Comparison principles for systems that embed in cooperative systems, with applications to diffusive Lotka-Volterra models.(English)Zbl 0885.35015

Let $L_i u_i= f_i(z, u_1,\dots, u_N),\quad i= 1,2,\dots, N,\tag{$$*$$}$ be a weakly coupled differential system, where $$z=(z_1,\dots, z_p)$$ for integral $$p\geq 1$$, and $$L_i$$ is a first- or second-order differential operator which admits a maximum principle. The system $$(*)$$ is said to be cooperative (quasimonotone) if $$\partial f_i/\partial u_j\geq 0$$ for $$i\neq j$$. Under the more general condition that either $$\partial f_i/\partial u_j\geq 0$$ or $$\partial f_i/\partial u_j\leq 0$$ for $$i\neq j$$ and for all $$u\in U\subseteq \mathbb{R}^N$$, the author shows that there is a $$2N$$ cooperative differential system on $$Ux(-U)\subseteq \mathbb{R}^{2N}$$ which is satisfied by $$(u,-u)$$ for $$u$$ a solution of $$(*)$$. Using properties of the $$2N$$ cooperative system, he deduces order preserving results about noncooperative parabolic or elliptic systems of $$N$$ equations. The interaction of spatially distributed populations of different species is then used to illustrate some of the results developed here.

MSC:

 35B50 Maximum principles in context of PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35K57 Reaction-diffusion equations 92D25 Population dynamics (general)