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 $a$, 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 $a$ -- 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