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In this paper, we study the minimal speed (spreading speed) selection mechanism of traveling waves to a periodic diffusive Lotka-Volterra model with monostable nonlinearity. The method of upper and lower solutions pair is applied to establish the existence of traveling waves. We prove that the nature of nonlinear selection is to find a lower solution pair with the first species decaying in a faster rate. By novel constructions of various upper and lower solutions, we obtain a number of new results on the minimal speed determinacy. Numerical simulations are carried out to illustrate all of our discoveries.

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.

In this paper we study speed selection for traveling wavefronts of the lattice Lotka-Volterra competition model. For the linear speed selection, by constructing new types of upper solutions to the system, we widely extend the results in the literature. We prove that, for the nonlinear speed selection, the wavefront of the first species decays with a faster rate at the far end. This enables us to construct novel lower solutions to establish the existence of pushed wavefronts, a topic that has been understudied. We raise a new conjecture related to the classical Hosono's version of the diffusive system and our numerical simulations help to confirm it, while our rigorous results only provide a partial answer.

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.

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.

In this paper, we investigate the speed determinacy (or selection mechanism) of traveling waves to a reaction-advection-diffusion stream-population model. We concentrate on how the spreading speed ...

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.

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.

This paper is devoted to investigating the propagation speed and direction of the bistable traveling wave for the Lotka–Volterra competition model with bistable nonlinearity. A range-estimation of the bistable wave speed is provided by comparing the bistable traveling wave with the monostable ones in the system. Comparison theorems on the speed and direction of wave propagation are first established by the method of upper and lower solutions. Through the construction of novel upper and lower solutions, we obtain some new criteria that explicitly determine the speed sign of the bistable traveling wave. Based on these results, we can predict which species will win in the competition for the same resources.

In this paper the minimal-speed determinacy of traveling wave fronts of a two-species competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upper-lower solution technique on the cooperative system to study the traveling waves as well as its minimal-speed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.

In this paper, we study the selection mechanism of the minimal wave speed for traveling waves to an abstract monotone semiflow. A necessary and sufficient condition for the nonlinear selection of t...

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 nonlinearilty., 15 pp

In this paper, we focus on a general state-dependent Kolmogorov predator-prey model subject to non-selective harvesting along with delivery. Certain criteria are established for the existence, non-existence and multiplicity of order-1 impulsive periodic solutions to the system. Based on the geometric phase analysis and the method of Poincare map or successor function with Bendixson domain theory, three typical types of Bendixson domains (i.e., Parallel Domain, Sub-parallel Domain and Semi-ring Domain) are introduced to deal with the discontinuity of the Poincare map or successor function. We incorporate two discriminants $ \Delta_1 $ and $ \Delta_2 $ to link with the existence, non-existence and multiplicity as well as the stability of order-1 periodic solutions. At the same time, we locate the order-1 periodic solutions with the help of three characteristic points and the parameters ratio of delivery over harvesting. The results show that there must exist at least one order-1 periodic solution when the trajectory, that is tangent to the mapping line, can hit the impulsive line. While the trajectory tangent to the mapping line cannot hit the impulsive line, there is not necessary the existence of an order-1 periodic solution, which means the impulsive control may be invalid after finite times stimulation or suppression. In conclusion, we reveal that the delivery can prevent the predator from extinction and stabilize the order-1 periodic solution.

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.

In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness as well as the asymptotic formula of the periodic solutions.

We study the existence, uniqueness, global attractivity and bifurcation of time-periodic patterns for a seasonal phytoplankton model with self-shading effect. By the comparison principle, we obtain the globally asymptotical stability of the zero solution when the principal eigenvalue λ 1 is less than zero. When λ 1 > 0 , by transforming the model into a new system, we successfully prove the conjecture in previous studies on the uniqueness and attractivity of the positive periodic solution. The positive periodic pattern bifurcating from the zero solution is a very interesting phenomenon. Here we apply the Crandall and Rabinowiz's theory to prove rigorously the existence of a bifurcation point. By way of asymptotic analysis, we derive an asymptotic formula for the positive periodic pattern. Based on the solution formula, we find the linear stability of this positive pattern. Finally, we provide a numerical scheme for the calculations of the principal eigenvalue and the simulations of the solution. The simulations corroborate our theoretical analysis.

