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A differential equation with state-dependent delay from cell population biology
Abstract
We analyze a differential equation with a state-dependent delay that is
implicitly defined via the solution of an ODE. The equation describes an
established though little analyzed cell population model. Based on theoretical
results of Hartung, Krisztin, Walther and Wu we elaborate conditions for the
model ingredients, in particular vital rates, that guarantee the existence of a
local semiflow. Here proofs are based on implicit function arguments.
To show global existence, we adapt a theorem from a classical book on
functional differential equations by Hale and Lunel, which gives conditions
under which - if there is no global existence - closed and bounded sets are
left for good, to the -topology, which is the natural setting when dealing
with state-dependent delays. The proof is based on an older result for
semiflows on metric spaces.
... For information on general results and techniques for DDEs we refer to [7,6]. The present article is an application of a recent finding from [10] to a generalized population model for stem cell growth that has been introduced in [8] and has subsequently been studied in articles such as [11,4]. We will prove well-posedness of the mathematical model as a system of DDEs. ...
... In doing so we improve on previous results. Indeed in [11] substantially more regularity on the individual functions is required, whereas in [4], complementing [11], the structural hypothesis of positivity needs to be imposed on initial prehistories. Here we simultaneously generalize both of these results. ...
... In doing so we improve on previous results. Indeed in [11] substantially more regularity on the individual functions is required, whereas in [4], complementing [11], the structural hypothesis of positivity needs to be imposed on initial prehistories. Here we simultaneously generalize both of these results. ...
We present an application of recent well-posedness results in the theory of delay differential equations for ordinary differential equations arXiv:2308.04730 to a generalized population model for stem cell maturation. The weak approach using Sobolev-spaces we take allows for a larger class of initial prehistories and makes checking the requirements for well-posedness of such a model considerably easier compared to previous approaches. In fact the present approach is a possible means to guarantee that the solution manifold is not empty, which is a necessary requirement for a -approach to work.
... Ultimately this is based on the contraction mapping principle in C 1 with a sensible analysis of the constants involved. In applications, checking the individual conditions can be quite technical, see, e.g., [9]. ...
... In Section 5 one finds a suitable generalisation to functional differential equations giving more details and a proof for Theorem 1.1. By means of applications in Section 6 we illustrate our theory with a classical example from [24] and an equation from mathematical biology taken from [9,4]. In this section, we will particularly show how the assumptions on unique existence of solutions in [4] can be used to obtain a local solution theory also within the present situation. ...
... In this section, we revisit the example from cell population biology, which has been analysed in [9,4]. Here, we will use the slightly more general theorem on the solution theory for a class of functional differential equations, see Theorem 5.1. ...
Classically, solution theories for state-dependent delay equations are developed in spaces of continuous or continuously differentiable functions. In applications, these two approaches naturally offer some limitations. The former can be technically challenging to apply in as much as suitably Lipschitz continuous extensions of mappings onto (open subsets of) the space of continuous functions are required; whereas the latter approach leads to potentially undue restrictions on the class of initial pre-histories. Here, we establish a solution theory (i.e., unique existence) for state-dependent delay equations for arbitrary Lipschitz continuous pre-histories and suitably Lipschitz continuous right-hand sides on the Sobolev space . The provided solution theory is independent of previous ones and is based on the contraction mapping principle on exponentially weighted spaces yielding global in time well-posedness right away. In particular, initial pre-histories are not required to belong to certain solution manifolds and the generality of the approach permits the consideration of a large class of functional differential equations even when the continuity of the right-hand side has constraints on the derivative. The solution theory provided implies the classical Picard--Lindel\"of theorem as a special case.
... The equations have been deduced via integration along the characteristics from a partial differential equation describing the "transport" of a density n(t, x) over the progenitor cell maturity x ∈ [x 1 , x 2 ]. See [7] and references therein for the latter and modeling background and [6] for biological background. We will refer to (1.1-1.4) as the cell SD-DDE. ...
... For discussion of the results we will also refer to C 1 := C 1 ([−h, 0], R n ), endowed with its standard norm defined by φ 1 := φ + φ . In [7] the authors have elaborated conditions to guarantee, via application of results of [13,21], that the cell SD-DDE is well-posed and the solutions define, for n = 2, a semiflow on the solution manifold, a continuously differentiable submanifold of C 1 , and that the semiflow is differentiable in the C 1 -topology. For general SD-DDE differentiability of the semiflow in the C 1 -topology implies a linearized stability theorem, see [13] and [19] for a criterion for, respectively, stability and instability of a supposed equilibrium. ...
