In this paper, the extinction problem for a class of distylous plant populations is considered within the framework of certain nonhomogeneous nearest-neighbor random walks in the positive quadrant. For the latter, extinction means absorption at one of the axes. Despite connections with some classical probabilistic models (standard two-type Galton–Watson process, two-urn model), exact formulae for the probabilities of absorption seem to be difficult to come by and one must therefore resort to good approximations. In order to meet this task, we develop potential-theoretic tools and provide various sub- and super-harmonic functions which, for large initial populations, provide bounds which in particular improve those that have appeared earlier in the literature.
Extinction probability Markov jump process Random walk Branching process Random environment Sub- and superharmonic function Potential theory
Mathematics Subject Classification
60J05 60H25 60K05
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Most of this work was done during mutual visits of the authors between 2015 and 2018 at their respective home institutions. Hospitality and excellent working conditions at these institutions are most gratefully acknowledged. The second author would also like to thank Viet Chi Tran for interesting discussions. We are indebted to two anonymous referees and an associate editor for very careful reading and many constructive remarks that helped to improve the original version of this article.
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