We study the existence of forced traveling waves and gap formations for the lattice Lotka–Volterra competition system in a shifting habitat. By virtue of upper–lower solution method, we establish that the system admits a forced wave provided that the shifting speed of climate change falls in a certain interval. When the shifting speed is outside the range (except for the two endpoints), gap formations are then shown by theoretical proofs.

This paper is concerned with linear or nonlinear selection mechanism for the minimal speed of traveling wave solutions to a reaction–diffusion–advection equation in a cylindrical domain with Fisher–KPP-type nonlinearity. By using the method of upper and/or lower solutions, we establish the speed selection results. Precisely, we obtain sufficient conditions under which the linear or nonlinear selection is realized when the model is prescribed by Neumann boundary conditions and Dirichlet boundary conditions respectively.

In this paper we re-visit the Lagerstrom problem y + n − 1 r y + eyy = 0, y(1) = 0, y(∞) = 1, where e 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 e → 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 = 1a ndr = ∞. An extension and a discussion of the problem for n ∈ (1, 2) is also given.

For delayed non-local reaction–diffusion equations arising from population biology, selection mechanisms of the speed sign for the bistable travelling wavefront have not been found. In this paper, based on the theory of asymptotic speeds of spread for monotone semiflows, we firstly provide an interval of values of wave speed and a novel general condition for determining the speed sign by applying the comparison principle and the globally asymptotic stability of the bistable travelling wave. Moreover, through constructing novel upper/lower solutions, we give explicit conditions for the speed sign to be positive or negative. The obtained results are efficiently applied to three classical forms of the kernel functions.

In this paper, we derive the conditions for the existence of stationary solutions (i.e., nonconstant steady states) of a volume-filling chemotaxis model with logistic growth over a bounded domain subject to homogeneous Neumann boundary conditions. At the same time, we show that the same system without the chemotaxis term does not admit pattern formations. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, we establish the stability criteria and find a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue. We show that all bifurcations except the one at the first location of the bifurcation parameter are unstable, and if the pattern is stable, then its principal wave mode must be a positive integer which minimizes the bifurcation parameter. For a special case where the carrying capacity is one half, we find a necessary and sufficient condition for th...

In this paper, we study the stability and convergence rate of traveling wavefronts for a nonlocal population model in a periodic habitat \[ \left\{ \begin{array}{ll} \displaystyle\frac{\partial u(t,x)}{\partial t}=D(x)\frac{\partial ^2u(t,x)}{% \partial x^2}-d(x,u(t,x))+\int_R\Gamma (\tau ,x,y)b(y,u(t-\tau ,y))dy, & \\ u(\theta ,x)=\varphi (\theta ,x),\theta \in [-\tau ,0],& \end{array} \right. \] where $D(x), d(x,\cdot ), b(x,\cdot ), \Gamma (\tau ,x,y)$ are L-periodic functions with respect to space $x$ (and $y$) for some positive real constant $L $. Using the analysis of the principal eigenvalue of a non-local linear operator, we show that all noncritical wavefronts are globally exponentially stable, as long as the initial perturbation is uniformly bounded in a weighted space. This result can be generalized to n-dimensional case and three applications of our main results are also presented.

In this paper, we revisit Sherratt’s avascular tumor growth system modeled in terms of the continuum densities of proliferating, quiescent and necrotic cells, together with the consideration of the generic nutrient supply from underlying tissue and cell movement of contact inhibition. By adopting a perturbation method combined with the Banach fixed point theorem, we theoretically justify the existence of the traveling wavefronts for this model.

In this paper a new approach based on a shooting method in a half line coupled with the technique of upper–lower solution pair is used to study the existence and nonexistence of monotone waves for one form of the delayed Fisher equation that does not have the quasimonotonicity property. A necessary and sufficient condition is provided. This new method can be extended to investigate many other nonlocal and non–monotone delayed reaction–diffusion equations.

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 [Formula: see text]. 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 [Formula: see text] 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.

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.

We apply the perturbation method and the Implicit Function Theorem to study the persistence of periodic patterns for two perturbed general biological oscillators: one weakly coupled ordinary differential equation and one reaction–diffusion system. The proof relies also on the formal adjoint equation theory and the Lyapunov–Schmidt method. For the perturbed reaction–diffusion model, spatial–temporal patterns such as rotating waves are shown to exist. Numerical simulations are presented to illustrate our analytical results.

