It is proved that every solution of the Neumann initial-boundary problem
[formula here]
converges to some equilibrium, if the system satisfies
(i) ∂Fi/∂uj [ges ] 0 for all
1 [les ] i ≠ j [les ] n,
(ii) F(u * g(s)) [ges ] h(s) [midast ] F(u)
whenever u ∈ ℝn+ and 0 [les ] s [les ] 1, where x * y
= (x1y1, …, xnyn)
and g, h : [0, 1] → [0, 1]n are
continuous functions satisfying gi(0)
= hi(0) = 0, gi(1) = hi(1)
= 1, 0 < gi(s); hi(s) < 1
for all s ∈ (0, 1) and i = 1, 2, …, n, and
(iii) the solution of the corresponding ordinary differential equation system is bounded in
ℝn+. We also study the convergence of the solution of the Lotka–Volterra system
[formula here]
where ri > 0, α [ges ] 0, and
aij [ges ] 0 for i ≠ j.