ISSN:

1531-3492

eISSN:

1553-524X

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## Discrete & Continuous Dynamical Systems - B

August 2019 , Volume 24 , Issue 8

Special issue in honor of Dr. Peter E. Kloeden on the occasion of his 70th birthday

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*+*[Abstract](1861)

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**Abstract:**

In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on

*+*[Abstract](1846)

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**Abstract:**

In this paper a drift-randomized Milstein method is introduced for the numerical solution of non-autonomous stochastic differential equations with non-differentiable drift coefficient functions. Compared to standard Milstein-type methods we obtain higher order convergence rates in the

*+*[Abstract](1599)

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**Abstract:**

Numerical techniques for solving some mathematical models related to a mining extraction project under uncertainty are proposed. The mine valuation is formulated as a complementarity problem associated to a degenerate second order partial differential equation (PDE), which incorporates the option to abandon the project. The probability of completion and the expected lifetime of the project are the respective solutions of problems governed by similar degenerated PDE operators. In all models, the underlying stochastic factors are the commodity price and the remaining resource. After justifying the required boundary conditions on the computational bounded domain, the proposed numerical techniques mainly consist of a Crank-Nicolson characteristics method for the time discretization to cope with the convection dominating setting and Lagrange finite elements for the discretization in the commodity and resource variables, with the additional use of an augmented Lagrangian active set method for the complementarity problem. Some numerical examples are discussed to illustrate the performance of the methods and models.

*+*[Abstract](1243)

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**Abstract:**

For pullback attractors of asymptotically autonomous dynamical systems we study the convergences of their components towards the global attractors of the limiting semigroups. We use some conditions of uniform boundedness of pullback attractors, instead of uniform compactness conditions used in the literature. Both forward convergence and backward convergence are studied.

*+*[Abstract](1481)

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**Abstract:**

To estimate the growth rate of matrix products

Given a set

*+*[Abstract](1223)

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**Abstract:**

In this work we improve the result presented by Kloeden-Simsen-Stefanello Simsen in [

*+*[Abstract](1399)

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**Abstract:**

In this paper we obtain the existence of global attractors for the dynamicalsystems generated by weak solution of the three-dimensional Navier-Stokesequations with damping. We consider two cases, depending on the values of the parameter *β* controlling the damping term. First, we prove that for *β*≥4 weaksolutions are unique and establish the existence of the global attractor forthe corresponding semigroup. Second, for 3≤*β*<4 we define amultivalued dynamical systems and prove the existence of the global attractoras well. Finally, some numerical simulations are performed.

*+*[Abstract](1399)

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**Abstract:**

In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter *β* as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter *β* is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.

*+*[Abstract](1292)

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**Abstract:**

This paper is concerned with a class of stochastic differential equations with Markovian switching. The Malliavin calculus is used to study the smoothness of the density of the solution under a Hörmander type condition. Furthermore, we obtain a Bismut type formula which is used to establish the strong Feller property.

*+*[Abstract](1355)

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**Abstract:**

The purpose of this work is to prove the existence and uniqueness of the solution for a Cahn-Hilliard/Allen-Cahn system with singular potentials (and, in particular, the thermodynamically relevant logarithmic potentials). We also prove the existence of the global attractor. Finally, we show further regularity results and we prove a strict separation property (from the pure states) in one space dimension.

*+*[Abstract](1366)

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**Abstract:**

We examine the invariance principle in the stability theory of differential equations, within a general singularly perturbed system. The limit dynamics of such a system is depicted by the evolution of a Young measure whose values are invariant measures of the fast equation. We establish an invariance principle for the limit dynamics, and examine the relations, at times subtle, with the singularly perturbed system itself.

*+*[Abstract](1302)

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**Abstract:**

This work focuses on optimal pollution controls. The main effort is devoted to obtaining approximation methods for optimal pollution control. To take into consideration of random environment and other random factors, the control system is formulated as a controlled switching diffusion. Markov chain approximation techniques are used to design the computational schemes. Convergence of the algorithms are obtained. To demonstrate, numerical experimental results are presented. A particular feature is that computation using real data sets is provided.

