On the formulation and analysis of general deterministic structured population models I. Linear Theory

—We define a linear physiologically structured population model by two rules, one for reproduction and one for “movement” and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R ₀ and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step. Received 26 July 1996; received in revised form 3 September 1997     

Steady-state analysis of structured population models

Our systematic formulation of nonlinear population models is based on the notion of the environmental condition. The defining property of the environmental condition is that individuals are independent of one another (and hence equations are linear) when this condition is prescribed (in principle as an arbitrary function of time, but when focussing on steady states we shall restrict to constant functions). The steady-state problem has two components: (i). the environmental condition should be such that the existing populations do neither grow nor decline; (ii). a feedback consistency condition relating the environmental condition to the community/population size and composition should hold. In this paper we develop, justify and analyse basic formalism under the assumption that individuals can be born in only finitely many possible states and that the environmental condition is fully characterized by finitely many numbers. The theory is illustrated by many examples. In addition to various simple toy models introduced for explanation purposes, these include a detailed elaboration of a cannibalism model and a general treatment of how genetic and physiological structure should be combined in a single model.     

Stochastic and deterministic models for the kinetic behavior of certain structured enzyme systems II: consecutive two enzyme systems.

This paper contains extensions of results from a previous paper regarding structured one enzyme systems to a more complicated structured two enzyme system. A stochastic model and a deterministic model are developed for such systems and their steady state reaction kinetics are compared. These comparisons are in the form of graphs of the reaction kinetics versus substrate concentration. Two quantities are proposed as indications of lack of agreement between the two models. This lack of agreement corresponds to situations in which the model systems are more highly non-linear, in accord with Jensen's inequalities. Implications of these results, relative to experimental procedures are briefly discussed.     

Stochastic analogues of deterministic single-species population models

Although single-species deterministic difference equations have long been used in modeling the dynamics of animal populations, little attention has been paid to how stochasticity should be incorporated into these models. By deriving stochastic analogues to difference equations from first principles, we show that the form of these models depends on whether noise in the population process is demographic or environmental. When noise is demographic, we argue that variance around the expectation is proportional to the expectation. When noise is environmental the variance depends in a non-trivial way on how variation enters into model parameters, but we argue that if the environment affects the population multiplicatively then variance is proportional to the square of the expectation. We compare various stochastic analogues of the Ricker map model by fitting them, using maximum likelihood estimation, to data generated from an individual-based model and the weevil data of Utida. Our demographic models are significantly better than our environmental models at fitting noise generated by population processes where noise is mainly demographic. However, the traditionally chosen stochastic analogues to deterministic models--additive normally distributed noise and multiplicative lognormally distributed noise--generally fit all data sets well. Thus, the form of the variance does play a role in the fitting of models to ecological time series, but may not be important in practice as first supposed.     

Numerical solution of structured population models

Numerical methods are presented for a general mass-structured population model with demographic rates that depend on individual mass, time, and total population mass. Several types of numerical methods are described, Eulerian methods, implicit methods, and the method of characteristics. These methods are compared for a sample problem with an exact solution. The preferred numerical technique combines the method of characteristics with an adaptive grid. Numerical solution of model equations developed for mosquitofish illustrates this method and demonstrates how seasons can play a dominant role in shaping population development.     

Numerical solution of structured population models

Numerical methods are presented for a general age-structured population model with demographic rates depending on age and the total population size. The accuracy of these methods is established by solving problems for which alternate solution techniques are available and are used for comparison. The methods reliably solve test problems with a variety of dynamic behavior. Simulations of a blowfly population exhibit cyclic fluctuations, whereas a simulated squirrel population reaches a stable age distribution and stable equilibrium population size. Life-history attributes are easily studied from the computed solutions, and are discussed for these examples. Recovery of a stressed population back to equilibrium is examined by computing the transition in age structure, and the transient behavior of other properties of the population such as the per capita growth rate, the average age, and the generation length.     

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.     

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.     

Some population genetic models combining artificial and natural selection pressures: II. Two-locus theory

Some population genetic systems of traits controlled principally at two or three main loci subject to the combined effects of artificial and natural selection are studied. A number of genotypic-phenotypic associations are formulated, and both random and selfing mating patterns are examined. The interaction of some forms of epistasis and linkage in these systems are evaluated.     

On simulating strongly interacting, stochastic population models. II. Multiple compartments

In Wick and Stelf [Math. Biosci. 187 (2004) 1], we showed how to simulate a pair of strongly interacting biological populations evolving stochastically over many orders-of-magnitude. Here we generalize the method to any (finite) number of compartments; transitions including births, deaths, progression through life-stages, and mitoses; and arbitrary rate functions. We illustrate the technique for a seven-compartment model of the cellular immune response to a viral infection.     

On the formulation of discrete-time epidemic models

Jacquez constructed a properly posed, more general model for the Reed-Frost epidemic process by assuming independent behaviors for the susceptibles and introducing the generating function for the number of contacts per person. An alternative approach is proposed here that relies on similar hypotheses for the infectives and allows the usual chain-binomial structure of the infection process to be extended. For this new model, the derivation of the final size and the threshold phenomenon becomes much simpler. A detailed analysis and its generalization to heterogeneous populations and continuous-time models will be the subject of a forthcoming paper.     

Unsteady beam-plasma discharge: II. Nonlinear theory

The mathematical model constructed in the first part of our paper is used to numerically investigate the development of the beam-plasma instability in a traveling-wave tube amplifier in the presence of a residual neutral gas. It is shown that the self-generation of ion acoustic waves in a plasma-filled amplifier can give rise to a modulation regime with a rigid excitation threshold. The dependence of the threshold for the self-modulation instability on the amplifier parameters is determined. The effect of self-modulation on the spectral and energy characteristics of the amplifier is analyzed.     

Well-posedness of physiologically structured population models for Daphnia magna

In this paper we heuristically discuss the well-posedness of three variants of the Kooijman/Metz model. Shortcomings concerning the uniqueness and continuous dependence on data of the solutions to one of the variants are traced back to an inconsistency in the biological concept of energy allocation in this model version. The conceptional consequences are discussed and an open question concerning energy allocation is pin-pointed.     

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.     

Periodic and quasi-periodic behavior in resource-dependent age structured population models

To describe the dynamics of a resource-dependent age structured population, a general non-linear Leslie type model is derived. The dependence on the resources is introduced through the death rates of the reproductive age classes. The conditions assumed in the derivation of the model are regularity and plausible limiting behaviors of the functions in the model. It is shown that the model dynamics restricted to its ω-limit sets is a diffeomorphism of a compact set, and the period-1 fixed points of the model are structurally stable. The loss of stability of the non-zero steady state occurs by a discrete Hopf bifurcation. Under general conditions, and after the loss of stability of the structurally stable steady states, the time evolution of population numbers is periodic or quasi-periodic. Numerical analysis with prototype functions has been performed, and the conditions leading to chaotic behavior in time are discussed.     

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.