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We analyze a singularly Kuramoto-Sivashinsky perturbed Camassa-Holm equation with methods of the geometric singular perturbation theory. Especially, we study the persistence of smooth and peaked solitons. Whether a solitary wave of the original Camassa-Holm equation is smooth or peaked depends on whether there is linear dispersion, i.e. whether 2 k = 0 . If 2 k > 0 , then a unique smooth solitary wave persists with selected wave speed under singular Kuramoto-Sivashinsky perturbation just as what happens in the KS-KdV equation. On the other hand, we show that if there is no linear dispersion, i.e. 2 k = 0 , then any observable peaked soliton fails to persist. This case is non-typical since the related slow manifold blows up and the classical geometric singular perturbation theory is not available.

In this paper, we study a classical two species stage structure Lotka-Volterra diffusion-advection system. By employing the Krein-Rutman theorem, analyzing the principal eigenvalue, and combining with the theory of monotone dynamical systems, a complete classification of the global dynamics is given for the weak competition case. 2020 MSC: 35B35, 35K57, 35Q92, 92D25

In this paper, we are concerned with the existence of traveling pulse solutions of one-dimensional generalized Keller-Segel system with nonlinear chemical gradients and small cell diffusion by using the dynamical systems approach, especially based on geometric singular perturbation theory and Poincare-Bendixson theorem. To show the existence of traveling pulse solutions, we first analyze the dynamics of the system by geometric singular perturbation theory. And then we seek an invariant region for the associated traveling wave equation. Finally, we apply Poincare-Bendixson theorem to analyze the flow on this invariant region to obtain the existence of traveling pulse solutions in this bounded invariant region. As applications, we present two examples to illustrate our main results.

In this paper, we consider the existence of traveling wave fronts in a Belousov-Zhabotinskii system with delay. By traveling wave transformation and time scale transformation, we change the Belousov-Zhabotinskii system with delay into a singularly perturbed differential system. By applying geometric singular perturbation theory, we construct a locally invariant manifold for the associated traveling wave equation and obtain the traveling wave fronts for the equation by using the Fredholm orthogonality. Finally, we discuss the asymptotic behaviors of traveling wave solutions by applying the asymptotic theory.

In this paper, we deal with Sel’kov model with saturation law which has been applied to numerous problems in chemistry and biology. We will study the stability of the unique constant steady state, existence and nonexistence of nonconstant steady states of such models. In particular, we prove that Turing pattern may occur when the saturation coefficient is small but will not occur when the coefficient becomes large. Therefore for a Sel’kov model with saturation law, it is the saturation law that determines the formation of spatial patterns.

In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium \begin{document}$ (0, 0) $\end{document} to some unknown positive steady state for wave speed \begin{document}$ c>c^* = \max\left\{2, 2\sqrt{dr}\right\} $\end{document} and there is no such traveling wave solutions for \begin{document}$ c , where \begin{document}$ d $\end{document} and \begin{document}$ r $\end{document} respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.

This paper is devoted to the existence of traveling, solitary and periodic waves for the perturbed generalized KdV by applying geometric singular perturbation, differential manifold theory and the regular perturbation analysis of Hamiltonian systems. Under the assumptions that the distributed delay kernel is the strong general one and the average delay is sufficiently small, traveling, solitary and periodic waves are shown to exist in the perturbed system. It is further proved that the wave speed is decreasing by analyzing the ratio of Abelian integrals, and we analyze these functions by using the theory of analytic functions and algebraic geometry. Moreover, the upper and lower bounds of the limit wave speed are presented. The relationship between wavelength and wave speed of traveling waves is also established.

In this paper, we consider an eco-epidemic predator–prey model with Crowley–Martin type functional response and disease in prey population, which is a branch of study in biomathematics which reflects both ecological and epidemiological cases simultaneously. By using Mawhin continuation theorem and constructing a suitable Lyapunov function, we obtained some sufficient conditions for the existence, uniqueness and global attractivity of a positive periodic solution for the predator–prey model.

In this paper, we study a nonlinear fifth-order KdV equation which has wide physical applications. We first establish the existence of solitary wave solutions for this equation without delay by using the Hamilton function method. Then we obtain the desired homoclinic orbits by constructing a locally invariant manifold. Finally we obtain the existence of solitary wave solutions for this equation with local and nonlocal delay convolution kernels by using the geometric singular perturbation analysis, implicit function theorem and Fredholm theory.

