Aims: To introduce students to problems arising in population biology that can be tackled using applied mathematics. Both mathematical modelling and mathematical analysis will be covered, and at all times the interplay between the mathematics and the underlying biology will be emphasised.
Students should be familiar with mathematical modelling issues for problems in population biology. They should be able to analyse models written in terms of ordinary differential equations or difference equations, give a qualitative and quantitative account of their solution, and interpret the results in terms of the original biological problem.
Mathematical modelling in biology, ordinary differential equations, difference equations.
Single species population dynamics: Models in discrete and continuous time: basic reproductive ratioR_0; compensatory and depensatory competition; transcritical, tangent and period doubling bifurcations, chaos. Harvesting: maximum sustainable yield; yield effort curves. Population dynamics of interacting species: host-parasitoid interactions: Nicholson-Bailey model; Jury conditions and Naimark-Sacker bifurcations. Predator-prey models: Lotka-Volterra model; phase plane analysis; Routh-Hurwitz conditions and Hopf bifurcations; Poincare-Bendixon theorem, Dulac condition; Lyapunov functions; Volterra's principle. Competition: Gause's principle of competitive exclusion. Infectious diseases: SIS disease: basic reproductive ratio R_0; threshold theorem. SIR epidemics and endemics: threshold theorem; size of the epidemic; eradication and control. Vector-borne diseases and sexually transmitted diseases.