Seminar Details
2015-06-18 (14.00) : Differential Equations Models in Biosciences and Reliable Numerics
At Euler building (A002)
Organized by Mathematical Engineering
Speaker :
Jean M-S Lubuma (University of Pretoria)
Abstract :
Biological processes that arise in science are very complex. A lot of effort has been and is being made to build differential equations based models
that aim at elucidating these phenomena. However, these equations cannot be completely solved by analytic techniques. Consequently, reliable numerical simulations
are of fundamental importance in gaining some useful insights on the solutions of the continuous models. Of paramount importance for the involved dynamical systems,
is the design of numerical methods that, apart from being convergent, replicate their underlying dynamics such as the positivity of solutions, the dissipativity of
the systems, the conservation laws, and the stability of equilibria. In this talk, we construct, analyse and implement Nonstandard Finite Difference (NSFD) Schemes
for some deterministic models defined by ordinary and partial differential equations. The NSFD schemes are reliable in three directions. They are topologically
dynamically consistent for one-dimensional models. They replicate the global asymptotic stability of the disease-free equilibrium of compartmental models in epidemiology
whenever the associated threshold quantity is less than unity. They preserve the positivity and boundedness of solutions of reaction-diffusion equations of Fisher-KPP
(Kolmogorov, Petrovsky, Piscounoff) type as well as of cross-diffusion equations that arise in the modelling of cancer growth and predator-prey pursuit-evasion interactions.