Density-dependent Population Model Research Articles

Sensitivity and elasticity of density-dependent population models

The sensitivity analysis of linear matrix population models is well-developed, but no comparable approach exists for the nonlinear models appropriate for density-dependent populations. This paper presents such an approach, using matrix differential calculus to obtain the derivatives of equilibria and cycles with respect to arbitrary demographic parameters. The method readily calculates the sensitivity and elasticity of the population vector or of weighted densities, ratios of age classes or stages, proportional structures or temporal means and variances calculated from the population vector. Examples are presented using data on flour beetles of the genus Tribolium. An attempt is made to extend the approach to the sensitivity of averages over strange attractors, but the resulting calculations fail to converge, apparently due to a positive Lyapunov exponent for the sensitivity equation.

Reactivity and transient dynamics of discrete-time ecological systems

Most studies of ecological models focus exclusively on the asymptotic stability properties of equilibria. However, short-term transient effects can be important, and can in some cases dominate the dynamics seen in experimental or field studies. The reactivity of a stable equilibrium point measures the potential for short-term amplification of perturbations. The reactivity of a fixed point in a discrete-time system is given by the natural logarithm of the largest singular value of the Jacobian matrix of the linear approximation near the fixed point. If the reactivity is positive, the fixed point is said to be reactive. Here we examine the reactivity of discrete-time predator–prey models and density-dependent matrix population models. We find reactivity to be common (but not universal) and sometimes extremely high. Predator–prey or food web models that include a predator whose per-capita growth rate depends on the density of its prey, but not on its own density, are a special case. Any positive equilibrium of such a model must be reactive. Reactivity of discrete-time models depends on the timing of the census relative to the timing of reproduction. Perturbation analysis of singular values can be used to calculate the sensitivity and elasticity of reactivity to changes in model parameters. We conclude that transient amplification of perturbations should be a common ecological phenomenon. The interaction of these transient effects with the asymptotic nonlinear dynamics warrants further study.

Red environmental noise and the appearance of delayed density dependence in age-structured populations.

Previous work suggests that red environmental noise can lead to the spurious appearance of delayed density dependence (DDD) in unstructured populations regulated only by direct density dependence. We analysed the effect of noise reddening on the pattern of spurious DDD in several variants of the density-dependent age-structured population model. We found patterns of spurious DDD in structured populations with either density-dependent fertility or density-dependent survival of the first age class, inconsistent with predictions from unstructured population models. Moreover, we found that nonspurious negative DDD always emerges in populations with deterministic chaotic dynamics, regardless of population structure or the type of environmental noise. The effect of noise reddening in generating spurious DDD is often negligible in the chaotic region of population deterministic dynamics. Our findings suggest that differences in species' life histories may exhibit different patterns of spurious DDD (owing to noise reddening) than predicted by unstructured models.

Elasticity analysis of density-dependent matrix population models: the invasion exponent and its substitutes

In density-independent models, the population growth rate λ measures population performance, and the perturbation analysis of λ (its sensitivity and elasticity) plays an important role in demography. In density-dependent models, the invasion exponent log λ I replaces λ as a measure of population performance. The perturbation analysis of λ I reveals the effects of environmental changes and management actions, gives the direction and intensity of density-dependent natural selection on life history traits, and permits calculation of the sampling variance of the invasion exponent. Because density-dependent models require more data than density-independent models, it is tempting to look for substitutes for the invasion exponent, the sensitivity and elasticity of which can be calculated from a density-independent model. Here we examine the accuracy of two such substitutes: the dominant eigenvalue of the projection matrix evaluated at equilibrium ( A n ̂ ) and the dominant eigenvalue of the matrix averaged over the attractor ( A ̄ ). Using a two-stage model that represents a wide range of life history types, we find that the elasticities of A n ̂ or A ̄ often agree to within less than 5% error with those of the invasion exponent, even when population dynamics are chaotic. The exceptions are for semelparous life histories, especially when density-dependence affects fertility. This suggests that the elasticity analysis of density-independent models near equilibrium, or averaged over the attractor, provides useful information about the elasticity of the invasion exponent in density-dependent models.

Abundance and Projected Control of Invasive House Crows in Singapore

The Indian house crow (Corvus splendens) has successfully invaded tropical and subtropical regions well beyond its native range, reaching pest proportions in many areas. The invasive population of house crows in Singapore (Southeast Asia) has increased at least 30-fold since 1985 and now numbers in excess of 130,000 birds. To understand the population ecology and behavior of the house crow in Singapore, we undertook regular population size and roost surveys, dissections of birds shot (to provide age structure and breeding status), detailed nestsite observations, and monitoring of coastal dispersal. Using a discrete-time, density-dependent population model to synthesize this information, we demonstrated that at least 41,000 crows will need to be culled in the first year of a control program, and equivalent effort committed each year thereafter, to be confident of suppressing the Singapore population from its 2001 density of 190 birds/km(2) to the management target of <10 birds/km(2) within a 10-year period. This figure drops to 32,000 if culling is combined with other management strategies such as resource limitation and nest destruction. Complete eradication of the house crow from Singapore may be an unrealistic goal due to potential difficulties in detecting crows at IOW population densities and influx of migrants from neighboring Malaysia. Our study has implications for pest-bird management in other cities in Asia, Africa, and the Middle East, and presents it surrogate population-dynamics management tool for use in regions where the house crow has become established as a pest species, but where limited local field data is available.

