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Stochastic diffusion is the noisy process through which dynamics like epidemics, or agents like animal species, disperse over a larger area. These processes are increasingly important to better prepare for pandemics and as species ranges shift in response to climate change. Unfortunately, modelling is mostly done with expensive computational simulations or inaccurate deterministic tools that ignore the randomness of dispersal. We introduce ‘mean-FLAME’ models, tracking stochastic dispersion using approximate master equations to follow the probability distribution over all possible states of an area of interest, up to states active enough to be approximated using a mean-field model. In the limit where we track all states, this approach is locally exact, and in the other limit collapses to traditional deterministic models. In predator–prey systems, we show that tracking a handful of states around key absorbing states is sufficient to accurately model extinction. In disease models, we show that classic mean-field approaches underestimate the heterogeneity of epidemics. And in nonlinear dispersal models, we show that deterministic tools fail to capture the speed of spatial diffusion. These effects are all important for marginal areas that are close to unsuitable for diffusion, like the edge of a species range or epidemics in small populations. 

It has been proposed that climate adaptation research can benefit from an evolutionary approach. But related empirical research is lacking. We advance the evolutionary study of climate adaptation with two case studies from contemporary United States agriculture. First, we define ‘cultural adaptation to climate change’ as a mechanistic process of population-level cultural change. We argue this definition enables rigorous comparisons, yields testable hypotheses from mathematical theory and distinguishes adaptive change, non-adaptive change and desirable policy outcomes. Next, we develop an operational approach to identify ‘cultural adaptation to climate change’ based on established empirical criteria. We apply this approach to data on crop choices and the use of cover crops between 2008 and 2021 from the United States. We find evidence that crop choices are adapting to local trends in two separate climate variables in some regions of the USA. But evidence suggests that cover cropping may be adapting more to the economic environment than climatic conditions. Further research is needed to characterize the process of cultural adaptation, particularly the routes and mechanisms of cultural transmission. Furthermore, climate adaptation policy could benefit from research on factors that differentiate regions exhibiting adaptive trends in crop choice from those that do not. This article is part of the theme issue ‘Climate change adaptation needs a science of culture’.