The convergence of a class of quasimonotone reaction-diffusion systems

It is proved that every solution of the Neumann initial-boundary problem {partial derivativeu(i)/partial derivativet = d(i)Deltau(i) + F-i(u) t > 0, x is an element of Omega, {partial derivativeu(i)/partial derivativen(t, x) = 0 t > 0, x is an element of Omega, i = 1, 2, ..., n, {u(i)(x, 0) = u(i),(0)(x) greater than or equal to 0 x is an element of Omega, converges to some equilibrium, if the system satisfies (i) partial derivativeF(i)/partial derivativeu(j) greater than or equal to 0 for all 1 less than or equal to i not equal j less than or equal to n, (ii) F(u*g(s)) greater than or equal to h(s) * F(u) whenever u is an element of R-+(n) and 0 less than or equal to s less than or equal to 1, where x * y = (x(1)y(1),..., x(n)y(n)) and g,h: [0,1] --> [0,1](n) are continuous functions satisfying g(i)(0) = h(i)(0) = 0, g(i)(1) = h(i)(1) = 1, 0 < g(i)(s), h(i)(s) < 1 for all s is an element of (0,1) and i = 1, 2,..., n, and (iii) the solution of the corresponding ordinary differential equation system is bounded in R-+(n). We also study the convergence of the solution of the Lotka-Volterra system {partial derivativeu(i)/partial derivativet = Deltau(i) + u(i) (r(i) + Sigma (n)(j=1)a(ij)u(j) t > 0, x is an element of Omega, {partial derivativeu(i)/partial derivativen+ alphau(i) = 0 t > 0, x is an element of partial derivative Omega, i = 1, 2,..., n, {u(i)(x,0) = u(i,0)(x) greater than or equal to 0 x is an element of Omega, where r(i) > 0, alpha greater than or equal to 0, and a(ij) greater than or equal to 0 for i not equal j.