The small-scale spatial distribution of an invading moth

We studied the spread of a small leaf-mining moth [Phyllonorycter leucographella (Zeller), Gracillariidae] after its accidental introduction into the British Isles. At large geographical scales, previous work had shown the spread to be well described by a travelling weve of constant velocity. Here, we report the pattern of spread at scales of 1 km². By locating all bushes of the insect's foodplant (Pyracantha spp.) within 1-km² quadrats, the precise pattern of colonisation at finer spatial scales could be established. Where the 1-km² site was colonised by moths from the main advancing front, no spatial pattern in the order that bushes were infested was found. If the source of colonisation was a single or small group of infested plants within the site, there was some evidence that nearby plants were colonised first. We found no evidence of population turnover after colonisation. We interpret the results in terms of a two-stage model of invasion that produces different patterns at small and large geographical scales.     

The impact of diffusion and stirring on the dynamics of interacting populations

We investigate the combined effects of diffusion and stirring on the dynamics of interacting populations which have spatial structure. Specifically we consider the marine phytoplankton and zooplankton populations, and model them as an excitable medium. The results are applicable to other biological and chemical systems. Under certain conditions the combination of diffusion and stirring is found to enhance the excitability, and hence population growth of the system. Diffusion is found to play an important role: too much and initial perturbations are smoothed away, too little and insufficient mixing takes place before the reaction is over. A key time-scale is the mix-down time, the time it takes for the spatial scale of a population to be reduced to that of a diffusively controlled filament. If the mix-down time is short compared to the reaction time-scale, then excitation of the system is suppressed. For intermediate values of the mix-down time the peak population can attain values many times that of a population without spatial structure. We highlight the importance of the spatial scale of the initial disturbance to the system.     

Population ecology of an afro-tropical savanna herb,Lapeirousia rivularis,in Zambia

Detailed demographic studies of herbaceuos plants in afro-tropicalsavannas are extremely rare in published literature. I studied phenology andpopulation dynamics of a perennial herb, Lapeirousiarivularis Wanntorp, at a savanna site in Zambia over a 4-yearperiod, from 1997 to 2001, using enumeration techniques in permanent andtemporary quadrants. The age of the plants was accurately determined frompersistent annual sheaths that accumulate around the corm throughout the lifeofthe plant while the estimated survivorship of the 1998 cohort was developedfroma mathematical model based on the observed mortality of the cohort over a3-yearperiod.L. rivularis completed its annual phenological cycle in asingle rainy season. Plants sprouted from perennating corms in early December,flowered, fruited and dispersed seeds by end of January. Seeds germinatedimmediately after dispersal and seedlings produced small corms before aerialparts died early in the dry season. L. rivularis has twomorphs: a vegetative morph and a reproductive morph; the latter bears severalleaves and flower stalks. In the vegetative morph, the corm is renewed annuallywhile the corms of the reproductive morph did not appear to be renewed. Insteadcorms grew larger and produced lateral daughter corms that became independentramets the following rainy season.Most L. rivularis plants reached reproductive maturitywhen they were 6–10 years old. Reproductive success, seedlingestablishment and recruitment varied from year to year in the grassland plotperhaps because of fluctuations in weather conditions and heavy episodic insectherbivory. The survivorship curve of L. rivularis wascharacteristically concave due to high juvenile mortality (0.2–0.5) andvery low adult mortality (0.03). Although about 3% of the plants live to be upto 30–35 years, the mean age of the population in 2001 was 6.4 yearsbecause of the predominance of juvenile plants. The production of a protectivesheath at the end of the rainy season and the accumulation of old sheathsaroundthe fleshy corm are apparent adaptations against desiccation during the longdryseason drought when the topsoil remains below wilting point. The high juvenilemortality during the dry season is probably caused by inadequate protection bythe few sheaths around the corm against desiccation.Population dynamics in L. rivularis were caused byvariable annual recruitment and high juvenile mortality. Population densitydoubled in 1999 due to good fruiting success in the previous season that wasassociated with good weather conditions and negligible herbivory. Althoughcultivation had a significant negative effect on the population of L.rivularis, it increased consistency in fruiting success andproduction of fruits per plant, presumably because of improvement insoil-moisture status, reduction in plant competition and by providing temporalescape from insect herbivory through delayed flowering. The phenology and lifehistory of L. rivularis exhibit adaptations to a savannaenvironment that is characterized by disturbance and stress caused by seasonaldrought, fire, episodic herbivory and cultivation.     

