Complexity in matrix population models: Polyvariant ontogeny and reproductive uncertainty

Linear matrix models of stage-structured population dynamics are widely used in plant and animal demography as a tool to evaluate the growth potential of a population in a given environment. The potential is identified with λ₁, the dominant eigenvalue of the projection matrix, which is compiled of stage-specific transition and fertility rates. Advanced botanical studies reveal polyvariant ontogeny in perennial plants, i.e., multiple different versions of individual development within a local population of a single species. This phenomenon complicates any standard, successive-stage, life cycle graph to a digraph defined on a 2D lattice in the age and stage dimensions, the pattern of projection matrix becoming more complex too. In a kind of experimental design, the transition rates can be calculated directly from the data for two successive time moments, but the age-stage-specific rates of reproduction still remain uncertain, adding more complexity to the calibration problem. Simple additional assumptions could technically eliminate the uncertainty, but they contravene the biology of a species in which polyvariant ontogeny is considered to be the major mechanism of adaptation. Given the data and expert constraints, the calibration can be reduced instead to a nonlinear maximization problem, yet with linear constraints. I prove that it has a unique solution to be attained at a vertex of the constraint polyhedral. To facilitate searching for the solution in practice, I use the net reproductive rate R₀, a well-known indicator for the principal property of λ₁ to be greater or less than 1. The method is exemplified with the calibration of a projection matrix in an age-stage-structured model (published elsewhere) for Calamagrostis canescens, a perennial herbaceous species with a complex (multivariant) life cycle that features unlimited growth when colonizing open areas.     

The structure and dynamics of woodreed Calamagrostis canescens population: a modelling approach

A scale of ontogenetic states has been developed for woodreed Calamagrostis canescens, a perennial species dominating the grass layer of fell forest areas. The population structure is considered as a set of age-stage groups of individuals differing both in the ontogenetic stage and the chronological age measured in years. to describe the dynamics through years a special kind of matrix formalism has been proposed which is reducible neither to the classic Leslie matrix for an age-structured population, nor to the well-known Lefkovitch matrix for a stage-structured one, and which does not suffer from excessiveness of the "two-dimensional" representation for the structure implying the projection matrix of a block pattern. It has been shown however that the protection matrix corresponding to C. canescens life-history graph embodies the canonical features of matrix formalism for structured population dynamics, such as the exponential population growth or decline, the convergence to a stable equilibrium structure, the calculable indicator of growth/decline/equilibrium (i.e., a measure of the population reproductive potential) as well as possibility to determine the relative reproductive value of each group. On the other hand, "left-sidedness of the age spectrum", a property that is often observed in real populations and is inherent in Leslie models of growing populations, may fail in the age-stage-structured model. The aggregation of age-stage groups into the age classes is possible only under special strict relationship among the age-stage-specific vital rates of the population. The both circumstances serve a methodical indication that an additional dimension such as the stages, for example, ought to be introduced into the age structure of the model population.     

The usefulness of sensitivity analysis for predicting the effects of cat predation on the population dynamics of their avian prey

Sensitivity analyses of population projection matrix (PPM) models are often used to identify life-history perturbations that will most influence a population's future dynamics. Sensitivities are linear extrapolations of the relationship between a population's growth rate and perturbations to its demographic parameters. Their effectiveness depends on the validity of the assumption of linearity. Here we assess whether sensitivity analysis is an appropriate tool to investigate the effect of predation by cats on the population growth rates of their avian prey. We assess whether predation by cats leads to non-linear effects on population growth and compare population growth rates predicted by sensitivity analysis with those predicted by a non-linear simulation. For a two-stage, age-classified House Sparrow Passer domesticus PPM slight non-linearity arose when PPM elements were perturbed, but perturbation to the vital rates underlying the matrix elements had a linear impact on population growth rate. We found a similar effect with a slightly larger three-stage, age-classified PPM for a Winter Wren Troglodytes troglodytes population perturbed by cat predation. For some avian species, predation by cats may cause linear or only slightly nonlinear impacts on population growth rates. For these species, sensitivity analysis appears to be a useful conservation tool. However, further work on multiple perturbations to avian prey species with more complicated life histories and higher-dimension PPM models is required.     

