Derivation of stochastic partial differential equations for size- and age-structured populations

Stochastic partial differential equations (SPDEs) for size-structured and age- and size-structured populations are derived from basic principles, i.e. from the changes that occur in a small time interval. Discrete stochastic models of size-structured and age-structured populations are constructed, carefully taking into account the inherent randomness in births, deaths, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the size and age intervals decrease, SPDEs are derived for size-structured and age- and size-structured populations. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations.     

Estimating fluctuating selection in age-structured populations

In age-structured populations, viability and fecundity selection of varying strength may occur in different age classes. On the basis of an original idea by Fisher of weighting individuals by their reproductive value, we show that the combined effect of selection on traits at different ages acts through the individual reproductive value defined as the stochastic contribution of an individual to the total reproductive value of the population the following year. The selection differential is a weighted sum of age-specific differentials that are the covariances between the phenotype and the age-specific relative fitness defined by the individual reproductive value. This enables estimation of weak selection on a multivariate quantitative character in populations with no density regulation by combinations of age-specific linear regressions of individual reproductive values on the traits. Demographic stochasticity produces random variation in fitness components in finite samples of individuals and affects the statistical inference of the temporal average directional selection as well as the magnitude of fluctuating selection. Uncertainties in parameter estimates and test power depend strongly on the demographic stochasticity. Large demographic variance results in large uncertainties in yearly estimates of selection that complicates detection of significant fluctuating selection. The method is illustrated by an analysis of age-specific selection in house sparrows on a fitness-related two-dimensional morphological trait, tarsus length and body mass of fledglings.     

Equilibrium and local stability in a logistic matrix model for age-structured populations

A logistic matrix model for age-structured population dynamics is constructed. This model discretizes a continuous, density-dependent model with age structure, i.e. it is an extension of the logistic model to the case of age-dependence. We prove the existence and uniqueness of its equilibrium and give a necessary and sufficient condition for the local stability of the equilibrium.     

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.     

Matrix population model with density-dependent recruitment for assessment of age-structured wildlife populations

A logistic density-dependent matrix model is developed in which the matrices contain only parameters and recruitment is a function of adult population density. The model was applied to simulate introductions of white-tailed deer into an area; the fitted model predicted a carrying capacity of 215 deer, which was close to the observed carrying capacity of 220 deer. The rate of population increase depends on the dominant eigenvalue of the Leslie matrix, and the age structure of the simulated population approaches a stable age distribution at the carrying capacity, which was similar to that generated by the Leslie matrix. The logistic equation has been applied to study many phenomena, and the matrix model can be applied to these same processes. For example, random variation can be added to life history parameters, and population abundances generated with random effects on fecundity show both the affect of annual variation in fecundity and a longer-term pattern resulting from the age structure.     

Effective size of harvested ungulate populations

The harvest of ungulate populations is often directed against certain sex or age classes to maximize the yield in terms of biomass, number of shot animals or number of trophies. Here we examine how such directional harvest affects the effective size of the population. We parameterize an age-specific model assumed to describe the dynamics of Fennoscandian moose. Based on expressions for the demographic variance     for a small subpopulation of heterozygotes Aa bearing a rare neutral allele a , we use this model to calculate how different harvest strategies influence the effective size of the population, given that the population remains stable after harvest. We show that the annual genetic drift, determined by     , increases with decreasing harvest rate of calves and increasing sex bias in the harvest towards bulls 1 year or older. The effective population size per generation decreased with reduced harvest of calves and increased harvest of bulls 1 year or older. The magnitude of these effects depends on the age-specific pattern of variation in reproductive success, which influences the demographic variance. This shows that the choice of harvest strategy strongly affects the genetic dynamics of harvested ungulate populations.     

Stability effects of a juvenile period in age-structured populations

Prior theoretical studies have shown that the juvenile period's length is an important determinant of local stability in age-structured population dynamics. For example, both short and long periods produce stability, but intermediate lengths can cause instability. Short juvenile periods significantly increase stability (compared to no juvenile period) if fecundity is independent of adult age. Here I re-examine these and other patterns, using a model which includes a variable juvenile period, juvenile mortality, density-dependent fecundity and adult mortality, and age-dependence is adult fecundity. Among other things, the results confirm the stable-unstable-stable pattern with increasing juvenile period length, but show that the stabilizing effect of short periods disappears when fecundity varies with adult age. Broadly speaking, the results suggest that age-dependence in adult fecundity has important dynamical consequences, and that models assuming that fecundity is independent of adult age may be unreliable guides to the dynamics of populations for which this assumption is not reasonably accurate.     

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.     

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.     

A computational model for populations of dividing cells.

A mathematical model of the cell movements due to cell division is presented. In the model we assume that every cell is a computational object with a given volume, and that the cell pushes the neighbouring cells in order to acquire the space for this volume. The Force that each cell exerts over the other cells is derived from a harmonic arbitrary Potential. The main parameter of the model is the average distance among the cells, that checks if the system is in spatial equilibrium or not. We show that just changing the physical constraints we can model two different systems, a two-dimensional culture on a plate and a three-dimensional early embryo. In both cases the patterns of the cell populations we obtain are similar to the real ones.     

