Positive Traveling Wave Research Articles

Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.

Traveling waves for nonlocal and non-monotone delayed reaction-diffusion equations

We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-diffusion equation. Based on the construction of two associated auxiliary reaction diffusion equations with monotonicity and by using the traveling wavefronts of the auxiliary equations, the existence of the positive traveling wave solutions for c ≥ c* is obtained. Also, the exponential asymptotic behavior in the negative infinity was established. Moreover, we apply our results to some reactiondiffusion equations with spatio-temporal delay to obtain the existence of traveling waves. These results cover, complement and/or improve some existing ones in the literature.

Asymptotic behaviour for a class of delayed cooperative models with patch structure

For a class of cooperative population models with patch structure and multiple discrete delays, we give conditions for the absolute global asymptotic stability of both the trivial solution and -- when it exists -- a positive equilibrium. Under a sublinearity condition, sharper results are obtained. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. As a by-product, we obtain a criterion for the existence of positive traveling wave solutions for an associated reaction-diffusion model with patch structure. Our results improve and generalize criteria in the recent literature.

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.

Non-linear determinacy of minimum wave speed for a Lotka–Volterra competition model

For a reaction–diffusion system that serves as a 2-species Lotka–Volterra diffusive competition model, suppose that the corresponding reaction system has one stable boundary equilibrium and one unstable boundary equilibrium. Then it is well known that there exists a positive number c ⁎ , called the minimum wave speed, such that, for each c larger than or equal to c ⁎ , the reaction–diffusion system has a positive traveling wave solution of wave speed c connecting these two equilibria if and only if c ⩾ c ⁎ . It has been shown that the minimum wave speed for this system is identical to another important quantity – the asymptotical speed of population spread towards the stable equilibrium. Hence to find the minimum wave speed c ⁎ not only is of the interest in mathematics but is of the importance in application. It has been conjectured that the minimum wave speed can be determined by studying the eigenvalues of the unstable equilibrium, called the linear determinacy. In this paper we will show that the conjecture on the linear determinacy is not true in general.

Positive travelling fronts for reaction–diffusion systems with distributed delay

We give sufficient conditions for the existence of positive travelling wave solutions for multi-dimensional autonomous reaction–diffusion systems with distributed delay. To prove the existence of travelling waves, we give an abstract formulation of the equation for the wave profiles in some suitable Banach spaces and apply known results about the index of some associated Fredholm operators. After a Lyapunov–Schmidt reduction, these waves are obtained via the Banach contraction principle, as perturbations of a positive heteroclinic solution for the associated system without diffusion, whose existence is proven under some requirements. By a careful analysis of the exponential decay of the travelling wave profiles at −∞, their positiveness is deduced. The existence of positive travelling waves is important in terms of applications to biological models. Our method applies to systems of delayed reaction–diffusion equations whose nonlinearities are not required to satisfy a quasi-monotonicity condition. Applications are given, and include the delayed Fisher–KPP equation.

Parametric instability of flexible disk rotating at periodically varying angular speed

Under considering the inertial force of angular acceleration and the shear stress due to the varying rotational speed, parametric instability of a flexible disk rotating at periodically varying angular speed is analyzed via an improved Hill’s method which is suitable for harmonic balance processing. The causation of parametric instability is interpreted innovatively, and new parameter region of instability is found. The analysis shows that the parametric instability occurs when a positive and a negative traveling wave intersect and overlap each other, and each intersection can result in its own instability region. The instability region is enlarged with increasing amplitude of the speed variation, but is reduced by increasing the difference between harmonic numbers of corresponding negative and positive traveling wave. And the shear stress enhances modes interaction of the disk and affects the instability regions considerably.

KPP type flame fronts in porous media

In this paper we study the model of pressure-driven flames in porous media proposed by Brailovsky et al (1997 Combust. Sci. Technol. 124 145–65). We show that, under the assumption of first order reaction with linear reaction kinetics (quadratic nonlinearity), the model admits a family of positive travelling wave solutions. Moreover, under the same assumption, we prove that propagation of disturbances in the system is fully determined by the rate of decay of the initial data at infinity. We also give an upper bound of the burning rate in the case of arbitrary chemical kinetics bounded by linear function.

Existence of standing waves for competition-diffusion equations

In this paper, we consider the competition-diffusion equation of Lotka-Volterra type, and establish an existence theorem of a one parameter family of positive standing waves which exponentially decay to a stable equilibrium point in the ODE sense. To do it, we study fundamental sets of solutions of the linearized problem around a positive travelling wave by using the comparison principle, and employ the bifurcation theory for homoclinic orbits which was shown in Kokubu [10].