Collective evolution under catastrophes. (English) Zbl 1530.92160
Summary: We introduce the following discrete time model. Each site of \(\mathbb{N}\) represents an ecological niche and is assigned a fitness in \((0,1)\). All the sites are updated simultaneously at every discrete time. At any given time the environment may be normal with probability \(p\) or a catastrophe may occur with probability \(1-p\). If the environment is normal the fitness of each site is replaced by the maximum of its current fitness and a random number. If there is a catastrophe the fitness of each site is replaced by a random number. We compute the joint fitness distribution of any finite number of sites at any fixed time. We also show convergence of this system to a stationary distribution. This too is computed explicitly.
MSC:
| 92D15 | Problems related to evolution |
| 60K10 | Applications of renewal theory (reliability, demand theory, etc.) |
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