The logistic model proposed by Courtillot and Gaudemer to describe the growth of biodiversity during geological ages is more explored here and further developed. A new parameterisation is first proposed. Another expression of this model is obtained by introducing a new variable representing the number of ecological niches. It appears that the rates of increase of biodiversity during Jurassic and Cretaceous periods is quite different from other ones. The classical literature essentially focuses on possible extinction mechanisms, but explosions in biodiversity must be more precisely explored. For this purpose, on the basis of data analysis through different expressions of the logistic model, different ecological mechanisms can be assumed (e.g., qualitative and quantitative niches changes, possible appearance of new kinds of ecological relationships, such as 'niche-sharing', which involves coexistence or cooperation), even if genetic processes must also be involved. Finally, we emphasise the astonishing speed of biological diversification following a 'catastrophic' mass extinction. We could refer to this feature as 'catastrophic biological diversification'.