The Convergence of a Class of Quasimonotone Reaction–Diffusion Systems

It is proved that every solution of the Neumann initial‐boundary problem{∂ u i / ∂ t = d i Δ u i + F i ( u )t > 0 ,   x ∈ Ω ,∂ u i / ∂ n ( t , x ) = 0t > 0 ,   x ∈ ∂ Ω ,   i = 1 , 2 , … , n ,u i ( x , 0 ) = u i , 0( x ) ⩾ 0x ∈ Ω ¯ ,converges to some equilibrium, if the system satisfies (i) ∂ F i /∂ u j ⩾ 0 for all 1 ⩽ i ≠ j ⩽ n , (ii) F ( u * g ( s )) ⩾ h ( s ) * F ( u ) whenever u ∈R + nand 0 ⩽ s ⩽ 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 ∈ (0, 1) and i = 1, 2, …, n , and (iii) the solution of the corresponding ordinary differential equation system is bounded inR + n . We also study the convergence of the solution of the Lotka–Volterra system{∂ u i / ∂ t = Δ u i + u i ( r i + ∑ j = 1 na i ju j)t > 0 ,     x ∈ Ω∂ u i / ∂ n + α u i = 0t > 0 ,     x ∈ ∂ Ω ,     i = 1 , 2 … , n ,u i ( x , 0 ) = u i , 0( x ) ⩾ 0x ∈ Ω ¯ ,where r i > 0, α ⩾ 0, and a ij ⩾ 0 for i ≠ j .