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Reaction-advection-diffusion equations, in periodic settings and with general type nonlinearities, admit a threshold known as the minimal speed of propagation. The minimal speed does not have an accessible formula when the nonlinearity is not of KPP type, for instance. The question becomes whether the minimal speed can be obtained through a linearization procedure or not. In this paper, we derive selection criteria for the minimal speed: a key feature of the nonlinear selection is unveiled. Moreover, we use upper/lower solution techniques in order to derive practical criteria determining the minimal speed in the presence of advection and a general type nonlinearity. [ABSTRACT FROM AUTHOR]

This paper is concerned with traveling waves to a time-periodic bistable Lotka–Volterra competition system with nonlocal dispersal. We first establish the existence, uniqueness and stability of traveling wave solutions for this system. By utilizing comparison principle and the stability property, the relationship among the bistable wave speed, the asymptotic propagation speeds of the associated monotone subsystems and the speed of upper/lower solutions is obtained. Explicit sufficient conditions for positive and negative bistable wave speeds are derived. Our explicit results are derived by constructing particular and novel upper/lower solutions with specific asymptotical behaviors, which can be seen as case studies applicable to further investigations and improvements. Finally, the theoretical results are corroborated under weak conditions by direct simulations of the underlying time-periodic system with nonlocal dispersal. The combined impact of competition, dispersal and seasonality on the invasion direction has shed new light on the modelings and analysis of population competition and species invasion in heterogeneous media. [ABSTRACT FROM AUTHOR]

This paper concerns the propagation direction of the bistable traveling wave for a time-periodic Lotka–Volterra reaction–diffusion competition system. The speed is first shown to be bounded by the asymptotic spreading speeds of the monostable subsystems. General results toward the propagation direction are obtained by establishing two comparison principles on wave speed. These results rely on the establishment of the speed sign of an upper (or lower) solution to the system of wave profile equations. In particular, by subtly constructing upper or lower solutions, we derive a set of explicit formulas that determine whether the speed sign is positive or negative, which indicates whether the bistable traveling wave spatially propagates to the left or to the right. Finally, numerical simulations are given to demonstrate the effects of system parameters to the propagation direction of the bistable traveling wave. [ABSTRACT FROM AUTHOR]

Global exponential stability of bistable traveling wave solutions to a periodic Lotka–Volterra system with nonlocal dispersal is investigated. We first establish refined squeezing upper and lower solutions based on the properties of the bistable wavefront. This important squeezing technique has been successfully applied to study the exponential convergence of solutions. [ABSTRACT FROM AUTHOR]

Cooperation in a population system can result in the occurrence of a coexistence (win–win) equilibrium. When diffusion is incorporated, individual species possibly invade into far ends with different spreading speeds (co-speed or fast–slow speed). Predicting or determining them becomes challenging. This paper is devoted to the complete understanding of propagation dynamics to a cooperative Lotka–Volterra system, in addition of the determinacy of these speeds. The first major contribution of this paper is to establish a necessary and sufficient condition for the existence of a single spreading speed. In the existence of a single spreading speed (i.e., the two species share a common invasion speed), nonnegative traveling wave profiles, connecting the extinction equilibrium, exist, if the wave speed is not less than the common speed. We find that predicting or determining the invasion speed can be linked to the linearized system at the extinction state, and linear/nonlinear selections are established. The existence of multiple spreading speeds indicates new connections of traveling wave profiles into certain intermediate states. Due to this, the determinacy of multiple spreading speeds focuses not only on the extinction state but also on the corresponding intermediate states. Based on our constructions of upper–lower solutions, we also establish analytic criteria that can determine the fast–slow invasive speeds. Our speed selection mechanism can also help scientists greatly understand the movement of stacked fronts in cooperative systems with multiple species. [ABSTRACT FROM AUTHOR]

