C. V. Pao

On nonlinear reaction-diffusion systems

Abstract This paper presents a qualitative analysis for a coupled system of two reaction-diffusion equations under various boundary conditions which arises from a number of physical problems. The nonlinear reaction functions are classified into three basic types according to their relative quasi-monotone property. For each type of reaction functions, an existence-comparison theorem, in terms of upper and lower solutions, is established for the time-dependent system as well as some boundary value problems. Three concrete physical systems arising from epidemics, biochemistry and engineering are taken as representatives of the basic types of reacting problems. Through suitable construction of upper and lower solutions, various qualitative properties of the solution for each system are obtained. These include the existence and bounds of time-dependent solutions, asymptotic behavior of the solution, stability and instability of nontrivial steady-state solutions, estimates of stability regions, and finally the blowing-up property of the solution. Special attention is given to the homogeneous Neumann boundary condition.

Coexistence and stability of a competition—diffusion system in population dynamics

Abstract The coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper, where the effect of dispersion (transportation) is taken into consideration. The mathematical problem involves a coupled system of Lotka-Volterra-type reaction-diffusion equations together with some initial and boundary conditions, including the Dirichlet, Neumann and third type. Necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given. For the stability problem, general criteria for the stability and instability of a steady-state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters. Also given are necessary and sufficient conditions for the existence of multiple steady-state solutions and the stability or instability of each of these solutions. Special attention is given to the Neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized. It is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable. The stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients.

Monotone iterative methods for finite difference system of reaction-diffusion equations

SummaryThis paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.

Models of genetic control by repression with time delays and spatial effects

Two models for cellular control by repression are developed in this paper. The models use standard theory from compartmental analysis and biochemical kinetics. The models include time delays to account for the processes of transcription and translation and diffusion to account for spatial effects in the cell. This consideration leads to a coupled system of reactiondiffusion equations with time delays. An analysis of the steady-state problem is given. Some results on the existence and uniqueness of a global solution and stability of the steady-state problem are summarized, and numerical simulations showing stability and periodicity are presented. A Hopf bifurcation result and a theorem on asymptotic stability are given for the limiting case of the models without diffusion.

Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions

Abstract This paper is concerned with some dynamical property of a reaction-diffusion equation with nonlocal boundary condition. Under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function, the asymptotic behavior of the time-dependent solution is characterized in relation to a finite or an infinite set of constant steady-state solutions. This characterization is determined solely by the initial function and it leads to the stability and instability of the various steady-state solutions. In the case of finite constant steady-state solutions, the time-dependent solution blows up in finite time when the initial function in greater than the largest constant solution. Also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain.

Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays

Abstract In the Lotka–Volterra competition system with N -competing species if the effect of dispersion and time-delays are both taken into consideration, then the densities of the competing species are governed by a coupled system of reaction–diffusion equations with time-delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to the competing rate constants to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the competing system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system.

Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory

Abstract In a nonlocal reaction-diffusion model in combustion theory the reaction function involves a physical parameter σ which is a measure of the strength of the reaction mechanism. The purpose of this paper is to show the existence of a critical value σ ∗ such that for σ ∗ a unique global time-dependent solution exists and converges to a steady-state solution as t → ∞, and for σ > σ ∗ the solution blows-up in finite time. A characterization as well as upper and lower bounds of σ ∗ are given.

Numerical Analysis of Coupled Systems of Nonlinear Parabolic Equations

This paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method. Three monotone iteration processes for the finite difference system are presented, and the sequences of iterations are shown to converge monotonically to a unique solution of the system, including an existence-uniqueness-comparison theorem. A theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given. Also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary-value problem. An application to a reaction-diffusion model in chemical engineering and combustion theory is given.

Numerical Methods for Semilinear Parabolic Equations

The method of upper-lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions. The idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem. The aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable. These two schemes are modified Jacobi method and Gauss–Seidel method for nonlinear algebraic equations. An advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration. On the other hand, the standard Picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations.

Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions

Abstract The purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction–diffusion equations with nonlocal boundary conditions. Using the finite-difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time-dependent and the steady-state finite-difference systems. Each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence-comparison theorem for the corresponding finite-difference system. The existence-comparison theorems are used to investigate the asymptotic behavior of the discrete time-dependent solution in relation to the discrete maximal and minimal solutions of the steady-state problem. Numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes.