Population dynamics of size-structured populations

[First paragraph]Many population cohorts are structured by different attributes: space and age are the most common of these. For many applications this structure is just as important as the temporal evolution with discrete or continuous time. For these applications modellers have to cope with this greatly increased complexity and cannot afford to neglect it. Different modellers proceed in a variety of ways to suit the circumstances and tools available. Nigel Barlow introduced me (in 1988) to the idea of using a limited number of spatial or age compartments, each of which was structurally uniform with well-defined interactive links between compartments. His view, if I interpret it correctly, was that this enabled us to avoid the technically difficult partial differential equation analysis. Of course, there are a wide variety of opinions about this, but I have recently used this approach in an analysis of a plankton– nutrient model, where Nigel was clearly right. We would have never succeeded in using the lovely path- following (in parameter space) computer algorithms had we not adopted Nigel’s approach. This paper addresses a situation where compartments were not, in fact, sufficient: where the population is structured by size or DNA content. Data sets of plant-root-cells, plankton, muscle cells, and cancer cells, often show a steady (in time) size distribution that appears asymptotically and structurally stable. (“Steady-size” means it is constant in shape as time changes.) About the same time as Nigel and I began (in Palmerston North around 1988) looking at a spatially compartmentalised model for tuberculosis in possums, another biologist, Dr Paul Gandar, asked whether we could explain this with a relatively simple model which, when validated and embedded in the cell physiological context, could have the capability of making robust predictions.     

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997     

Temperature-driven regime shifts in the dynamics of size-structured populations

Global warming impacts virtually all biota and ecosystems. Many of these impacts are mediated through direct effects of temperature on individual vital rates. Yet how this translates from the individual to the population level is still poorly understood, hampering the assessment of global warming impacts on population structure and dynamics. Here, we study the effects of temperature on intraspecific competition and cannibalism and the population dynamical consequences in a size-structured fish population. We use a physiologically structured consumer-resource model in which we explicitly model the temperature dependencies of the consumer vital rates and the resource population growth rate. Our model predicts that increased temperature decreases resource density despite higher resource growth rates, reflecting stronger intraspecific competition among consumers. At a critical temperature, the consumer population dynamics destabilize and shift from a stable equilibrium to competition-driven generation cycles that are dominated by recruits. As a consequence, maximum age decreases and the proportion of younger and smaller-sized fish increases. These model predictions support the hypothesis of decreasing mean body sizes due to increased temperatures. We conclude that in size-structured fish populations, global warming may increase competition, favor smaller size classes, and induce regime shifts that destabilize population and community dynamics.     

Existence of solutions of a two-dimensional boundary value problem for a system of nonlinear equations arising in growing cell populations

In the paper [A. Ben Amar, A. Jeribi, and B. Krichen, Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg's model type, to appear in Math. Slovaca. (2014)], the existence of solutions of the two-dimensional boundary value problem (1) and (2) was discussed in the product Banach space L_p×L_p for p∈(1, ∞). Due to the lack of compactness on L₁ spaces, the analysis did not cover the case p=1. The purpose of this work is to extend the results of Ben Amar et al. to the case p=1 by establishing new variants of fixed-point theorems for a 2×2 operator matrix, involving weakly compact operators.     

A model of physiologically structured population dynamics with a nonlinear individual growth rate

In this article we consider a size structured population model with a nonlinear growth rate depending on the individual's size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.This work was partially supported by DGICYT PB90-0730-C02-01 and PB91-0497.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

A size-structured model of bacterial growth and reproduction

We consider a size-structured bacterial population model in which the rate of cell growth is both size- and time-dependent and the average per capita reproduction rate is specified as a model parameter. It is shown that the model admits classical solutions. The population-level and distribution-level behaviours of these solutions are then determined in terms of the model parameters. The distribution-level behaviour is found to be different from that found in similar models of bacterial population dynamics. Rather than convergence to a stable size distribution, we find that size distributions repeat in cycles. This phenomenon is observed in similar models only under special assumptions on the functional form of the size-dependent growth rate factor. Our main results are illustrated with examples, and we also provide an introductory study of the bacterial growth in a chemostat within the framework of our model.     

