On synchronization in semelparous populations

Synchronization, i.e., convergence towards a dynamical state where the whole population is in one age class, is a characteristic feature of some population models with semelparity. We prove some rigorous results on this, for a simple class of nonlinear one- population models with age structure and semelparity: (i) the survival probabilities are assumed constant, and (ii) only the last age class is reproducing (semelparity), with fecundity decreasing with total population. For this model we prove: (a) The synchronized, or Single Year Class (SYC), dynamical state is always attracting. (b) The coexistence equilibrium is often unstable; we state and prove simple results on this. (c) We describe dynamical states with some, but not all, age classes populated, which we call Multiple Year Class (MYC) patterns, and we prove results extending (a) and (b) into these patterns.Acknowledgement Boris Kruglikov contributed the nonlinear part of the formulation as well as the proof of Theorem 1. The authors are grateful for critical and constructive comments by N. Davydova and O. Diekmann. E.M. is also grateful for discussions with Marius Overholt concerning problems of proving Theorem 2.     

Delayed maturation in temporally structured populations with non-equilibrium dynamics

In this paper we study the evolutionary dynamics of delayed maturation in semelparous individuals. We model this in a two-stage clonally reproducing population subject to density-dependent fertility. The population dynamical model allows multiple — cyclic and/or chaotic — attractors, thus allowing us to illustrate how (i) evolutionary stability is primarily a property of a population dynamical system as a whole, and (ii) that the evolutionary stability of a demographic strategy by necessity derives from the evolutionary stability of the stationary population dynamical systems it can engender, i.e., its associated population dynamical attractors. Our approach is based on numerically estimating invasion exponents or “mutant fitnesses”. The invasion exponent is defined as the theoretical long-term average relative growth rate of a population of mutants in the stationary environment defined by a resident population system. For some combinations of resident and mutant trait values, we have to consider multi-valued invasion exponents, which makes the evolutionary argument more complicated (and more interesting) than is usually envisaged. Multi-valuedness occurs (i) when more than one attractor is associated with the values of the residents' demographic parameters, or (ii) when the setting of the mutant parameters makes the descendants of a single mutant reproduce exclusively either in even or in odd years, so that a mutant population is affected by either subsequence of the fluctuating resident densities only. Non-equilibrium population dynamics or random environmental noise selects for strategists with a non-zero probability to delay maturation. When there is an evolutionarily attracting pair of such a strategy and a population dynamical attractor engendered by it, this delaying probability is a Continuously Stable Strategy, that is an Evolutionarily Unbeatable Strategy which is also Stable in a long term evolutionary sense. Population dynamical coexistence of delaying and non-delaying strategists is possible with non-equilibrium dynamics, but adding random environmental noise to the model destroys this coexistence. Adding random noise also shifts the CSS towards a higher probability of delaying maturation.     

Approach to equilibrium in age structured populations with an increasing recruitment process

This paper considers a model for the dynamics of an age structured population subject to a density dependent factor which regulates the recruitment. Certain properties of biological interest are obtained and the stability of the equilibrium age distributions is investigated. Finally some applications to known fishery models are considered.Work done under the contract 80.02333.01 of C.N.R.     

Discontinuous Galerkin method for piecewise regular solutions to the nonlinear age-structured population model

A discontinuous Galerkin approximation of the nonlinear Lotka-McKendrick equation is considered in the frequent case when the solution is only piecewise regular. An O(h(r+1/2)) error estimate for rth order polynomial finite elements is proved, as well as piecewise H(1)-regularity of the exact solution which guarantees the error estimate for r=0. Certain implementational details which improve the robustness of the method are also addressed.     

Understanding contributions of cohort effects to growth rates of fluctuating populations

1. Understanding contributions of cohort effects to variation in population growth of fluctuating populations is of great interest in evolutionary biology and may be critical in contributing towards wildlife and conservation management. Cohort-specific contributions to population growth can be evaluated using age-specific matrix models and associated elasticity analyses. 2. We developed age-specific matrix models for naturally fluctuating populations of stoats Mustela erminea in New Zealand beech forests. Dynamics and productivity of stoat populations in this environment are related to the 3-5 year masting cycle of beech trees and consequent effects on the abundance of rodents. 3. The finite rate of increase (lambda) of stoat populations in New Zealand beech forests varied substantially, from 1.98 during seedfall years to 0.58 during post-seedfall years. Predicted mean growth rates for stoat populations in continuous 3-, 4- or 5-year cycles are 0.85, 1.00 and 1.13. The variation in population growth was a consequence of high reproductive success of females during seedfall years combined with low survival and fertility of females of the post-seedfall cohort. 4. Variation in population growth was consistently more sensitive to changes in survival rates both when each matrix was evaluated in isolation and when matrices were linked into cycles. Relative contributions to variation in population growth from survival and fertility, especially in 0-1-year-old stoats, also depend on the year of the cycle and the number of transitional years before a new cycle is initiated. 5. Consequently, management strategies aimed at reducing stoat populations that may be best during one phase of the beech seedfall cycle may not be the most efficient during other phases of the cycle. We suggest that management strategies based on elasticities of vital rates need to consider how population growth rates vary so as to meet appropriate economic and conservation targets.     

