Global Dynamics of a Novel Delayed Logistic Equation Arising from Cell Biology

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 nontrivial 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.

Synchronised oscillations in growing cell populations are explained by demographic noise

“…See Section section S.4 of Supplementary Materials for full details. transient-oscillatory regime and asymptotic-exponential regime) is a common feature of many structured growing population [Jafarpour, 2019, Jafarpour et al, 2018, Pirjol et al, 2017, Baker and Röst, 2019. It is not surprising, therefore, that these two phases play distinct but critically important roles in the dynamics of a growing cell population.…”

“…However the individual variations in the cell-cycle time tend to break the correlation of synchronised cells, resulting in a gradual desynchronisation the population ( transient regime). The presence of these two regimes (a transient-oscillatory regime and an asymptotic-exponential regime) is a common feature of many structured growing population [9, 10, 20]. It is not surprising, therefore, that these two phases play distinct but critically important roles in the dynamics of a growing cell population.…”

Convergence of Solutions in a Mean-Field Model of Go-or-Grow Type with Reservation of Sites for Proliferation and Cell Cycle Delay

“…Here we consider a strong simplification of this phenomenon by assuming that (differently from [3]) motile cells stop for a fixed period of time to complete cell division, upon which they immediately switch back into the migratory phenotype. We study the mathematical properties of a mean-field approximation of an individual based model describing this process, and this note complements our other ongoing works [4,5] where we investigate in detail a range of biological hypotheses with the corresponding individual-based as well as mean-field, analytically tractable, models.…”

A Continuous Model of Three Scenarios of the Infection Process with Delayed Immune Response Factors

“…The Hutchinson equation was not derived by Hutchinson himself in 1948, in contrast to what may often be found in publications (old works have been digitalized and are available, and classic works are now possible to read rather than to refer to from habit). Hutchinson [37] briefly outlined the hypothesis that earlier states affect the reproduction efficiency; the idea was not the main idea of the work. The model was ascribed to Wright in [38], while Wright [39] provided a somewhat different form:…”