Delay Differential Equations: With Applications In Population Dynamics

Delay partial differential equations arise from variousapplications, like biology, medicine, control theory, climate models, and many others (see e.g. Wu (1996) and the references therein). Their independent variables are time $t$ and one or more dimensional variable $x\ ,$ which usuallyrepresents position in space but may also represent relative DNA content, size of cells, or their maturation level,or other values. The solutions (dependent variables) of delay partial differential equations may represent temperature, voltage, orconcentrations or densities of various particles, for example cells, bacteria,chemicals, animals and so on.

Equation (1) is a parabolic delay partial differential equation with two delays $\tau_1>0$ and$\tau_2>0\ .$It has been proposed by Wang (1963) to consideran automatically controlled furnace, see Fig.1 in Wang (1963). From this reference,the furnace is fed by the material strip that has to be heat-treated witha controlled temperature.The furnace temperature is varied by means of a heater actuated by a heater controller.The control objective is to maintain a desired spatial temperature distributionin the incoming material, which is fed into the furnace by rollers, thespeed of which is regulated by a speed controller. This may be accomplished byplacing temperature transducers along the material strip. The transducersprovide information for a computer, which generates the appropriate control signals for the heater and feed-roller speed controllers.

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Equation (3) is a parabolic delay partial differential equation with an integral over past time intervals $[t-\tau ,t]\ .$ It has been proposed by Green & Stech (1981)for a class of single-species population models, where species can diffuse. From Green & Stech (1981), the unknown function $u(t,x)$ represents a size of a population at a time $t$ and position $x\ .$ The constants $D$ and $r$ are positive and determine rates of diffusion and growth, respectively. The initial and boundary conditions for (3) are illustrated in Figure 3.

In the first step, the partial derivatives with respect to $x$ are replaced by some approximations. For example, application of the finite difference method to (3) means replacing the partial derivative with respect to $x$ by the approximating operator\[\fracu(t,x_i-1)-2u(t,x_i)+u(t,x_i+1)h^2 \approx \frac\partial^2 \partial x^2 u(t,x_i).\]Here, $h$ is a step-size in $x$-direction and $x_i$ are grid-points defined by\[x_i=ih, \quad i=0,1,\dots ,N, \quad h=\frac\piN.\]This discretization in $x$ results in the following system of ordinary delay differential equations:\[\tag16\frac\partial u(t,x_i)\partial t=D\fracu(t,x_i-1)-2u(t,x_i)+u(t,x_i+1)h^2+r u(t,x_i) \Bigg(1-\int_-\tau ^0u(t+s,x_i)d\eta (s)\Bigg),\]

The finite difference methods (e.g. illustrated by (16)) are the most often applied numerical methods for the process of semi-discretization of delay partial differential equations, see e.g. van der Houwen et al (1986), Higham & Sardar (1995), and Zubik-Kowal & Vandewalle (1999). However, the Galerkin finite elements method has been successfully applied to (2) by Rey & Mackey (1993).Moreover, recent papers by Zubik-Kowal (2000), Mead & Zubik-Kowal (2005), and Jackiewicz & Zubik-Kowal (2006) show that pseudospectral methods solve delay partial differential equations with exponential accuracy; that is, their errors decay at exponential rates. Therefore, the accuracies of the finite difference and finite elements methods do not even come close to this exponential accuracy. Pseudospectral methods for non-delay and non-functional partial differential equations are investigated e.g. by Canuto et al. (1988).

Numerical methods for delay partial differential equations bring specific difficulties, which do not appear for equations without delays. For example, because of the delays in(1), previously computed approximations to $u(\eta ,\xi)\ ,$ for all $\xi $ and$\bart-\max\ \tau_1,\tau_2\\leq \eta \leq \bart\ ,$ have to be stored so that the next approximations for $t\geq \bart$ can be computed. Since the transformed $x$-variable in (2) may not be included in the $x$-mesh,additional approximations have to be constructed for $u(t,\alpha x)\ .$ The integrals in (3) have to be approximated for all grid-points in the $(t,x)$-domain.The infinite domain of the integration in (5) has to be decomposed by using the structure of the kernel $\gamma $ so that the infinite integrals can be approximated by integrals over finite intervals. More details about numerical solutions for (4)-(5) are described by Zubik-Kowal (2006).

