Positive Stationary Solutions Research Articles

A priori estimates and existence of positive stationary solutions for semilinear parabolic equations

We study the boundedness of global solutions to the semilinear parabolic equations with combined nonlinearities. Firstly, we establish a sufficient condition for the initial data using the potential well method, ensuring the global existence of solutions. Subsequently, we proceed to demonstrate an a priori estimate for all global solutions. Finally, the potential well theory, in conjunction with the derived a priori estimate, we show that there is a sub-convergent sequence of the global flow, which converges the positive solution to the corresponding stationary equation.

Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].

Influence of Human Behavior on COVID-19 Dynamics Based on a Reaction-Diffusion Model.

To investigate the influence of human behavior on the spread of COVID-19, we propose a reaction-diffusion model that incorporates contact rate functions related to human behavior. The basic reproduction number is derived and a threshold-type result on its global dynamics in terms of is established. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if ; while there exists a positive stationary solution and the disease is uniformly persistent if . By the numerical simulations of the analytic results, we find that human behavior changes may lower infection levels and reduce the number of exposed and infected humans.

Existence of Pulses for Monotone Reaction-Diffusion Systems

Existence of pulse solutions, that is, positive stationary solutions with zero limit at infinity is studied for monotone reaction-diffusion systems in the bistable case. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray--Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.

Biological populations as stationary distributions in the space of genotypes

In this work, we develop a new model describing population density distribution with respect to genotype considered as a continuous variable. The model represents a reaction-diffusion equation with a special drift term arising due to random mutations of the genotype. We prove the existence and stability of a continuous family of positive stationary solutions of this equation decaying at infinity. The minimal solution of this family resembles a normal distribution with over-exponential decay rate, while all other solutions have a polynomial decay. Furthermore, the minimal, quasi-normal, symmetric stationary solution is an obligatory ultimate destination for each evolutionary process starting with the realistic compact initial data.

Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

<abstract><p>In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $ d_2 $ or cross-diffusion $ d_4 $ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $ d_2 $ and the fractional type cross-diffusion $ d_4 $ can create spatial patterns.</p></abstract>

Dynamics of lineages in adaptation to a gradual environmental change

We investigate a simple quantitative genetics model subjet to a gradual environmental change from the viewpoint of the phylogenies of the living individuals. We aim to understand better how the past traits of their ancestors are shaped by the adaptation to the varying environment. The individuals are characterized by a one-dimensional trait. The dynamics -- births and deaths -- depend on a time-changing mortality rate that shifts the optimal trait to the right at constant speed. The population size is regulated by a nonlinear non-local logistic competition term. The macroscopic behaviour can be described by a PDE that admits a unique positive stationary solution. In the stationary regime, the population can persist, but with a lag in the trait distribution due to the environmental change. For the microscopic (individual-based) stochastic process, the evolution of the lineages can be traced back using the historical process, that is, a measure-valued process on the set of continuous real functions of time. Assuming stationarity of the trait distribution, we describe the limiting distribution, in large populations, of the path of an individual drawn at random at a given time $T$. Freezing the non-linearity due to competition allows the use of a many-to-one identity together with Feynman-Kac's formula. This path, in reversed time, remains close to a simple Ornstein-Uhlenbeck process. It shows how the lagged bulk of the present population stems from ancestors once optimal in trait but still in the tail of the trait distribution in which they lived.

Dynamical analysis on a size-structured population model of Daphnia with delayed birth process

In this paper, we investigate a size-structured population model of Daphnia coupled with an unstructured algal food source. We introduce a delay into the boundary condition of this model, where the delay describes the effect of competition for finding food or resource in short supply juvenile period. Following the semigroup theory, we transform our model into the abstract boundary delay problem and obtain the existence and uniqueness of a positive stationary solution. By means of the spectral analysis and the characteristic equation technique, the instability and linear stability results of stationary solutions are formulated in terms of the basic reproduction number ℛ(F). Some numerical simulation examples are presented to illustrate the feasibility of our main results.

Bifurcation and Turing instability for genetic regulatory networks with diffusion

In this paper, a diffusive genetic regulatory network under Neumann boundary conditions is considered. First, the criteria for the local stability and diffusion-driven instability of the positive stationary solution without and with diffusion are investigated, respectively. Moreover, Turing regions and pattern formation are obtained in the plane of diffusion coefficients. Second, the existence and multiplicity of spatially homogeneous/nonhomogeneous non-constant steady-states are studied by using the Lyapunov–Schmidt reduction. Finally, some numerical simulations are carried out to illustrate the theoretical results.

Fisher–KPP equation with Robin boundary conditions on the real half line

We consider the Fisher–KPP equation with Robin boundary conditions on the half line. We show that this problem has even number of nonnegative stationary solutions, which are ordered, a stable one is sandwiched by two unstable ones. Then we construct several types of entire solutions between them, each entire solution connects an unstable stationary solution with a stable one. In particular, we construct two types of entire solutions Uc(x,t) and U(x,t) connecting 0 and the smallest positive stationary solution. In addition, Uc tend to the traveling wave solution with speed c as t→−∞ in a moving frame, and U(x,t) enjoys convexity. This paper extends the recent results of Lou, Lu and Morita in Lou et al. (2020) from Dirichlet boundary condition to Robin boundary condition.

Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential

We consider the Schrödinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty.