This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period \begin{document}$ T $\end{document} of traveling wave when the small positive parameter \begin{document}$ \tau\rightarrow 0 $\end{document} are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation.

In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation.

In this work, we deal with the n -dimensional reaction-diffusive predator–prey systems with the discrete time delay, which have numerous applications in biology and ecology. By applying an asymptotic analysis, the Schauder’s fixed point theorem as well as the upper and lower solution method, we investigate the existence, non-existence, exponentially asymptotic stability and asymptotic behaviors of traveling wave solutions with the appropriate small delay when the wave speed c varies. Moreover, we derive the minimal wave speed c ∗ based on the well-established properties of the Fisher equation, and prove the existence of traveling wave solutions when the wave speed c ≥ c ∗ , which provides a sharp contrast to the case of c c ∗ . The obtained results extend the existing ones in the literature to arbitrary finite dimensional systems with the discrete time delay and demonstrate asymptotic behaviors of traveling wave solutions.

In this paper, we discuss a reaction–diffusion predator–prey model with nonlocal delays. By using the theory of dynamical systems, specifically based on geometric singular perturbation theory, center manifold theorem and Fredholm theory, we construct an invariant manifold for the associated predator–prey equation and use this invariant manifold to obtain a heteroclinic orbit between two non-negative equilibrium points. Furthermore, we establish the existence result of traveling wave solution for the predator–prey model.

In this paper, we are concerned with the existence and asymptotic behavior of traveling wave fronts in a modified vector-disease model. We establish the existence of traveling wave solutions for the modified vector-disease model without delay, then explore the existence of traveling fronts for the model with a special local delay convolution kernel by employing the geometric singular perturbation theory and the linear chain trick. Finally, we deal with the local stability of the steady states, the existence and asymptotic behaviors of traveling wave solutions for the model with the convolution kernel of a special non-local delay.

In this paper, we discuss a neutral delay predator–prey model with Hassel–Varley type functional response and impulse is investigated. By using Mawhin coincidence degree theory, we obtain some sufficient conditions for the existence of positive periodic solutions. We extend some known work.

This paper is concerned with the dynamics of traveling wave solutions for a reaction-diffusion predator-prey model with a nonlocal delay. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper-lower solutions which are easy to construct in practice. We also investigate the asymptotic behavior of traveling wave solutions by employing the standard asymptotic theory.

In this paper, we consider a two-species competitive and diffusive system with nonlocal delays. We investigate the existence of traveling wave fronts of the system by employing linear chain techniques and geometric singular perturbation theory. The existence of the traveling wave fronts analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a six-dimensional system without delay.

For analytic differential systems x ˙ = A x + f ( x ) in F n with F = R or C , we study the varieties of their partial integrability in a neighborhood of the origin. We also prove the existence of analytic normalizations of partially integrable differential systems.

This paper deals with the existence of traveling wave solutions for n-dimensional delayed reaction–diffusion systems. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper–lower solutions that are easy to construct. As an application, we apply our main results to a four-dimensional delayed predator–prey system and obtain the existence of traveling wave solutions. Copyright © 2016 John Wiley & Sons, Ltd.

By the use of a constructing adequate real functional and choosing an appropriate admissible function space, the existence and multiplicity of solutions to a three-point second-order differential system is proved via Mountain Pass Lemma. The interesting point is the boundary value conditions are imposed on admissible space rather than the functionals.

D. Zeidan In this paper, we consider the existence of traveling waves in a generalized nonlinear dispersive–dissipative equation, which is found in many areas of application including waves in a thermoconvective liquid layer and nonlinear electromagnetic waves. By using the theory of dynamical systems, specifically based on geometric singular perturbation theory and invariant manifold theory, Fredholm theory, and the linear chain trick, we construct a locally invariant manifold for the associated traveling wave equation and use this invariant manifold to obtain the traveling waves for the nonlinear dispersive–dissipative equation. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we discuss the existence and asymptotic behavior of traveling waves for a generalized Burgers–KdV equation. We show the heteroclinic orbits of the associated ordinary differential equations for the generalized Burgers–KdV equation with a special convolution kernel and then establish the existence result of traveling wave solutions for the Burgers–KdV equation by employing geometric singular perturbation theory and the linear chain trick. And the asymptotic behavior of traveling waves is obtained by using the standard asymptotic theory.