*+*[Abstract](1323)

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**Abstract:**

In this paper we study a stationary problem arising from population dynamics with a local and nonlocal variable diffusion coefficient. We show the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global structure of this continuum depends on the value of the nonlocal diffusion at infinity and the relative position of the refuge of the species and of the sets where it diffuses locally and not locally, respectively.

*+*[Abstract](1592)

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**Abstract:**

In this paper, we study a Navier-Stokes delay differential inclusion with time fractional derivative of order

*+*[Abstract](1333)

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**Abstract:**

Convergence of particle systems to the Vlasov-Navier-Stokes equations is a difficult topic with only fragmentary results. Under a suitable modification of the classical Stokes drag force interaction, here a partial result in this direction is proven. A particle system is introduced, its interaction with the fluid is modelled and tightness is proved, in a suitable topology, for the family of laws of the pair composed by solution of Navier-Stokes equations and empirical measure of the particles. Moreover, it is proved that every limit law is supported on weak solutions of the Vlasov-Navier-Stokes system. Open problems, like weak-strong uniqueness for this system and its relevance for the convergence of the particle system, are outlined.

*+*[Abstract](1373)

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**Abstract:**

We give an new proof of the well-known competitive exclusion principle in the chemostat model with

*+*[Abstract](1205)

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**Abstract:**

This work investigates the existence of both martingale and pathwise solutions of the single layer shallow water equations on a bounded domain

*+*[Abstract](1421)

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**Abstract:**

This work establishes local existence and uniqueness as well as blow-up criteria for solutions

holds for

*+*[Abstract](1105)

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**Abstract:**

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium.

The contraction metric is described by a matrix-valued function

*+*[Abstract](1177)

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**Abstract:**

We study the Mandelbrot-van Ness representation of fractional Brownian motion

*+*[Abstract](1509)

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**Abstract:**

The multilevel Monte Carlo path simulation method introduced by Giles (*Operations Research*, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

*+*[Abstract](1212)

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**Abstract:**

In this paper we demonstrate how multiobjective optimal control problems can be solved by means of model predictive control. For our analysis we restrict ourselves to finite-dimensional control systems in discrete time. We show that convergence of the MPC closed-loop trajectory as well as upper bounds on the closed-loop performance for all objectives can be established if the ‘right’ Pareto-optimal control sequence is chosen in the iterations. It turns out that approximating the whole Pareto front is not necessary for that choice. Moreover, we provide statements on the relation of the MPC performance to the values of Pareto-optimal solutions on the infinite horizon, i.e. we investigate on the inifinite-horizon optimality of our MPC controller.

*+*[Abstract](1019)

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**Abstract:**

In this paper we analyze some regularity properties of a double time-delayed 2D-Navier-Stokes model, that includes not only a delay force but also a delay in the convective term. The interesting feature of the model -from the mathematical point of view- is that being in dimension two, it behaves similarly as a 3D-model without delay, and extra conditions in order to have uniqueness were required for well-posedness. This model was previously studied in several papers, being the existence of attractor in the **34** (2014), 4085-4105]. Here regularization properties of the solutions and existence of (regular) attractors for several associated dynamical systems are established. Moreover, relationships among these objects are also provided.

*+*[Abstract](979)

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**Abstract:**

This paper provides a dynamical frame to study non-autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type Ⅱ over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.

