The thesis is an expansion of the work Gerald R. North did in 1975 on an energy balance climate model. By considering a similar model on an infinite line, and allowing the heat diffusion coefficient to vary on the line, more complicated behaviour arose from the model. Much of Norths work was recreated on the infinite lines, but a lot of new discoveries were made. Among what was found were spontaneous symmetry breaking, possible infinitely many solutions for a specific solar irradiation constant, bistability and in some cases tristability. Unconnected bifurcation diagrams were also found for some choices of unsymmetric heat diffusion function.