Chemotaxis is an observed phenomenon in which an organism moves preferentially toward a relatively high concentration, which is contrary to the process of natural diffusion. In this paper, we study a volume-filling model of chemotaxis and provide a valid approach to establishing the existence of traveling wavefronts via the Banach fixed point theorem. Rigorous results hold either when the chemotactic sensitivity is relatively small or when the wave speed is large. Numerical simulations are presented to illustrate the main results, and comparisons of wave patterns in different parameters are demonstrated.

Cancer is a disease characterized by abnormal division of cells combined with the malignant behaviour of these cells. Malignant cancer cells tend to spread, either by direct growth into adjacent tissue through invasion and dispersal or by implantation into distant sites by metastasis. In this paper, we will concentrate on the first type of tumour growth and consider a reaction–diffusion model for competition cells with nonlinear diffusion term modelling contact inhibition between normal and tumour cell populations for which wave propagation is usually observed in clinical data. Mathematically, based on a combination of perturbation methods, the Fredholm theory and the Banach fixed point theorem, we theoretically justify the existence of the travelling wave solution. Numerical simulations are finally illustrated to confirm our rigorous results.

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.

In this paper we develop a new method to establish the existence of traveling wavefronts for a food-limited population model with nonmonotone delayed nonlocal effects. Our approach is based on a combination of perturbation methods, the Fredholm theory, and the Banach fixed point theorem. We also develop and theoretically justify Canosa's asymptotic method for the wavefronts with large wave speeds. Numerical simulations are provided to show that there exists a prominent hump when the delay is large. 1. Introduction. There has been some success in establishing the existence of traveling wavefronts for the reaction-diffusion equation with nonlocal delayed nonlin- earity. When the nonlinearity is monotone, the existence of traveling wavefronts can be obtained by extension of the methods of the super/subsolution pair (1), (7), (29), homotopy (3), and Leray-Schauder degree (28). Unfortunately, when the delayed non- linearity is no longer monotone, very little has been achieved (except for the work in (9)). While one suspects that the method developed by Wu and Zou (29) and based on a nonstandard ordering could be applicable, the construction of a supersolution and subsolution pair is nontrivial, and it is almost as difficult as solving the original given equations. In this paper we develop a new approach to establish the existence of trav- eling wavefronts in the case when the delayed nonlinearity is nonmonotone. We shall demonstrate this approach by considering the following food-limited reaction-diffusion equation

An alternative proof is provided for Littlewood's asymptotic expression arising from Lorentz's problem (1911) on the adiabatic invariance of a simple pendulum. Our approach is based on a standard WKB approximation. Our proof is simpler than those of both Littlewood (1963) and Wasow (1973). If the coefficient function in their differential equation is analytic, then Littlewood's asymptotic expression can even be replaced by an exponentially small term. To cite this article: C.H. Ou, R. Wong, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Metastable dynamics for a nonlocal PDE modeling the upwards propagation of a flame-front interface in a vertical channel is analyzed in the one-dimensional case where the channel cross-section is taken to be the slab -1

We consider the spatio‐temporal patterns of disease spread involving structured populations. We start with a general model framework in population biology and spatial ecology where the individual’s spatial movement behaviors depend on its maturation status, and we show how delayed reaction diffusion equations with nonlocal interactions arise naturally. We then consider the impact of this delayed nonlocal interaction on the disease spread by revisiting the spatial spread of rabies in continental Europe during the period between 1945 and 1985. We show how the distinction of territorial patterns between juvenile and adult foxes, the main carriers of the rabies under consideration, yields a class of partial differential equations involving delayed and nonlocal terms that are implicitly defined by a hyperbolic‐parabolic equation, and we show how incorporating this distinction into the model leads to a formula describing the relation of the minimal wave speed and the maturation time of foxes. We show how the ho...

We consider a class of hyperbolic–parabolic equation with time delay and a non-local feedback due to the maturation for the adult population density of a single species population, and we show that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. We show the existence and uniqueness of a travelling wavefront of the hyperbolic–parabolic system in the so-called bistable case, by considering the same problem for a properly parametrized parabolic system, and then by considering the continuous dependence of the wave speed on the parameter involved.