Carbohydrate dynamics and growth rate of natural phytoplankton populations

Diurnal changes in carbohydrate content of natural phytoplankton populations differed from those found for cyanobacteria grown in continuous cultures. The carbohydrate accumulation rate was not constant during the light period. Also in contrast to results obtained using continuous cultures the photosynthetic characteristics changed during the light period. A close correlation was observed between changes in carbohydrate accumulation rate and the efficiency of photosynthesis over 24 hours. Seasonal changes in carbohydrate consumption rate over the dark period were proportional to changes in growth rate.     

Self-thinning and community persistence in a simple size-structured dynamical model of plant growth

This paper presents a size-structured dynamical model of plant growth. The model takes the form of a partial differential-integral equation and includes the effects of self- shading by leaves. Closed form solutions are presented for the equilibrium size density distribution. Analytic conditions are derived for community persistence, and the self-thinning exponent is obtained as a function of species characteristics and environmental conditions.     

Asynchronous exponential growth for a two-phase size-structured population model and comparison with the corresponding one-phase model

In this paper we study a two-phase size-structured population model with distributed delay in the birth process. This model distinguishes individuals by ‘active’ or ‘resting’ status. The individuals in the two life-stages have different growth rates. Only individuals in the ‘active’ stage give birth to the individuals in the ‘active’ stage or the ‘resting’ stage. The size of all the newborns is 0. By using the method of semigroups, we obtain that the model is globally well-posed and its solution possesses the property of asynchronous exponential growth. Moreover, we give a comparison between this two-phase model with the corresponding one-phase model and show that the asymptotic behaviours of the sum of the densities of individuals in the ‘active’ stage and the ‘resting’ stage and the solution of the corresponding one-phase model are different.     

Developing tests of successional hypotheses with size-structured populations,and an assessment using long-term data from a Ugandan rain forest

In 1947, W. J. Eggeling published an account of forest succession at Budongo, Uganda. This interpretation was based on a large-scale comparative plot study, performed in the 1930s and 1940s. This account, with its implication that species richness declines in late succession, endures as a controversial corner-stone in theories and disputes about community diversity. Data have now been collected over six decades from five of Eggeling's original plots.This paper evaluates Eggeling's successional interpretation of the Budongo vegetation. The first set of analyses assesses the consistency of the original data with the predictions of compositional progression and convergence implicit in Eggeling's model. The second analyses do the same for the time-series observations. A logical approach shows how temporal information may be derived from both between plot, and within plot, evaluations using size-structured data. A Detrended-Correspondence-Analysis (DCA) of canopy-tree composition, from the original data, ranks the plots in perfect correspondence to Eggeling's successional sequence. A development-scoring procedure is developed using passive-ordination against this sequence; this is then applied to composition by plot and stem-size class.Eggeling's original data are consistent with each prediction assessed. The analyses show compositional progression and apparent convergence across the plot series, and also progression and convergence within each plot. A monodominant-Cynometra forest is the natural end-point of this progression. The time-series results, though in apparent agreement for one early successional plot, do not generally accord with Eggeling's ideas. The analyses illustrate a general means for evaluating explicit and implicit compositional trends in communities with structured populations.     

Predator-prey interactions in size-structured fish communities: implications of prey growth

Predator-prey interactions among size-structured populations may be strongly influenced by factors which affect growth rates of prey. I examined the importance of prey growth in the interaction between large-mouth bass (Micropterus salmoides) and their prey, bluegill (Lepomis macrochirus), by analyzing diets and growth rates of bass in a set of seven lakes in south-central Wisconsin. Sizes of bluegill consumed by bass changed dramatically across a gradient of bluegill growth, which resulted in differing patterns of bass growth. In lakes with slow bluegill growth, small bass fed on the youngest bluegill cohort, but large bass were capable of feeding on several age classes. Consequently, bass growth rates were strongly size-dependent; small bass ate small prey and had low growth, but growth rates increased substantially with size as bass ate progressively larger prey. When bluegill had high growth rates, they quickly reached a size refuge from predation and bass of all sizes were restricted to feeding on the youngest/smallest prey. In these lakes, bass growth rates were more uniform across bass sizes. Because growth rates influence population size-distributions, variation in bluegill growth can have strong effects on the structure of bass populations. These effects could potentially feed back to further influence the interaction between predator and prey.     

