Spreading speed for a nonlocal dispersal vaccination model with general incidence

This paper is concerned with the spreading speed for a nonlocal dispersal vaccination model with general incidence. We first prove the existence and uniform boundedness of solutions for this model by using the Schauder’s fixed point theorem. Then, applying comparison principle, we establish the existence of spreading speed for the infective individuals. According to our result, one can see that the spreading speed coincides with the critical speed of traveling wave solution connecting the disease-free and endemic equilibria. In addition, the diffusion rate of the infected individuals can increase the spread of infectious diseases, while the vaccination rate reduces the spread of infectious diseases.     

Propagation dynamics of a nonlocal periodic organism model with non-monotone birth rates

This work is concerned with a nonlocal reaction–diffusion system modeling the propagation dynamics of organisms owning mobile and stationary states in periodic environments. We establish the existence of the asymptotic speed of spreading for the model system with monotone birth function via asymptotic propagation theory of monotone semiflow, and then discuss the case for non-monotone birth function by using the squeezing technique. In terms of the truncated problem on a finite interval, we apply the method of super- and sub-solutions and the fixed point theorem combined with regularity estimation and limit arguments to obtain the existence of time periodic traveling waves for the model system without quasi-monotonicity. The non-existence proof is to use the results of the spreading speed. Finally, as an application, we study the spatial dynamics of the model with the birth rate function of Ricker type and numerically demonstrate analytic results.     

The principal eigenvalue for periodic nonlocal dispersal systems with time delay

The theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time-periodic nonlocal dispersal cooperative system with time delay. Then we apply it to a Nicholson's blowflies population model and obtain a threshold type result on its global dynamics.     

Propagation dynamics of a time periodic diffusion equation with degenerate nonlinearity

This paper is on study of traveling wave solutions and asymptotic spreading of a class of time periodic diffusion equations with degenerate nonlinearity. The asymptotic behavior of traveling wave solutions is investigated by using auxiliary equations and a limit process. In addition, the monotonicity and uniqueness, up to translation, of traveling wave solution with critical speed are determined by sliding method. Finally, combining super and sub-solutions and the stability of steady states, some sufficient conditions on asymptotic spreading are given, which indicates that the success or failure of asymptotic spreading are dependent on the degeneracy of nonlinearity as well as the size of compact support of initial value.     

Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats

The current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established.     

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems.     

Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay

This paper studies a nonautonomous Lotka-Volterra dispersal systems with infinite time delay which models the diffusion of a single species into n patches by discrete dispersal. Our results show that the system is uniformly persistent under an appropriate condition. The sufficient condition for the global asymptotical stability of the system is also given. By using Mawhin continuation theorem of coincidence degree, we prove that the periodic system has at least one positive periodic solution, further, obtain the uniqueness and globally asymptotical stability for periodic system. By using functional hull theory and directly analyzing the right functional of almost periodic system, we show that the almost periodic system has a unique globally asymptotical stable positive almost periodic solution. We also show that the delays have very important effects on the dynamic behaviors of the system.     

Boundedness and persistence of traveling wave solutions for a non-cooperative lattice-diffusion system with time delay

In this paper we study traveling wave solutions of a non-cooperative lattice-diffusion system with time delay, which includes predator–prey models and disease-transmission models. Minimal wave speed of traveling wave solutions is given. Schauder’s fixed-point theorem is applied to show the existence of semi-traveling wave solutions. The boundness and persistence of traveling wave solutions are overcome by using rescaling method and Laplace transform, where the application of Laplace transform to persistence is very novel and creative. The traveling wave solutions for some specific models are shown to connect to a positive equilibrium by using Lyapunov function and LaSalle’s invariance principle.     

On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space

This paper is concerned with the existence and uniqueness of time periodic solutions in the whole‐space for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |u |^{q ? 1}u + f (x ,t ), with , N > 2. We show that there exists a unique time periodic solution when the source term f is small. In fact, is a critical exponent; when , there is no time periodic solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and stability of periodic solutions for parabolic systems with time delays

This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.     

Solitary Waves, periodic peakons, pseudo-peakons and compactons given by three ion-acoustic wave models in electron plasmas

The nonlinear ion-acoustic oscillations models are governed by three partial differential equation systems. Their travelling wave equations are three first class singular traveling wave systems depending on different parameter groups, respectively. By using the method of dynamical system and the theory of singular traveling wave systems, in this paper, it is shown that there exist parameter groups such that these singular systems have solitary wave solutions, pseudo-peakons, periodic peakons and compactons as well as kink and anti-kink wave solutions. The results of this paper complete the studies of three papers [5,13] and [14].     

Traveling waves in nonlocal diffusion systems with delays and partial quasi-monotonicity

This paper is devoted to the study of a three-dimensional delayed system with nonlocal diffusion and partial quasi-monotonicity. By developing a new definition of upper-lower solutions and a new cross iteration scheme, we establish some existence results of traveling wave solutions. The results are applied to a nonlocal diffusion model which takes the three-species Lotka-Volterra model as its special case.     

Explicit exact periodic wave solutions and their limit forms for a long waves-short waves model

A long waves-short waves model is studied by using the approach of dynamical systems. The sufficient conditions to guarantee the existence of solitary wave, kink and anti-kink waves, and periodic wave in different regions of the parametric space are given. All possible explicit exact parametric representations of above traveling waves are presented. When the energy of Hamiltonian system corresponding to this model varies, we also show the convergence of the periodic wave solutions, such as the periodic wave solutions converge to the solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions, respectively.     

Spreading speeds and traveling waves for a time periodic DS-I-A epidemic model

This paper is devoted to studying the speed of asymptotic spreading and minimal wave speed of traveling wave solutions for a time periodic and diffusive DS-I-A epidemic model, which describes the propagation threshold of disease spreading. The main feature of this model is the possible deficiency of the classical comparison principle such that many known results do not directly work. The speed of asymptotic spreading is estimated by constructing auxiliary equations and applying the classical theory of asymptotic spreading for Fisher type equation. The minimal wave speed is established by proving the existence and nonexistence of the nonconstant traveling wave solutions. Moreover, some numerical examples are presented to model the propagation dynamics of this system.     

Spreading and generalized propagating speeds of discrete KPP models in time varying environments

The current paper deals with spatial spreading and front propagating dynamics for spatially discrete KPP (Kolmogorov, Petrovsky and Paskunov) models in time recurrent environments, which include time periodic and almost periodic environments as special cases. The notions of spreading speed interval, generalized propagating speed interval, and traveling wave solutions are first introduced, which are proper modifications of those introduced for spatially continuous KPP models in time almost periodic environments. Among others, it is then shown that the spreading speed interval in a given direction is the minimal generalized propagating speed interval in that direction. Some important upper and lower bounds for the spreading and generalized propagating speed intervals are provided. When the environment is unique ergodic and the so called linear determinacy condition is satisfied, it is shown that the spreading speed interval in any direction is a singleton (called the spreading speed), which equals the classical spreading speed if the environment is actually periodic. Moreover, in such a case, a variational principle for the spreading speed is established and it is shown that there is a front of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction.