Structure of positive solutions for quasilinear elliptic systems—degenerate ecological models

We study the degenerate ecological models$$ \matrix{{-\Delta_pu}=a\vert u\vert^{p-2}u\vert1-u\vert^{\alpha-1}(1-u)- b\vert u\vert ^{p-2}uv,& x\in \Omega \cr {-\Delta_qv}=c\vert v\vert^{q-2}v\vert1-v\vert^{\beta-1}(1-v)- d\vert v\vert ^{q-2}uv, & x\in \Omega\cr {u=v=0,}\hfill & {x\in\partial \Omega}}$$where ${p,q>1, {\Delta_pu}={{\rm div}(\vert Du\vert^{p-2}Du)},{{\Delta_q}v={{\rm div}(\vert Dv\vert^{q-2}Dv)}}}, a,b,c,d,\alpha, \beta$ are positive numbers. The structure of positive solutions of the models is discussed via bifurcation theory and monotone techniques. Copyright © 2004 John Wiley & Sons, Ltd.