A matrix model for density-dependent selection in stage-classified populations, with application to pesticide resistance in Tribolium

The study of eco-evolutionary dynamics is based on the idea that ecological and evolutionary processes may operate on the same, or very similar, time scales, and that interactions of ecological and evolutionary processes may have important consequences. Here we develop a model that combines Mendelian population genetics with nonlinear demography to create a truly eco-evolutionary model. We use the vec-permutation matrix approach, classifying individuals by stage and genotype. The demographic component is female dominant and density-dependent. The genetic component includes random mating by stage and genotype, and arbitrary effects of genotype on the demographic phenotype. Mutation is neglected. The result is a nonlinear matrix population model that projects the stage × genotype distribution. We show that the results can include bifurcations of population dynamics driven by the response to selection. We present analytical criteria that determine whether one allele excludes the other or if they persist in a protected polymorphism. The results are based on local stability analysis of the homozygous boundary equilibria. As an example, we use a density-dependent stage-classified model of the flour beetle Tribolium castaneum. Our model permits arbitrary life-cycle complexity and nonlinearity. Tribolium has developed resistance to the pesticide malathion due to a dominant allele at a single autosomal locus. Using parameters reported from laboratory experiments, we show that the model successfully describes the dynamics of both resistant and susceptible homozygotes, and the outcome of a selection experiment containing both alleles. Stability analysis of the boundary equilibria confirms that the resistant allele excludes the susceptible allele, even in the absence of malathion, agreeing with previously reported results

A matrix model for density-dependent selection in stage-classified populations, with application to pesticide resistance in Tribolium

The study of eco-evolutionary dynamics is based on the idea that ecological and evolutionary processes may operate on the same, or very similar, time scales, and that interactions of ecological and evolutionary processes may have important consequences. Here we develop a model that combines Mendelian population genetics with nonlinear demography to create a truly eco-evolutionary model. We use the vec-permutation matrix approach, classifying individuals by stage and genotype. The demographic component is female dominant and density-dependent. The genetic component includes random mating by stage and genotype, and arbitrary effects of genotype on the demographic phenotype. Mutation is neglected. The result is a nonlinear matrix population model that projects the stage × genotype distribution. We show that the results can include bifurcations of population dynamics driven by the response to selection. We present analytical criteria that determine whether one allele excludes the other or if they persist in a protected polymorphism. The results are based on local stability analysis of the homozygous boundary equilibria. As an example, we use a density-dependent stage-classified model of the flour beetle Tribolium castaneum. Our model permits arbitrary life-cycle complexity and nonlinearity. Tribolium has developed resistance to the pesticide malathion due to a dominant allele at a single autosomal locus. Using parameters reported from laboratory experiments, we show that the model successfully describes the dynamics of both resistant and susceptible homozygotes, and the outcome of a selection experiment containing both alleles. Stability analysis of the boundary equilibria confirms that the resistant allele excludes the susceptible allele, even in the absence of malathion, agreeing with previously reported results

Lattice effects observed in chaotic dynamics of experimental populations

Animals and many plants are counted in discrete units. The collection of possible values (state space) of population numbers is thus a nonnegative integer lattice. Despite this fact, many mathematical population models assume a continuum of system states. The complex dynamics, such as chaos, often displayed by such continuous-state models have stimulated much ecological research; yet discretestate models with bounded population size can display only cyclic behavior. Motivated by data from a population experiment, we compared the predictions of discrete-state and continuous-state population models. Neither the discrete- nor continuous-state models completely account for the data. Rather, the observed dynamics are explained by a stochastic blending of the chaotic dynamics predicted by the continuous-state model and the cyclic dynamics predicted by the discretestate models. We suggest that such lattice effects could be an important component of natural population fluctuations. The discovery that simple deterministic population models can display complex aperiodi

Experimental Beetle Metapopulations Respond Positively to Dynamic Landscapes and Reduced Connectivity

Interactive effects of multiple environmental factors on metapopulation dynamics have received scant attention. We designed a laboratory study to test hypotheses regarding interactive effects of factors affecting the metapopulation dynamics of red flour beetle, Tribolium castaneum. Within a four-patch landscape we modified resource level (constant and diminishing), patch connectivity (high and low) and patch configuration (static and dynamic) to conduct a 23 factorial experiment, consisting of 8 metapopulations, each with 3 replicates. For comparison, two control populations consisting of isolated and static subpopulations were provided with resources at constant or diminishing levels. Longitudinal data from 22 tri-weekly counts of beetle abundance were analyzed using Bayesian Poisson generalized linear mixed models to estimate additive and interactive effects of factors affecting abundance. Constant resource levels, low connectivity and dynamic patches yielded greater levels of adult beetle abundance. For a given resource level, frequency of colonization exceeded extinction in landscapes with dynamic patches when connectivity was low, thereby promoting greater patch occupancy. Negative density dependence of pupae on adults occurred and was stronger in landscapes with low connectivity and constant resources; these metapopulations also demonstrated greatest stability. Metapopulations in control landscapes went extinct quickly, denoting lower persistence than comparable landscapes with low connectivity. When landscape carrying capacity was constant, habitat destruction coupled with low connectivity created asynchronous local dynamics and refugia within which cannibalism of pupae was reduced. Increasing connectivity may be counter-productive and habitat destruction/recreation may be beneficial to species in some contexts

Nonlinear Demographic Dynamics: Mathematical Models, Statistical Methods, and Biological Experiments

Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2—cycles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three—dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2—cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to understanding nonlinear ecological dynamics

Evolution lab : lab manual for biology labs on-line

11 p .; 30 cm

Data from: Temporal scale of environmental correlations affects ecological synchrony

Population densities of a species, measured in different locations are often correlated over time, a phenomenon referred to as synchrony. Synchrony results from dispersal of individuals among locations and spatially correlated environmental variation, among other causes. Synchrony is often measured by a correlation coefficient. However, synchrony can vary with timescale. We demonstrate theoretically and experimentally that the timescale-specificity of environmental correlation affects the overall magnitude and timescale-specificity of synchrony, and that these effects are modified by population dispersal. Our laboratory experiments linked populations of flour beetles by changes in habitat size and dispersal. Linear filter theory, applied to a metapopulation model for the experimental system, predicted the observed timescale-specific effects. The timescales at which environmental covariation occurs can affect the population dynamics of species in fragmented habitats

Species Competition: Uncertainty on a Double Invariant Loop

The Tribolium (flour beetle) competition experiments conducted by Park have been highly influential in ecology. We have previously shown that the dynamics of single-species Tribolium populations can be well-described by the discrete-time, 3-dimensional larva-pupa-adult (LPA) model. Motivated by Park\u27s experiments, we explore the dynamics of a 6-dimensional competition LPA model consisting of two LPA models coupled through cannibalism. The model predicts a double-loop coexistence attractor, as well as an unstable exclusion equilibrium with a 5-dimensional stable manifold that plays an important role in causing one of the species to go extinct in the presence of stochastic perturbations. We also present a stochastic version of the model, using binomial and Poisson distributions to describe the aggregation of demographic events within life stages. A novel stochastic outcome diagram, the stochastic counterpart to a bifurcation diagram, summarizes the model-predicted dynamics of uncertainty on the double-loop. We hypothesize that the model predictions provide an explanation for Park\u27s data. This stochastic double-loop hypothesis is accessible to experimental verification. © 2005 Taylor & Francis Group Ltd