In this thesis, we study a spatio–temporal prey–predator model of plankton. This model has spatial interaction terms which represent a plankton dynamics that includes info–chemical mediated trophic interactions. We consider both a simplified two species model which has been studied in the literature (mostly in biological terms) and an extended, four-species model. In the latter, the grazing pressure of microzooplankton (M) on phytoplankton (P) is controlled through external infochemical (C) mediated predation by copepods (Z). We undertake a stability analysis of both the two species model and the four species model and compare the system dynamics. In relation to this, the critical conditions for Turing instability are derived; these are necessary and sufficient. Furthermore, we consider the degenerated situation wherein Turing bifurcation and Hopf bifurcation occur simultaneously. We also consider under what conditions Turing patterns are exhibited and under what conditions spatiotemporal patterns are observed generally. The Transient Turing instability of spatial interactions –exhibited by the two species model–is introduced and investigated in a number of ways. We also study the effects of the paradox of enrichment. This paradox led to a loss of stability in the four species model after this was derived from the two species model by expansion and by the addition of resources. Further, a numerical continuation technique was used to determine the existence of multiple stationary patterns
