Mathematical Medicine and Biology - current issue

Pest management of a prey-predator model with sexual favoritism

Pei, Y., Yang, Y., Li, C., Chen, L. — 2009-06-02

Although sex of prey is an important factor for the risk of predating, few articles consider the consequences of sexual favoritism and the corresponding effects on the impulsive predator–prey dynamics and its utility in biological control. This paper investigates the pest management strategy of a prey–predator system model with sexual favoritism. An impulsive differential equation which models the process of periodically releasing natural enemies and spraying pesticides at different fixed time for pest control is proposed and investigated. It is proved that the pest-eradication periodic solution is globally asymptotically stable under the assumption that the release amount of the predator is greater than some critical value. Permanent conditions are established under the assumption that the release amount of the predator is less than another critical value. In particular, two single control strategies are proposed. Furthermore, we compare three pest control strategies and find that if we choose narrow-spectrum pesticides that targeted to a specific pest's life cycle to kill the pest, then the combined strategy is preferable. Finally, the corresponding system with no sexual favoritism is investigated. The results indicate that we can release fewer amount of the predators to eliminate the preys with sexual favoritism than without and any strong sexual favoritism will drive the pest towards extinction. In view of the biological meaning, the sexual favoritism plays a more active role in suppressing insect pests.

A bistable genetic switch which does not require high co-operativity at the promoter: a two-timescale model for the PU.1-GATA-1 interaction

Bokes, P., King, J. R., Loose, M. — 2009-06-02

The transcription factors PU.1 and GATA-1 antagonize each other in common myeloid progenitors and their relative abundance is thought to decide whether the cell follows the erythrocyte/megakaryocyte lineage or the granulocyte/macrophage lineage. We propose a kinetic model for the PU.1–GATA-1 interaction, analyse its phase space and interpret the results of our analysis. The conclusions have broader implications for the modelling of cell-fate selection.

Modelling the growth and stabilization of cerebral aneurysms

Watton, P. N., Ventikos, Y., Holzapfel, G. A. — 2009-06-02

Experimental and theoretical guidance is needed to understand how the collagen fabric evolves during the development of aneurysms. In this paper, we model the development of an aneurysm as a cylindrical/spherical membrane subject to 1D enlargement; these conceptual models reflect the development of fusiform and saccular cerebral aneurysms. The mechanical response is attributed to the elastin and collagen. We introduce variables which define the elastin and collagen fibre concentration; these evolve to simulate growth/atrophy of the constituents. A hypothetical aneurysm model is analysed: collagen stretch is constant, elastin degrades and collagen fibre concentration can adapt to maintain mechanical equilibrium. An analytic expression for the rate of evolution of the fibre concentration is derived. The functional form is dependent on (i) the current collagen fibre concentration, (ii) the deviations in the collagen fibre stretch from the attachment stretch, (iii) the rate of change of fibre stretch, (iv) the rate of loss of elastin and (v) the ratio of load borne by elastinous and collagenous constituents. Finally, numerical examples of aneurysm development are considered. Suitable candidates for the fibre concentration evolution equations are identified that yield stabilization of the aneurysm even when there is complete loss of elastin. This theoretical analysis provides the basis for the development of physiologically realistic models of aneurysm development.

A mathematical model for the deformation of the eyeball by an elastic band

Keeling, S. L., Propst, G., Stadler, G., Wackernagel, W. — 2009-06-02

In a certain kind of eye surgery, the human eyeball is deformed sustainably by the application of an elastic band. This article presents a mathematical model for the mechanics of the combined eye/band structure along with an algorithm to compute the model solutions. These predict the immediate and the lasting indentation of the eyeball. The model is derived from basic physical principles by minimizing a potential energy subject to a volume constraint. Assuming spherical symmetry, this leads to a two-point boundary-value problem for a non-linear second-order ordinary differential equation that describes the minimizing static equilibrium. By comparison with laboratory data, a preliminary validation of the model is given.