*+*[Abstract](1225)

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**Abstract:**

During epithelium tissue maintenance, lineages of cells differentiate and proliferate in a coordinated way to provide the desirable size and spatial organization of different types of cells. While mathematical models through deterministic description have been used to dissect role of feedback regulations on tissue layer size and stratification, how the stochastic effects influence tissue maintenance remains largely unknown. Here we present a stochastic continuum model for cell lineages to investigate how both layer thickness and layer stratification are affected by noise. We find that the cell-intrinsic noise often causes reduction and oscillation of layer size whereas the cell-extrinsic noise increases the thickness, and sometimes, leads to uncontrollable growth of the tissue layer. The layer stratification usually deteriorates as the noise level increases in the cell lineage systems. Interestingly, the morphogen noise, which mixes both cell-intrinsic noise and cell-extrinsic noise, can lead to larger size of layer with little impact on the layer stratification. By investigating different combinations of the three types of noise, we find the layer thickness variability is reduced when cell-extrinsic noise level is high or morphogen noise level is low. Interestingly, there exists a tradeoff between low thickness variability and strong layer stratification due to competition among the three types of noise, suggesting robust layer homeostasis requires balanced levels of different types of noise in the cell lineage systems.

*+*[Abstract](957)

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**Abstract:**

In this work the theory of generalized processes is used to describe the dynamics of a nonautonomous multivalued problem and, through this approach, some conditions for the existence of trajectory attractors are proved. By projecting the trajectory attractor on the phase space, the uniform attractor for the multivalued process associated to the problem is obtained and some conditions to guarantee the invariance of the uniform attractor are given. Furthermore, the existence of the uniform attractor for a class of

*+*[Abstract](1020)

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**Abstract:**

In this paper, we prove that the solution constructed in [

*+*[Abstract](1170)

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**Abstract:**

We study a fractional time porous medium equation with fractional potential pressure. The initial data is assumed to be a bounded function with compact support and fast decay at infinity. We establish existence of weak solutions for which we determine whether the property of compact support is conserved in time depending on some parameters of the problem. Special attention is paid to the property of finite propagation for specific values of the parameters.

*+*[Abstract](1246)

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**Abstract:**

The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative Itô noise is studied. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the trivial steady state. This differs from previous works on the stabilisation by noise of parabolic PDEs, where the noise acts inside the domain and stabilisation typically occurs for an infinite range of noise intensities. To the best of our knowledge, this is the first result on the stabilisation of PDEs by boundary noise.

*+*[Abstract](916)

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**Abstract:**

Causal relationships are important to understand the dynamics of coupled processes and, moreover, to influence or control the effects by acting on the causes. Among the different approaches to determine cause-effect relationships and, in particular, coupling directions in interacting random or deterministic processes, we focus in this paper on information-theoretic measures. So, we study in the theoretical part the difference between directionality indicators based on transfer entropy as well as on its dimensional reduction via transcripts in algebraic time series representations. In the applications we consider specifically the lowest dimensional case, i.e., 3-dimensional transfer entropy, which is currently one of the most popular causality indicators, and the (2-dimensional) mutual information of transcripts. Needless to say, the lower dimensionality of the transcript-based indicator can make a difference in practice, where datasets are usually small. To compare numerically the performance of both directionality indicators, synthetic data (obtained with random processes) and real world data (in the form of biomedical recordings) are used. As happened in previous related work, we found again that the transcript mutual information performs as good as, and in some cases even better than, the lowest dimensional binned and symbolic transfer entropy, the symbols being ordinal patterns.

*+*[Abstract](1316)

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**Abstract:**

Given an unstable hybrid stochastic differential equation (SDE, also known as an SDE with Markovian switching), can we design a *delay* feedback control to make the controlled hybrid SDE become asymptotically stable? The paper [*delay* feedback controls for hybrid SDEs, though the stabilization by *non-delay* feedback controls had been well studied. A critical condition imposed in [*delay* feedback controls in order to stabilise a class of highly nonlinear hybrid SDEs whose coefficients satisfy the polynomial growth condition.