Incorporation of changes in habitat quantity and quality into density-dependent population models

The use of dynamic age-structured models is becoming more prevalent in environmental assessments. We review these models and discuss the application of density-dependent population models with varying intrinsic rate of increase (r) and carrying capacity (K). Application of these models is described in terms of the model selection process, evaluation of r and K and how they are influenced by habitat suitability, the calculation of habitat suitability indices, and the selection of competition type (scramble vs contest). A short example is given. We conclude that these models offer valuable insight into the understanding of population responses to environmental impacts and mitigation.

SAMPLING-VARIANCE EFFECTS ON DETECTING DENSITY DEPENDENCE FROM TEMPORAL TRENDS IN NATURAL POPULATIONS

Monte Carlo simulations were conducted to evaluate robustness of four tests to detect density dependence, from series of population abundances, to the addition of sampling variance. Population abundances were generated from random walk, stochastic exponential growth, and density-dependent population models. Population abundance es- timates were generated with sampling variances distributed as lognormal and constant coefficients of variation (CV) from 0.00 to 1.00. In general, when data were generated under a random walk, Type I error rates increased rapidly for Bulmer's R, Pollard et al.'s, and Dennis and Taper's tests with increasing magnitude of sampling variance for n . 5y r and all values of process variation. Bulmer's R* test maintained a constant 5% Type I error rate for n . 5 yr and all magnitudes of sampling variance in the population abundance estimates. When abundances were generated from two stochastic exponential growth models (R 5 0.05 and R 5 0.10), Type I errors again increased with increasing sampling variance; magnitude of Type I error rates were higher for the slower growing population. Therefore, sampling error inflated Type I error rates, invalidating the tests, for all except Bulmer's R* test. Comparable simulations for abundance estimates generated from a density-dependent growth rate model were conducted to estimate power of the tests. Type II error rates were influenced by the relationship of initial population size to carrying capacity ( K), length of time series, as well as sampling error. Given the inflated Type I error rates for all but Bulmer's R*, power was overestimated for the remaining tests, resulting in density depen- dence being detected more often than it existed. Population abundances of natural popu- lations are almost exclusively estimated rather than censused, assuring sampling error. Therefore, because these tests have been shown to be either invalid when only sampling variance occurs in the population abundances (Bulmer's R, Pollard et al.'s, and Dennis and Taper's tests) or lack power (Bulmer's R* test), little justification exists for use of such tests to support or refute the hypothesis of density dependence.

EML: A modeling environment for exploring landscape dynamics

This paper describes a modeling environment, EML (Environmental Modeling Language), for exploring landscape dynamics. The strengths of three existing environmental modeling techniques — Geographical Information Systems (GIS), System Dynamics (SD), and Cellular Automata (CA) — are combined to provide spatio-temporal modeling capabilities. We review the strengths and weaknesses of each of the three existing approaches, describe our composite approach, and demonstrate how a user constructs a model using EML. Also, we show, using a simple fire spread model, Schelling's segregation model, and a density-dependent population model, how EML can be used to explore landscape dynamics. EML's integration of GIS, SD, and CA techniques provides a broader perspective and a richer set of methods than each individual modeling technique affords. EML also provides users an integrated modeling environment to support the iterative construction of models and exploration of landscape dynamics.

Cycles in nonlinear age-structured models I. Renewal equations

A variety of density-dependent population models can be described by nonlinear renewal equations. This paper develops analytical tools for such models to study the sustained population cycles which arise by bifurcation. The results obtained describe explicitly the direction of bifurcation, and the period, form, and dynamic stability of sustained cycles. The results are illustrated by application to a cohort-controlled model of human populations which has been proposed as a formalization of the Easterlin effect.

Régulation par le recrutement chez la mouette rieuse Larus ridibundus

The Black-headed Gull population of the Forez (central France), under study since 1976, is remarkably stable. This stability, or slow growth at the very most, implies a change in some demographic parameters of the population. Indeed, large differences in breeding success were found between colonies established on traditional sites, successfully occupied year after year (where breeding success is high), and those established on sites that are occupied discontinuously (where breeding success is low). Such differences suggest that the gull population has reached saturation at the best sites. Most of our work was carried out in one of the traditional colonies (with more than 2 000 breeding pairs). Here, the saturation hypothesis is supported by the strong competition taking place for nest sites, which leads the younger breeders to make use of nest sites low over water, therefore highly vulnerable to wave action, and to variations of water level. A study of mortality and recruitment, using a capture-resighting technique, provided a more direct evidence of population saturation on this colony : full breeding was only reached at five years of age. But the strongest evidence of density-dependence in recruitment is brought forward by the estimation of the proportion of birds locally recruited is each cohort : this proportion varies inversely with the initial fecundity of the cohort. Only a limited number of breeders can therefore be recruited each year in this colony. A density-dependent population model confirms that density-dependence is strong enough to induce a stabilization in numbers of the colony members. Theoretical and practical consequences of this mode of population regulation are discussed.

Optimal fishery policy for size-specific, density-dependent population models

Optimal harvest policy is derived for a size-specific population model based on the continuity equation. In this model both growth and recruitment rates can depend on either the number of individuals of specific sizes in the population or the level of food available. Necessary conditions for values of fishing pressure and size limit that maximize the present value of the resource are obtained and are interpreted in economic terms. The general solution obtained here reduces to solutions obtained previously for some special cases: the single age-class model, and the linear age-dependent model. Solutions involving constant fishery policy are sought for several different, specific versions of the general model. In each of these versions a constant policy solution is not optimal. This implies that for a general, realistic model the policy that maximizes present value is a time-varying or “pulse fishing” policy. The theoretical and practical implications of the results are discussed in the light of existing results.

The dynamics of density dependent population models

The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bofurcations away from the equilibrium point is investigated and the existence of a "strange attractor" is demonstrated--i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).