Upper-lower Solutions Research Articles

The Stability Analysis of Two-Species Competition Model with Stage Structure and Diffusion Terms

In this paper, the author proposed and considered a reaction-diffusion equation with diffusion terms and stage structure. We discussed the stability of the positive equilibrium. By using the upper-lower solutions and monotone iteration technique, we obtained the zero steady state and the boundary equilibrium were linear unstable and the unique positive steady state was globally asymptotic stability. The traditional results are improved and this result applies to broader frameworks.

Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term

The authors of this paper study some singular phenomena (blowing-up or vanishing in finite time) of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation involving the $p(x,t)$-Laplace operator and a nonlinear source. The variable exponent $p(x,t)$ leads to the failure of some techniques, such as upper-lower solutions technique and the scaling method etc., in studying the problem; it also leads to the lack of some valuable properties such as the monotonicity of the energy integral etc. The authors construct a suitable control functional, improve the regularity of the approximate solutions and obtain a new energy inequality to prove that the solution of the problem with a positive initial energy blows up in finite time. Furthermore, under some appropriate conditions, the authors study the vanishing property and the extinction rate estimate of the solutions to the problem by establishing some inequalities the solutions satisfy. It is worth pointing out that the results are obtained with the assumption that $p_{t}(x,t)$ is only negative and integrable which is weaker than those the most of the other papers required.

TRAVELLING WAVES OF A DELAYED PREDATOR-PREY MODEL WITH STAGE STRUCTURE

This paper investigates the existence of travelling wave solutions of a stage-structured predator-prey model with spatial diffusion and time delay. By using the cross iteration method and Schauder's fixed point theorem, we reduce the existence of travelling wave solutions to the existence of a pair of upper-lower solutions.

TRAVELING WAVEFRONTS OF A DELAYED LATTICE REACTION-DIFFUSION MODEL

We investigate a system of delayed lattice differential system which is a model of pioneer-climax species distributed on one dimensional discrete space. We show that there exists a constant $c^*>0$, such that the model has traveling wave solutions connecting a boundary equilibrium to a co-existence equilibrium for $c\geq c^*$. We also argue that $c^*$ is the minimal wave speed and the delay is harmless. The Schauder's fixed point theorem combining with upper-lower solution technique is used for showing the existence of wave solution.

Traveling wave solutions in n-dimensional delayed nonlocal diffusion system with mixed quasimonotonicity

This paper is devoted to the study of an n-dimensional delayed system with nonlocal diffusion and mixed quasimonotonicity. By developing a new definition of upper–lower solutions and a new cross iteration scheme, we establish some existence results of traveling wave solutions. These results are applied to a nonlocal diffusion model which takes the four-species Lotka–Volterra model as its special case.

SMONOTONE METHOD FOR COUPLED SYSTEM OF FINITE DIFFERENCE EQUATIONS OF TIME DEGENERATE PARABOLIC PROBLEMS

The aim of this paper is to develop monotone iterative scheme using the notion of upper-lower solutions for weakly coupled system of finite difference equations which corresponds to weakly coupled system of semilinear time degenerate parabolic Dirichlet initial boundary value problem (IBVP). Using upper and lower solutions as distinct initial iterations, two monotone convergent sequences from linear system are constructed. It is shown that these two sequences converge monotonically from above and below to maximal and minimal solutions respectively which lead to the existence-comparison and uniqueness results for the solution of the discrete nonlinear Dirichlet IBVP. Positivity lemma is the main ingredient used in the proof of these results. AMS Subject Classification: 65Q10

Existence of Traveling Waves for a Delayed SIRS Epidemic Diffusion Model with Saturation Incidence Rate

This paper is concerned with the existence of traveling waves for a delayed SIRS epidemic diffusion model with saturation incidence rate. By using the cross-iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By careful analyzsis, we derive the existence of traveling waves connecting the disease-free steady state and the endemic steady state through the establishment of the suitable upper-lower solutions.

Traveling Wave Solutions for a Delayed SIRS Infectious Disease Model with Nonlocal Diffusion and Nonlinear Incidence

A delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence is investigated. By constructing a pair of upper-lower solutions and using Schauder's fixed point theorem, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.

Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay

This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with Beddington-DeAngelis functional response and a discrete time delay. By introducing a partial quasi-monotonicity condition and constructing a pair of upper-lower solutions, we establish the existence of traveling wave solutions. Moreover, a numerical simulation is carried out to illustrate the theoretical results.

Behavioral analysis of a nonlinear three-staged population model with age-size-structure

We present and analyze a nonlinear egg-juvenile-adult model, in which eggs and juveniles are structured by age, while adults by size. The model consists of three first-order partial differential equations with initial and boundary conditions. Existence and uniqueness of nonnegative solutions to the state system are investigated by the method of characteristics and a comparison principle. Extinction or persistence conditions of the population are obtained via the upper-lower solution technique. Our investigation extends some results in the literature.