... The cell SD-DDE (1.1-1.4) may feature a unique positive equilibrium emerging from the zero equilibrium in a transcritical bifurcation: q may decrease to a negative value and q(0) should increase from negative to positive upon variation of the bifurcation parameter, see [6,8]. A combination of the discussed results of [13,19,7] facilitated a local stability analysis of equilibria for the cell SD-DDE in [8]. ...
We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.
... The equations have been deduced via integration along the characteristics from a partial differential equation describing the "transport" of a density n(t, ξ) over the progenitor cell maturity ξ ∈ [x 1 , x 2 ]. See [9] and references therein for the latter and [7,25,6] for modelling and biological background. ...
... In [9], the smoothness conditions are guaranteed for the functional inducing the cell SD-DDE and, by application of the general result, well-posedness and the existence of a maximal differentiable semiflow has been established for the cell SD-DDE. In the same paper a criterion for global existence for general DDE is established and applied to the cell SD-DDE, such that for the latter also associability of a global differentiable semiflow is proven. ...
... The analytical stability results are based on linearised stability theorems for the cell SD-DDE. These have been established by combining the differentiability of the functional proven in [9] with the linearised stability theorems for general DDE in [14] (stability) and [20] (instability). ...
In this paper we study, at different levels of generality, certain systems of delay differential equations (DDE). One focus and motivation is a system with state-dependent delay (SD-DDE) that has been formulated to describe the maturation of stem cells. We refer to this system as the cell SD-DDE. In the cell SD-DDE, the delay is implicitly defined by a threshold condition. The latter is specified by the time at which the (also implicitly defined) solution of an external nonlinear ordinary differential equation (ODE), which is parametrised by a component of the SD-DDE, meets a given threshold value. We focus on the dynamical properties global asymptotic stability (GAS) of the zero equilibrium, persistence and dissipativity/ultimate boundedness.
... By guaranteeing differentiability of the functional inducing the DDE and an application of a theoretical result for SD-DDE of Walther (2003), Getto and Waurick (2016) showed that the model is well posed and that a linear variational equation can be associated with the solutions. The stem cell model has a trivial equilibrium and a positive equilibrium, which emerges from the trivial in a transcritical bifurcation. ...
... Regarding analytical methods, in this paper we combine the linear variational equation derived by Getto and Waurick (2016) with theoretical results on linearised stability for SD-DDE by Hartung et al. (2006) and Stumpf (2016), to guarantee that the principle of linearised stability holds for the stem cell model in the DDE formulation. Then, we use the linear variational equation to derive characteristic equations for arbitrary, trivial and positive equilibria, respectively. ...
... We consider a model for cell maturation studied in PDE formulation by Doumic et al. (2011) and in DDE formulation by Getto and Waurick (2016). A schematic illustration of the model is given in Fig. 1. ...
We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.
... In [7], the authors have elaborated conditions to guarantee via application of results of [13,22] that the solutions of the cell population equation define a differentiable semiflow on the solution manifold, for n = 2 a submanifold of C 1 := C 1 ([−h, 0], R n ). An advantage of the approach in [13,22] is the associability of a linear variational equation, from which a characteristic equation, which allows to analyze local stability of equilibria, can be deduced. ...
... As in many fixed point arguments, also in [14] convexity and compactness of the domain is used, properties the solution manifold in general does not have. Next, note that differentiability of the semiflow in the C 1 -topology as established in [7] implies continuous dependence on initial values in C 1 , i.e., convergence of sequences of solution segments in C 1 , if sequences of initial histories converge in C 1 . The latter however can appear as too strong in applications, see again the discussion section. ...
... As common in delay differential equations (DDE) we use the notation x t (s) := x(t + s), s < 0, for functions x defined in t + s ∈ R. The system describes the dynamics of a stem cell population (w) regulated by the mature cell population (v). We refer to [7] and references therein, in particular [6], for biological background of the model. The SD-DDE can be deduced via integration along the characteristics from a partial differential equation of transport type which features a progenitor cell maturity density and maturity structure, see [7]. ...
We establish variants of existing results on existence, uniqueness and continuous dependence for a class of delay differential equations (DDE). We apply these to continue the analysis of a differential equation from cell biology with state-dependent delay, implicitly defined as the time when the solution of a nonlinear ODE, that depends on the state of the DDE, reaches a threshold. For this application, previous results are restricted to initial histories belonging to the so-called solution manifold. We here generalize the results to a set of nonnegative Lipschitz initial histories which is much larger than the solution manifold and moreover convex. Additionally, we show that the solutions define a semiflow that is continuous in the state-component in the topology, which is a variant of established differentiability of the semiflow in . For an associated system we show invariance of convex and compact sets under the semiflow for finite time.