Competition stems from the fact that resources are limited. When multiple competitive species are involved with spatial diffusion, the dynamics becomes even complex and challenging. In this paper, we investigate the invasive speed to a diffusive three species competition system of Lotka-Volterra type. We first show that multiple species share a common spreading speed when initial data are compactly supported. By transforming the competitive system into a cooperative system, the determinacy of the invasive speed is studied by the upper-lower solution method. In our work, for linearly predicting the invasive speed, we concentrate on finding upper solutions only, and don't care about the existence of lower solutions. Similarly, for nonlinear selection of the spreading speed, we focus only on the construction of lower solutions with fast decay rate. This greatly develops and simplifies the ideas of past references in this topic. [ABSTRACT FROM AUTHOR]

We studied the existence, uniqueness, and stability of forced traveling waves for the Lotka–Volterra competition system in a shifting habitat. Based on the asymptotic behaviors of the wave profiles and by means of upper–lower solution method coupled with sliding technique, we showed that the forced wave for the system exists and is unique, when the forced speed lies in a specific interval. Explicit expressions of the two end pointsof this interval were derived and our finding indicated that they are related to the Fisher‐KPP‐type invasion speed. Furthermore, we established a squeezing theorem to show the local stability of the forced waves. Finally, with the aid of comparison principle, we established the global stability of the forced waves when the initial data were properly assigned. [ABSTRACT FROM AUTHOR]

A general reaction–diffusion equation with nonlinear advection term having neither monotonicity nor variational structure is considered. This work is concerned with traveling waves for this type of equations. By constructing an invariant region with a lower curve smoothly connecting two fixed points where a heteroclinic orbit exists, we establish several theorems on the existence of wave fronts. The challenging problem on linear or nonlinear determinacy of the minimal speed is investigated. A necessary and sufficient condition for the nonlinear selection mechanism is obtained by a geometrical argument coupled with a perturbation method in a weighted functional space. On this basis, explicit sufficient conditions for the linear and nonlinear selection are derived. As applications, the claims or conjectures proposed in some important literature are rigorously proved. [ABSTRACT FROM AUTHOR]

In this paper, the speed sign of the traveling wave to the bistable Lotka–Volterra competitive lattice system is investigated via the upper–lower solution method as well as the comparison principle. We provide an interval estimation for the bistable speed firstly. Two comparison principles are further established to obtain new conditions to the determinacy of the sign of the bistable speed. To our knowledge, this is the first investigation to the lattice system for the propagation direction. [ABSTRACT FROM AUTHOR]

This work deals with the speed sign of travelling waves to the Lotka-Volterra model with diffusion and bistable nonlinearity. We obtain new conditions for the determinacy of the sign of the bistable wave speed by constructing upper or lower solutions with an inside parameter to be adjusted. The established conditions improve or supplement the results in the references and give insight into the combined effect of system parameters on the propagation direction of the bistable wave. [ABSTRACT FROM AUTHOR]

We study propagation direction of the traveling wave for the diffusive Lotka–Volterra competition system with bistable nonlinearity in a periodic habitat. By directly proving the strong stability of two semitrivial equilibria, we establish a new and sharper result on the existence of traveling wave. Using the method of upper and lower solutions, we provide two comparison theorems concerning the direction of traveling wave propagation. Several explicit sufficient conditions on the determination of the speed sign are established. In addition, an interval estimation of the bistable-wave speed reveals the relations among the bistable speed and the spreading speeds of two monostable subsystems. Biologically, our idea and insight provide an effective approach to find or control the direction of wave propagation for a system in heterogeneous environments. [ABSTRACT FROM AUTHOR]

The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species. [ABSTRACT FROM AUTHOR]