EXISTENCE, ASYMPTOTICS AND UNIQUENESS OF TRAVELING WAVES FOR NONLOCAL DIFFUSION SYSTEMS WITH DELAYED NONLOCAL RESPONSE

In this paper, we deal with the existence, asymptotic behavior and uniqueness of traveling waves for nonlocal diffusion systems with delay and global response. We first obtain the existence of traveling wave front by using upper-lower solutions method and Schauder's fixed point theorem for $c\gt c_*$ and using a limiting argument for $c=c_*$. Secondly, we find a priori asymptotic behavior of (monotone or non-monotone) traveling waves with the help of Ikehara's Theorem by constructing a Laplace transform representation of a solution. Thirdly, we show that the traveling wave front for each given wave speed is unique up to a translation. Last, we apply our results to two models with delayed nonlocal response.

Periodic solutions of nonlinear finite difference systems with time delays

In this paper a coupled system of nonlinear finite difference equations corresponding to a class of periodic-parabolic systems with time delays and with nonlinear boundary conditions in a bounded domain is investigated. Using the method of upper-lower solutions two monotone sequences for the finite difference system are constructed. Existence of maximal and minimal periodic solutions of coupled system of finite difference equations with nonlinear boundary conditions is also discussed. The proof of existence theorem is based on the method of upper-lower solutions and its associated monotone iterations. It is shown that the sequence of iterations converges monotonically to unique solution of the nonlinear finite difference system with time delays under consideration.

Sublinear and superlinear Ambrosetti–Prodi problems for the Dirichlet [formula omitted]-Laplacian

We deal with an Ambrosetti–Prodi problem driven by the p-Laplace differential operator, with a “crossing” reaction which can be sublinear or superlinear (in the positive direction). Using variational methods based on the critical point theory, together with upper–lower solutions, truncation and comparison techniques and critical groups, we show the existence of a unique critical parameter value λ∗ such that for λ<λ∗ there are at least two nontrivial solutions, for λ=λ∗ there is at least one nontrivial solution, and for λ>λ∗ no solutions exist. We extend several recent results on this problem.

Traveling wave solutions of a diffusive SI model with strong Allee effect

We investigate a diffusive SI epidemic model with strong Allee effect. We establish the existence of traveling wave solutions connecting a disease-free equilibrium to the co-existence equilibrium for wave speed c>c∗. A pair of upper–lower solutions are constructed delicately such that the application of Schauder fixed point theorem and iteration technique are available in a convex set restricted by the upper-lower solutions.

Influence of weight functions to a nonlocal p-Laplacian evolution equation with inner absorption and nonlocal boundary condition

In this paper, we investigate an initial boundary value problem for a nonlocal p-Laplacian evolution equation with an inner absorption and weighted linear nonlocal boundary and initial conditions. By using the modified comparison principle and the method of upper-lower solution, we find the influence of weighted function on determining blow-up or not of nonnegative solutions.

Existence and asymptotics of traveling wave fronts for a delayed nonlocal diffusion model with a quiescent stage

In this paper, we propose a delayed nonlocal diffusion model with a quiescent stage and study its dynamics. By using Schauder’s fixed point theorem and upper-lower solution method, we establish the existence of traveling wave fronts for speed c⩾c∗(τ), where c∗(τ) is a critical value. With the method of Carr and Chmaj (PAMS, 2004), we discuss the asymptotic behavior of traveling wave fronts and then get the nonexistence of traveling wave fronts for c<c∗(τ).

Monotone traveling waves of a population model with non-monotone terms

In this paper, we first investigate monotone traveling waves for general lattice equations with delays by the monotone iteration technique starting from a lower solution in the set of profiles. Last, our main aim is to establish monotone traveling waves for population models with non-monotone birth functions by choosing a pair of suitable upper-lower solutions, which was left open in some recent works.

Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion

We study a competing pioneer-climax species model with nonlocal diffusion. By constructing a pair of upper-lower solutions and using the iterative technique, we establish the existence of traveling wavefronts connecting the pioneer-existence equilibrium and the coexistence equilibrium. We also discuss the asymptotic behavior of the wave tail for the traveling wavefronts ass=x+ct→−∞.

Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System

The dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey system subject to Neumann boundary conditions are considered. By choosing the ratio of intrinsic growth rates of predators to preys as a bifurcation parameter, the existence and stability of spatially homogeneous and nonhomogeneous Hopf bifurcations and steady state bifurcation are investigated in detail. Meanwhile, we show that Turing instability takes place at a certain critical value; that is, the stationary solution becomes unstable induced by diffusion. Particularly, the sufficient conditions of the global stability of the positive constant coexistence are given by the upper-lower solutions method.

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c>c∗(τ) and using a limiting argument for c=c∗(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ∂2f(0,0)>0 and the delay is independent of the minimal wave speed for ∂2f(0,0)=0.