... If the matrix is negative definite then the constructed functional satisfies all conditions of Theorem and therefore the zero solution of the equation (4) is asymptotically mean square stable. Using the matrices and defined in (6), one can represent the matrix in the form (12) Via Schur complement the matrix is negative definite if and only if the LMI (6) holds. The proof is completed. ...
... Remark 2.1 Using Lemma with another representation for a and it is possible to get another LMI in Theorem 2.1. For example, putting for and for , we have Choosing the additional functional in the form instead of (12) and (6) we obtain the matrix and the LMI in another form (13) Using other different representations for a and b, one can get other different forms of the LMI in Theorem . ...
This chapter is devoted to stability investigation of systems with state-dependent delays under stochastic perturbations. Sufficient conditions of asymptotic mean square stability for the zero solution of a linear stochastic differential equation with distributed delays are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Besides delay-independent and delay-dependent conditions of stability in probability are obtained for two equilibria of a nonlinear stochastic differential equation with delay and exponential nonlinearity. The negative definiteness of matrices in the obtained LMIs is checked using the special MATLAB program. It is noted that the proposed research method can be used for the study of other types of linear and nonlinear systems with state-dependent delays. Numerical simulation of solutions of the considered stochastic differential equations with state-dependent delays illustrate the presented here theoretical results and open to readers attention a new unsolved problem of the obtained stability conditions improving.
... From a modelling perspective the cell cycle duration is a positive time delay between two sequential cell proliferation events. There are two main types of models that incorporate time delays: one involves functional differential equations (Mackey and Rudnicki 1994;Byrneo 1997;Baker et al. 1997Baker et al. , 1998Villasana and Radunskaya 2003;Getto and Waurick 2016;Getto et al. 2019;Cassidy and Humphries 2020), of which delay differential equations are a specific type; and multi-stage models (Yates et al. 2017;Simpson et al. 2018;Vittadello et al. 2018Vittadello et al. , 2019Gavagnin et al. 2019). Models incorporating time delays are consistent with the kinetics of cell proliferation, and can result in a better qualitative and quantitative fit of the model to experimental data (Baker et al. 1998). ...
... Delay differential equations are often used when the evolution of the process to be modelled depends on the history of the process, represented as a time delay which may be discrete (Lu 1991;Engelborghs et al. 2000;Sun 2006), distributed (McCluskey 2010Khasawneh and Mann 2011;Huang et al. 2016;Kaslik and Neamtu 2018;Cassidy and Humphries 2020), or, more generally, state-dependent (Getto and Waurick 2016). We employ a system of two coupled nonlinear delay differential equations to model the transient dynamics of cell proliferation in a population consisting of slowand fast-proliferating cells. ...
We present a novel mathematical model of heterogeneous cell proliferation where the total population consists of a subpopulation of slow-proliferating cells and a subpopulation of fast-proliferating cells. The model incorporates two cellular processes, asymmetric cell division and induced switching between proliferative states, which are important determinants for the heterogeneity of a cell population. As motivation for our model we provide experimental data that illustrate the induced-switching process. Our model consists of a system of two coupled delay differential equations with distributed time delays and the cell densities as functions of time. The distributed delays are bounded and allow for the choice of delay kernel. We analyse the model and prove the nonnegativity and boundedness of solutions, the existence and uniqueness of solutions, and the local stability characteristics of the equilibrium points. We find that the parameters for induced switching are bifurcation parameters and therefore determine the long-term behaviour of the model. Numerical simulations illustrate and support the theoretical findings, and demonstrate the primary importance of transient dynamics for understanding the evolution of many experimental cell populations.
... As a second example we consider a model for cell maturation, originally proposed in [21] and recently studied numerically in [24] using the pseudospectral discretization method applied to the state-dependent delay differential formulation of the model [25]. Cells are divided into stem cells, progenitor cells, which are structured by a maturity indicator x ∈ [x b , x m ], and fully mature cells. ...
... We finally study the convergence of the environmental conditions in the stem cell model, assuming that q, r : R + → R and g, δ : [x b , x m ] × R + → R are continuously differentiable. These assumptions are motivated by the derivation of the linear variational equation associated to equilibria in [25]. In light of Remark 1, since w is considered as environmental condition and the birth rate as inhomogeneous in m, we cannot in principle simplify the constant b from the equation for the birth rate. ...
Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.
... State-dependent delay (SDD) is all around us [19][20][21][22]. With limited natural resources, Antarctic whales and seals tend to mature longer if their populations are large [19]. ...
... In the problem of car following, it is inevitable to encounter the phenomenon that the time delay changes with the state, which contains physiological time delay, mechanical time delay, and motion time delay [20,21]. In addition, in the blood circulation system, the concentration of nutrients regulates the mitotic cycle of hematopoietic stem cells; thus, the mitotic cycle of stem cells is affected by the concentration of cells in the region [22]. In these cases, in order to describe change and evolution of things more accurately and make the research results more realistic and modest with nuanced understanding, we must adopt differential equations with SDD. ...
