Abstract
We first study the eigenvalue problem for a system of coupled integral operators and investigate existence and monotonicity of the principal eigenvalue and asymptotic behavior of the principal eigenpair. As applications of the spectral theory, we investigate a single genotype stem cell regeneration model with epigenetic transitions, and multiple genotype stem cell regeneration models with epigenetic transitions, with or without gene mutations. These models give rise to three classes of nonlinear integro-differential evolution equations. We consider existence, uniqueness and multiplicity of positive steady states to these equations, as well as the long time behavior of time-dependent solutions. Various explicit formulae for threshold values for tissue development, degeneration and abnormal growth are obtained.
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