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This paper focuses on the stability analysis of a delay differential system encountered in modelling immune dynamics during imatinib treatment for chronic myelogenous leukemia. A simple algorithm is proposed for the analysis of delay effects on the stability. Such an algorithm takes advantage of the particular structure of the dynamical interconnections of the model. The analysis shows that the model yields three fixed points, two of which are always unstable and one of which is sometimes stable. The stable fixed point corresponds to an equilibrium solution in which the leukemia population is kept below the cytogenetic remission level. This result implies that, during imatinib treatment, the resulting anti-leukemia immune response can serve to control the leukemia population. However, the rate of approach to the stable fixed point is very slow, indicating that the immune response is largely ineffective at driving the leukemia population towards the stable fixed point. To extend the stability analysis with respect to the delay parameter, we conduct a global non-linear analysis to demonstrate the existence of unbounded solutions. We provide sufficient conditions based on initial cell concentrations that guarantee unbounded solutions and comment on how these conditions can serve to predict whether imatinib treatment will result in a sustained remission based on a patient's initial leukemia load and initial anti-leukemia T cell concentration. [ABSTRACT FROM PUBLISHER]

We consider two regions with a fish population that is dispersing between the two areas, and fishing takes place only in region 2, with region 1 established as no-fishing zone. Marine protected areas (MPAs) have been promoted as conservation and fishery management tools, and at present, there are over 1300 MPAs in the world. A new mathematical model of an MPA that reflects the complexity of the natural setting is presented. The resulting model of an age-structured fish population belongs to a class of non-linear systems of differential equations with delay. New easily verifiable sufficient conditions for the existence, boundedness, permanence and stability of the positive internal steady-state solutions are obtained. From the point of view of fishery managers, the existence of stable solutions is necessary for planning harvesting strategies and sustaining the fishing grounds. Numerical simulations illustrate qualitative behaviour of the model, including stability switches. [ABSTRACT FROM PUBLISHER]

In this paper we propose a class of models that describe the mutual interaction between tumour growth and the development of tumour vasculature and that generalize existing models. The study is mainly focused on the effect of a therapy that induces tumour vessel loss (anti-angiogenic therapy), with the aim of finding conditions that asymptotically guarantee the eradication of the disease under constant infusion or periodic administration of the drug. Furthermore, if tumour and/or vessel dynamics exhibit time delays, we derive conditions for the existence of Hopf bifurcations. The destabilizing effect of delays on achieving the tumour eradication is also investigated. Finally, global conditions for stability and eradication in the presence of delays are given for some particular cases. [ABSTRACT FROM AUTHOR]