The differential equations with state-dependent delay are very important equations because they can describe some problems in the real world more accurately. Due to the complexity of state-dependent delay, it also brings challenges to the research. The value of delay varying with the state is the difference between state-dependent delay and time-dependent delay. It is impossible to know exactly in advance how far historical state information is needed, and then the problem of state-dependent delay is more complicated compared with time-dependent delay. The dominating work of this paper is to solve the stability problem of neural networks equipped with state-dependent state delay. We use the purely analytical method to deduce the sufficient conditions for local exponential stability of the zero solution. Finally, a few numerical examples are presented to prove the availability of our results.
... This can be argued successfully, for example, in machining when the tool has nearly infinite stiffness perpendicular to the cutting direction [75], or in laser dynamics where light travels over a fixed distance [40]. On the other hand, in many contexts, including in biological systems and in control problems [9,10,11,21,36,38,68,82], the delays one encounters are not actually constant. In particular, they may depend on the state in a significant way, that is, change dynamically during the time-evolution of the system. ...
We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincar\'e section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds. This allows us to study transitions through resonance tongues and the breakup of a 1:4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.
... The literature on functional differential equations with delay presents some works on integro-differential and integral problems with time integral intervals depending of the state. In particular, we mention the interesting papers [4,10,11,33,34], where some finite-dimensional population models with SDD are studied via an integral equation with state-dependent integration intervals. We also note the papers [6,24,32,36] on integral and integro-differential abstract problems in finite-dimensional spaces and the interesting papers by Angelov et al. [1][2][3], where some finite-dimensional neutral explicit differential equations with SDD are studied via integral equations with state-dependent integration intervals. ...
In this work, we study a new class of integro‐differential equations with delay, where the informations from the past are represented as an average of the state over state‐dependent integration intervals. We establish results on the local and global existence and qualitative properties of solutions. The models presented and the ideas developed will allow the generalization of an extensive literature on different classes of functional differential equations. The last section presents some examples motivated by integro‐differential equations arising in the theory of population dynamics.
... SDD differential equations have applications in population growth models, two-body problem in electrodynamics, delay adoption in neutral networks, hematological disorders, mathematical epidemiology, and queuing processes; see [6][7][8][9][10] and the references therein for a brief of natural processes, which are formulated via SDD equations. For SDD differential equations in finite dimension, we refer to [9,[11][12][13][14]. For abstract evolution equations, we refer to [1, 3-5, 15, 16]. ...
We establish the existence and uniqueness of mild solutions of a first‐order neutral differential equation with state‐dependent delay and almost sectorial operator by using the theory of almost sectorial operators and fixed point theorems and show that under certain conditions mild solution becomes the classical solution. Later, we also discuss the local well‐posedness of the associated Cauchy problem.
... State-dependent delay arises naturally in several areas of scientific interest, including cell biology [17,32], structured population models [13], infectious diseases [45], electromagnetism [12] and turning processes [23]. In this setting, there is a fairly mature theory of solutions, with the most robust perhaps being the solution manifold approach originally developed by Walther [39]. ...
We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds.
... A variety of problems in areas such as physics [1,2], chemistry [3][4][5], economics [6,7], biology [8,9], etc., can be modeled using ordinary differential equations (ODEs), systems of differential equations (SYSODEs) and partial differential equations (PDEs). Due to the importance of differential equations, several methods have appeared in the relevant literature, such as Runge-Kutta methods [10][11][12] or Predictor-Corrector methods [13,14]. ...
A new method for solving differential equations is presented in this work. The solution of the differential equations is done by adapting an artificial neural network, RBF, to the function under study. The adaptation of the parameters of the network is done with a hybrid genetic algorithm. In addition, this text presents in detail the software developed for the above method in ANSI C++. The user can code the underlying differential equation either in C++ or in Fortran format. The method was applied to a wide range of test functions of different types and the results are presented and analyzed in detail.
... But in many cases it is inevitable to consider the statedependent delays in impulsive effects. For example, it is natural that the control signal is transferred less frequently when system states are small and more frequently when states are large [29,20]. Recently, fundamental theory of impulsive differential systems with SDDIs have been reported in [78], and impulsive control and impulsive disturbance problems have been studied in [59,60,124]. ...
