Travelling waves in a model of species migration

A model of species migration is presented which takes the form of a reaction-diffusion system. We consider special limits of this model in which we demonstrate the existence of travelling wave solutions. These solutions can be used to describe the migration of cells, bacteria, and some organisms. © 2000 Elsevier Science Ltd. All rights reserved

Термостат с двухвременной конкурентной динамикой термостатирования

С целью исследовать методом математического моделирования неравновесное динамическое поведение макромолекул в реалистичных условиях часто используются детерминированные термостаты. В частности, детерминированная (не стохастическая) Nosé-Hoover (NH) динамика. Понимая такой механизм термостатирования как детерминированную имитацию представительной выборочной реализации траектории динамической системы, взаимодействующей с тепловым резервуаром, мы собираем и исследуем детерминированный термостат с двумя конкурирующими шкалами времени. Эти шкалы в тесной аналогии с парадигмой неравновесной статистической физики относятся к релаксационным процессам в импульсном и конфигурационном пространствах. Доказано теоретически и проверено численным симулированием, что дополнительная шкала времени, связанная с изменениями в конфигурационном пространстве, − эффективный контрольный параметр, который помогает сопоставить результат симулирования с известными особенностями неравновесного динамического поведения. Разумно ожидать, что предложенный термостат подходит для моделирования специфических процессов медленной конформационной динамики протеинов и нуклеиновых кислот. Проанализирована возможность гамильтоновой реформулировки термостатирующей динамики.Determinate thermostats are frequently used to investigate the nonequilibrium dynamic behavior of macromolecules in real conditions by using the mathematical modelling method. In particular, the determinate (nonstochastic) Nosé−Hoover (NH) dynamics. Taking such thermostatting mechanism for a determinate imitation of the representative selective realization of trajectory of a dynamic-system interacting with a thermal reservoir, we construct and investigate a determinate thermostat with two competing time scales. The scales, in close analogy with the paradigm of nonequilibrium statistical physics, refer to relaxation processes in pulsed and configurational spaces. It has been proved theoretically and checked by numerical simulation that the additional time scale related with changes in configurational space is an effective control parameter which helps in comparing the simulation result with the known features of nonequilibrium dynamic behavior. It is reasonable to expect that the proposed thermostat is suitable for the modelling of specific processes of slow conformational dynamics of proteins and nucleic acids. A possibility of the Hamiltonian reformulation of thermostatting dynamics has been analysed

A Lotka–Volterra type food chain model with stage structure and time delays

AbstractA three-species Lotka–Volterra type food chain model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators (immature top predators) do not have the ability to feed on prey (predator). By using some comparison arguments, we first discuss the permanence of the model. By means of an iterative technique, a set of easily verifiable sufficient conditions are established for the global attractivity of the nonnegative equilibria of the model

On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities

We study a non-local variant of a diffuse interface model proposed by Hawkins--Darrud et al. (2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn--Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth potentials, we derive well-posedness results, which are the non-local analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015). Furthermore, we establish existence of weak solutions for the case of degenerate mobilities and singular potentials, which serves to confine the order parameter to its physically relevant interval. Due to the non-local nature of the equations, under additional assumptions continuous dependence on initial data can also be shown.Comment: 28 page

A generative approach for image-based modeling of tumor growth

22nd International Conference, IPMI 2011, Kloster Irsee, Germany, July 3-8, 2011. ProceedingsExtensive imaging is routinely used in brain tumor patients to monitor the state of the disease and to evaluate therapeutic options. A large number of multi-modal and multi-temporal image volumes is acquired in standard clinical cases, requiring new approaches for comprehensive integration of information from different image sources and different time points. In this work we propose a joint generative model of tumor growth and of image observation that naturally handles multi-modal and longitudinal data. We use the model for analyzing imaging data in patients with glioma. The tumor growth model is based on a reaction-diffusion framework. Model personalization relies only on a forward model for the growth process and on image likelihood. We take advantage of an adaptive sparse grid approximation for efficient inference via Markov Chain Monte Carlo sampling. The approach can be used for integrating information from different multi-modal imaging protocols and can easily be adapted to other tumor growth models.German Academy of Sciences Leopoldina (Fellowship Programme LPDS 2009-10)Academy of Finland (133611)National Institutes of Health (U.S.) (NIBIB NAMIC U54-EB005149)National Institutes of Health (U.S.) (NCRR NAC P41- RR13218)National Institutes of Health (U.S.) (NINDS R01-NS051826)National Institutes of Health (U.S.) (NIH R01-NS052585)National Institutes of Health (U.S.) (NIH R01-EB006758)National Institutes of Health (U.S.) (NIH R01-EB009051)National Institutes of Health (U.S.) (NIH P41-RR014075)National Science Foundation (U.S.) (CAREER Award 0642971

A new ghost cell/level set method for moving boundary problems:application to tumor growth

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth

Continuous and discrete mathematical models of tumor-induced angiogenesis

summary:We study a class of parabolic-ODE systems modeling tumor growth, its mathematical modeling and the global in time existence of the solution obtained by the method of Lyapunov functions

The mathematical modelling of tumour angiogenesis and invasion

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. Capillary sprouts are formed which migrate towards the tumour eventually penetrating it and permitting vascular growth to take place. It is during this stage of growth that the insidious process of invasion of surrounding tissues can and does take place. A model mechanism for angiogenesis is presented which includes the diffusion of the TAF into the surrounding host tissue and the response of the endothelial cells to the chemotactic stimulus. Numerical simulations of the model are shown to compare very well with experimental observations. The subsequent vascular growth of the tumour is discussed with regard to a classical reaction-diffusion pre-pattern model