In this paper we review the concepts of the Gibbs and conditional entropies and examine their dynamic behaviour when the underlying dynamics are described by ordinary differential equations or stochastic differential equations. We then go on to introduce techniques for the analogous examination when the dynamics involve delays and noise. It is found that the effects of stochastic perturbations and/or delayed dynamics may be such that the approach of the entropies to equilibrium are not necessarily monotone and are dependent on system parameters.

We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov-Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable.

We review the behaviour of the Gibbs' and conditional entropies in deterministic and stochastic systems with the added twist of a formulation appropriate for a stochastically perturbed system with delayed dynamics. The underlying question driving these investigations: “Is the origin of the universally observed unidirectionality of time in our universe connected to the behaviour of entropy?" We focus on temporal entropic behaviour with a review of previous results in deterministic and stochastic systems. Our emphasis is on the temporal behaviour of the Gibbs' and conditional entropies as they give equilibrium results in concordance with experimental findings. In invertible deterministic systems both entropies are temporally constant as has been well known for decades. The addition of stochastic perturbations (Wiener process) leads to an indeterminate (either increasing or decreasing) behaviour of the Gibbs' entropy, but the conditional entropy monotonically approaches equilibrium with increasing time. The presence of delays in the dynamics, whether stochastically perturbed or not, leads to situations in which the Gibbs' and conditional entropies evolution can be oscillatory and not monotone, and may not approach equilibrium.

In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.

We present an investigation of stochastic evolution in which a family of evolution equations in $L^1$ are driven by continuous-time Markov processes. These are examples of so-called piecewise deterministic Markov processes (PDMP's) on the space of integrable functions. We derive equations for the first moment and correlations (of any order) of such processes. We also introduce the mean of the process at large time and describe its behaviour. The results are illustrated by some simple, yet generic, biological examples characterized by different one-parameter types of bifurcations.

Although the theory of density evolution in maps and ordinary differential equations is well developed, the situation is far from satisfactory in continuous time systems with delay. This paper reviews some of the work that has been done numerically, the interesting dynamics that have emerged, and the largely unsuccessful attempts that have been made to analytically treat the evolution of densities in differential delay equations. We also present a new approach to the problem and illustrate it with a simple example.

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.

Purpose of Review Hematopoietic stem cells (HSCs) produce all blood cells via a tightly controlled production system. Disruptions to control mechanisms can induce serious disorders, including leukemias. In this review, we provide an overview of how mathematical modelling has contributed to our understanding of normal and pathological HSC biology. Recent Findings Through the increased availability of a variety of experimental and clinical data, new approaches to mathematically modelling HSCs have revealed how clonality is regulated in the hematopoietic system over time, how increasingly clonal hematopoietic and leukemic stem cell populations contribute to the development of acute myeloid leukemia, and the mechanisms and kinetics of HSC regulation. Summary Mathematical modelling is a complementary tool to quantitatively explore HSC and hematopoietic regulation. Studies combining experimental, clinical, and theoretical approaches have deepened our understanding of HSC biology and aid future investigations to reveal the mechanisms of HSC maintenance and production.

This entire chapter is taken from an unpublished manuscript (S.R. Taylor, Liouville-like equations and invariant densities for delay differential equations, 2011).

This entire section was originally published in Losson and Mackey (Phys Rev E 52(1):115–128, 1995). In this chapter we examine the potential use of a variety of approximations, or reductions, of a differential delay equation to a system of ordinary differential equations in the first instance, and reducing the delay differential equation to a high-dimensional map in the second.

Consider the augmented differential delay equation initial value problem (with τ ≡ 1) $$\displaystyle \begin {aligned}{} &x'(t) = \begin {cases} \mathcal {G}\big ( x(t) \big ) & t \in [0,1) \\ \mathcal {F}\big ( x(t), x(t-1) \big ) & t \geq 1 \end {cases} \\ &x(0) = x_0, \end {aligned} $$ with $$x(t) \in \mathbb {R}$$ , and suppose that an ensemble of initial values x 0 is specified with density f 0. We would like to derive an evolution equation for the density f(x, t) of the corresponding ensemble of solutions x(t).

This paper summarizes the evidence supporting the classification of cyclic neutropenia as a dynamical disease and periodic chronic myelogenous leukemia is also considered. The unsatisfactory state of knowledge concerning the genesis of cyclic thrombocytopenia and periodic autoimmune hemolytic anemia is detailed.

In spite of the recent focus on the development of novel targeted drugs to treat cancer, cytotoxic chemotherapy remains the standard treatment for the vast majority of patients. Unfortunately, chemotherapy is associated with high hematopoietic toxicity that may limit its efficacy. We have previously established potential strategies to mitigate chemotherapy-induced neutropenia (a lack of circulating neutrophils) using a mechanistic model of granulopoiesis to predict the interactions defining the neutrophil response to chemotherapy and to define optimal strategies for concurrent chemotherapy/prophylactic granulocyte colony-stimulating factor (G-CSF). Here, we extend our analyses to include monocyte production by constructing and parameterizing a model of monocytopoiesis. Using data for neutrophil and monocyte concentrations during chemotherapy in a large cohort of childhood acute lymphoblastic leukemia patients, we leveraged our model to determine the relationship between the monocyte and neutrophil nadirs during cyclic chemotherapy. We show that monocytopenia precedes neutropenia by 3 days, and rationalize the use of G-CSF during chemotherapy by establishing that the onset of monocytopenia can be used as a clinical marker for G-CSF dosing post-chemotherapy. This work therefore has important clinical applications as a comprehensive approach to understanding the relationship between monocyte and neutrophils after cyclic chemotherapy with or without G-CSF support.

For background material see Lasota and Mackey (Chaos, fractals, and noise: Stochastic aspects of dynamics, Applied Mathematical Sciences, vol. 97, Springer-Verlag, New York, 1994).