This thesis mainly focusses on the behaviour of humans and animals when competing (or cooperating) in a shared environment. We relate their behaviour to a selection of games, namely, the hawk-dove game and the prisoner's dilemma. We first aim to give an overview of the link between species interactions and game theory whilst introducing the spatial and non-spatial interaction with memory. We then analyse the temporal dynamics of a large population of spatially distributed agents who recall each other's previous actions whilst assuming that individuals are anti-conformists. We extend a recently published analysis of memory effects in the hawk-dove gameby first introducing an alternative derivation of the results and then extending to new types of memory dynamics. We explore two different types of trapline dynamics, Cross dynamics of space use and an iterative model of trapline foraging, to explore movement patterns of individuals while capturing their memory and interactions. Finally, we explore the birth-death process whilst considering the two dimensional case.