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Consider the advective Fisher-KPP equation u t = u x x − β u x + f (u) on the half line [ 0 , ∞) with Dirichlet boundary condition at x = 0. In a recent paper [10] , the authors considered the problem without advection (i.e., β = 0) and constructed a new type of entire solution U (x , t) , which, under the additional assumption f ″ (u) ≤ 0 , is concave and U (∞ , t) = 1 for all t ∈ R. In this paper, we consider the equation with advection and without the additional assumption f ″ (u) ≤ 0. In case β = 0 , using a quite different approach from [10] we construct an entire solution U ˜ which is similar as U in the sense that U ˜ (∞ , t) ≡ 1 and U ˜ (⋅ , t) is asymptotically flat as t → − ∞ , but different from U in the sense that it does not have to be concave. Our result reveals that the asymptotically flat (as t → − ∞) property rather than the concavity is more essential for such entire solutions. In case β < 0 , we construct another new entire solution U ˆ which is completely different from the previous ones in the sense that U ˆ (∞ , t) increases from 0 to 1 as t increasing from −∞ to ∞. [ABSTRACT FROM AUTHOR]

In this paper we consider the spreading phenomena in the Fisher-KPP equation in a high dimensional cone with Dirichlet boundary condition. We show that any solution starting from a nonnegative and compact supported initial data spreads and converges to the unique positive steady state. Moreover, the asymptotic spreading speeds of the front in all directions pointing to the opening are c 0 (which is the minimal speed of the traveling wave solutions of the 1-dimensional Fisher-KPP equation). Surprisingly, they do not depend on the shape of the cone, the propagating directions and the boundary condition. [ABSTRACT FROM AUTHOR]

We consider a time-periodic competition system with a free boundary, which can be used to describe two species that are struggling on the free boundary to obtain their own habitats and the final competition result is the regional partition of two species. In [15] , we have studied the (classical) traveling wave solutions of the temporal homogeneous problems. In this paper, we consider the time-periodic case and study periodic traveling wave solutions of the problem. More precisely, we will prove the existence and uniqueness of the periodic traveling wave solution, and show that the speed of the periodic traveling wave is uniquely determined by and monotonically dependent on the time-periodic terms in the system. Moreover, we will present some sufficient conditions to guarantee that the wave travels with a positive speed. [ABSTRACT FROM AUTHOR]

We study a diffusive predator–prey model of Lotka–Volterra type functional response in which both species obey the logistic growth such that the carrying capacity of the predator is proportional to the prey population and the one for prey is a constant. Both continuous and discrete diffusion are addressed. Our aim is to see whether both species can survive eventually, if an alien invading predator is introduced to the habitat of an existing prey. The answer to this question is positive under certain restriction on the parameter. Applying Schauder's fixed point theory with the help of suitable upper and lower solutions, the existence of traveling wave solutions for this model is proven. Furthermore, by deriving the non-existence of traveling wave solutions, we also determine the minimal speed of traveling waves for this model. This provides an estimation of the invasion speed. [ABSTRACT FROM AUTHOR]

The problem of two-dimensional arbitrary amplitude low-frequency electrostatic oscillation in a quasi-neutral quantum plasma is solved exactly by elementary means. In such quantum plasmas we have treated electrons quantum mechanically and ions classically. The exact analytical solution of the nonlinear system exhibits the formation of dark and black solitons. Numerical simulation also predicts the possible periodic solution of the nonlinear system. Nonlinear analysis reveals that the system does have a bifurcation at a critical Mach number that depends on the angle of propagation of the wave. The small-amplitude limit leads to the formation of weakly nonlinear Kadomstev–Petviashvili solitons. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of non-cooperative diffusion–reaction systems, which include prey–predator models and disease-transmission models. The concept of weak traveling wave solutions is proposed. The necessary and sufficient conditions for the existence of such solutions are obtained by the Schauder's fixed-point theorem and persistence theory. The introduction of persistence theory is very technical and crucial. The LaSalle's invariance principle is applied to show that traveling wave solutions connect two equilibria. The nonexistence of traveling wave solutions is proved by introducing a negative one-sided Laplace transform. The results are applied to a prey–predator model and a disease-transmission model with specific interaction functions. We find that the profile of traveling wave solutions may depend on different eigenvalues according to the corresponding condition, which is a new phenomenon. [ABSTRACT FROM AUTHOR]

This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa–Holm equation. The set of these solutions can be parametrized using the wave height a (or “peak-to-peak amplitude”). Our main result establishes monotonicity properties of the map a ⟼ λ ( a ) , i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviors of λ ( a ) , namely monotonicity and unimodality. The key point is to relate λ ( a ) to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems. [ABSTRACT FROM AUTHOR]

This paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatio-temporal delay, i.e., solutions that are defined in the whole space and for all time t ∈ R . We first derive a particular model for a single species and show how such systems arise from population biology. Then we construct some new types of entire solutions other than traveling wave solutions and equilibrium solutions of the equation under consideration with quasi-monotone and non-quasi-monotone nonlinearities. Various qualitative properties of the entire solutions are also investigated. In particular, the relationship between the entire solutions and the traveling wave fronts which they originated is considered. Our main arguments are based on the comparison principle, the method of super- and sub-solutions, and the construction of auxiliary control systems. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study the spatial dynamics of some delayed nonlocal reaction–diffusion systems in whole space. We first establish a series of comparison theorems to investigate the global attractivity of the equilibria for a delayed nonlocal reaction–diffusion system with and without quasi-monotonicity. Then we show that the spreading speed of a general system without quasi-monotone conditions is coincident with the minimal wave speed. Applying a fluctuation method, we further provide some sufficient conditions to ensure the upward convergence of the spreading speed and traveling wave solutions. Finally, we point out the effects of the delay and nonlocality on the spreading speed of the non-quasi-monotone systems. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the entire solutions for a monostable age-structured population model in a 2D lattice strip, i.e., solutions defined in the whole space and for all time . In the quasi-monotone case, we first establish the existence and asymptotic behavior of solutions of the equation without () variable. Combining traveling wave fronts with different speeds and a solution without variable, the existence and qualitative features of entire solutions are then proved. In the non-quasi-monotone case, we introduce two auxiliary quasi-monotone equations and establish a comparison argument for the three systems. Some new entire solutions are then constructed by using the comparison argument, the traveling wave fronts and a solution without variable of the auxiliary equations. [Copyright &y& Elsevier]

Abstract: This paper extends the classical result of Fisher (1937) from the case of two alleles to the case of multiple alleles. Consider a population living in a homogeneous one-dimensional infinite habitat. Individuals in this population carry a gene that occurs in k forms, called alleles. Under the joint action of migration and selection and some additional conditions, the frequencies of the alleles, , satisfy a system of differential equations of the form (1.2). In this paper, we first show that under the conditions is the most fit among the homozygotes, (1.2) is cooperative, the state that only allele A 1 is present in the population is stable, and the state that allele A 1 is absent and all other alleles are present in the population is unstable, then there exists a positive constant, c ⁎, such that allele A 1 propagates asymptotically with speed c ⁎ in the population as . We then show that traveling wave solutions connecting these two states exist for . Finally, we show that under certain additional conditions, there exists an explicit formula for c ⁎. These results allow us to estimate how fast an advantageous gene propagates in a population under selection and migration forces as . Selection is one of the major evolutionary forces and understanding how it works will help predict the genetic makeup of a population in the long run. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the spreading speeds and traveling wave solutions of a nonlocal dispersal equation with degenerate monostable nonlinearity. We first prove that the traveling wave solution with critical minimal speed decays exponentially as , while other traveling wave solutions with do not decay exponentially as . Then the monotonicity and uniqueness (up to translation) of traveling wave solution with critical minimal speed is established. Finally, we prove that the critical minimal wave speed coincides with the asymptotic speed of spread. [Copyright &y& Elsevier]

Abstract: We study the existence of traveling wave solutions for partially degenerate cooperative reaction–diffusion systems that can have three or more equilibria. We show via integral systems that there exist traveling wave solutions in a partially degenerate reaction–diffusion system with speeds above two well-defined extended real numbers. We prove that the two numbers are the same and may be characterized as the spreading speed as well as the slowest speed of a class of traveling wave solutions provided that the linear determinacy conditions are satisfied. We demonstrate our theoretical results by examining a partially degenerate Lotka–Volterra competition model with advection terms. [Copyright &y& Elsevier]

Abstract: Individual learning and social learning are two primary abilities supporting cultural evolution. Conditions for their evolution have mostly been studied by investigating gene frequency dynamics, which essentially implies constant population size. Predictions from such “static” models may only be of partial relevance to the evolution of advanced individual learning in modern humans, because modern humans have experienced rapid population growth and range expansion during “out-of-Africa.” Here we model the spatial population dynamics of individual and social learners by a reaction–diffusion system. One feature of our model is the inclusion of the possibility that social learners may fail to find an exemplar to copy in regions where the population density is low. Due to this attenuation effect, the invasion speed of social learners is diminished, and various kinds of invasion dynamics are observed. Our primary findings are: (1) individual learners can persist indefinitely when invading environmentally homogeneous infinite space; (2) the occurrence of individual learners at the front may inhibit the spread of social learners. These results suggest that “out-of-Africa” may have driven the evolution of advanced individual learning ability in modern humans. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder''s fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka–Volterra systems. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, by using bifurcation method, we successfully find the Fornberg–Whitham equation has a type of traveling wave solutions called kink-like wave solutions and antikink-like wave solutions. They are defined on some semifinal bounded domains and possess properties of kink waves and anti-kink waves. Their implicit expressions are obtained. For some concrete data, the graphs of the implicit functions are displayed, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation. [Copyright &y& Elsevier]

Abstract: Entire solutions for monostable reaction–diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as in term of appropriate subsolutions and supersolutions. Two models of reaction–diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results. [Copyright &y& Elsevier]