Control of relative growth rate by application of the relative addition rate technique to a traditional solution culture system

The relative addition rate (RAR) technique allows the nutritional control of plant relative growth rate (RGR) by the provision of nutrients at exponential supply rates. The technique, however, was developed with technologically sophisticated aeroponic systems. In this paper, we report on experiments used to adapt the RAR technique to a conventional solution culture system. A background concentration requirement of 36 μM nitrogen (N), with other nutrients supplied in proportion to N, was necessary to produce a constant RGR of Triticum aestivum L. (wheat) at a low RAR. Solution pH changes were reduced by increasing the percentage of NH₄ in the nitrogen supply, but the plants exhibited dry weight reductions and symptoms of toxicity above 30% NH₄. For wheat, a ratio of 25/75 NH₄/NO₃ was optimum for minimizing pH changes within the nontoxic range. A test of the effectiveness of the RAR technique using this background concentration and NH₄/NO₃ ratio showed that RGR increased with RAR with a linear slope of 0.55 and an intercept of 0.07 d^-1. Although the relationship between growth rate and nutrient supply was less than the one-to-one dependence of RGR on RAR that has been obtained with more sophisticated apparatus, application of the RAR technique to a conventional solution culture system still affords considerable control of RGR and presents a simple method for growing plants at different levels of nutrient stress and at distinct RGRs.     

An epidemiological model with a delay and a nonlinear incidence rate

An epidemiological model with both a time delay in the removed class and a nonlinear incidence rate is analysed to determine the equilibria and their stability. This model is for diseases where individuals are first susceptible, then infected, then removed with temporary immunity and then susceptible again when they lose their immunity. There are multiple equilibria for some parameter values, and, for certain of these, periodic solutions arise by Hopf bifurcation from the large nontrivial equilibrium state.Research supported in parts by Centers for Disease Control Contract 200-87-0515Research supported in part by NSERC A-8965     

A monotone approximation for a size-structured population model with a generalized environment

We study a nonlinear size-structured population model with an environment general enough to include hierarchy. We also remove the standard requirement that individuals have nonnegative growth rates, which allows the modeling of populations in which individuals may experience a reduction in size. To show existence and uniqueness of the solution to the model, we establish a comparison principle and construct monotone sequences. A fully discretized numerical scheme based on these monotone sequences is presented and utilized to provide some numerical examples.     

Measuring selection in human populations using the growth rate per generation

Estimates of the speed of evolution between generations depend on the association between individual traits and a measure of fitness. The two most frequently used measures of fitness are the net reproduction rate and the 1-year growth factor implied by the fertility and mortality rates. Results based on the two lead to very different results. The reason is that the 1-year growth factor is not a measure of change between generations. Therefore, studies of changes between generations should use the amount of growth over the length of a generation. This is especially important for studies of human populations because of the long length of generation. In addition, estimates based on a single year''s growth are overly sensitive to data on individuals who fail to reproduce. The effects of using a generational measure are demonstrated using data from Kenya and Ukraine. These results demonstrate that using a 1-year growth rate to measure fitness leads to estimates that understate the rate at which evolution changes the characteristics of a human population.     

Life history parameters as indicators of growth rate in three salp populations

A strong correlation exists between individual growth rate andfive life history parameters in the fast growing salp Thaliademocratica. The five parameters are number of aggregate budsper chain; body size of reproductive solitaries; the ratio ofaggregate offspring to their solitary parents; the solitaryto aggregate ratio; and the relative proportions of juvenilesand adults in the solitary generation. Correlation coefficientsaveraging 0.68 and ranging up to 0.85 were found between theestimates of these parameters in 37 samples of salp blooms takenover 15 years in the western Pacific and eastern Indian oceans.Growth rate, estimated previously in 12 of these samples, alsocorrelated strongly with each of the five parameters with coefficientsranging from 0.88 to 0.96. After analysis of partial correlationswe selected parameters to form regression equations with growth.These equations are excellent indicators of growth in the sequentiallysampled salp populations. Because of the significant correlationbetween the parameters in a wider range of samples, they arealso likely to be indicators of growth in general salp populations.