摘要
In this paper we discuss the existence and uniqueness of solutions for the boundary value problem \\\\\\\\\\\\\\\\[ \\\\\\\\\\\\\\\\begin{gathered} \\\\\\\\\\\\\\\\begin{array}{*{20}l} {u''(x) = F(u(x))v(x),} \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ {\\\\\\\\\\\\\\\\lambda v''(x) = - [\\\\\\\\\\\\\\\\kappa F(u(x)) - \\\\\\\\\\\\\\\\theta ]v(x),} \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\end{array} \\\\\\\\\\\\\\\\qquad 0 \\\\\\\\\\\\\\\\leq x \\\\\\\\\\\\\\\\leq 1,\\\\\\\\\\\\\\\\qquad \\\\\\\\\\\\\\\\lambda ,\\\\\\\\\\\\\\\\kappa ,\\\\\\\\\\\\\\\\theta > 0, \\\\\\\\\\\\\\\\hfill \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ u'(0) = 0,\\\\\\\\\\\\\\\\qquad u(1) = 1, \\\\\\\\\\\\\\\\hfill \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ v'(0) = 0,\\\\\\\\\\\\\\\\qquad v'(1) = 0, \\\\\\\\\\\\\\\\hfill \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\end{gathered}\\\\\\\\\\\\\\\\] which arises in microbial ecology. The growth rate $F(u)$ of bacteria satisfies $F(0) = 0,F'(u) > 0$. We study this problem by using Rabinowitz’s global bifurcation theorem and the maximum principle.