This survey addresses stability analysis for impulsive systems with delayed impulses, which constitute an important generalization of delayed impulsive systems. Fundamental issues such as the concept of a solution to an impulsive system with delayed impulses and methods to determine impulse instants are revisited and discussed. In view of the types of delays in impulses, impulsive systems with delayed impulses are classified into two categories including systems with time-dependent delayed impulses and systems with state-dependent delayed impulses. Then more efforts are devoted to the stability analysis of these two classes of impulsive systems, where corresponding Lyapunov-function-based sufficient conditions for Lyapunov stability, asymptotic stability, exponential stability, input-to-state stability and finite-time stability are presented, respectively. Moreover, the double effects of time-dependent delayed impulses on system performance are reemphasized, and recent applications of delayed impulses in synchronization control are discussed in detail. Several challenges are suggested for future works.
... Delay differential equations (DDEs) are very important and useful in many areas of science and engineering, for instance, physics, biomathematics, medicine, economics, chemistry, etc [7,8,13,16]. Firstly, DDEs occur in modeling effects and interactions between cancer cells, namely tumor population [17]. ...
... which is also an important class of FDE and many dynamic phenomena can be characterized by it, see, e.g., population ecology ( [22]), physiology ( [3]), economics ( [18]), engineering ( [19]), materials science ( [29]), electrodynamics ( [8]), neural networks ( [5]) and control theory ( [20]). In particular, when T (t, x(t)) < t in a vicinity of initial time t 0 , SDM-FDE (2) is the well-known state-dependent delay FDE (see, e.g., [2,7,13,16,17,21,24,28,37]). Currently, there also exist many articles on SDM-FDE (2) which mainly study the existence of solutions. ...
In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation \begin{document}\end{document} As \begin{document}\end{document}, this equation can be regarded as a mixed-type functional differential equation with state-dependence \begin{document}\end{document} of a special form but, being a nonlinear operator, \begin{document} n \end{document}-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.
... Les Équations Différentielles Ordinaires correspondent à la forme dÉquations Différentielles la plus utilisée pour construire des modèles biologiques dynamiques (CHASSAGNOLE et al. (2002), TYSON et al. (2003), RIZZI et al. (1997), CHEN et al. (1999) et ZÚÑIGA et al. (2014). Néanmoins des EDP peuvent également être utilisées pour décrire plus finement la cinétique d'un processus dans le temps et dans l'espace (GETTO et WAURICK (2016), FLEGG et al. (2012) et WAYNE et DAVID (2007). ...
Ces travaux de thèse ont montré que les Equations aux Dérivées Partielles (EDP) sont des outils très intéressants pour construire d’efficientes Intelligences Artificielles, et tout particulièrement pour traiter des problématiques liées à l’élevage de précision. L’Assimilation de Données semble être l’outil qui permettra de piloter les élevages de demain. Or pour réaliser de l’Assimilation de Données d’élevage il est nécessaire de disposer d’outils ayant une très forte capacité d’apprentissage et qui soient capables de prédire l’évolution de variables biologiques. Pour atteindre ces objectifs nous avons construit des outils de Statistical Learning biomimétiques basés sur des systèmes d’EDP embarquant l’expression mathématique de processus biologiques clés. Ces équations permettent d’embarquer une modélisation synthétique de la dynamique interne de l’animal. Elles contiennent des paramètres, associés à des facteurs biologiques, qui peuvent être appris sur des données.Au cours de ces travaux de thèse nous avons montré que ces outils se distinguent des outils existants par leur capacité à s’ajuster sur très peu de données, sans surajuster les données d’apprentissage. Les outils construits se distinguent également par leur capacité d’extrapolation et leur capacité à intégrer des informations au cours du temps.
... State-dependent delay arises naturally in problems in various areas, including cell biology [17,32], structured population models [13], infectious diseases [45], electromagnetism [12] and turning processes [23]. In this setting, there is a fairly mature theory of solutions, with the most robust perhaps being the solution manifold approach originally developed by Walther [39]. ...
We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds
... Hou and Guo [14] dealt with the Hopf bifurcation problem for a class of predator-prey equations with state-dependent delayed feedback according to the Hopf bifurcation theory. Recently, Getto and Waurick [5] obtained the existence of a local semiflow and a new general sufficient criterion of saturated existence for a class of differential equations with SD from cell population biology. For more results concerning the application of such differential equations we refer the interested reader to [1,4,9,12,19] and references therein. ...
... Time delays have been incorporated by many researchers into biological models to represent resource regeneration times, maturity periods, feeding times, reaction times, etc. There has been a substantial amount of work related to this topic, as one can see consulting for example [16], [6], [27], [5] and the bibliography therein. ...
