Crustose lichen communities on rocks exhibit fascinating spatial mosaics resembling political maps of nations or municipalities. Although the establishment and development of biological populations are important themes in ecology, our understanding of the formation of such patterns on the rocks is still in its infancy. Here, we present a novel model of the concurrent growth, establishment and interaction of lichens. We introduce an inverse technique based on Monte Carlo simulations to test our model on field samples of lichen communities. We derive an expression for the time needed for a community to cover a surface and predict the historical spatial dynamics of field samples. Lichens are frequently used for dating the time of exposure of rocks in glacial deposits, lake retreats or rock falls. We suggest our method as a way to improve the dating.