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Conceived and designed the experiments: EJ AN GH ET JM. Performed the experiments: EJ AN GH ET JM. Analyzed the data: EJ AN GH ET JM. Contributed reagents/materials/analysis tools: EJ AN GH ET JM. Wrote the paper: EJ AN GH ET JM.

The authors have declared that no competing interests exist.

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.

Lichens are composite organisms of fungi and algae (mycobionts and photobionts) living in a symbiotic relationship

The collective growth of adjacent crustose lichens form a complex pattern with similarities to the patterns formed by Voronoi tessellations of space, see

A) Photograph of a partially covered rock which has been exposed for approximately 60 years (when a nearby road was built). The diameter of the lens cap is approximately 6 cm. B) Picture of a rock completely covered by lichens. Both rocks have been located in an open pine forest at 700 m altitude in Oppland, Norway.

On sufficiently smooth surfaces, lichens grow radially from their points of establishment and may after some possible transient for small radii

New lichens are nucleated in the space between existing lichens at a rate

The first term on the right hand side describes the continuous shrinking of gaps, the second term describes the number of gaps of size

Here we are interested in knowing the time it takes to cover fully an area, which is the same as knowing the time it takes for the unoccupied area to vanish. The total gap size

From this equation we find that a given coverage

If the dynamics is now happening on a two dimensional surface, the nucleation rate

The characteristic time

We compare the predicted evolution of covered area in the field sample shown in

Panel A), synthetic image of a part of the pattern in

Species | Color | Relative velocity |

yellow | 0.80 | |

cyan | 1.00 | |

purple | 0.55 | |

grey | 0.98 | |

red | 0.63 | |

green | 0.71 | |

blue | 0.86 |

The final pattern of a community is determined by the nucleation points (

The nucleation sites, nucleation times and growth rates of individual lichens in a natural sample e.g. the one shown in

The evolution is estimated from applying the inverse method outlined in the text to a real sample of lichens. The snapshots are taken at A) 30%, B) 40%, C) 60% and D) 80% of the time needed for total coverage.

Even though our model only has two parameters and one parameter is fixed by the average velocity, we find that it fits the temporal evolution of the area coverage very well. Using the coverage time computed by the Monte Carlo simulation, we are able to estimate the nucleation rate

In general, the spatial evolution of crustose lichen communities is controlled by species specific growth rates and a rate for how often new lichens are established in unoccupied areas. We find that on the scale of 10 cm an isotropic nucleation rate provides a very good fit to field samples. In other words, the establishment of new lichens is not dominated by spores dispersed by nearest neighbors. In general, dispersal mechanisms such as wind and water flows operate over long distances and can account for floristic affinities between even remote landmasses

Although crustose lichens, if undisturbed in their growth, cover areas proportional to the square of their age, we find that the uncovered area retracts super-exponentially in time due to the establishment of new lichens. In lichenometric dating, the age of a community is usually estimated by measuring the diameters of individual lichens

In the Monte Carlo method, the spatial configuration of a given lichen community is computed from the estimated nucleation points (

In addition to the algorithm used in the Monte Carlo simulation, we have implemented an off-lattice stochastic computer model inspired by the Eden growth model. In this model, lichens are made up of small non-overlapping spherical particles with a radius