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.


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)