The field of stochastic growth encompasses various different processes which are ubiquitously seen across the physical world. In many systems, stochasticity appears quite naturally, where inherent randomness provides the right setting for the tone of motion and interaction, whose symphony leads to the surprising emergence of interesting patterns and structure. Although on the microscopic scale one can be overwhelmed by the randomness arising from the fluctuating interactions between components, on the macroscopic scale, however, one is mesmerized by the emergence of mathematical beauty and symmetry, leading to complex structures with fractal architecture. Competition between components adds an extra degree of complexity and leads to the possibility of critical behaviour and phase transitions. It is an important aspect of many systems, and in order to provide a full explanation of many natural phenomena, we have to understand the role it plays on modifying behaviour. The combination of stochastic growth and competition leads to the emergence of interesting complex patterns. They occur in various systems and in many forms, and thus we treat competition in growth models driven by different laws for the stochastic noise. As a consequence our results are widely applicable and we encourage the reader to find good use for them in their respective field. In this thesis we study stochastic systems containing interacting particles whose motion and interplay lead to directed growth structures on a particular geometry. We show how the effect of the overall geometry in many growth processes can be captured elegantly in terms of a time dependent metric. A natural example we treat is isoradial growth in two dimensions, with domain boundaries of competing microbial species as an example of a system with a homogeneously changing metric. In general, we view domain boundaries as space-time trajectories of particles moving on a dynamic surface and map those into more easily tractable systems with constant metric. This leads to establishing a generic relation between locally interacting, scale invariant stochastic space-time trajectories under constant and time dependent metric. Indeed "the book of nature is written in the language of mathematics" (Galileo Galilei) and we provide a mathematical framework for various systems with various interactions and our results are backed with numerical confirmation.