This is the second part of the author in the study of the competition model
where and , denotes a smooth bounded in , , and are positive constants. Moreover, the author supposes that and satisfy homogeneous Dirichlet boundary conditions on . The aim of the author is to understand the effects of the degeneracy of on (1). Here, based on results obtained in Part I [ibid. 181, 92–132 (2002; Zbl 1042.35016)] the author shows that both the classical and generalized steady-states of (1) occur naturally as the limits, when , of the positive classical solutions of the perturbed system
where is a constant.