Delay Differential Equations emphasizes the global analysis of full nonlinear equations or systems. The book treats both autonomous and nonautonomous systems with various delays. Key topics addressed are the possible delay influence on the dynamics of the system, such as stability switching as time delay increases, the long time coexistence of populations, and the oscillatory aspects of the dynamics. The book also includes coverage of the interplay of spatial diffusion and time delays in some diffusive delay population models. The treatment presented in this monograph will be of great value in the study of various classes of DDEs and their multidisciplinary applications.

There is no doubt that some of the recent developments in the theory of DDEs have enhanced our understanding of the qualitative behavior of their solutions and have many applications in mathematical biology and other related fields. Both theory and applications of DDEs require a bit more mathematical maturity than their ODEs counterparts. The mathematical description of delay dynamical systems will naturally involve the delay parameter in some specified way. Nonlinearity and sensitivity analysis of DDEs have been studied intensely in recent years in diverse areas of science and technology, particularly in the context of chaotic dynamics [8, 9].

This special issue aims at creating a multidisciplinary forum of discussion on recent advances in differential equations with memory such as DDEs or fractional-order differential equations (FODEs) in biological systems as well as new applications to economics, engineering, physics, and medicine. It provides an opportunity to study the new trends and analytical insights of the delay differential equations, existence and uniqueness of the solutions, boundedness and persistence, oscillatory behavior of the solutions, stability and bifurcation analysis, parameter estimations and sensitivity analysis, and numerical investigations of solutions.

In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:[1]

Delay differential equations (DDEs) arise in a variety of areas, notably population dynamics, epidemiology, and control theory [1,2]. DDEs are typically first order initial value problems of the form $x'(t) = f(t, x(t), x(\tau(t)))$ with $\tau(t) \le t$, but there are also applications involving other delay types, as well as second- and higher-order DDEs and DDE boundary value problems [3,4].

The biophysics of an organism span multiple scales from subcellular to organismal and include processes characterized by spatial properties, such as the diffusion of molecules, cell migration, and flow of intravenous fluids. Mathematical biology seeks to explain biophysical processes in mathematical terms at, and across, all relevant spatial and temporal scales, through the generation of representative models. While non-spatial, ordinary differential equation (ODE) models are often used and readily calibrated to experimental data, they do not explicitly represent the spatial and stochastic features of a biological system, limiting their insights and applications. However, spatial models describing biological systems with spatial information are mathematically complex and computationally expensive, which limits the ability to calibrate and deploy them and highlights the need for simpler methods able to model the spatial features of biological systems.

We developed and demonstrate a method for generating spatiotemporal, multicellular models from non-spatial population dynamics models of multicellular systems. We envision employing our method to generate new ODE model terms from spatiotemporal and multicellular models, recast popular ODE models on a cellular basis, and generate better models for critical applications where spatial and stochastic features affect outcomes.

The ability to derive cell-based, spatiotemporal models from ordinary differential equation (ODE) models would enhance the utility of both types of models. Cell-based, spatiotemporal models can explicitly describe cellular and spatial mechanisms neglected by ODE models that affect the emergent dynamics and properties of multicellular systems, such as the influence of dynamic aggregate shape on diffusion-limited growth dynamics [16] and individual infected cells on the progression of viral infection [17]. Likewise, ODE models can inform cell-based, spatiotemporal models with efficient parameter fitting to experimental data, and can appropriately describe dynamics at coarser scales and distant locales with respect to a particular multicellular domain of interest (e.g., the population dynamics of a lymph node when explicitly modeling local viral infection). One such example is the approach of Murray and Goyal to derive discrete stochastic dynamics from continuous dynamical descriptions using the Poisson distribution in their multiscale modeling work on hepatitis B virus infection [18]. Likewise Figueredo et al. compared derived representations for mechanisms associated with early-stage cancer using non-spatial agent-based, ODE and stochastic differential equation modeling approaches and demonstrated the feasibility of generating equivalent mechanistic models [19]. However, to our knowledge, no well-defined general formalism describes systematic translation of models to the cellular scale from coarser scales at which spatially homogeneous, population dynamics models using ODEs appropriately describe a biological system. In the very least, the lack of consistent translation of model terms and parameters between spatial and non-spatial models severely inhibits the potential to apply the vast amount of available information and resources in non-spatial modeling, such as those available in BioModels [20], to spatial contexts, and to share validated, reproducible spatial models [21].

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