In this paper, we are presenting a new method based on operator-valued Fourier multipliers to characterize the existence and uniqueness of ℓp-solutions for discrete time fractional models in the form. δαu(n,x)=Au(n,x)+∑j=1kβju(n-τj,x)+f(n,u(n,x)),n∈Z,x∈Ω⊂RN,βj∈Randτj∈Z, where A is a closed linear operator defined on a Banach space X and δα denotes the Grünwald-Letnikov fractional derivative of order α>0. If X is a UMD space, we provide this characterization only in terms of the R-boundedness of the operator-valued symbol associated to the abstract model. To illustrate our results, we derive new qualitative properties of nonlinear difference equations with shiftings, including fractional versions of the logistic and Nagumo equations.
... More particularly, singularly perturbed delay differential initial or boundary value problems (BVPs) arise in modelling of the study of human pupil light reflex [26], first-exit problems in neurobiology [32], models of physiological processes and diseases [27], optimal control theory [12], optically bistable devices [5] and signal transmission [7], and other models [8]. However, DDEs in the regular case often appear in various phenomena in science and engineering such as dynamical diseases, population ecology, economics, biosciences [4,6,10,11,13,25,33]. Furthermore, some existence and uniqueness of solution to DDEs are discussed in [3,14,15,16,22,23,29]. ...
... Delay differential equations (DDEs) often appear in various phenomena in science and engineering such as dynamical diseases, population ecology, economics, control theory, biosciences [3,8,10,12,14,20,23]. Besides, neutral delay differential equations (NDDEs) are connected to real world life problems. ...
In this paper, the initial-value problem for a linear first order neu- tral delay differential equation is considered. To solve this problem numerically, we construct a fitted difference scheme on a uniform mesh which is succeeded by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. Also, the method is first-order convergent in the discrete maximum norm. Furthermore, numerical illustrations provide support of the theoretical results. © International Journal of Mathematics and Computer Science, 2017.
... Time delays have been incorporated by many researchers into biological models to represent resource regeneration times, maturity periods, feeding times, reaction times, etc. There has been a substantial amount of work related to this topic, as one can see consulting for example [16,6,27,5] and the bibliography therein. ...
... The statement of Theorem 1.1 consists of three parts and we show them in a series of propositions. But before doing so, we prove the following auxiliary result which is of similar type as Theorem 1.6 in Getto & Waurick [4]: Proposition 3.3. Given f as in Theorem 1.1, there is some b > 0 such that if φ ∈ X f and if the associated solution x φ : [−h, t + (φ)) → R n of Eq. (1.1) is bounded by b, that is, x φ t C 1 ≤ b for all 0 ≤ t < t + (φ), then t + (φ) = ∞. ...
In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.
The paper discusses the analytical stability and numerical stability of differential equations with piecewise constant arguments with matrix coefficients. It is proved that the Runge-Kutta method can keep the convergence order. The recurrence relation of the Runge-Kutta method applied to the equations is obtained. Then, the stability conditions of the numerical solution under different matrix coefficients are given by Pade approximation and order star theory. Finally, the conclusions are verified by several numerical experiments.
In physical science, nonlinear singular Lane–Emden and pantograph delay differential equations (LE–PDDEs) have abundant applications and thus are of great interest for the researchers. The presented investigation is related to the development of a new application of intelligent computing for the solution of the LE–PDDEs-based system introduced recently by merging the essence of delay differential equation of Pantograph type and standard second-order Lane–Emden equation. Intelligent computing is exploited through Levenberg–Marquardt backpropagation networks (LMBNs) and Bayesian regularization backpropagation networks (BRBNs) to provide the solutions to nonlinear second-order LE–PDDEs. The performance of design LMBNs and BRBNs is substantiated on three different case studies through comparative analysis from known exact/explicit solutions. The correctness of the designed solvers for LE–PDDEs is further certified by accomplishing through assessment on error histograms, regression measures and index of mean squared error.
Stability of a linear stochastic differential equation with distributed and state-dependent delays is investigated. Sufficient conditions of asymptotic mean square stability are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Numerical simulations illustrate the theoretical results and open a new unsolved problem of the obtained stability conditions improving.
Journal of Applied Mathematics and Computation, 2020, 4(4), 181-188
In this paper, the system of fractional differential equations with delay is studied. The computational method based on the hybrid functions of the general block-pulse functions and Legendre polynomials is presented to obtain the numerical results for the solution of our problem. In fact, the system under study is converted to the corresponding nonlinear algebraic system of equations, which is resolved by using the collocation technique. Also, the error estimate for the suggested computational method that verifies the convergence is presented. Finally, we examine two models arising in engineering and applied science, including the Chen and human immunodeficiency viruses systems, to illustrate the efficiency of the method.
Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; the result is also proved theoretically under some regularity conditions on the model ingredients.
We study a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. In the presence of the state dependence of the two delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We determine the type of the Hopf-Hopf bifurcation points by determining the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it allows us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincar\'e section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1:4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.
