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The delayed logistic equation (also known as Hutchinson's equation orWright's equation) was originally introduced to explain oscillatory phenomenain ecological dynamics. While it motivated the development of a large number ofmathematical tools in the study of nonlinear delay differential equations, italso received criticism from modellers because of the lack of a mechanisticbiological derivation and interpretation. Here we propose a new delayedlogistic equation, which has clear biological underpinning coming from cellpopulation modelling. This nonlinear differential equation includes terms withdiscrete and distributed delays. The global dynamics is completely described,and it is proven that all feasible nontrivial solutions converge to thepositive equilibrium. The main tools of the proof rely on persistence theory,comparison principles and an $L^2$-perturbation technique. Using localinvariant manifolds, a unique heteroclinic orbit is constructed that connectsthe unstable zero and the stable positive equilibrium, and we show that thesethree complete orbits constitute the global attractor of the system. Despiteglobal attractivity, the dynamics is not trivial as we can observe long-lastingtransient oscillatory patterns of various shapes. We also discuss thebiological implications of these findings and their relations to other logistictype models of growth with delays.

Global dynamics of a novel delayed logistic equation arising from cell biology