Integral projection models perform better for small demographic data sets than matrix population models: a case study of two perennial herbs

1. Matrix population models are widely used to describe population dynamics, conduct population viability analyses and derive management recommendations for plant populations. For endangered or invasive species, management decisions are often based on small demographic data sets. Hence, there is a need for population models which accurately assess population performance from such small data sets.
2. We used demographic data on two perennial herbs with different life histories to compare the accuracy and precision of the traditional matrix population model and the recently developed integral projection model (IPM) in relation to the amount of data.
3. For large data sets both matrix models and IPMs produced identical estimates of population growth rate (λ). However, for small data sets containing fewer than 300 individuals, IPMs often produced smaller bias and variance for λ than matrix models despite different matrix structures and sampling techniques used to construct the matrix population models.
4. Synthesis and applications . Our results suggest that the smaller bias and variance of λ estimates make IPMs preferable to matrix population models for small demographic data sets with a few hundred individuals. These results are likely to be applicable to a wide range of herbaceous, perennial plant species where demographic fate can be modelled as a function of a continuous state variable such as size. We recommend the use of IPMs to assess population performance and management strategies particularly for endangered or invasive perennial herbs where little demographic data are available.     

General relationships between species diversity and stability in competitive systems

Investigating the effect of biodiversity on the stability of ecological communities is complicated by the numerous ways in which models of community interactions can be formulated. This has led to differences in conclusions and interpretations of how the number of species in a community affects its stability. Here, we derive a simple, general relationship between the coefficient of variation (CV) of combined species densities and the environmentally driven variability in species' per capita population growth rates. For a given level of environmentally driven variability in per capita population growth rates, increasing the number of species in a community decreases the CV of combined species densities, provided that species do not respond to environmental fluctuations in a perfectly correlated way. Thus, a community with more species of competitors will be more stable (have lower CV in combined species densities for a given level of environmental variability) than a species-poor community, provided that the species in both communities show equal variability in per capita population growth rates and provided that species within each community do not show strongly correlated responses to environmental fluctuations. This conclusion also applies to "noninteractive" models in which there is no competition between species.     

Pitfalls to avoid in nonlinear perturbation analysis of structured population models

Despite the ubiquity of nonlinear functional relationships in nature we tend to characterize mechanisms in science using more tractable linear functions. In demographic modeling, transfer function analysis is used to calculate the nonlinear response of population growth rate to a theoretical perturbation of one or more matrix elements. This elegant approach is not yet popular in ecology. Inconveniently, using transfer function without care can produce erroneous results without warning. We used a large matrix projection model database to explore the potential pitfalls to be avoided in using transfer function analysis. We asked a fundamental population control question, what matrix element perturbation would be needed to reach a minimum goal of replacement population growth? We then tracked instances in which transfer function yields erroneous output and explored these cases in detail to measure how frequently it occurs. We developed a phylogenetically-corrected mixed effects logistic regression model in a Bayesian framework to test the effect of species traits and the identity of matrix elements on the probability that transfer function yields errors. We found in 16% of cases the transfer function yielded erroneous outcomes. These errors were more likely when perturbing demographic stasis and also for shrubs more than any other life form. Errors in transfer function analysis were often due to perturbing matrix elements beyond their biological limits, even when this is still mathematically correct. To use transfer function analysis properly in demographic modeling and avoid erroneous results, input must be carefully selected to include only a biologically admissible set of perturbations.     

