Why would a climax community have high species diversity?Watch
Why would a climax community such as oak woodland have high species diversity ?
These types of community tend to be dominated by large, long-lived organisms (like oak trees) rather than smaller, quick reproducers like young grasslands. In theory, climax communities like an oak woodland provide lots more ecological niches for other species - for example, there might be a think layer of leaf mould on the ground, an understorey of small plants, the tree trunks and branches of the oaks, and the open canopy of leaves. This diversity of niches means lots of different species can make a living there. (compare that with a garden lawn, where there is just a few millimetres of ground-level roots and stems, plus a few centimetres of grass stalks).
This diversity makes it more stable, and in turn, this stability allows more species to find a place in the community, until a steady state is reached.
In real life, this theory of the 'climax community' is probably untrue, or at least things are more complicated. Communities where there is an intermediate level of disturbance are probably more diverse. But that that's the theory that is still often taught in elementary biology.