Incorporating the temporal autocorrelation of demographic rates into structured population models

Population dynamics are typically temporally autocorrelated: population sizes are positively or negatively correlated with past population sizes. Previous studies have found that positive temporal autocorrelation increases the risk of extinction due to ‘inertia’ that prolongs downward fluctuations in population size. However, temporal autocorrelation has not yet been analyzed at the level of life cycle transitions. We developed an R package, colorednoise, which creates stochastic matrix population projections with distinct temporal autocorrelation values for each matrix element. We used it to analyze long-term demographic data on 25 populations from the COMADRE and COMPADRE databases and simulate their stochastic dynamics. We found a broad range of temporal autocorrelation across species, populations and life cycle stages. The number of stage-classes in the matrix strongly affected the temporal autocorrelation of the growth rate. In the plant populations, reproduction transitions had more negative temporal autocorrelation than survival transitions, and matrices dominated by positive temporal autocorrelation had higher extinction risk, while in animal populations transition type was not associated with noise color. Our results indicate that temporal autocorrelation varies across life cycle transitions, even among populations of the same species. We present the colorednoise package for researchers to analyze the temporal autocorrelation of structured demographic rates.     

The dynamics of hierarchical age-structured populations

An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.     

Estimating sex-specific dispersal rates with autosomal markers in hierarchically structured populations

Abstract –A recent study suggests that sex-specific dispersal rates can be quantitatively estimated on the basis of sex- and state-specific (pre- vs. postdispersal) F -statistics. In the present paper, we extend this approach to account for the hierarchical structure of natural populations, and we validate it through individual-based simulations. The model is applied to an empirical data set consisting of 536 individuals (males, females, and predispersal juveniles) of greater white-toothed shrews ( Crocidura russula ), sampled according to a hierarchical design and typed for seven autosomal microsatellite loci. From this dataset, dispersal is significantly female biased at the local scale (breeding-group level), but not at the larger scale (among local populations). We argue that selective pressures on dispersal are likely to depend on the spatial scale considered, and that short-distance dispersal should mainly respond to kin interactions (inbreeding or kin competition avoidance), which exert differential pressure on males and females.     

Relationships among recent models for insect population dynamics with variable rates of development

A classification scheme for those population models which allow variation in development rates is proposed, based on two ways of modifying standard age-structured models. The resulting classes of models are termed development index models and sojourn time models. General formulations for the two classes of models are developed from two basic balance equations, and numerous specific models from the literature are shown to fit into the scheme. Concepts from competing risks theory are shown to be important in understanding the interplay between mortality and maturation. Relationships among the classes are investigated both for the most general forms of the models and for the simpler forms often used. The scheme can provide guidance in developing appropriate insect population models for specific modelling situations.Contribution 3878871     

Using statistics to design and estimate vital rates in matrix population models for a perennial herb

Matrix population models are widely used to assess population status and to inform management decisions. Despite existing theories for building such models, model construction is often partially based on expert opinion. So far, model structure has received relatively little attention, although it may affect estimates of population dynamics. Here, we assessed the consequences of two published matrix structures (a 4 × 4 matrix based on expert opinion and a 10 × 10 matrix based on statistical modeling) for estimates of vital rates and stochastic population dynamics of the long-lived herb Astragalus scaphoides. We explored the ways in which choice of model structure alters the accuracy (i.e., mean) and precision (i.e., variance) of predicted population dynamics. We found that model structure had a negligible effect on the accuracy and precision of vital rates and stochastic stage distribution. However, the 10 × 10 matrix produced lower estimates of stochastic population growth rates than the 4 × 4 matrix, and more accurately predicted the observed trends in population abundance for three out of four study populations. Moreover, estimates of realized variation in population growth rate due to fluctuations in population stage structure over time were occasionally sensitive to matrix structure, suggesting differential roles of transient dynamics. Our study indicates that statistical modeling for choosing categories in matrix models might be preferable over expert opinion to accurately predict population trends and can provide a more objective way for model construction when the biological knowledge of the species is limited.     

