Parametric Inference in Biological Systems in a Random Environment

Controlled Branching Processes

1 Long‐term demographic field studies combined with simulation models that incorporated two types of stochasticity were used to investigate the relative impact of reproduction and survival on the persistence of populations of the North American woodland herb, Asarum canadense. 2 Field data were collected over 7 years in replicate plots located in early and late successional forest habitat. By following marked ramets from year to year it proved possible to investigate the demography both of ramets and of groups of genetically identical ramets (clones). 3 A. canadense exhibited considerable temporal and spatial variation in reproductive success, survivorship, and population growth rates. Populations of ramets and clones in late successional forest habitats held their own, while those in early successional forest habitats declined. 4 Sexual and clonal reproduction had less impact on population growth than did survival. Nonetheless, seedling recruitment was important, as indicated by the relatively high rates of genet turnover in A. canadense populations. 5 Results from simulations indicated that the vast majority of genets in both habitat types failed to reproduce. Over their lifetime, simulated genets produced a maximum of 44 and 77 seedlings in early and late successional forest habitat, respectively. 6 Although A. canadense genets are potentially immortal, the maximum life span for 2000 simulated genets was 104 years. Simulated genets lived a mean of 3.0 and 4.8 years in early and late successional forest habitat, respectively. 7 Simulated A. canadense populations only persisted in late successional forest habitat. Within simulated late successional forest populations, the number of ramets tended to decrease over time, while the number of genets tended to increase over time. 8 The minimum viable population size for A. canadense ramets in late successional forest was 25 individuals with environmental stochasticity only, and 1000 individuals with both environmental and demographic stochasticity; for genets in late successional forest, the corresponding numbers were 20 and 25, respectively. 9 Ramet and genet dynamics differed greatly in A. canadense, highlighting the importance of long‐term demographic studies at each of these levels.

We study large deviation probabilities for bisexual branching process in a random (i.i.d.) environment. Under several conditions on the mating function (which may depend on the environment) we introduce the associated random walk of the process. We also assume Cramer condition for the step of the walk and moment conditions on the number of descendants of one pair. Under Cramer condition for the steps of the walk and moment conditions on the number of descendants of a pair we find an exact asymptotics of probabilities P(ln Nn ∈ [x, x + Δn )) as n → ∞ for x/n varying in some domain for all sequences Δn , tending to zero sufficiently slowly. Similar results we obtain for bisexual branching process with immigration in a random environment.

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.

Nonadditive genetic variation and genetic disequilibrium are two important factors that influence the evolutionary trajectory of natural populations. We assayed quantitative genetic variation in a temporary-pond-dwelling population of Daphnia pulex over a full season to examine the role of nonadditive genetic variation and genetic disequilibrium in determining the short-term evolutionary trajectory of a cyclic parthenogen. Quantitative traits were influenced by three factors: (1) clonal selection significantly changed the population mean phenotype during the course of the growing season; (2) sexual reproduction and recombination led to significant changes in life-history trait means and the levels of expressed genetic variation, implying the presence of substantial nonadditive genetic variation and genetic disequilibrium; and (3) Egg-bank effects were found to be an important component of the realized year-to-year change. Additionally, we examined the impact of genetic disequilibria induced by clonal selection on the genetic (co)variance structure with a common principal components model. Clonal selection caused significant changes in the (co)variance structure that were eliminated by a single bout of random mating, suggesting that a build-up of disequilibria was the primary source of changes in the (co)variance structure. The results of this study highlight the complexity of natural selection operating on populations that undergo alternating phases of sexual and asexual reproduction.

Stochastic simulation algorithms provide a powerful means to understand complex biochemical processes as well as to solve the inverse problem of reconstructing hidden states and parameters from experimental single-cell data. At presence, a repertoire of efficient algorithms for simulating and calibrating stochastic reaction networks is available. However, most of these approaches do not account for the fact that each cell of a clonal population is exposed to a random extrinsic environment, i.e., the agglomerate of so-called extrinsic factors such as cell size, shape or cell cycle stage. We recently proposed a dynamic description of stochastic chemical kinetics in random but unknown extrinsic environments, reflected by a stochastic process where uncertain parameters are marginalized out. In this work we further investigate that process and provide additional analytical results. We demonstrate the marginalization using several biologically relevant parameter distributions and derive exact waiting-time distributions. We further show that the marginalized process model can achieve a variance reduction in the context of parameter inference.

On the surviving probability of an annihilating branching process and application to a nonlinear voter model

Let Zn be the number of particles at time n in a critical branching process in random environment and Zm,n be the number of particles in this process at time m ≤ n which have non-empty offspring at time n. We prove limit theorems for the process {Z[nt],n, t ∈ (0, 1]} conditioned on the event {Zn > 0}. Quenched and annealed approaches are considered.*

Branching processes in a random environment are natural generalisations of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in an iid random environment, where the offspring expectation converges from above to 1. We prove that Haldane’s asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical/subcritical regime. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.

Let $\{\mathbf{Z}_n\}$ be a multitype branching process in a random environment (MBPRE) which grows to infinity with positive probability for almost all environmental sequences. Under some conditions involving the first two moments of the environmental sequence, it is shown that dividing the $\{\mathbf{Z}_n\}$ components by their environment-conditioned expectations yields a sequence convergent in $L^2$ to a random vector with equal components.

We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real- valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.

A branching process $Z(n)$, $n=0,1,\ldots,$ is considered which evolves in a random environment generated by a sequence of independent identically distributed generating functions $f_0(s),f_{1}(s),\ldots\;$. Let $S_0=0$, $S_k=\log f_0^{\prime}(1)+\cdots +\log f_{k-1}^{\prime }(1)$, $k\ge 1$, be the associated random walk and let $\tau (n)$ be the leftmost point of minimum of $\{S_k\}_{k\ge 0}$ on the interval $[0,n]$. Assuming that the random walk satisfies the Spitzer condition $n^{-1}\sum_{k=1}^{n}{\bf P}\{S_k>0\}\to\rho\in(0,1)$, $n\to \infty$, we show (under the quenched approach) that for each fixed $t\in (0,1]$ and $m=0,\pm 1,\pm 2,\ldots,$ the distribution of $Z(\tau (nt)+m)$ given $Z(n)>0$ converges as $n\to \infty$ to a (random) discrete distribution. Thus, in contrast to fixed points of the form $nt$, where the size of the population is large (even exponentially large; see [V. A. Vatutin and E. E. Dyakonova, Theory Probab. Appl., 49 (2005), pp. 275–309]), the size of the population at (random) p...

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

Denote by αt(μ)the probability law ofAt(μ)= ∫0texp(2(Bs+μs))dsfor a Brownian motion {Bs,s≥ 0}. It is well known that αt(μ)is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0)and αt(1), as well as an equally remarkable relationship between αt(μ)and αt(ν)for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for theprocesses{At(μ),t≥ 0}.

Propriétés asymptotiques des processus de branchement en environnement aléatoire