

Traveling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays

This paper is concerned with a diffusive and cooperative Lotka–Volterra model with distributed delays and nonlocal spatial effect. By using an iterative technique recently developed by Wang, Li and Ruan (Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006), 185–232), sufficient conditions are established for the existence of traveling wave front solutions connecting the zero and the positive equilibria by choosing different kernels. The result is an extension of an existing result for Fisher-KPP equation with nonlocal delay and is somewhat parallel to the existing result for diffusive and cooperative Lotka–Volterra system with discrete delays.     

时滞依赖状态的非自治多值偏积分微分方程

利用预解算子理论结合Leray-Schauder型多值映射不动点定理,在公理化定义的相空间上,得到了一类时滞依赖状态的非自治多值一阶偏积分微分方程适度解的存在性.     

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial‐homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non‐existence of traveling waves is obtained by two‐sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.     

Traveling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays

This paper is concerned with a diffusive and cooperative Lotka–Volterra model with distributed delays and nonlocal spatial effect. By using an iterative technique recently developed by Wang, Li and Ruan (Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006), 185–232), sufficient conditions are established for the existence of traveling wave front solutions connecting the zero and the positive equilibria by choosing different kernels. The result is an extension of an existing result for Fisher-KPP equation with nonlocal delay and is somewhat parallel to the existing result for diffusive and cooperative Lotka–Volterra system with discrete delays. Supported by the NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.     

Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.     

三物种竞争-扩散系统双稳行波解的波速符号

在双稳竞争-扩散模型中,由于行波解的波速符号可以预测哪些物种更具有优势并最终占据整个栖息地,因此研究行波解的波速符号具有重要的生物学意义.首先将三物种种群Lotka-Volterra竞争-扩散系统转化为合作系统.然后运用比较原理得到双稳波速与波廓方程特定上下解波速的比较原理.最后根据比较原理以及构造合适的上下解,得到一些判断双稳行波解波速符号的充分条件.这些结果能够更好地预测和控制生物种群的竞争结果.     

Approximate Controllability of Fractional Differential Equations with State-Dependent Delay

The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays. The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay. The concept of controllability plays an important role in physics and mathematics. In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays. Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators. A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus. In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable. Also, an example is presented to illustrate the applicability of the obtained theory.     

Spreading Speeds and Traveling Waves for Non-cooperative Reaction�CDiffusion Systems

Much has been studied on the spreading speed and traveling wave solutions for cooperative reaction–diffusion systems. In this paper, we shall establish the spreading speed for a large class of non-cooperative reaction–diffusion systems and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. Our results are applied to a partially cooperative system describing interactions between ungulates and grass.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

EXISTENCE RESULTS FOR IMPULSIVE NEUTRAL EVOLUTION DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

This paper is mainly concerned with the existence of mild solutions to a first order impulsive neutral evolution differential equations with state-dependent delay. By suitable fixed point theorems combined with theories of evolution systems,we prove some existence theorems. As an application,an example is also given to illustrate the obtained results.     

Global Hopf bifurcation of differential equations with threshold type state-dependent delay

We develop global Hopf bifurcation theory of differential equations with state-dependent delay using the S¹$S^1$-equivariant degree and investigate a two-degree-of-freedom mechanical model of turning processes. For the model of turning processes we show that the extreme points of each vibration component of the non-constant periodic solutions can be embedded into a manifold with explicit algebraic expression. This observation enables us to establish analytical upper and lower bounds of the amplitudes of the periodic solutions in terms of the system parameters and to exclude certain periods. Using the achieved global bifurcation theory we reveal that if the relative frequency between the natural frequency and the turning frequency varies in a certain interval, then generically every bifurcated continuum of periodic solutions must terminate at a bifurcation point. This termination means that the underlying system with parameters in the stability region near the vertical asymptotes of the stability lobes is less subject to chatter. In the process, several sufficient conditions for the non-existence of non-constant periodic solutions are also obtained.     

一类具状态依赖时滞的微分系统解的渐近行为

研究了一类具有状态时滞的微分方程系统解的渐近行为,获得了该系统每一个有界解当t→∞时都趋于常向量,所获得的结果改进和扩展了已有文献的相关结果.     

Periodic solutions in periodic state-dependent delay equations and population models

Sufficient and realistic conditions are obtained for the existence of positive periodic solutions in periodic equations with state-dependent delay. The method involves the application of the coincidence degree theorem and estimations of uniform upper bounds on solutions. Applications of these results to some population models are presented. These application results indicate that seasonal effects on population models often lead to synchronous solutions. In addition, we may conclude that when both seasonality and time delay are present and deserve consideration, the seasonality is often the generating force for the often observed oscillatory behavior in population densities.

     

Stability and travelling waves for a time-delayed population system with stage structure

We study a time-delayed population system with stage structure for the interaction between two species, the adult members of which are in competition. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. The global stability results are established for each equilibrium. The criteria for global convergence to each equilibrium are sharp and involve these delays. By using lower and upper travelling wave solutions, we show that the model has travelling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.     

Propagation of second order integrodifference equations with local monotonicity

This paper is concerned with the spreading speeds and traveling wavefronts of second order integrodifference equations with local monotonicity. By introducing two auxiliary integrodifference equations, the spreading speed and traveling wave solutions are studied. In particular, we obtain the nonexistence of monotone traveling wave solutions for an example if it is local monotone. These results are applied to a model which is obtained by introducing the spatial variable to a difference equation used by the International Whaling Commission.     

Existence results for an impulsive neutral functional differential equation with state-dependent delay

In this article, we study the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay. The results are obtained by using Leray–Schauder Alternative fixed point theorem.