S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension

We study exact multiplicity and bifurcation diagrams of positive solutions for a multiparameter spruce budworm population steady-state problem in one space dimension{u″(x)+λ(ru(1−uq)−u21+u2)=0,−10λ>0 is a bifurcation parameter. Assume that either r⩽η1qr⩽η1q and (q,r)(q,r) lies above the curve Γ1={(q,r):q(a)=2a3a2−1,r(a)=2a3(a2+1)2,1