Periodic Solutions and Positive Solutions of First and Second Order Logistic Type ODEs with Harvesting

It was recently shown that the nonlinear logistic type ODE with periodic harvesting has a bifurcation on the periodic solutions with respect to the parameter : u′ = f(u)− h(t). Namely, there exists an 0 such that for 0 < < 0 there are two periodic solutions, for = 0 there is one periodic solution, and for > 0 there are no periodic solutions, provided that ∫ T 0 h(t)dt > 0. In this paper we look at some numerical evidence regarding the behavior of this threshold for various types of harvesting terms, in particular we find evidence in the negative for a conjecture regarding the behavior of this threshold value. Additionally, we look at analagous steady states for the reaction-diffusion IBVP with logistic growth and positive harvesting: −u′′ = f(u)− h(u), u(0) = u(L) = 0. Using phase plane arguments we show that there is a threshold value of such that this BVP has no positive solutions.