Towards novel approaches to modelling biotic interactions in multispecies assemblages at large spatial extents

Aim Biotic interactions – within guilds or across trophic levels – have widely been ignored in species distribution models (SDMs). This synthesis outlines the development of ‘species interaction distribution models’ (SIDMs), which aim to incorporate multispecies interactions at large spatial extents using interaction matrices. Location Local to global. Methods We review recent approaches for extending classical SDMs to incorporate biotic interactions, and identify some methodological and conceptual limitations. To illustrate possible directions for conceptual advancement we explore three principal ways of modelling multispecies interactions using interaction matrices: simple qualitative linkages between species, quantitative interaction coefficients reflecting interaction strengths, and interactions mediated by interaction currencies. We explain methodological advancements for static interaction data and multispecies time series, and outline methods to reduce complexity when modelling multispecies interactions. Results Classical SDMs ignore biotic interactions and recent SDM extensions only include the unidirectional influence of one or a few species. However, novel methods using error matrices in multivariate regression models allow interactions between multiple species to be modelled explicitly with spatial co‐occurrence data. If time series are available, multivariate versions of population dynamic models can be applied that account for the effects and relative importance of species interactions and environmental drivers. These methods need to be extended by incorporating the non‐stationarity in interaction coefficients across space and time, and are challenged by the limited empirical knowledge on spatio‐temporal variation in the existence and strength of species interactions. Model complexity may be reduced by: (1) using prior ecological knowledge to set a subset of interaction coefficients to zero, (2) modelling guilds and functional groups rather than individual species, and (3) modelling interaction currencies and species’ effect and response traits. Main conclusions There is great potential for developing novel approaches that incorporate multispecies interactions into the projection of species distributions and community structure at large spatial extents. Progress can be made by: (1) developing statistical models with interaction matrices for multispecies co‐occurrence datasets across large‐scale environmental gradients, (2) testing the potential and limitations of methods for complexity reduction, and (3) sampling and monitoring comprehensive spatio‐temporal data on biotic interactions in multispecies communities.     

Volatility of network indices due to undersampling of intraspecific variation in plant–insect interactions

Appropriate sampling effort of interaction networks is necessary to extract robust indices describing the structure of species interactions. Here we show that time-invariant variation in the composition and diversity of interaction partners of plant individuals of the same species explains volatility in aggregate network statistics due to undersampling. Within a multi-species pollinator–plant interaction matrix, we replaced the interactions observed on multiple individuals of a single plant species (Sinapis arvensis, pooled interactions) with the plant–insect interactions observed on a single plant individual. In the resampling approach, we considered the interactions of 1 to 84 S. arvensis individuals in different combinations. For each resampled network, several commonly applied aggregated statistics were calculated to test how intraspecific variation affects the properties of a multi-species network. Our results showed that aggregate statistics are sensitive towards qualitative and quantitative intraspecific variation of flower–visitor interactions within a multi-species network, which may affect the ecological interpretation about the properties of a community. These findings challenge the robustness of commonly applied network indices, confirm the urge for a sufficient and representative sampling of interactions, and emphasize the significance of intraspecific variation in the context of communities and networks.     

Modeling vital rates improves estimation of population projection matrices

Population projection matrices are commonly used by ecologists and managers to analyze the dynamics of stage-structured populations. Building projection matrices from data requires estimating transition rates among stages, a task that often entails estimating many parameters with few data. Consequently, large sampling variability in the estimated transition rates increases the uncertainty in the estimated matrix and quantities derived from it, such as the population multiplication rate and sensitivities of matrix elements. Here, we propose a strategy to avoid overparameterized matrix models. This strategy involves fitting models to the vital rates that determine matrix elements, evaluating both these models and ones that estimate matrix elements individually with model selection via information criteria, and averaging competing models with multimodel averaging. We illustrate this idea with data from a population of Silene acaulis (Caryophyllaceae), and conduct a simulation to investigate the statistical properties of the matrices estimated in this way. The simulation shows that compared with estimating matrix elements individually, building population projection matrices by fitting and averaging models of vital-rate estimates can reduce the statistical error in the population projection matrix and quantities derived from it.     