Coinfection and superinfection in RNA virus populations: a selection-mutation model

In this paper, we present a general selection-mutation model of evolution on a one-dimensional continuous fitness space. The formulation of our model includes both the classical diffusion approach to mutation process as well as an alternative approach based on an integral operator with a mutation kernel. We show that both approaches produce fundamentally equivalent results. To illustrate the suitability of our model, we focus its analytical study into its application to recent experimental studies of in vitro viral evolution. More specifically, these experiments were designed to test previous theoretical predictions regarding the effects of multiple infection dynamics (i.e., coinfection and superinfection) on the virulence of evolving viral populations. The results of these experiments, however, did not match with previous theory. By contrast, the model we present here helps to understand the underlying viral dynamics on these experiments and makes new testable predictions about the role of parameters such the time between successive infections and the growth rates of resident and invading populations.     

Implementing vertex dynamics models of cell populations in biology within a consistent computational framework

The dynamic behaviour of epithelial cell sheets plays a central role during development, growth, disease and wound healing. These processes occur as a result of cell adhesion, migration, division, differentiation and death, and involve multiple processes acting at the cellular and molecular level. Computational models offer a useful means by which to investigate and test hypotheses about these processes, and have played a key role in the study of cell–cell interactions. However, the necessarily complex nature of such models means that it is difficult to make accurate comparison between different models, since it is often impossible to distinguish between differences in behaviour that are due to the underlying model assumptions, and those due to differences in the in silico implementation of the model. In this work, an approach is described for the implementation of vertex dynamics models, a discrete approach that represents each cell by a polygon (or polyhedron) whose vertices may move in response to forces. The implementation is undertaken in a consistent manner within a single open source computational framework, Chaste, which comprises fully tested, industrial-grade software that has been developed using an agile approach. This framework allows one to easily change assumptions regarding force generation and cell rearrangement processes within these models. The versatility and generality of this framework is illustrated using a number of biological examples. In each case we provide full details of all technical aspects of our model implementations, and in some cases provide extensions to make the models more generally applicable.     

Evolution of the rate of biological aging using a phenotype based computational model

In this work I introduce a simple model to study how natural selection acts upon aging, which focuses on the viability of each individual. It is able to reproduce the Gompertz law of mortality and can make predictions about the relation between the level of mutation rates (beneficial/deleterious/neutral), age at reproductive maturity and the degree of biological aging. With no mutations, a population with low age at reproductive maturity R stabilizes at higher density values, while with mutations it reaches its maximum density, because even for large pre-reproductive periods each individual evolves to survive to maturity. Species with very short pre-reproductive periods can only tolerate a small number of detrimental mutations. The probabilities of detrimental (P_d) or beneficial (P_b) mutations are demonstrated to greatly affect the process. High absolute values produce peaks in the viability of the population over time. Mutations combined with low selection pressure move the system towards weaker phenotypes. For low values in the ratio P_d/P_b, the speed at which aging occurs is almost independent of R, while higher values favor significantly species with high R. The value of R is critical to whether the population survives or dies out. The aging rate is controlled by P_d and P_b and the amount of the viability of each individual is modified, with neutral mutations allowing the system more “room” to evolve. The process of aging in this simple model is revealed to be fairly complex, yielding a rich variety of results.     

Trait Substitution Sequence process and Canonical Equation for age-structured populations

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin–McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.      

Stability in an age-structured metapopulation model

We present a discrete model for a metapopulation of a single species with overlapping generations. Based on the dynamical behavior of the system in absence of dispersal, we have shown that a migration mechanism which depends only on age can not stabilize a previously unstable homogeneous equilibrium, but can drive a stable uncoupled system to instability if the migration rules are strongly related to age structure.     

Inbreeding dynamics in reintroduced, age-structured populations of highly fecund species

Reintroduction programs aim at reinstalling a self-sustained population into the wild via a period of supplementation with captive-bred individuals. This procedure can rapidly generate inbreeding among offspring because of the mating scheme and this inbreeding might be further enhanced by the reintroduction scenario. First, we used simulations to assess the consequences of breeding designs on mean inbreeding index F among offspring when the genetic diversity of breeders, the number and sex ratios of breeders, and the proportion of successful crosses vary. A high number of breeders, a balanced sex ratio, a high proportion of effective crosses and a genetically diverse source population generally contribute to lower F values. However, moderately high (≥20) numbers of breeders combined with all but the most biased sex ratios produced mean F values near minimal values. The variability in F was negligible in all parameter combinations except for a very small number of breeders (5) and very biased sex ratios (≤ 1M : 19F). We also simulated the long-term inbreeding dynamics in the introduced population under various demographic scenarios. Our main finding was that the annual number of introduced offspring is a decisive factor in establishing long-term F values in the supplemented population. Low supplementation levels (10²) quickly generated an almost completely inbred population whereas high levels (≥10⁴) produced stable F values close to that of the introduced offspring. Simulations were run based on the life history and specific demographics of the bloater (Coregonus hoyi), whose reintroduction in Lake Ontario is being considered.     

Threshold and stability results for an age-structured epidemic model

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.