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.     

Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

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.     

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.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

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.     

Stability of steady-state solutions to a prey-predator system with cross-diffusion

This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions.     

Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation

We study the global existence of solutions to a parabolic-parabolic system for chemotaxis with a logistic source in a two-dimensional domain, where the degradation order of the logistic source is weaker than quadratic. We introduce nonlinear production of a chemoattractant, and show the global existence of solutions under certain relations between the degradation and production orders.     

Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.     

Steady-state solutions of a reaction–diffusion system arising from intraguild predation and internal storage

Intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat. At first, we established the sharp a priori estimates for nonnegative solutions of the system, which assure that all of nonnegative solutions belong to a special cone. The selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions. Secondly, existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone. It turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive.     

Asymptotic behaviour of travelling waves for the delayed Fisher–KPP equation

In this work we study the behaviour of travelling wave solutions for the diffusive Hutchinson equation with time delay. Using a phase plane analysis we prove the existence of travelling wave solution for each wave speed c?2$c?2$. We show that for each given and admissible wave speed, such travelling wave solutions converge to a unique maximal wavetrain. As a consequence the existence of a nontrivial maximal wavetrain is equivalent to the existence of travelling wave solution non-converging to the stationary state u=1$u=1$.     

Existence,uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller–Segel system in the plane

In the present article we consider several issues concerning the doubly parabolic Keller–Segel system  and  in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. In particular, we prove that there exist integral solutions of any mass, provided that ε>0$\epsilon >0$ is sufficiently large. With those results at hand, we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term (α=0)$(\alpha =0)$ the solutions behave like self-similar solutions, while in the presence of the degradation term (α>0)$(\alpha >0)$ the global solutions behave like the heat kernel.     

Positive solutions for a diffusive Bazykin model in spatially heterogeneous environment

In this paper, we investigate a diffusive Bazykin model in a spatially heterogeneous environment. We obtain some results on nonexistence and existence of positive solutions of the model. Moreover, the asymptotic behavior of positive solutions with respect to certain parameters is also studied.     

Bifurcation of positive solutions to scalar reaction–diffusion equations with nonlinear boundary condition

The bifurcation of non-trivial steady state solutions of a scalar reaction–diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.     

On a quasilinear parabolic–elliptic chemotaxis system with logistic source

This paper deals with a quasilinear parabolic–elliptic chemotaxis system with logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. For the case of positive diffusion function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, if the diffusion function is zero at some point, or a positive diffusion function and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.     

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.     

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, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks

The author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks. Previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations. He applies fixed point theorems to prove the existence of the traveling waves. Then, he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves.     

Blowup in infinite time of radial solutions for a parabolic–elliptic system in high-dimensional Euclidean spaces

We consider radial solutions blowing up in infinite time to a parabolic–elliptic system in N$N$-dimensional Euclidean space. The system was introduced to describe the gravitational interaction of particles. In the case where N≥2$N\ge 2$, we can find positive and radial solutions blowing up in finite time. In the present paper, in the case where N≥11$N\ge 11$, we find positive and radial solutions blowing up in infinite time and investigate those blowup speeds, by using the so-called asymptotic matched expansion techniques and parabolic regularity.     

Rates of convergence to zero for a semilinear parabolic equation with a critical exponent

We study nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of Joseph and Lundgren. We establish the rate of convergence to zero of solutions that start from initial data which are near the singular steady state. In the critical case, this rate contains a logarithmic term which does not appear in the supercritical case and makes the calculations more delicate.