If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting MRCA age process has be
en studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright--Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on $mathbb{R}_{++}timesmathbb{R}_{++}$ with intensity measures of the form $lambdaotimesmu$ where $lambda$ is Lebesgue measure, and $mu$ (the family lifetime measure) is an arbitrary, absolutely continuous measure satisfying $mu((0,infty))=infty$ and $mu((x,infty))<infty$ for all $x>0$. Special cases of this construction describe the time evolution of the MRCA age in $(1+beta)$-stable continuous state branching processes conditioned on nonextinction--a particular case of which, $beta=1$, is Fellers continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist.
New models for evolutionary processes of mutation accumulation allow hypotheses about the age-specificity of mutational effects to be translated into predictions of heterogeneous population hazard functions. We apply these models to questions in the
biodemography of longevity, including proposed explanations of Gompertz hazards and mortality plateaus, and use them to explore the possibility of melding evolutionary and functional models of aging.
We consider the asymptotic behavior as $ntoinfty$ of the spectra of random matrices of the form [frac{1}{sqrt{n-1}}sum_{k=1}^{n-1}Z_{nk}rho_n ((k,k+1)),] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $
rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $rho_n$ of the symmetric group on ${1,2,...,n}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $rho_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on ${1,2,...,n}$ are indexed by partitions $lambda_n$ of $n$. A consequence of the results we establish is that if $lambda_{n,1}gelambda_{n,2}ge...ge0$ is the partition of $n$ corresponding to $rho_n$, $mu_{n,1}gemu_{n,2}ge >...ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $mu_n$ is the transpose of the Young diagram of $lambda_n$), $lim_{ntoinfty}frac{lambda_{n,i}}{n}=p_i$ for each $ige1$, and $lim_{ntoinfty}frac{mu_{n,j}}{n}=q_j$ for each $jge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $theta Z$ and variance $1-theta^2$, where $theta$ is the constant $sum_ip_i^2-sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.
We study a continuous-time dynamical system that models the evolving distribution of genotypes in an infinite population where genomes may have infinitely many or even a continuum of loci, mutations accumulate along lineages without back-mutation, ad
ded mutations reduce fitness, and recombination occurs on a faster time scale than mutation and selection. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete time models, were presented in earlier work by Evans, Steinsaltz, and Wachter for quite general selective costs. Here we study a special case where the selective cost of a genotype with a given accumulation of ancestral mutations from a wild type ancestor is a sum of costs attributable to each individual mutation plus successive interaction contributions from each $k$-tuple of mutations for $k$ up to some finite ``degree. Using ideas from complex chemical reaction networks and a novel Lyapunov function, we establish that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions. That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.
W. D. Hamiltons 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 Hamiltons 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. Haldanes Principle for genetic load has an exact but unfamiliar generalization.
As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and
provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between $2M$ points for some integer $M$. The role of matrices is now played by $2M$-dimensional hypercubic arrays, and the determinant is replaced by a suitable generalization of it to such arrays -- Cayleys first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.