Demography when history matters: construction and analysis of second-order matrix population models

History matters when individual prior conditions contain important information about the fate of individuals. We present a general framework for demographic models which incorporates the effects of history on population dynamics. The framework incorporates prior condition into the i-state variable and includes an algorithm for constructing the population projection matrix from information on current state dynamics as a function of prior condition. Three biologically motivated classes of prior condition are included: prior stages, linear functions of current and prior stages, and equivalence classes of prior stages. Taking advantage of the matrix formulation of the model, we show how to calculate sensitivity and elasticity of any demographic outcome. Prior condition effects are a source of inter-individual variation in vital rates, i.e., individual heterogeneity. As an example, we construct and analyze a second-order model of Lathyrus vernus, a long-lived herb. We present population growth rate, the stable population distribution, the reproductive value vector, and the elasticity of λ to changes in the second-order transition rates. We quantify the contribution of prior conditions to the total heterogeneity in the stable population of Lathyrus using the entropy of the stable distribution.     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.     

Stress and disturbance determine the demographic dynamics of two grass species along a grazing gradient in Southern Mexico

Chronic anthropogenic disturbance (CAD), characterized by low-intensity but high frequency, is a major driver of environmental degradation in developing countries. CAD is a mixture of disturbance sensu stricto (DSS), that is, plant biomass removal and stress that reduces biomass production due to changes in environmental conditions. However, we still lack data on the separate effects of both components and their interaction in nature. We analyze the demographic effects of DSS and stress on two grass species in an area heavily affected by livestock raising (a widespread cause of CAD) during the last 500 year. We compared areas exposed to DSS and stress with areas without grazing but that continue experiencing a gradient of stress. Using matrix and integral projection models, we analyzed DSS and stress effects on population growth rates (λ) of two grass species and determined the relative importance of different vital rates and states for the change on λ. Disturbance and stress affected different individuals and processes. For example, changed conditions due to stress increased seedling mortality, but DSS reduced size (growth) of large plants through grazing. CAD had highly nonlinear and species-specific effects on population size structures, λ and elasticities. Such complex behavior is seemingly due to changes in the components of CAD as it intensified and synergic interactions between disturbance and stress. Given CAD's multivariate nature, these results are not surprising. Nevertheless, grouping this multitude of factors into two broad categories, namely DSS and stress, may prove a useful conceptual tool for analysis.     

Stochastic stable population growth in integral projection models: theory and application

Stochastic matrix projection models are widely used to model age- or stage-structured populations with vital rates that fluctuate randomly over time. Practical applications of these models rest on qualitative properties such as the existence of a long term population growth rate, asymptotic log-normality of total population size, and weak ergodicity of population structure. We show here that these properties are shared by a general stochastic integral projection model, by using results in (Eveson in D. Phil. Thesis, University of Sussex, 1991, Eveson in Proc. Lond. Math. Soc. 70, 411-440, 1993) to extend the approach in (Lange and Holmes in J. Appl. Prob. 18, 325-344, 1981). Integral projection models allow individuals to be cross-classified by multiple attributes, either discrete or continuous, and allow the classification to change during the life cycle. These features are present in plant populations with size and age as important predictors of individual fate, populations with a persistent bank of dormant seeds or eggs, and animal species with complex life cycles. We also present a case-study based on a 6-year field study of the Illyrian thistle, Onopordum illyricum, to demonstrate how easily a stochastic integral model can be parameterized from field data and then applied using familiar matrix software and methods. Thistle demography is affected by multiple traits (size, age and a latent "quality" variable), which would be difficult to accommodate in a classical matrix model. We use the model to explore the evolution of size- and age-dependent flowering using an evolutionarily stable strategy (ESS) approach. We find close agreement between the observed flowering behavior and the predicted ESS from the stochastic model, whereas the ESS predicted from a deterministic version of the model is very different from observed flowering behavior. These results strongly suggest that the flowering strategy in O. illyricum is an adaptation to random between-year variation in vital rates.     

Projection matrices in population biology

Projection matrix models are widely used in population biology to project the present state of a population into the future, either as an attempt to forecast population dynamics, or as a way to evaluate life history hypotheses. These models are flexible and mathematically relatively easy. They have been applied to a broad range of plants and animals. The asymptotic properties of projection matrices have clearly defined biological interpretations, and the analysis of the effects of perturbations on these asymptotic properties offers new possibilities for comparative life history analysis. The connection between projection matrix models and the secondary theorem of natural selection opens life cycle phenomena to evolutionary interpretation.     

