具有非局部反应的时滞扩散Nicholson方程的行波解

对具有扩散项的时滞Nicholson方程的行波解进行了研究．特别是考虑到生物个体在空间位置上的迁移,研究了具有非局部反应的时滞扩散模型．对于弱生成时滞核,运用几何奇异摄动理论,在时滞充分小的情况下,证明了行波解的存在性．     

Asymptotic behaviour of travelling waves for the delayed Fisher–KPP equation

In this work we study the behaviour of travelling wave solutions for the diffusive Hutchinson equation with time delay. Using a phase plane analysis we prove the existence of travelling wave solution for each wave speed c?2$c?2$. We show that for each given and admissible wave speed, such travelling wave solutions converge to a unique maximal wavetrain. As a consequence the existence of a nontrivial maximal wavetrain is equivalent to the existence of travelling wave solution non-converging to the stationary state u=1$u=1$.     

Existence of travelling waves in a modified vector-disease model

In this paper, we consider travelling wave solutions for a modified vector-disease model. Special attention is paid to the model in which a susceptible vector can receive the infection not only from the infectious host but also from the infectious vector. For the strong generic delay kernel, we show that travelling wave solutions exist using the geometric singular perturbation theory.     

On the Diffusive Nicholson’s Blowflies Equation with Nonlocal Delay

This paper is concerned with the diffusive Nicholson’s blowflies model with nonlocal (or spatiotemporal) delay. When the spatial variable is one-dimensional, we establish the existence of travelling wave-front solutions by using the approach developed by Wang, Li, and Ruan (J. Differ. Equ. 222, 185–232, 2006) on the existence of travelling front solutions of reaction–diffusion systems with nonlocal delay. Moreover, we consider the dependence of the minimal wave speed on the delay and the mobility of the population. Our main finding here is that delay can induce slow travelling wave-fronts and the mobility of the population can increase fast travelling wave-fronts. In particular, if we choose some special kernel forms, then our results include and improve some known results.      

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.     

Travelling Waves and Numerical Approximations in a Reaction Advection Diffusion Equation with Nonlocal Delayed Effects

Summary. In this paper, we consider the growth dynamics of a single-species population with two age classes and a fixed maturation period living in a spatial transport field. A Reaction Advection Diffusion Equation (RADE) model with time delay and nonlocal effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of travelling waves for the delay model with three birth functions which appeared in the well-known Nicholson's blowflies equation, and we consider and analyze numerical solutions of the travelling wavefronts from the wave equations for the problems with nonlocal temporally delayed effects. In particular, we report our numerical observations about the change of the monotonicity and the possible occurrence of multihump waves. The stability of the travelling wavefront is numerically considered by computing the full time-dependent partial differential equations with nonlocal delay.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Travelling waves of a delayed SIRS epidemic model with spatial diffusion

This paper is concerned with the existence of travelling waves to an SIRS epidemic model with bilinear incidence rate, spatial diffusion and time delay. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder’s fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Existence of travelling fronts in a diffusive vector disease model with spatio-temporal delay

In this paper, we study travelling front solutions of a vector disease model incorporating time delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate the associated non-local spatial terms which account for the drift of individuals to their present positions from their possible positions at previous times. We shall show that such fronts exist for the weak generic delay kernel and sufficiently small delays by using geometric singular perturbation theory. Then, for the discrete-delay case, following Canosa’s asymptotic analysis method, we give some information on travelling front solutions.     

非线性系统的精确解

Based on the computerized symbolic, a new generalized tanh functions method is used for constructing exact travelling wave solutions of nonlinear partial differential equations （PDES） in a unified way. The main idea of our method is to take full advantage of an auxiliary ordinary differential equation which has more new solutions. At the same time, we present a more general transformation, which is a generalized method for finding more types of travelling wave solutions of nonlinear evolution equations （NLEEs）. More new exact travelling wave solutions to two nonlinear systems are explicitly obtained.