Building integral projection models with nonindependent vital rates

Population dynamics are functions of several demographic processes including survival, reproduction, somatic growth, and maturation. The rates or probabilities for these processes can vary by time, by location, and by individual. These processes can co‐vary and interact to varying degrees, e.g., an animal can only reproduce when it is in a particular maturation state. Population dynamics models that treat the processes as independent may yield somewhat biased or imprecise parameter estimates, as well as predictions of population abundances or densities. However, commonly used integral projection models (IPMs) typically assume independence across these demographic processes. We examine several approaches for modelling between process dependence in IPMs and include cases where the processes co‐vary as a function of time (temporal variation), co‐vary within each individual (individual heterogeneity), and combinations of these (temporal variation and individual heterogeneity). We compare our methods to conventional IPMs, which treat vital rates independent, using simulations and a case study of Soay sheep (Ovis aries). In particular, our results indicate that correlation between vital rates can moderately affect variability of some population‐level statistics. Therefore, including such dependent structures is generally advisable when fitting IPMs to ascertain whether or not such between vital rate dependencies exist, which in turn can have subsequent impact on population management or life‐history evolution.     

A general structured model for a sequential hermaphrodite population

This paper introduces and analyzes a model of sequential hermaphroditism in the framework of continuously structured population models with sexual reproduction. The model is general in the sense that the birth, transition (from one sex to the other) and death processes of the population are given by arbitrary functions according to a biological meaningful hypotheses. The system is reduced to a single equation introducing the intrinsic sex-ratio subspace. The steady states are analyzed and illustrated for several cases. In particular, neglecting the competition for resources we have explicitly found a unique non-trivial equilibrium which is unstable.     

非线性高阶阻尼时滞微分方程的振动定理

研究具有阻尼项的非线性时滞微分方程,给出了使得方程的一切解振动的两个充分条件.一些实例说明,本文的结果在判定非线性阻尼方程的振动性时较文献中的结果更为有效.     

Differential equations models for interacting wild and transgenic mosquito populations

We formulate and study continuous-time models, based on systems of ordinary differential equations, for interacting wild and transgenic mosquito populations. We assume that the mosquito mating rate is either constant, proportional to total mosquito population size, or has a Holling-II-type functional form. The focus is on the model with the Holling-II-type functional mating rate that incorporates Allee effects, in order to account for mating difficulty when the size of the total mosquito populations is small. We investigate the existence and stability of both boundary and positive equilibria. We show that the Holling-II-type model is the more realistic and, by means of numerical simulations, that it exhibits richer dynamics.     

Modelling Hematopoiesis Mediated by Growth Factors With Applications to Periodic Hematological Diseases

Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to understand some blood diseases characterized by very long period oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is proposed and analyzed. The existence of a Hopf bifurcation at a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long-period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors, for reasonable cell cycle durations. These oscillations can be related to observations on some periodic hematological diseases (such as chronic myelogenous leukemia, for example). ¹Research was partially supported by the INRIA Futurs, ANUBIS Team. ²Research was partially supported by the NSF and the University of Miami.     

Models for Bounded Systems with Continuous Dynamics

Summary .  Models for natural nonlinear processes, such as population dynamics, have been given much attention in applied mathematics. For example, species competition has been extensively modeled by differential equations. Often, the scientist has preferred to model the underlying dynamical processes (i.e., theoretical mechanisms) in continuous time. It is of both scientific and mathematical interest to implement such models in a statistical framework to quantify uncertainty associated with the models in the presence of observations. That is, given discrete observations arising from the underlying continuous process, the unobserved process can be formally described while accounting for multiple sources of uncertainty (e.g., measurement error, model choice, and inherent stochasticity of process parameters). In addition to continuity, natural processes are often bounded; specifically, they tend to have nonnegative support. Various techniques have been implemented to accommodate nonnegative processes, but such techniques are often limited or overly compromising. This article offers an alternative to common differential modeling practices by using a bias-corrected truncated normal distribution to model the observations and latent process, both having bounded support. Parameters of an underlying continuous process are characterized in a Bayesian hierarchical context, utilizing a fourth-order Runge–Kutta approximation.     

A model with 'growth retardation' for the kinetic heterogeneity of tumour cell populations

In the present paper we propose a continuous cell population model based on Shackney's idea of growth retardation. Cells are characterized by two state variables: the cell maturity x, 0 < or = x < or = 1, and a state variable T that identifies the rate of maturation along cell cycle. During their life span, cells can change T at random by jump transitions to T values corresponding to slower maturation rates, while at each jump the maturity x is conserved. Both the time evolution of the population and the exponential stationary solution are numerically computed. The distribution of the cell cycle transit time in asynchronous exponential growth is investigated by Monte Carlo simulation. An approximated formula for the distribution of cell cycle time is also provided.     

Heterogeneous capture-recapture models with covariates: a partial likelihood approach for closed populations

In practice, when analyzing data from a capture-recapture experiment it is tempting to apply modern advanced statistical methods to the observed capture histories. However, unless the analysis takes into account that the data have only been collected from individuals who have been captured at least once, the results may be biased. Without the development of new software packages, methods such as generalized additive models, generalized linear mixed models, and simulation-extrapolation cannot be readily implemented. In contrast, the partial likelihood approach allows the analysis of a capture-recapture experiment to be conducted using commonly available software. Here we examine the efficiency of this approach and apply it to several data sets.