*+*[Abstract](1164)

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**Abstract:**

In this paper, the dynamical behavior of weakly damped wave equations with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then we obtain the existence of the

*+*[Abstract](1119)

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**Abstract:**

We study some time-related properties of the random attractor for the stochastic wave equation on an unbounded domain with time-varying coefficient and force. We assume that the coefficient is bounded and the time-dependent force is backward tempered, backward complement-small, backward tail-small, and then prove both existence and backward compactness of a random attractor on the universe of all backward tempered sets. By using the Egoroff and Lusin theorems, we show the measurability of the absorbing set although it is the union of some random sets over an uncountable index set. Moreover, we obtain the backward compactness of the attractor if the force is periodic, and obtain the periodicity of the attractor if both force and coefficient are periodic.

*+*[Abstract](809)

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**Abstract:**

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution.

We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first,

The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

*+*[Abstract](689)

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**Abstract:**

This paper concerns optimal control of a nonconvex perturbed sweeping process and its applications to optimization of the planar crowd motion model of traffic equilibria. The obtained theoretical results allow us to investigate a dynamic optimization problem for the microscopic planar crown motion model with finitely many participants and completely solve it analytically in the case of two participants.

*+*[Abstract](767)

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**Abstract:**

In this paper we analyze the asymptotic behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating region with reaction terms concentrated in a neighborhood of the oscillatory boundary $\theta_\varepsilon \subset\Omega_{\varepsilon }\subset \mathbb{R}^2$ when a small parameter $\varepsilon >0$ goes to zero. Our main result is concerned with the upper and lower semicontinuity of the set of solutions in $H^1$. We show that the solutions of our perturbed equation can be approximated with one defined in a fixed limit domain, which also captures the effects of reaction terms that take place in the original problem as a flux condition on the boundary of the limit domain.

*+*[Abstract](758)

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**Abstract:**

The *γ*-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic *γ*-basin of attraction of the equilibrium.

*+*[Abstract](719)

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**Abstract:**

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Lévy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.

*+*[Abstract](433)

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**Abstract:**

In this paper, we investigate the convergence in probability of a stochastic symplectic scheme for stochastic nonlinear Schrödinger equation with quadratic potential and an additive noise. Theoretical analysis shows that our symplectic semi-discretization is of order one in probability under appropriate regularity conditions for the initial value and noise. Numerical experiments are given to simulate the long time behavior of the discrete averaged charge and energy as well as the influence of the external potential and noise, and to test the convergence order.

*+*[Abstract](450)

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**Abstract:**

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.

*+*[Abstract](515)

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**Abstract:**

We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS is a matrix-valued random variable whose expectation corresponds to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy of a step, and the entropy production rate of the MC are discussed. These three concepts, as key anchoring points in applications of stochastic dynamics, characterize respectively the uncertainties of a system at instant time

*+*[Abstract](460)

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**Abstract:**

We consider a special discrete time Leslie-Gower competition models for three species:

*+*[Abstract](450)

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**Abstract:**

In such different domains as statistical physics, neurosciences, spin glasses, social science, economics and finance, large ensemble of interacting individuals evolving following (mainstream) or against (hipsters) the majority are ubiquitous. Moreover, in a variety of applications, interactions between agents occur after specific delays that depends on the time needed to transport, transmit or take into account information. This paper focuses on the role of opposition to majority and delays in the emerging dynamics in a population composed of mainstream and anti-conformist individuals. To this purpose, we introduce a class of simple statistical system of interacting agents taking into account (ⅰ) the presence of mainstream and anti-conformist individuals and (ⅱ) delays, possibly heterogeneous, in the transmission of information. In this simple model, each agent can be in one of two states, and can change state in continuous time with a rate depending on the state of others in the past. We express the thermodynamic limit of these systems as the number of agents diverge, and investigate the solutions of the limit equation, with a particular focus on synchronized oscillations induced by delayed interactions. We show that when hipsters are too slow in detecting the trends, they will consistently make the same choice, and realizing this too late, they will switch, all together to another state where they remain alike. Another modality synchronizing hipsters are asymmetric interactions, particularly when the cross-interaction between hipsters and mainstreams aree prominent, i.e. when hipsters radically oppose to mainstream and mainstreams wish to follow the majority, even when led by hipsters. We demonstrate this phenomenon analytically using bifurcation theory and reduction to normal form. We find that, in the case of asymmetric interactions, the level of randomness in the decisions themselves also leads to synchronization of the hipsters. Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding synchronization of nerve cells, investment strategies in finance, or emergent dynamics in social science, domains in which delays of communication and the geometry of information accessibility are prominent.

