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The path of species diversification is commonly observed by inspecting the fossil record. Yet, how species diversity changes at geological timescales relate to lower-level processes remains poorly understood. Here we use mathematical models of spatially structured populations to show that natural selection and gradual environmental change give rise to discontinuous phenotype changes that can be connected to speciation and extinction at the macroevolutionary level. In our model, new phenotypes arise in the middle of the environmental gradient, while newly appearing environments are filled by existing phenotypes shifting their adaptive optima. Slow environmental change leads to loss of phenotypes in the middle of the extant environmental range, whereas fast change causes extinction at one extreme of the environmental range. We compared our model predictions against a well-known yet partially unexplained pattern of intense hoofed mammal diversification associated with grassland expansion during the Late Miocene. We additionally used the model outcomes to cast new insight into Cope's law of the unspecialized. Our general finding is that the rate of environmental change determines where generation and loss of diversity occur in the phenotypic and physical spaces.

We present a model for the coevolution of seed size and germination time within a season when both affect the ability of the seedlings to compete for space. We show that even in the absence of a morphological or physiological constraint between the two traits, a correlation between seed size and germination time is nevertheless likely to evolve. This raises the more general question to what extent a correlation between any two traits should be considered as an a priori constraint or as an evolved means (or "instrument") to actually implement a beneficial combination of traits. We derive sufficient conditions for the existence of a positive or a negative correlation. We develop a toy model for seed and seedling survival and seedling growth and use this to illustrate in practice how to determine correlations between seed size and germination time.

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