We develop a structured population model for the maturation pro-cess of stem cells in the form of a state-dependent delay differential equation. Moreover, results on existence, uniqueness and positivity of solutions as well as conditions of existence for equilibria and representations of these are es-tablished. We give biological interpretations for the conditions of existence of equilibria.
We introduce and analyze several aspects of a new model for cell differentiation. It assumes that differentiation of progenitor cells is a continuous process. From the mathematical point of view, it is based on partial differential equations of transport type. Specifically, it consists of a structured population equation with a nonlinear feedback loop. This models the signaling process due to cytokines, which regulate the differentiation and proliferation process. We compare the continuous model to its discrete counterpart, a multicompartmental model of a discrete collection of cell subpopulations recently proposed by Marciniak-Czochra et al. [Stem Cells Dev., 18 (2009), pp. 377–386] to investigate the dynamics of the hematopoietic system. We obtain uniform bounds for the solutions, characterize steady state solutions, and analyze their linearized stability. We show how persistence or extinction might occur according to values of parameters that characterize the stem cells' self-renewal. We also perform numerical simulations and discuss the qualitative behavior of the continuous model vis à vis the discrete one.
Read More: http://epubs.siam.org/doi/abs/10.1137/100816584
We study two- and three-compartment models of a hierarchical cell production system with cell division regulated by the level of mature cells. We investigate the structure of equilibria with respect to parameters as well as local stability properties for the equilibria. To interpret the results we adapt the concept of reproduction numbers, which is well known in ecology, to stem cell population dynamics. In the two-compartment model, the positive equilibrium is stable wherever it exists. In the three-compartment model, we find that the intermediate stage of differentiation is responsible for the emergence of an instability region in the parameter plane. Moreover, we prove that this region shrinks as the mortality rate for mature cells increases and discuss this result.
We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov–Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state.
We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023-1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman-Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254-274, 1984; de Roos et al. in J Math Biol 28:609-643, 1990) and a model introduced by Gurney-Nisbet (Theor Popul Biol 28:150-180, 1985) and Jones et al. (J Math Anal Appl 135:354-368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.
This paper is as much about a certain modelling methodology, as it is about the constructive definition of future population states from a description of individual behaviour and an initial population state. The key idea is to build a nonlinear model in two steps, by explicitly introducing the environmental condition via the requirement that individuals are independent from one another (and hence equations are linear) when this condition is prescribed as a function of time. A linear physiologically structured population model is defined by two rules, one for reproduction and one for development and survival, both depending on the initial individual state and the prevailing environmental condition. In Part I we showed how one can constructively define future population state operators from these two ingredients. A nonlinear model is a linear model together with a feedback law that describes how the environmental condition at any particular time depends on the population size and composition at that time. When applied to the solution of the linear problem, the feedback law yields a fixed point problem. This we solve constructively by means of the contraction mapping principle, for any given initial population state. Using subsequently this fixed point as input in the linear population model, we obtain a population semiflow. We then say that we solved the nonlinear problem.
Mathematical modeling is a powerful technique to address key questions and paradigms in a variety of complex biological systems and can provide quantitative insights into cell kinetics, fate determination and development of cell populations. The chapter is devoted to a review of modeling of the dynamics of stem cell-initiated systems using mathematical methods of ordinary differential equations. Some basic concepts and tools for cell population dynamics are summarized and presented as a gentle introduction to non-mathematicians. The models take into account different plausible mechanisms regulating homeostasis. Two mathematical frameworks are proposed reflecting, respectively, a discrete (punctuated by division events) and a continuous character of transitions between differentiation stages. Advantages and constraints of the mathematical approaches are presented on examples of models of blood systems and compared to patients data on healthy hematopoiesis.
We develop global Hopf bifurcation theory of differential equations with state-dependent delay using the S1S1-equivariant degree and investigate a two-degree-of-freedom mechanical model of turning processes. For the model of turning processes we show that the extreme points of each vibration component of the non-constant periodic solutions can be embedded into a manifold with explicit algebraic expression. This observation enables us to establish analytical upper and lower bounds of the amplitudes of the periodic solutions in terms of the system parameters and to exclude certain periods. Using the achieved global bifurcation theory we reveal that if the relative frequency between the natural frequency and the turning frequency varies in a certain interval, then generically every bifurcated continuum of periodic solutions must terminate at a bifurcation point. This termination means that the underlying system with parameters in the stability region near the vertical asymptotes of the stability lobes is less subject to chatter. In the process, several sufficient conditions for the non-existence of non-constant periodic solutions are also obtained.
We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C0, 1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall–Liggett type for the semigroup is obtained in a dense subset of (C, ‖·‖∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations.
We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators ([25]) to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed.
