Abstract
The delayed logistic equation (also known as Hutchinson’s equation or Wright’s equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here, we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes terms with discrete and distributed delays. The global dynamics is completely described, and it is proven that all feasible non-trivial solutions converge to the positive equilibrium. The main tools of the proof rely on persistence theory, comparison principles and an L^2-perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit is constructed that connects the unstable zero and the stable positive equilibrium, and we show that these three complete orbits constitute the global attractor of the system. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes. We also discuss the biological implications of these findings and their relations to other logistic-type models of growth with delays.
Highlights
The well-known logistic differential equation N (t) = r N (t) (1 − N (t)/K ) was proposed by Verhulst in 1838 to resolve the Malthusian dilemma of unbounded growth (Bacaër 2011)
Lindström’s equation shares the same dynamical properties as the alternative delayed logistic equation from Arino et al (2006), both generating a monotone semiflow, unlike Hutchinson’s equation. In addition to these biological applications, the delayed logistic equation motivated the development of a large number of analytical and topological tools, including local and global Hopf bifurcation analysis for delay differential equations (Chow and Mallet-Paret 1977; Faria 2006; Hassard et al 1981; Nussbaum 1975), asymptotic analysis Fowler (1982), 3/2-type stability criteria (Ivanov et al 2002; Wright 1955) and the study of slowly oscillatory solutions (Lessard 2010)
Its biological validity has been questioned, despite the fact that it was introduced by Hutchinson to explain observations of oscillatory behaviour in ecological systems
Summary
This model behaves to the classical logistic equation, in the sense that it cannot sustain periodic oscillations; large delays cause the extinction of the population In another recent work, Lindström studied chemostat models with time lag in the conversion of the substrate into biomass (Lindström 2017) and obtained equations of logistic type with delays as a limiting case. Lindström’s equation shares the same dynamical properties as the alternative delayed logistic equation from Arino et al (2006), both generating a monotone semiflow, unlike Hutchinson’s equation In addition to these biological applications, the delayed logistic equation motivated the development of a large number of analytical and topological tools, including local and global Hopf bifurcation analysis for delay differential equations (Chow and Mallet-Paret 1977; Faria 2006; Hassard et al 1981; Nussbaum 1975), asymptotic analysis Fowler (1982), 3/2-type stability criteria (Ivanov et al 2002; Wright 1955) and the study of slowly oscillatory solutions (Lessard 2010).
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