二、課程之大綱：

 

Lecture 1

Title: Introduction to the applications of delay differential equations

Abstract: This talk will explore some of the applications of simple systems of delay differential equations in the fields of number theory, ecology, epidemiology, and engineering applications such as machine tool vibration.The talk will explore the details of why the time delays are important, and how models containing delays are derived, and I will discuss simple analytical techniques that can be applied to the study of such systems. 

 

Lecture 2

Title: Wolbachia infection in a sex-structured mosquito population carrying West Nile virus

Abstract: Wolbachia is possibly the most studied reproductive parasite of arthropod species. It appears to be a promising candidate for biocontrol of some mosquito borne diseases. I will present a sex-structured model for a Wolbachia infected mosquito population. The model incorporates the key effects of Wolbachia infection including cytoplasmic incompatibility and male killing. I also allow the possibility of reduced reproductive output, incomplete maternal transmission, and different mortality rates for uninfected/infectedmale/female individuals. Existence and local stability of equilibria, including the biologically relevant boundary equilibria, will be discussed. For some biologically relevant parameter regimes there may be multiple coexistence steady states including a coexistencesteady state in which Wolbachia infected individuals dominate. The model can be extended to an SEI model incorporating West Nile virus (WNv) dynamics. Recent evidence suggests that a particular strain of Wolbachia infection significantly reduces WNv replication in Aedes aegypti. If the mosquito population consists mainly of Wolbachia infected individuals, WNv eradication is likely if WNv replication in Wolbachia infected individuals is sufficiently reduced.

 

Lecture 3 

Title: Delay equation models for populations that experience competition at immature life stages

Abstract: Intra-specific competition in insect and amphibian species is often experienced in completely different ways in their distinct life stages. Competition among larvae is important because it can impact on adulttraits that affect disease transmission, yet mathematical models often ignore larval competition. We present two models of larval competition in the form of delay differential equations for the adult population derived from age-structured models that includelarval competition. We present a simple prototype equation that models larval competition in a simplistic way. Recognising that individual larvae experience competition from other larvae at various stages of development, we then derive a more complex equationcontaining an integral with a kernel that quantifies the competitive effect of larvae of all ages on larvae of a particular age. In some parameter regimes, this model and the famous spruce budworm model have similar dynamics, with the possibility of multiple co-existing equilibria.

 

Lecture 4

Title: Differential equations with variable time delays.

Abstract: In this talk I will discuss the fact that the realistic incorporation of time delays may require the delay itself to be a function of time, or even of the state of the system (i.e. the delay may depend on theunknown solution of the differential equation). For example the developmental time of an insect larva may depend strongly on temperature and therefore on the time of year. Or, it may depend on how many larvae are present, since greater numbers of them willhamper the ability of an individual larva to find enough food and this will slow down its growth. In some modern lathes and milling machines the spindle speed can be made to vary sinusoidally, as a vibration elimination measure, and this also gives rise todifferential equations with time-dependent delays.