Asymptotic behavior of an age-structured prey-predator system with distributed delay

This paper devotes to the study on the asymptotic behavior of an age-structured prey-predator model with distributed delay. The existence of positive stationary solutions of the considered model is first discussed and the nonlinear system is linearized around a positive stationary solution. Then the resulting linear system is formulated as an abstract non-densely defined Cauchy problem so that the well-posedness is obtained for the system. Following that the problems of locally asymptotic stability/instability, asynchronous exponential growth as well as Hopf bifurcations for the linearized system are investigated with the help of operator semigroups theory and spectral analysis under some conditions. Finally, some numerical simulations are provided to illustrate the obtained results.

Combined effects of nonlinear proliferation and logistic damping in a three-component chemotaxis system for alopecia areata

This work considers the zero Neumann initial–boundary problem to the chemotaxis system ut=Δu−χ1∇⋅(u∇w)+w−μ1u2,vt=Δv−χ2∇⋅(v∇w)+w+ruv−μ2v2,0=Δw+u+v−w,in a bounded domain Ω⊂Rn,n≥1, with smooth boundary, which was initially proposed by Dobreva et al. (2020) to describe the process of alopecia areata lesions that involves the complicated interplay between CD4+ T cells, CD8+ T cells and interferon-gamma. Compared with the well-known Keller–Segel-type systems, the appearance of the nonlinear proliferation term ruv constituted by the product of two immune cell densities along with a new production term w activated by the chemical in the first two equations forms a novel feature of this model.For any given suitably regular initial data, it is firstly shown that if the positive parameters χ1,χ2, μ1,μ2 and r satisfy the assumptions μ1−r2>(n−2)+n(2χ1+χ22)andμ2−r>(n−2)+n(2χ2+χ12),then the problem admits a global and bounded classical solution. Moreover, we identify a set of parameter conditions: μ1<μ2<3μ1,r=μ2−μ1andχ12+χ22<2μ1(3μ1−μ2),under which any global smooth solution will stabilize to the constant coexistence equilibrium as the time goes to infinity. In addition, in the case of μ1>μ2, the nonlinear stability of the unique positive stationary solution to the corresponding ODE system is also discussed.

Positive stationary solutions of convection-diffusion equations for superlinear sources

We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation \[\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,\] for \(x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}\), \(n\gt 2\). The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when \(f(x,\cdot)\) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient \(g\) without radial symmetry.

Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel

&lt;p style='text-indent:20px;'&gt;This paper deals with front propagation for nonlocal monostable equations in spatially periodic habitats. In the authors' earlier works, assuming the existence of principal eigenvalue, it is shown that there are periodic traveling wave solutions to a spatially periodic nonlocal monostable equation with symmetric and compact kernel connecting its unique positive stationary solution and the trivial solution in every direction with all propagating speeds greater than the spreading speed in that direction. In this paper, first assuming the existence of principal eigenvalue, we extend the results to the case that the kernel is asymmetric and supported on a non-compact region. In addition, without the assumption of the existence of principal eigenvalue, we explore the existence of semicontinuous traveling wave solutions.&lt;/p&gt;

Nonlinear nonlocal reaction-diffusion problem with local reaction

&lt;p style='text-indent:20px;'&gt;In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.&lt;/p&gt;

Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal

We are concerned with the persistence of both predator and prey in a diffusive predator-prey system with a climate change effect, which is modeled by a spatial-temporal heterogeneity depending on a moving variable. Moreover, we consider both the cases of nonlocal and local dispersal. In both these situations, we first prove the existence of forced waves, which are positive stationary solutions in the moving frames of the climate change, of either front or pulse type. Then we address the persistence or extinction of the prey and the predator separately in various moving frames, and achieve a complete picture in the local diffusion case. We show that the survival of the species depends crucially on how the climate change speed compares with the minimal speed of some pulse type forced waves.

Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin–Ayala competition model with impulsive disturbance. The conclusion obtained in this paper reduces the conservatism of the algorithm compared with the known literature, for the impulsive disturbance is not limited to impulsive control.

Effects of dispersal and spatial heterogeneity on nonlocal logistic equations

In the present paper, we are interested in the nonlocal dispersal logistic equations in heterogeneous environment. This leads to study the spectrum theory and asymptotic behavior of nonlocal dispersal problems. We establish the existence and uniqueness of positive stationary solutions. By employing continuous lemma and nonlocal estimates, we obtain the limiting behavior of positive solutions when the dispersal rate is small or large. We also analyze the effect of spatial heterogeneity on the long time behavior of evolution equations.

Existence, Uniqueness, and Input-to-State Stability of Ground State Stationary Strong Solution of a Single-Species Model via Mountain Pass Lemma

In this study, the authors utilize mountain pass lemma, variational methods, regularization technique, and the Lyapunov function method to derive the unique existence of the positive classical stationary solution of a single-species ecosystem. Particularly, the geometric characteristic of saddle point in the mountain pass lemma guarantees that the equilibrium point is the ground state stationary solution of the ecosystem. Based on the obtained uniqueness result, the authors use the Lyapunov function method to derive the globally exponential stability criterion, which illuminates that under some suitable conditions, a certain internal competition is conducive to the global stability of the population, and a certain amount of family planning is conducive to the overall stability of the population. Most notably, the regularity technique of weak stationary solution employed in this study can also be applied to some existing literature related with time-delays reaction-diffusion systems for the purpose of regularization of weak solutions. Finally, an illuminative numerical example shows the effectiveness of the proposed methods.