We investigate the behaviour space of meta-heuristic optimisation algorithms automatically generated by Large Language Model driven algorithm discovery methods. Using the Large Language Evolutionary Algorithm (LLaMEA) framework with a GPT o4-mini LLM, we iteratively evolve black-box optimisation heuristics, evaluated on 10 functions from the BBOB benchmark suite. Six LLaMEA variants, featuring different mutation prompt strategies, are compared and analysed. We log dynamic behavioural metrics including exploration, exploitation, convergence and stagnation measures, for each run, and analyse these via visual projections and network-based representations. Our analysis combines behaviour-based projections, Code Evolution Graphs built from static code features, performance convergence curves, and behaviour-based Search Trajectory Networks. The results reveal clear differences in search dynamics and algorithm structures across LLaMEA configurations. Notably, the variant that employs both a code simplification prompt and a random perturbation prompt in a 1+1 elitist evolution strategy, achieved the best performance, with the highest Area Over the Convergence Curve. Behaviour-space visualisations show that higher-performing algorithms exhibit more intensive exploitation behaviour and faster convergence with less stagnation. Our findings demonstrate how behaviour-space analysis can explain why certain LLM-designed heuristics outperform others and how LLM-driven algorithm discovery navigates the open-ended and complex search space of algorithms. These findings provide insights to guide the future design of adaptive LLM-driven algorithm generators.