Non-constant positive steady states of the Sel'kov model

This paper deals with the reaction-diffusion system known as the Sel'kov model with the homogeneous Neumann boundary condition. This model has been applied to various problems in chemistry and biology. We first give a priori estimates (positive upper and lower bounds) of positive steady states, and then study the non-existence, bifurcation and global existence of non-constant positive steady states as the parameters λ and θ are varied.     

Loops and branches of coexistence states in a Lotka-Volterra competition model

A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate μ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of μ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.     

Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|^p−2∇u), 1<q?p<∞, f(s)∼|s|^θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.     

Turing pattern of the Oregonator model

The Oregonator model is the mathematical dynamics which describes the Field-Körös-Noyes mechanics of the famous Belousov-Zhabotinskii? reaction. In this work, we establish some fundamental analytic properties of this dynamics and its corresponding steady state. Under various conditions on the parameters and the size of the reactor, we examine the existence and non-existence of non-constant steady states. In particular, for some properly chosen parameter ranges, we prove the occurrence of the Turing pattern generated by this Oregonator model. Our results exhibit interesting and very different roles of the diffusion rates and the reactor in the formation of the Turing pattern. Our mathematical analysis mainly relies on a priori estimates and the topological degree argument.     

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular solutions to the equation −div(|Du|^p−2Du)=|u|^q−1u under the form u(r,θ)=r−βω(θ), r>0, θ∈SN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.     

Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations

We first study the initial value problem for a general semilinear heat equation. We prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low (n?10) or if the steady state is flat enough at infinity: the solution of the heat equation either becomes unbounded as t approaches the lifespan, or eventually stays above or below another bounded radial steady state, depending on if the initial value is above or below the first steady state; moreover, the second steady state must be a constant if n?10.Using this instability result, we then prove that every nonconstant radial steady state of the generalized Fisher equation is a hair-trigger for two kinds of dynamical behavior: extinction and spreading. We also prove more criteria on initial values for these types of behavior. Similar results for a reaction-diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained.     

Pattern formation in the Brusselator system

In the paper, we deal with a reaction-diffusion system well known as the Brusselator model and some improved results for the steady states of this model are presented. We first give an a priori estimates (positive upper and lower bounds) of positive steady states. Then, we obtain the non-existence and existence of positive non-constant steady states as the parameters λ, θ and b are varied, which means some certain conditions under which the pattern formation occurs or not.     

Turing patterns of a strongly coupled predator-prey system with diffusion effects

The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator-prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction-diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction-diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability.     

Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces

In this paper, we study Keller-Segel systems with fractional diffusion and a nonlocal term. We establish the global existence, uniqueness and stability of solutions for systems with small initial data in critical Besov spaces. Our main tools are the L^p−L^q estimates for in Besov spaces and the perturbation of linearization.     

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).     

Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton–Kantorovich theorem. It allows to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we obtain as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.     

Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge

We study a predator-prey model with Holling type II functional response incorporating a prey refuge under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions depending on the constant m∈(0,1], which provides a condition for protecting (1−m)u of prey u from predation. Moreover, we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions.     

Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ₁(Ω), the solution decays to zero as t→∞; while for λ>λ₁(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions.     

Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

We consider two simple conservative systems of parabolic-elliptic and parabolic-degenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the norm of initial data is small enough, where d 2 is the space dimension, then there is a global (in time) weak solution that stays in all the L^p spaces with max This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L¹) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.Lecture by B. Perthame held at the Presentation of MJM, Milano, October 18, 2002Received: March, 2003     

Existence and stability of nonconstant positive steady states of morphogenesis models

In this paper, we study a one‐dimensional morphogenesis model considered by C. Stinner et al. (Math. Meth. Appl. Sci.2012;35: 445–465). Under homogeneous boundary conditions, we prove the existence of nonconstant positive steady states through local bifurcation theories. Then we rigorously study the stability of these nonconstant solutions when the sensitivity functions are chosen to be linear and logarithmic, respectively. Finally, we present numerical solutions to illustrate the formation of stable inhomogeneous spatial patterns. Our numerical simulations show that this model can develop very complicated and interesting structures even over one‐dimensional finite domains. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics of Nonconstant Steady States of the Sel’kov Model with Saturation Effect

In this paper, we deal with Sel’kov model with saturation law which has been applied to numerous problems in chemistry and biology. We will study the stability of the unique constant steady state, existence and nonexistence of nonconstant steady states of such models. In particular, we prove that Turing pattern may occur when the saturation coefficient is small but will not occur when the coefficient becomes large. Therefore for a Sel’kov model with saturation law, it is the saturation law that determines the formation of spatial patterns.

     

Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p  -Laplacian (1<p<∞$1<p<\infty$) obtained by Matei (2000) [19] and Takeuchi (1998) [22], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet p-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricci curvature bounded from below w.r.t. some point. In the rest of this paper, we derive an upper and lower bound for the heat kernel of geodesic balls of complete manifolds with specified curvature constraints, which can supply new ways to prove the most part of two generalized eigenvalue comparison results given by Freitas, Mao and Salavessa (2013) [9].     

Homoclinic bifurcations at the onset of pulse self-replication

We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray-Scott system: u^″=uv², v^″=v−uv², which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray-Scott model. Specifically, we establish the existence, and study properties, of solution branches in the (α,β)-plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u(x) and v(x), respectively. We prove bounds for these solution branches, study their behavior as α→∞, and establish a series of geometric properties of these branches which are valid throughout the (α,β)-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches.     

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R² with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1.