This paper concerns ecological invasion phenomenon of species based on the diffusive Lotka–Volterra competition model. We investigate the spreading speed (or the minimal wave speed of traveling waves) selection to the model and concentrate on the conjecture raised by Roques et al. (J Math Biol 71(2):465–489, 2015). By using an abstract implicit function theorem in a weighted functional space coupled with a perturbation technique, we not only prove this conjecture, but also show that the fast decay behavior of the first species is necessary and sufficient for the nonlinear speed selection of the whole system. This may lead to further significant results on the answer to the original Hosono's conjecture, a problem that has been outstanding for more than twenty years. [ABSTRACT FROM AUTHOR]

In this paper, we study the speed selection mechanism for traveling wave solutions to a two-species Lotka–Volterra competition model. After transforming the partial differential equations into a cooperative system, the speed selection mechanism (linear vs. nonlinear) is investigated for the new system. Hosono conjectured that there is a critical value r c of the birth rate so that the speed selection mechanism changes only at this value. In the absence of diffusion for the second species, we obtain the speed selection mechanism and successfully prove a modified version of the Hosono's conjecture. Estimation of the critical value is given and some new conditions for linear or nonlinear selection are established. [ABSTRACT FROM AUTHOR]

We study the existence, uniqueness and asymptotic expansions to perturbed Poisson-Boltzmann equations on an unbounded domain in R2 or R3. First, a shooting method is applied to prove the existence and uniqueness of the exact solution. For the approximation to the regularly perturbed Poisson-Boltzmann equation, the solution via the classical method fails. We develop a novel approximate solution in terms of generalized asymptotic expansions. For the singularly perturbed problem, we show that a formula of asymptotic expansions with a boundary layer near the left end point provides a valid approximation. All our results are proved rigorously. [ABSTRACT FROM AUTHOR]

In this paper we re-visit the Lagerstrom problemy" + (n-1)/r y' + εyy' = 0,y(1) = 0, y(∞) = 1,where ε is a small positive real number and n is a positive integer (or any real number greater than 2). Using rigorous analysis, a generalized asymptotic expansion, as ε→0, is derived for the solution of this problem. A trans-series expansion of the solution for large values of r is also presented; the leading term coefficient is determined by a connection formula between the values of the solution at the two points r=1 and r=∞. An extension and a discussion of the problem for n∈[1,2) is also given. [ABSTRACT FROM AUTHOR]

In this paper we investigate the population dynamics of a species with age structure in the case where the diffusion and death rates of the matured population are both age-dependent. We develop a new application of the age-structure technique in terms of an integral equation. For unbounded spatial domains, we study the existence of travelling waves, whilst in bounded domains, we investigate the existence of positive steady-state solutions and their stability. [ABSTRACT FROM PUBLISHER]

For a class of nonlocal time-delayed reaction-diffusion equations, we prove that all noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally algebraically stable when the initial perturbations around the wavefront decay to zero exponentially near the negative infinity regardless of the magnitude of time delay. This work also improves and develops the existing stability results for local and nonlocal reaction-diffusion equations with delays. Our approach is based on the combination of the weighted energy method and the Green function technique. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials $C_n^{(a)}(z)$. We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou ([8]) can be applied. Finally, we produce three Airy-type asymptotic expansions for $C_n^{(a)}(z)$ in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller ([3]), there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong ([4]). Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials. [ABSTRACT FROM AUTHOR]

We investigate the effect and the impact of predator-prey interactions, diffusivity and chemotaxis on the ability of survival of multiple consumer levels in a predator-prey microbial food chain. We aim at answering the question of how many consumer levels can survive from a dynamical system point of view. To solve this standing issue on food-chain length, first we construct a chemotactic food chain model. A priori bounds of the steady state populations are obtained. Then under certain sufficient conditions combining the effect of conversion efficiency, diffusivity and chemotaxis parameters, we derive the co-survival of all consumer levels, thus obtaining the food chain length of our model. Numerical simulations not only confirm our theoretical results, but also demonstrate the impact of conversion efficiency, diffusivity and chemotaxis behavior on the survival and stability of various consumer levels. [ABSTRACT FROM AUTHOR]