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The effect of time delay and Hopf bifurcation in a tumor-immune system competition model with negative immune response

Radouane Yafia (2009)

Applicationes Mathematicae

We consider a system of delay differential equations modelling the tumor-immune system competition with negative immune response and three positive stationary points. The dynamics of the first two positive solutions are studied in terms of the local stability. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence and stability of a limit cycle bifurcating from the second positive stationary point, when the delay (taken as a parameter) crosses some...

The Effects of HIV-1 Infection on Latent Tuberculosis

Amy L. Bauer, Ian B. Hogue, Simeone Marino, Denise E. Kirschner (2008)

Mathematical Modelling of Natural Phenomena

Tuberculosis is the leading cause of death due to infectious diseases in the world today, and it is increasing due to co-infection with HIV-1, the causative agent of AIDS. Here, we examine the impact that HIV-1 infection has on persons with latent tuberculosis. Based on previous work, we develop a mathematical model of an adaptive immune response in the lung which considers relevant immune effectors such as macrophages, various sub-populations of T-cells, and key cytokines to predict which mechanisms...

The Second Half-With a Quarter of a Century Delay

O. Diekmann, M. Gyllenberg (2008)

Mathematical Modelling of Natural Phenomena

We show how results by Diekmann et al. (2007) on the qualitative behaviour of solutions of delay equations apply directly to a resource-consumer model with age-structured consumer population.

Time discrete 2-sex population model

C. O. A. Sowunmi (2003)

Banach Center Publications

A time-discrete 2-sex model with gestation period is analysed. It is significant that the conditions for local stability of a nontrivial steady state do not require that the expected number of female offspring per female equal unity. This is in contrast to results obtained by Curtin and MacCamy [4] and the author [10].

Uniqueness and local existence of solutions to an approximate system of a 1D simplified tumor invasion model

Maciej Cytowski, Akio Ito, Marek Niezgódka (2009)

Banach Center Publications

In the present paper, we consider an approximate system of one-dimensional simplified tumor invasion model, which was originally proposed by Chaplain and Anderson in [chaplain-anderson-03]. The simplified tumor invasion model is composed of PDE and ODE. Actually, the PDE is the balance equation of the density of tumor cells and the ODE describes the dynamics of concentration of extracellular matrix. In this model, we take into account that the random motility of the density of tumor cells is given...

Uniqueness and stability properties of monostable pulsating fronts

François Hamel, Lionel Roques (2011)

Journal of the European Mathematical Society

We prove the uniqueness, up to shifts, of pulsating traveling fronts for reaction-diffusion equations in periodic media with Kolmogorov–Petrovskiĭ–Piskunov type nonlinearities. These results provide in particular a complete classification of all KPP pulsating fronts. Furthermore, in the more general case of monostable nonlinearities, we also derive several global stability properties and convergence to pulsating fronts for solutions of the Cauchy problem with front-like initial data. In particular,...

Uniqueness of limit cycles bounded by two invariant parabolas

Eduardo Sáez, Iván Szántó (2012)

Applications of Mathematics

In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas.

Usefulness of Biocontrol of Pests in Tea: A Mathematical Model

A. Maiti, A. K. Pal, G. P. Samanta (2008)

Mathematical Modelling of Natural Phenomena

Nowadays there has been a growing consciousness among the tea industry to reduce the use of the chemical pesticides for pest control. Predators are beneficial insects that feed on harmful insects and mites, which incur considerable loss of production of tea. In this paper we have considered a tritrophic model consisting of tea plant, pest and predator to analyze different field observations. The effect of discrete time-delay on the tritrophic model is studied critically. The dynamical behaviours...

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