Stable oscillations in single species growth models with hereditary effects

Abstract

We study single species growth models incorporating hereditary effects. Detailed calculations are carried out for a specific model with one delay parameter T as T varies in the entire range T>0. Using perturbation and bifurcation techniques, we show that the effect of the hereditary term is that the equilibrium state, which is stable for small values of T (say T<T₁), is unstable for T₁<T<T₂, and regains its stability for large delays T>T₂. We show that for T₁<T<T₂ a stable oscillatory state, exists which bifurcates from the equilibrium state through an exchange of stability at T=T₁ and T=T₂. Numerical computations and graphs of the solutions are given for the solutions in all ranges of T.