61 research outputs found
Pollicott's Algorithm for Markovian Products of Positive Matrices
In a recent paper by M. Pollicott, an efficient algorithm was proposed,
applying Ruelle's theory of transfer operators and Grothendieck's classical
work on nuclear operators, to compute the Lyapunov exponent associated with the
i.i.d. products of positive matrices. In this article, we generalise the result
to Markovian products and correct some minor mistakes in Pollicott's original
paper.Comment: 27 page
Damage segregation at fissioning may increase growth rates: A superprocess model
A fissioning organism may purge unrepairable damage by bequeathing it
preferentially to one of its daughters. Using the mathematical formalism of
superprocesses, we propose a flexible class of analytically tractable models
that allow quite general effects of damage on death rates and splitting rates
and similarly general damage segregation mechanisms. We show that, in a
suitable regime, the effects of randomness in damage segregation at fissioning
are indistinguishable from those of randomness in the mechanism of damage
accumulation during the organism's lifetime. Moreover, the optimal population
growth is achieved for a particular finite, non-zero level of combined
randomness from these two sources. In particular, when damage accumulates
deterministically, optimal population growth is achieved by a moderately
unequal division of damage between the daughters. Too little or too much
division is sub-optimal. Connections are drawn both to recent experimental
results on inheritance of damage in protozoans, to theories of the evolution of
aging, and to models of resource division between siblings.Comment: Version 2 had significant conceptual and organizational changes,
though only minor changes to the mathematics. Version 3 has minor
proofreading corrections, and a few new references. The paper will appear in
Theoretical Population Biolog
Quasistationary distributions for one-dimensional diffusions with killing
We extend some results on the convergence of one-dimensional diffusions
killed at the boundary, conditioned on extended survival, to the case of
general killing on the interior. We show, under fairly general conditions, that
a diffusion conditioned on long survival either runs off to infinity almost
surely, or almost surely converges to a quasistationary distribution given by
the lowest eigenfunction of the generator. In the absence of internal killing,
only a sufficiently strong inward drift can keep the process close to the
origin, to allow convergence in distribution. An alternative, that arises when
general killing is allowed, is that the conditioned process is held near the
origin by a high rate of killing near infinity. We also extend, to the case of
general killing, the standard result on convergence to a quasistationary
distribution of a diffusion on a compact interval.Comment: 40 pages, final version accepted for Trans. Amer. Math. Soc. except
for a graphi
Derivatives of the Stochastic Growth Rate
We consider stochastic matrix models for population driven by random
environments which form a Markov chain. The top Lyapunov exponent , which
describes the long-term growth rate, depends smoothly on the demographic
parameters (represented as matrix entries) and on the parameters that define
the stochastic matrix of the driving Markov chain. The derivatives of --
the "stochastic elasticities" -- with respect to changes in the demographic
parameters were derived by \cite{tuljapurkar1990pdv}. These results are here
extended to a formula for the derivatives with respect to changes in the Markov
chain driving the environments. We supplement these formulas with rigorous
bounds on computational estimation errors, and with rigorous derivations of
both the new and the old formulas.Comment: 35 page
Random time changes for sock-sorting and other stochastic process limit theorems
E l e c t r o
A generalized model of mutation-selection balance with applications to aging
A probability model is presented for the dynamics of mutation-selection
balance in a haploid infinite-population infinite-sites setting sufficiently
general to cover mutation-driven changes in full age-specific demographic
schedules. The model accommodates epistatic as well as additive selective
costs. Closed form characterizations are obtained for solutions in finite time,
along with proofs of convergence to stationary distributions and a proof of the
uniqueness of solutions in a restricted case. Examples are given of
applications to the biodemography of aging, including instabilities in current
formulations of mutation accumulation.Comment: 20 pages Updated to include more historical comment and references to
the literature, as well as to make clear how our non-linear, non-Markovian
model differs from previous linear, Markovian particle system and
measure-valued diffusion models. Further updated to take into account
referee's comment
An approximation scheme for quasi-stationary distributions of killed diffusions
In this paper we study the asymptotic behavior of the normalized weighted
empirical occupation measures of a diffusion process on a compact manifold
which is killed at a smooth rate and then regenerated at a random location,
distributed according to the weighted empirical occupation measure. We show
that the weighted occupation measures almost surely comprise an asymptotic
pseudo-trajectory for a certain deterministic measure-valued semiflow, after
suitably rescaling the time, and that with probability one they converge to the
quasi-stationary distribution of the killed diffusion. These results provide
theoretical justification for a scalable quasi-stationary Monte Carlo method
for sampling from Bayesian posterior distributions.Comment: v2: revised version, 29 pages, 1 figur
The Age-Specific Force of Natural Selection and Walls of Death
W. D. Hamilton's celebrated formula for the age-specific force of natural
selection furnishes predictions for senescent mortality due to mutation
accumulation, at the price of reliance on a linear approximation. Applying to
Hamilton's setting the full non-linear demographic model for mutation
accumulation of Evans et al. (2007), we find surprising differences. Non-linear
interactions cause the collapse of Hamilton-style predictions in the most
commonly studied case, refine predictions in other cases, and allow Walls of
Death at ages before the end of reproduction. Haldane's Principle for genetic
load has an exact but unfamiliar generalization.Comment: 27 page
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