Evolution of populations with strategy-dependent time delays.

We study the effects of strategy-dependent time delays on the equilibria of evolving populations. It is well known that time delays may cause oscillations in dynamical systems. Here we report a novel behavior.

A mathematical model describes the malignant transformation of low grade gliomas: Prognostic implications.

Gliomas are the most frequent type of primary brain tumours. Low grade gliomas (LGGs, WHO grade II gliomas) may grow very slowly for the long periods of time, however they inevitably cause death due to the phenomenon known as the malignant transformation. This refers to the transition of LGGs to more aggressive forms of high grade gliomas (HGGs, WHO grade III and IV gliomas).

Mathematical model for path selection by ants between nest and food source.

Several models have been proposed to describe the behavior of ants when moving from nest to food sources. Most of these studies where based on numerical simulations with no mathematical justification. In this paper, we propose a mechanism for the formation of paths of minimal length between two points by a collection of individuals undergoing reinforced random walks taking into account not only the lengths of the paths but also the angles (connected to the preference of ants to move along straight lines).

Angiogenesis model with Erlang distributed delays.

We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.

Deterministic and Stochastic Study for a Microscopic Angiogenesis Model: Applications to the Lewis Lung Carcinoma.

Angiogenesis modelling is an important tool to understand the underlying mechanisms yielding tumour growth. Nevertheless, there is usually a gap between models and experimental data. We propose a model based on the intrinsic microscopic reactions defining the angiogenesis process to link experimental data with previous macroscopic models.

A simple model of carcinogenic mutations with time delay and diffusion.

In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.

Gompertz model with delays and treatment: mathematical analysis.

In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy.

Model of tumour angiogenesis -- analysis of stability with respect to delays.

In the paper we consider the model of tumour angiogenesis process proposed by Bodnar and Fory (2009). The model combines ideas of Hahnfeldt et al. (1999) and Agur et al.

New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model.

In the paper we propose a new methodology in modeling of antiangiogenic treatment on the basis of well recognized model formulated by Hahnfeldt et al. in 1999. On the basis of the Hahnfeldt et al.

Stochastic models of gene expression with delayed degradation.

In many biochemical reactions occurring in living cells, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous, there exist natural time delays in the evolution of cell states. It is a challenge to develop a systematic and rigorous treatment of stochastic dynamics with time delays and to investigate combined effects of stochasticity and delays in concrete models.

Time delay in necrotic core formation.

A simple model of avascular solid tumor dynamics is studied in the paper. The model is derived on the basis of reaction-diffusion dynamics and mass conservation law. We introduce time delay in a cell proliferation process.