On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.     

The time-periodic diffusive competition models with a free boundary and sign-changing growth rates

To understand the spreading of invasive and native species, in this paper we consider the diffusive competition models with a free boundary in the heterogeneous time-periodic environments, in which the variable intrinsic growth rates of these two species change signs and may be very negative in some large regions. We study the spreading–vanishing dichotomy, long-time dynamical behavior of solution, sharp criteria for spreading and vanishing, and estimates of the asymptotic spreading speed of the free boundary. Moreover, we establish the existence of positive solutions to a T-periodic boundary value problem of the diffusive competition system with sign-changing growth rates in the half line.     

Spreading and vanishing in a diffusive intraguild predation model with intraspecific competition and free boundary

This paper is concerned with the spreading and vanishing phenomena in a diffusive intraguild (IG) predation model with intraspecific competition and free boundary in one dimensional space. The main objective is to obtain the asymptotic behavior of spread of an invasive or new IG prey species via a free boundary. In two cases, we prove a spreading‐vanishing dichotomy for this model, specifically, the IG prey species either successfully spreads to infinity as t→∞ at the front and survives in the new environment or spreads within a bounded area and dies out in the long run. The long time behavior of (R,N,P) and criteria for spreading and vanishing are also obtained. And then, we estimate the asymptotic spreading speed of the free boundary when spreading happens. Besides, two numerical examples are given to illustrate the impacts of initial occupying area and expanding capability on the free boundary.     

Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c^*, the existence of traveling waves with the wave speed c?c^*, and the nonexistence of traveling waves with c<c^* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.     

A diffusive Lotka–Volterra model with Robin boundary condition and sign-changing growth rates in time-periodic environment

In this paper, a Lotka–Volterra model with Robin and free boundary conditions is considered in the heterogeneous time-periodic environment. We mainly consider the changes of local growth rates of native and invasive species that might be negative in some large regions. We study the spreading–vanishing dichotomy. When vanishing occurs, a native species cannot spread successfully as time goes to infinity. However, for an invasive species, in the long run, either it will go extinct or converge to the unique positive solution of time-periodic boundary value problem of logistic equation. When spreading occurs, both native and invasive species have upper and lower bounds. We also obtain the criteria for spreading and vanishing, and estimate of the asymptotic spreading speed.     

On a reaction–advection–diffusion equation with Robin and free boundary conditions

ABSTRACT

A reaction–advection–diffusion equation with variable intrinsic growth rate, Robin and free boundary conditions is investigated in this paper. Firstly, we present a spreading–vanishing dichotomy for the asymptotic behavior of the solutions of the equation. Then, we obtain criteria for spreading and vanishing, and get an estimate for the asymptotic spreading speed of the spreading front. Moreover, numerical simulation is also given to illustrate the impact of the expansion capacity on the free boundary.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

A free boundary problem for Aedes aegypti mosquito invasion

An advection–reaction–diffusion model with free boundary is proposed to investigate the invasive process of Aedes aegypti mosquitoes. By analyzing the free boundary problem, we show that there are two main scenarios of invasive regime: vanishing regime or spreading regime, depending on a threshold in terms of model parameters. Once the mortality rate of the mosquito becomes large with a small specific rate of maturation, the invasive mosquito will go extinct. By introducing the definition of asymptotic spreading speed to describe the spreading front, we provide an estimate to show that the boundary moving speed cannot be faster than the minimal traveling wave speed. By numerical simulations, we consider that the mosquitoes invasive ability and wind driven advection effect on the boundary moving speed. The greater the mosquito invasive ability or advection, the larger the boundary moving speed. Our results indicate that the mosquitoes asymptotic spreading speed can be controlled by modulating the invasive ability of winged mosquitoes.     

A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids

This paper deals with a mathematical model describing the cell cycle dynamics and chemotactic driven cell movement in a multicellular tumor spheroid. Tumor cells consist of two types of cells: proliferating cells and quiescent cells, which have different chemotactic responses to an extracellular nutrient supply. The model is a free boundary problem for a nonlinear system of reaction-diffusion-advection equations, where the free boundary is the outer boundary of the spheroid. The free boundary condition is quite novel due to different velocity of two types of cells. The global existence and uniqueness of solutions to the model is proved. The proof is based on a fixed point argument, together with the Lp-theory for parabolic equations with the third boundary condition.     

Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment

In this study, we consider the population dynamics of an invasive species and a resident species, which are modeled as a diffusive competition process in a radially symmetric setting with a free boundary. We assume that the resident species undergoes diffusion and growth in Rⁿ${\mathbb{R}}^n$, while the invasive species initially exists in a finite ball, but invades the environment with a spreading front evolving according to a free boundary. When the invasive species is inferior, we show that if the resident species is already well established initially, then the invader can never invade deep into the underlying habitat, thus it dies out before its invading front reaches a certain finite limiting position. When the invasive species is superior, a spreading–vanishing dichotomy holds, and sharp criteria for spreading and vanishing with _d1$d_1$, μ  , and _u0$u_0$ as variable factors are obtained, where _d1$d_1$, μ  , and _u0$u_0$ are the dispersal rate, expansion capacity, and initial number of invaders, respectively. In particular, we obtain some rough estimates of the asymptotic spreading speed when spreading occurs.     

Asymptotic patterns of a structured population diffusing in a two-dimensional strip

In this paper, we derive a population model for the growth of a single species on a two-dimensional strip with Neumann and Robin boundary conditions. We show that the dynamics of the mature population is governed by a reaction–diffusion equation with delayed global interaction. Using the theory of asymptotic speed of spread and monotone traveling waves for monotone semiflows, we obtain the spreading speed c^∗$c^{\ast }$, the non-existence of traveling waves with wave speed 0<c<c^∗$0<c<c^{\ast }$, and the existence of monotone traveling waves connecting the two equilibria for c≥c^∗$c\ge c^{\ast }$.     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

Dynamics for a nonlocal competition system with a free boundary

In this paper, we investigate a nonlocal reaction–diffusion competition model with a free boundary and discuss the long time behavior of species. The main objective is to understand the effect of the nonlocal term in the form of an integral convolution on the dynamics of competing species. Specially, for the weak competition case, when spreading occurs, we provide some sufficient conditions to prove that two competing species stabilize at a positive constant equilibrium state. Furthermore, for the case of successful spreading, we estimate the asymptotic spreading speed of the free boundary.     

Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations

This paper investigates the effects of a degenerate diffusion term in reaction-diffusion models u_t=[D(u)u_x]_x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c∗ and we show the appearance of a sharp-type profile when c=c∗. These results solve recent conjectures formulated by Sánchez-Garduño and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339).     

A diffusive predator-prey model with a protection zone

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue of the Laplacian operator over Ω₀ with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator-prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.     

Weak and strong attractors for the 3D Navier-Stokes system

We study in this paper the asymptotic behaviour of the weak solutions of the three-dimensional Navier-Stokes equations. On the one hand, using the weak topology of the usual phase space H (of square integrable divergence free functions) we prove the existence of a weak attractor in both autonomous and nonautonomous cases. On the other, we obtain a conditional result about the existence of the strong attractor, which is valid under an unproved hypothesis. Also, with this hypothesis we obtain continuous weak solutions with respect to the strong topology of H.