复杂性-稳定性研究: 数学模型的进展

对自然生态系统的观察给人们以复杂的群落更稳定的直观印象, 但数学模型却得出了截然相反的结论。这一“悖论”使得复杂性-稳定性研究自20世纪70年代以来成为长期的热点。本文对这一领域的数学模型研究进行简要综述。首先对这一论题进行概念剖析, 然后将各类模型分为线性和非线性两大类, 前者即群落矩阵法, 后者包括相互作用矩阵法、复杂网络数值模拟法和食物网构件动力学法。它们分别基于不同的群落构建方法和稳定性判断标准, 探求各物种是如何相互作用并实现共存的。总体而言, 在随机构建的群落模型中, 多样性和连接度的增长不利于系统稳定; 而在更接近真实自然群落的模型中, 相互作用方式、网络拓扑结构、相互作用强度分布等方面的机制提供了稳定效应, 按此组织的生态网络可达到很高的复杂度。然而, 复杂性-稳定性的研究还远未结束, 当前的模型仍不足以反映自然群落中的复杂相互作用, 稳定性的概念也有待拓展。对这一议题的深入研究在生态学理论和生态系统管理实践方面都具有重大价值。     

Richards-like two species population dynamics model

The two-species population dynamics model is the simplest paradigm of inter- and intra-species interaction. Here, we present a generalized Lotka–Volterra model with intraspecific competition, which retrieves as particular cases, some well-known models. The generalization parameter is related to the species habitat dimensionality and their interaction range. Contrary to standard models, the species coupling parameters are general, not restricted to non-negative values. Therefore, they may represent different ecological regimes, which are derived from the asymptotic solution stability analysis and are represented in a phase diagram. In this diagram, we have identified a forbidden region in the mutualism regime, and a survival/extinction transition with dependence on initial conditions for the competition regime. Also, we shed light on two types of predation and competition: weak, if there are species coexistence, or strong, if at least one species is extinguished.     

Monotone dependence of the spectral bound on the transition rates in linear compartment models

For linear compartment models or Leslie-type staged population models with quasi-positive matrix the spectral bound of the matrix (the eigenvalue determining stability) is studied in the situation where particles or individuals leave a compartment or stage with some rate and enter another with the same rate. Then the matrix carries the rate with a positive sign in some off-diagonal entry and with a negative sign in the corresponding diagonal entry. Hence the matrix does not depend on the rate in a monotone way. It is shown, however, that the spectral bound is a monotone function of the rate. It is all the time strictly increasing or strictly decreasing or it is constant. A simple algebraic criterion distinguishes between the three cases. The results can be applied to linear systems and to the stability of stationary states in non-linear systems, in particular to models for the transmission of infectious diseases, and in population dynamics.     

Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator–prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator–prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator–prey model.      

Population dynamics of the Mexican cycad Dioon edule Lindl. (Zamiaceae): life history stages and management impact

The demographic dynamics of three populations of Dioon edule Lindl. (Zamiaceae) were studied in a fragmented landscape using projection matrix modelling. Compared with other plant species, D. edule behaves like a tree life-form species. Density and spatial distribution patterns varied among populations according to models for animal-dispersed tree species. In all scenarios, λ was most sensitive to changes in abundance of adult plants. The elasticity reproductive component (F) for the three populations was zero and stasis values (L) were higher, this being a function of the permanence of non-reproductive individuals. It was detected that disturbance influences the population dynamics of D. edule as a function of adult plant persistence. This observation suggests that the conservation of adult plants is critical for D. edule and perhaps for all cycads species. Adult plant decapitation should be halted at the 'Monte Oscuro' population, subjected to sustainable management since 1990, if higher seed production is needed in rural nurseries.  © 2008 The Linnean Society of London, Botanical Journal of the Linnean Society , 2008, 157 , 381–391.