*+*[Abstract](486)

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**Abstract:**

Gene regulatory networks, which are complex high-dimensional stochastic dynamical systems, are often subject to evident intrinsic fluctuations. It is deemed reasonable to model the systems by the chemical Langevin equations. Since the mRNA dynamics are faster than the protein dynamics, we have a two-time scales system. In general, the process of protein degradation involves time delays. In this paper, we take the system memory into consideration in which we consider a model with a complete memory represented by an integral delay from

*+*[Abstract](486)

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**Abstract:**

Motivated by the recent investigation of a predator-prey model in heterogeneous environments [

is a strictly monotone decreasing function of the diffusion rate

*+*[Abstract](510)

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**Abstract:**

In this paper we first give a criterion on stability of equilibrium solutions for autonomous systems with constraints. Then we discuss the relationship between asymptotic behaviors of an asymptotically autonomous system with constraint and its limit system. Finally as an example, we revisit an extreme ideology model proposed in the literature and give a more detailed description on the dynamics of the system.

*+*[Abstract](477)

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**Abstract:**

The Alekseev-Gröbner formula is a well known tool in numerical analysis for describing the effect that a perturbation of an ordinary differential equation (ODE) has on its solution. In this article we provide an extension of the Alekseev-Gröbner formula for Banach space valued ODEs under, loosely speaking, mild conditions on the perturbation of the considered ODEs.

*+*[Abstract](549)

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**Abstract:**

This paper first establishes a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations with non-globally Lipschitz coefficients. Then two novel explicit schemes are designed and their convergence rates are exactly identified via the fundamental theorem. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate appropriate assumptions to allow for its super-linear growth. However, we require that the Lévy measure is finite. New arguments are developed to handle essential difficulties in the convergence analysis, caused by the super-linear growth of the jump coefficient and the fact that higher moment bounds of the Poisson increments

*+*[Abstract](467)

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**Abstract:**

Detailed models of structured populations are spatial and involve nonlocal effects. These features lead to a broad class of population models structured by a physiological parameter and space. Our focus of interest is on the well-posedness of their initial value problems. In more detail, we specify sufficient conditions on the coefficient functions for existence, positivity, uniqueness of weak solutions and their continuous dependence on the given data. The solutions considered here have their values in

*+*[Abstract](500)

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**Abstract:**

We deduce cell population models describing the evolution of a tumor (possibly interacting with its environment of healthy cells) with the aid of differential equations. Thereby, different subpopulations of cancer cells allow accounting for the tumor heterogeneity. In our settings these include cancer stem cells known to be less sensitive to treatment and differentiated cancer cells having a higher sensitivity towards chemo- and radiotherapy. Our approach relies on stochastic differential equations in order to account for randomness in the system, arising e.g., due to the therapy-induced decreasing number of clonogens, which renders a pure deterministic model arguable. The equations are deduced relying on transition probabilities characterizing innovations of the two cancer cell subpopulations, and similarly extended to also account for the evolution of normal tissue. Several therapy approaches are introduced and compared by way of tumor control probability (TCP) and uncomplicated tumor control probability (UTCP). A PDE approach allows to assess the evolution of tumor and normal tissue with respect to time and to cell population densities which can vary continuously in a given set of states. Analytical approximations of solutions to the obtained PDE system are provided as well.

2018 Impact Factor: 1.008

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