Periodic solutions of a logistic type population model with harvesting

We consider a bifurcation problem arising from population biologydu(t)dt=f(u(t))−εh(t), where f(u)f(u) is a logistic type growth rate function, ε⩾0ε⩾0, h(t)h(t) is a continuous function of period T   such that ∫0Th(t)dt>0. We prove that there exists an ε0>0ε0>0 such that the equation has exactly two T  -periodic solutions when 0<ε<ε00<ε<ε0, exactly one T  -periodic solution when ε=ε0ε=ε0, and no T  -periodic solution when ε>ε0ε>ε0.