We study the singularly perturbed state-dependent delay-differential equation(⁎)εx˙(t)=−x(t)−kx(t−r),r=r(x(t))=1+x(t), which is a special case of the equationεx˙(t)=g(x(t),x(t−r)),r=r(x(t)). One knows that for every sufficiently small ε>0, Eq. (⁎) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as ε→0. In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size O(ε) for small ε. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small ε.
We consider a class of autonomous delay-differential equationsz˙(t)=f(zt) which includes equations of the form(⁎)z˙(t)=g(z(t),z(t−r1),…,z(t−rn)),ri=ri(z(t))for1⩽i⩽n, with state-dependent delays ri(z(t))⩾0. The functions g and ri satisfy appropriate smoothness conditions.We assume there exists a periodic solution z=x(t) which is linearly asymptotically stable, namely with all nontrivial characteristic multipliers μ satisfying |μ|1. We prove that the appropriate nonlinear stability properties hold for x(t), namely, that this solution is asymptotically orbitally stable with asymptotic phase, and enjoys an exponential rate of attraction given in terms of the leading nontrivial characteristic multiplier.A principal difficulty which distinguishes the analysis of equations such as (⁎) from ones with constant delays, is that even with g and ri smooth, the associated function f is not smooth in function space. Techniques of Hartung, Krisztin, Walther, and Wu are employed to resolve these issues.
We study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future. The main result is existence of a periodic solution in case the equilibrium is linearly unstable. The proof employs the ejective fixed point principle on a compact convex set K0⊂C([−h,0],R) of Lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite-dimensional submanifold of the space C1([−h,0],R).
This chapter illustrates the recent work on equations with state dependent delays, with emphasis on particular models and on the emerging theory from the dynamical systems point of view. Several new results are presented. The chapter describes examples of differential equations with state dependent delays which arise in physics, automatic control, neural networks, infectious diseases, population growth, and cell production. Some of these models differ considerably from others, and most of them do not look simple. Typically the delay is not given explicitly as a function of what seems to be the natural state variable; the delay may be defined implicitly by a functional, integral or differential equation and should often be considered as part of the state variables.
We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S1-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.
Let h>0, open, and continuously differentiable. If f satisfies two mild additional smoothness conditions then the setis a C1-submanifold of codimension n in , the maximal solutions xφ of the initial value problemsdefine a continuous semiflow F on X, and all operators F(t,·) are continuously differentiable. Their derivatives D2F(t,φ) are given in the usual way by solutions v to the variational equation along xφ, with segments vt in the tangent spaces TF(t,φ)X. The additional conditions on f are motivated by properties of differential equations with state-dependent delay, and are verified for an example.
The paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations. We carry out continuation of local solutions to maximal solutions in a functional analytic setting. For maximal solutions we establish global existence via exponential boundedness and by a contraction argument, adapted to derive uniform existence time. Moreover, within the setting of dual Banach spaces, we derive results on continuous dependence with respect to time and initial state.To achieve generality the paper is organized top down, in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold.
In this paper we consider a class of nonlinear neutral differential equations with state-dependent delays. We study well-posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite-dimensional parameters in a pointwise sense and also in the C- and W1,∞-norms.
Thesis (Ph. D.)--University of Texas at Dallas, 1995. Includes vita. Includes bibliographical references (leaves 169-172).
By numerical continuation of equilibria, we study a size-structured model for the dynamics of a cannibalistic fish population and its alternative resource. Because we model the cannibalistic interaction as dependent on the ratio of cannibal length and victim length, a cannibal experiences a size distribution of potential victims which depends on its own body size. We show how equilibria of the resulting infinite-dimensional dynamical system can be traced with an existing method for numerical continuation for physiologically structured population models. With this approach we found that cannibalism can induce bistability associated with a fold (or, saddle-node) bifurcation. The two stable states can be qualified as 'stunted' and 'piscivorous', respectively. We identify a new ecological mechanism for bistability, in which the energy gain from cannibalism plays a crucial role: Whereas in the stunted population state cannibals consume their victims, on average, while they are very small and yield little energy, in the piscivorous state cannibals consume their victims not before they have become much bigger, which results in a much higher mean yield of cannibalism. We refer to this mechanism as the 'Hansel and Gretel' effect. It is not related to any individual 'choice' or 'strategy', but depends purely on a difference in population size distribution. We argue that studying dynamics of size-structured population models with this new approach of equilibrium continuation extends the insight that can be gleaned from numerical simulations of the model dynamics.
- H Amann
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A structured population model of cell differentiation
- M Doumic
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- B Perthame
- J P Zubello
M. Doumic, A. Marciniak-Czochra, B. Perthame, J.P. Zubello, A structured population
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