The Bistability Theorem in a Model of Metastatic Cancer

Abstract

The main theorem of the present paper is the bistability theorem for a four dimensional cancer model, in the variables representing primary cancer C, metastatic cancer , growth factor GF and growth inhibitor GI, respectively. It says that for some values of the para- meters this system is bistable, in the sense that there are exactly two positive singular points of this vector field. And one is stable and the other unstable. We also find an expression for for the discrete model T of the introduction, with variables , where C is cancer, are growth factors and growth inhibitors respectively. We find an affine vector field Y whose time one map is T 2 and then compute , where is an integral curve of Y through . We also find a formula for the first escape time for the vector field associated to T, see section four.