Several mathematical models have been used to describe spatial-temporal patterns observed in nature, such as the skin pigmentation patterns in some fish species and the tiger stripe-like skin pattern. These models can be computationally implemented using available numerical techniques, among which the finite differences method and the finite elements method are mostly preferred. The aim of this article is to describe the implementation of three mathematical models regarding biological systems by using the finite elements method. These three models are suitable to describe morphogenesis (skin patterns), pattern formation following a chemical reaction, and cell movement (migration). The numerical results obtained are favourably compared to those reported elsewhere using finite differences method. Therefore, we conclude that the numerical technique used here is suitable to implement the mathematical models at hand, allowing them to be extended to the formulation and implementation of mathematical models of complex biological systems such as tissue growth and formation, and cell and tissue development.