Periodic solutions of a logistic type population model with harvesting

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