Some Problems In Biomathematics

Biomathematics is an interdisciplinary research field between biology and mathematics.For one thing,it can solve many biological problems by using mathematical tools;for the other,it can broaden the mathematical research span.Biomathematics has been proved to be a great boost to both biology and mathematics.In this thesis,we will probe into some problems in group testing theory and competition number problem,which are both belong to biomathematics.Group testing theory has been flourishing since the World War Two.In 1943,Dorfman first proposed the group testing problem,that is,how to detect syphilis from the blood samples of millions of draftees efficiently.Over the past 60 years,group testing theory has been proved to be very useful in many fields,such as blood testing,electric shorting detection,multi-access channel communication and computational biology. Part one is made up of the first three chapters in this thesis.In chapter one,we will first present the basic properties of separating matrices, and then we will present the error-tolerant version of separating matrices,including our correction of one theorem in Du and Hwang and some extention work.In chapter two,we will give a brief survey on the construction of separating matrices,using tools from combinatorics, linear space over finite fields and so on.In chapter three,we will devote to random group testing,in this chapter we will first do some computational work on RID-model,then we will give a survey on the work by Cheng and Du about the great use of probabilistic method in the construction of separating matrices and matrices bound estimation.The competition number problem was first proposed by biologists when they do research into food chain problem in ecosystem.Those research will be helpful for people to know more about the structure of ecosystem and something others.Part two is the last chapter of this thesis,we will first list some results on competition number problem of Opsut,then we will generalize one of his result on line graph to any graph.