Bubbles for a Class of Delay Differential Equations

Abstract

We analyze the effect of increasing mortality in usual delayed recruitment models of the form x′(t) = −δx(t) + f (x(t − τ)). We consider constant effort harvesting, and discuss the phenomenon of bubbling, which appears when for parameters δ in an interval I there exists a unique positive stationary point K(δ) in the phase space \({C([-\tau,0],\mathbb{R})}\), and there exist δ1 < δ2 in I such that K(δ) is locally stable for \({\delta\in I{\setminus}[\delta_1,\delta_2]}\), and K(δ) is unstable for \({\delta\in(\delta_1,\delta_2)}\). We give a definition of bubbling in a more abstract setting, and show that it naturally appears in a variety of equations. For some nonlinearities rigorous proofs are available for the description of bubbles, for other nonlinearities we give numerical results to indicate the phenomenon of bubbling.