Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: For solid tumours to grow and metastise, they need to pass through two distinct stages: the avascular growth phase in which the tumour remains in a limited diffusion size and the vascular growth phase where the invasion may take place. In order to accomplish the transition from the former to the latter growth phase, a solid tumour may secrete a substance known as tumour angiogenesis factor (TAF) into the surrounding tissues to stimulate its own blood vessels. Once the tumour has its own blood supply, it can invade other parts of the body destroying healthy tissues organs by secreting the matrix degrading enzymes (MDE). During the invasion, the adhesion both cell-cell and cell-matrix play an extremely important role. In this work, we review some mathematical models dealing with various stages of development of solid tumours and the resulting reaction diffusion equations are solved using the Crank-Nicolson finite differences scheme. We also present a system of reaction-diffusion-taxis partial differential equations, with nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. We then investigate the local and global existence of the solution of the previous model using the semigroup method and Sobolev embeddings. / AFRIKAANSE OPSOMMING: Daar is twee afsonderlike fases nodig vir soliede kanker gewasse om te groei en kwaadaardig te word: die avaskulêre groeifase waarin die gewas tot ’n sekere diffusie grootte beperk word en die vaskulêre groei fase waar die indringing plaasvind. Ten einde die oorgang tussen die twee fases te bewerkstellig, skei die soliede gewas ân stof in die omliggende weefsel af wat bekend staan as âtumor angiogenese factorâ (TAF). Dit stimuleer die vorming van die gewas se eie bloedvate. Wanneer die gewas sy eie bloedtoevoer het, kan dit ander dele van die liggaam indring en gesonde orgaanweefsel vernietig deur die afskeiding van die âmatrix degrading enzymesâ (MDE). Gedurende hierdie proses speel die sel-sel en sel-matriks interaksies ân belangrike rol. In hierdie werk het ons ân paar wiskundige modelle vergelyk wat die verskillende stadiums van die ontwikkeling van soliede gewasse beskryf. Die gevolglike diffusiereaksie vergelykings is opgelos deur gebruik te maak van die âCrank-Nicolson finite differences schemeâ. Ons bied ook ’n stelsel van âreaction-diffusion-taxisâ, met nie-lokale (integrale) terme wat die interaksies tussen kankerselle en die gasheerweefsel beskryf. Ons stel dan ondersoek in na die lokale en globale bestaan van die oplossing van die vorige model, met behulp van die semi-groep metode en die Sobolev ingebeddings.