Quantifying time-inhomogeneous stochastic introgression processes with hazard rates

Introgression is the permanent incorporation of genes from one population into another through hybridization and backcrossing. It is currently of particular concern as a possible mechanism for the spread of modi ed crop genes to wild populations. The hazard rate is the probability per time unit that such an escape takes place, given that it has not happened before. It is a quantitative measure of introgression risk that takes the stochastic elements inherent in introgression processes into account. We present a methodology to calculate the hazard rate for situations with time-varying gene ow from a crop to a large recipient wild population. As an illustration, several types of time-inhomogeneity are examined, including deterministic periodicity as well as random variation. Furthermore, we examine the e ects of an extended tness bottleneck of hybrids and backcrosses in combination with time-varying gene ow. It is found that bottlenecks decrease the hazard rate, but also slow down and delay its changes in reaction to changes in gene ow. Furthermore, we nd that random variation in gene ow generates a lower hazard rate than analogous deterministic variation. We discuss the implications of our ndings for crop management and introgression risk assessment.This research was funded through the research program 'Ecology Regarding Genetically modified Organisms (ERGO)', commissioned by four Dutch ministries. This funding program is managed by the Earth and Life Sciences Council (ALW) of the Netherlands Organisation for Scientific Research (NWO). P. Haccou's research is additionally supported by the NDNS (Nonlinear Dynamics of Natural Systems) program of NWO. M.C. Serra would like to thank the Fundacao para a Ciencia e Tecnologia for financial support through the scholarship SFRH/BPD/47615/2008. We thank Marije Stoops and Prof. Baorong Lu for discussions and comments on a previous version

The likelihood ratio test for the change point problem for exponentially distributed random variables

AbstractLet x1,…, xn+1 be independent exponentially distributed random variables with intensity λ1 for i ⩽ τ and λ2 for i > τ, where τ as well as λ1 and λ2 are unknown. By application of theorems concerning the normed uniform quantile process it is proved that the asymptotic null-distribution of the likelihood ratio statistic for testing λ1 = λ2 (or, equivalently, τ = 0 or n + 1) is an extreme value distribution.Change point problems occur in a variety of experimental sciences and therefore have considerabla attention of applied statisticians. The problems are non-standard since the usual regularity conditions are not satisfied. Explicit asymptotic distributions of likelihood ratio tests have until now only been derived for a few cases. The method of proof used in this paper is based on the ‘strong invariance principle’.Furthermore it is shown that the test is optimal in the sense of Bahadur, although the Pitman efficiency is zero. However, simulation results indicate a good power for values of n that are relevant for most applications.The likelihood ratio test is compared with another test which has the same asymptotic null-distribution. This test has Bahadur efficiency zero. The simulation results confirm that the likelihood ratio test is superior to the latter test

On the First Crossing of Two Boundaries by an Order Statistics Risk Process

We derive a closed form expression for the probability that a non-decreasing, pure jump stochastic risk process with the order statistics (OS) property will not exit the strip between two non-decreasing, possibly discontinuous, time-dependent boundaries, within a finite time interval. The result yields new expressions for the ruin probability in the insurance and the dual risk models with dependence between the claim severities or capital gains respectively

Establishment versus population growth in spatio-temporally varying environments

We consider situations where repeated invasion attempts occur from a source population into a receptor population over extended periods of time. The receptor population contains two locations that provide different expected off- spring numbers to invaders. There is demographic stochasticity in offspring numbers. In addition, temporal variation causes local invader fitnesses to vary. We show that effects of environmental autocorrelation on establishment success depend on spatial covariance of the receptor subpopulations. In situ- ations with a low spatial covariance this effect is positive, whereas high spatial covariance and/or high migration probabilities between the subpopulations causes the effect to be negative. This result reconciles seemingly contradictory results from the literature concerning effects of temporal variation on popu- lation dynamics with demographic stochasticity. We study an example in the context of genetic introgression, where invasions of cultivar plant genes occur through pollen flow from a source population into wild-type receptor populations, but our results have implications in a wider range of contexts, such as the spread of exotic species, metapopulation dynamics and epidemics.Global Challenges (FGGA

Evolutionary branching in a stochastic population model with discrete mutational steps