In this paper, we are concerned with a singularly perturbed higher-order KdV equation, which is considered as a paradigm in nonlinear science and has many applications in weakly nonlinear and weakly dispersive physical systems. Based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of the solitary wave solution for the singularly perturbed KdV equation is investigated by using the geometric singular perturbation theory and dynamical systems approach when the perturbation parameter is suitably small.

This paper deals with a generalized KdV–mKdV equation with time delay. By employing the geometrical singular perturbation theory and the linear chain trick, we establish the existence result of solitary wave solutions when the average delay is sufficiently small, for a special convolution kernel.

In this paper, a class of neutral delayed Lotka-Volterra competition system with impulsive perturbations is investigated. We explore the existence of positive periodic solutions of the system by using Mawhin coincidence degree theory and some analytical approaches. From our results, we find that a part of n species of the system is driven to extinction due to competition exclusion and weak adaption to environment characterized by impulsive perturbations.

The paper deals with the existence of solutions for the dynamic equation on time scales $$ \begin{array}{cc}\hfill {u}^{\varDelta \varDelta \varDelta \varDelta}(t)= f\left( t, u\left(\sigma (t)\right),{u}^{\varDelta \varDelta}(t)\right),\hfill & \hfill t\in {\left[0,1\right]}_T,\hfill \end{array} $$ with the multipoint boundary conditions $$ \begin{array}{cccc}\hfill u(0)=0,\hfill & \hfill u\left(\sigma (1)\right)={\displaystyle \sum_{i=1}^{m-2}{a}_i u\left({\xi}_i\right),}\hfill & \hfill {u}^{\varDelta \varDelta}(0)=0,\hfill & \hfill {u}^{\varDelta \varDelta}\left(\sigma (1)\right)={\displaystyle \sum_{j=1}^{n-2}{b}_j{u}^{\varDelta \varDelta}\left({\eta}_j\right),}\hfill \end{array} $$ where T is a time scale [0, 1] T = {t ∈ T | 0 ≤ t ≤ 1}, a i > 0, i = 1, 2, …, m − 2, b j > 0, j = 1, 2, …, n − 2, 0 < ξ1 < ξ2 < … < ξ m−2 < ρ(1), and 0 < η 1 < η 2 < … < η n−2 < ρ(1). The existence result is given by using Green’s function, the method of upper and lower solutions, and the monotone iterative technique. We also give an example to illustrate our result.

In this paper, we consider a neutral predator-prey model with the Beddington-DeAngelis functional response and impulsive effect. Sufficient conditions are obtained for the existence of positive periodic solutions by a systematic qualitative analysis. Some known results in the literature are generalized.

In this paper, we study a delayed predator-prey model with the Leslie-Gower Holling-type II functional response and harvesting terms. The existence of multiple positive periodic solutions for the system and the permanence of the predator-prey model are obtained by means of the generalized Mawhin coincidence degree theory.

By using Mawhin coincidence degree theory, we investigate the existence of solutions for a class of second order nonlinear differential equations with generalized Sturm-Liouville integral boundary conditions at resonance. The results extend some known conclusions of integral boundary value problem at resonance for nonlinear differential equations

This paper deals with a third order nonlinear differential equations with multi-point boundary conditions. By using the coincidence degree theory, we establish some existence results of the problem at resonance under some appropriate conditions. The emphasis here is that the dimension of the linear operator is equal to three. We also give an example to demonstrate our results.

In this paper, we study the uniqueness and local behavior at the origin of positive solutions to [Formula: see text] where [Formula: see text], [Formula: see text]) is a bounded smooth domain and [Formula: see text], and [Formula: see text] is a nonnegative continuous function over [Formula: see text]. In the two cases: (i) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text]; (ii) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text], we establish the uniqueness of positive solutions and the exact blow-up rate of positive solutions at the origin. This paper seems to be the first one to deal with the latter case.

In this paper, we are concerned with a delayed multispecies competition predator-prey dynamic system with Beddington-DeAngelis functional response. Some sufficient conditions which guarantee the existence of a positive periodic solution for the system are obtained by applying the Mawhin coincidence theory. The interesting thing is that the result is related to the delays, which is different from the corresponding ones known from literature (the results are delay-independent).

A class of singularly perturbed initial boundary value problems of reaction diffusion equations for nonlinear boundary conditions is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and asymptotic behavior of solution for initial boundary value problem are studied. Moreover, the obtained solution indicates that there are initial and boundary layers, and the thickness of the initial layer is less than that of the boundary layer.