Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models

The theory of asymptotic speeds of spread and monotone traveling waves is generalized to a large class of scalar nonlinear integral equations and is applied to some time-delayed reaction and diffusion population models.     

Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage

This paper is concerned with the monotonicity and uniqueness of traveling waves for a reaction-diffusion model with quiescent stage. We first obtain the exponential decay rate of wave profiles, and then we show that any profile is strictly monotone by using the strong comparison principle. Furthermore, we prove the uniqueness (up to translation) of all traveling waves including even the waves with minimal speed.     

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

This paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained.     

Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:

u_t(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),     

Existence and uniqueness of traveling waves for non-monotone integral equations with application

This paper is concerned with the traveling waves in a class of non-monotone integral equations. First we establish the existence of traveling waves. The approach is based on the construction of two associated auxiliary monotone integral equations and a profile set in a suitable Banach space. Then we show that the traveling waves are unique up to translations under some reasonable assumptions. The exact asymptotic behavior of the profiles as ξ→−∞ and the existence of minimal wave speed are also obtained. Finally, we apply our results to an epidemic model with non-monotone “force of infection”.     

Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay

This paper is concerned with the existence, asymptotic stability and uniqueness of traveling wavefronts in a nonlocal diffusion equation with delay. By constructing proper upper and lower solutions, the existence and asymptotic behavior of traveling wavefronts are established. Then the asymptotic stability with phase shift as well as the uniqueness up to translation of traveling wavefronts are proved by applying the idea of squeezing technique.     

Regular traveling waves for a nonlocal diffusion equation

In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures.     

Traveling waves for non-local delayed diffusion equations via auxiliary equations

In this paper, we study the existence of traveling wave solutions for a class of delayed non-local reaction-diffusion equations without quasi-monotonicity. The approach is based on the construction of two associated auxiliary reaction-diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauder's fixed point theorem, we then show that there exists a constant c_∗>0 such that for each c>c_∗, the equation under consideration admits a traveling wavefront solution with speed c, which is not necessary to be monotonic.     

Spreading speeds and traveling waves for periodic evolution systems

The theory of spreading speeds and traveling waves for monotone autonomous semiflows is extended to periodic semiflows in the monostable case. Then these abstract results are applied to a periodic system modeling man-environment-man epidemics, a periodic time-delayed and diffusive equation, and a periodic reaction-diffusion equation on a cylinder.     

Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks

The author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks. Previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations. He applies fixed point theorems to prove the existence of the traveling waves. Then, he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves.     

Partial functional differential equation with an integral condition and applications to population dynamics

In this work we consider a semilinear functional partial differential equation with an integral condition. We apply the method of semidiscretization in time, to establish the existence and uniqueness of solutions. We also study the continuation of the solution to the maximal interval of existence. Finally we give examples to demonstrate the applications of our results.     

Existence of solutions to Hamilton-Jacobi functional differential equations

This paper deals with the Cauchy problem for nonlinear first order partial functional differential equations. The unknown function is the functional variable in the equation and the partial derivatives appear in a classical sense. A theorem on the local existence of a generalized solution is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions of this system is proved by using a method of successive approximations. A method of bicharacteristics and integral inequalities are applied.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

A transfer integral technique for solving a class of linear integral equations: Convergence and applications to DNA

An eigenvalue problem, the convergence difficulties that arise and a mathematical solution are considered. The eigenvalue problem is motivated by simplified models for the dissociation equilibrium between double-stranded and single-stranded DNA chains induced by temperature (thermal denaturation), and by the application of the so-called transfer integral technique. Namely, we extend the Peyrard–Bishop model for DNA melting from the original one-dimensional model to a three-dimensional one, which gives rise to an eigenvalue problem defined by a linear integral equation whose kernel is not in L²$L^2$. For the one-dimensional model, the corresponding kernel is not in L²$L^2$ either, which is related to certain convergence difficulties noticed by previous researchers. Inspired by methods from quantum scattering theory, we transform the three-dimensional eigenvalue problem, obtaining a new L²$L^2$ kernel which has improved convergence properties.     

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.     

Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity

This paper is concerned with the global stability of wavefronts with minimal speeds for degenerate nonlocal dispersal equations. By the method of super- and subsolutions and a squeezing technique, the wavefront with minimal speed is proved to be globally asymptotically stable with phase shift.     

Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ₁(Ω), the solution decays to zero as t→∞; while for λ>λ₁(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions.     

Persistence of wavefronts in delayed nonlocal reaction-diffusion equations

We develop a perturbation argument based on existing results on asymptotic autonomous systems and the Fredholm alternative theory that yields the persistence of traveling wavefronts for reaction-diffusion equations with nonlocal and delayed nonlinearities, when the time lag is relatively small. This persistence result holds when the nonlinearity of the corresponding ordinary reaction-diffusion system is either monostable or bistable. We then illustrate this general result using five different models from population biology, epidemiology and bio-reactors.