Pulsating dynamics of slow-fast population models with delay

TL;DR

This paper investigates how cascades of Hopf bifurcations in slow-fast delayed population models lead to periodic pulsating solutions, with insights into conditions, asymptotics, and scaling near bifurcation thresholds.

Contribution

It demonstrates that cascades of Hopf bifurcations can generate pulsating solutions in population models, extending previous findings from laser models and providing asymptotic analysis.

Findings

Periodic pulses have periods close to the delay time.

Conditions for pulsation depend on bifurcation proximity and competition.

Asymptotic approximations describe the pulsating solutions.

Abstract

We discuss a bifurcation scenario which creates periodic pulsating solutions in slow-fast delayed systems through a cascade of almost simultaneous Hopf bifurcations. This scenario has been previously associated with formation of pulses in a delayed model of mode-locked semiconductor lasers. In this work, through a case study of several examples, we establish that a cascade of Hopf bifurcations can produce periodic pulses, with a period close to the delay time, in population dynamics models and explore the conditions that ensure the realization of this scenario near a transcritical bifurcation threshold. We derive asymptotic approximations for the pulsating solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realization of the bifurcation scenario is highlighted.