Evolutionary branching is analysed in a stochastic, individual-based population model under mutation and selection. In such models, the common assumption is that individual reproduction and life career are characterised by values of a trait, and also by population sizes, and that mutations lead to small changes in trait value. Then, traditionally, the evolutionary dynamics is studied in the limit of vanishing mutational step sizes. In the present approach, small but non-negligible mutational steps are considered. By means of theoretical analysis in the limit of infinitely large populations, as well as computer simulations, we demonstrate how discrete mutational steps affect the patterns of evolutionary branching. We also argue that the average time to the first branching depends in a sensitive way on both mutational step size and population size.Comment: 12 pages, 8 figures. Revised versio

Optimal Patch-Leaving Behaviour: A Case Study Using The Parasitoid Cotesia rebecula

1. Parasitoids are predicted to spend longer in patches with more hosts, but previous work on Cotesia rubecula (Marshall) has not upheld this prediction. Tests of theoretical predictions may be affected by the definition of patch leaving behaviour, which is often ambiguous. 2. In this study whole plants were considered as patches and assumed that wasps move within patches by means of walking or flying. Within-patch and between-patch flights were distinguished based on flight distance. The quality of this classification was tested statistically by examination of log-survivor curves of flight times. 3. Wasps remained longer in patches with higher host densities, which is consistent with predictions of the marginal value theorem (Charnov 1976). Under the assumption that each flight indicates a patch departure, there is no relationship between host density and leaving tendency. 4. Oviposition influences the patch leaving behaviour of wasps in a count down fashion (Driessen et al. 1995), as predicted by an optimal foraging model (Tenhumberg, Keller & Possingham 2001). 5. Wasps spend significantly longer in the first patch encountered following release, resulting in an increased rate of superparasitism

Limiting Similarity Revisited

We reinvestigate the validity of the limiting similarity principle via numerical simulations of the Lotka-Volterra model. A Gaussian competition kernel is employed to describe decreasing competition with increasing difference in a one-dimensional phenotype variable. The simulations are initiated by a large number of species, evenly distributed along the phenotype axis. Exceptionally, the Gaussian carrying capacity supports coexistence of all species, initially present. In case of any other, distinctly different, carrying capacity functions, competition resulted in extinction of all, but a few species. A comprehensive study of classes of fractal-like carrying capacity functions with different fractal exponents was carried out. The average phenotype differences between surviving species were found to be roughly equal to the competition width. We conclude that, despite the existence of exceptional cases, the classical picture of limiting similarity and niche segregation is a good rule of thumb for practical purposes

Prospects & Overviews Bet hedging or not? A guide to proper classification of microbial survival strategies

Bacteria have developed an impressive ability to survive and propagate in highly diverse and changing environments by evolving phenotypic heterogeneity. Phenotypic heterogeneity ensures that a subpopulation is well prepared for environmental changes. The expression bet hedging is commonly (but often incorrectly) used by molecular biologists to describe any observed phenotypic heterogeneity. In evolutionary biology, however, bet hedging denotes a risk-spreading strategy displayed by isogenic populations that evolved in unpredictably changing environments. Opposed to other survival strategies, bet hedging evolves because the selection environment changes and favours different phenotypes at different times. Consequently, in bet hedging populations all phenotypes perform differently well at any time, depending on the selection pressures present. Moreover, bet hedging is the only strategy in which temporal variance of offspring numbers per individual is minimized. Our paper aims to provide a guide for the correct use of the term bet hedging in molecular biology

Single-crossover dynamics: finite versus infinite populations

Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarise and adapt a deterministic approach, as valid for infinite populations, which assumes continuous time and single crossover events. The corresponding nonlinear system of differential equations permits a closed solution, both in terms of the type frequencies and via linkage disequilibria of all orders. To include stochastic effects, we then consider the corresponding finite-population model, the Moran model with single crossovers, and examine it both analytically and by means of simulations. Particular emphasis is on the connection with the deterministic solution. If there is only recombination and every pair of recombined offspring replaces their pair of parents (i.e., there is no resampling), then the {\em expected} type frequencies in the finite population, of arbitrary size, equal the type frequencies in the infinite population. If resampling is included, the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.Comment: 21 pages, 4 figure