Stabilizing effects in spatial parasitoid-host and predator-prey models: a review

We review the literature on spatial host-parasitoid and predator-prey models. Dispersal on its own is not stabilizing and can destabilize a stable local equilibrium. We identify three mechanisms whereby limited dispersal of hosts and parasitoids combined with other features, such as spatial and temporal heterogeneity, can promote increased persistence and stability. The first mechanisms, "statistical stabilization", is simply the statistical effect that summing a number of out-of-phase population trajectories results in a relatively constant total population density. The second mechanism involves decoupling of immigration from local density, such that limited dispersal between asynchronous patches results in an effect that mimics density-dependence at the local patch level. The third mechanism involves altering spatially averaged parameter values resulting from spatial heterogeneity in density combined with non-linear responses to density. Persistence in spatially explicit models with local dispersal frequently associated with self-organized spatial patterning.     

Spatial variation in springtime food resources influences the winter body mass of roe deer fawns

It is well established that the dynamics of mammalian populations vary in time, in relation to density and weather, and often in interaction with phenotypic differences (sex, age and social status). Habitat quality has recently been identified as another significant source of individual variability in vital rates of deer, including roe deer where spatial variations in fawn body mass were found to be only about a tenth of temporal variations. The approach used was to classify the habitat into blocks a priori, and to analyse variation in animal performance among the predefined areas. In a fine-grained approach, here we use data collected over 24 years on 1,235 roe deer fawns captured at known locations and the plant species composition sampled in 2001 at 578 sites in the Chizé forest to determine the spatial structure at a fine scale of both vegetation and winter body mass of fawns, and then to determine links between the two. Space and time played a nearly equal role in determining fawn body masses of both sexes, each accounting for about 20% of variance and without any interaction between them. The spatial distribution of fawn body mass was perennial over the 24 years considered and predicted values showed a 2 kg range according to location in the reserve, which is much greater than suggested in previous work and is enough to have strong effects on fawn survival. The spatial distribution and the range of predicted body masses were closely similar in males and females. The result of this study is therefore consistent with the view that the life history traits of roe deer are only weakly influenced by sexual selection. The occurrence of three plant species that are known to be important food items in spring/summer roe deer diets, hornbeam (Carpinus betulus), bluebell (Hyacinthoides sp.) and Star of Bethlehem (Ornithogalum sp.) was positively related to winter fawn body mass. The occurrence of species known to be avoided in spring/summer roe deer diets [e.g. butcher's broom (Ruscus aculeatus) and beech (Fagus sylvatica)], was negatively related to fawn body mass. We conclude that the spatial variation in the body mass of fawns in winter in this forest is as important as the temporal variation, and that the distribution of plant species that are actively selected during spring and summer is an important determinant of spatial variation in winter fawn body mass. The availability of these plants is therefore likely to be a key factor in the dynamics of roe deer populations.     

Polymorphism in multiallelic migration–selection models with dominance

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.     

Effects of noise on some dynamical models in ecology

We investigate effects of random perturbations on the dynamics of one-dimensional maps (single species difference equations) and of finite dimensional flows (differential equations for n species). In particular, we study the effects of noise on the invariant measure, on the correlation dimension of the attractor, and on the possibility of detecting the nonlinear deterministic component by applying reconstruction techniques to the time series of population abundances. We conclude that adding noise to maps with a stable fixed-point obscures the underlying determinism. This turns out not to be the case for systems exhibiting complex periodic or chaotic motion, whose essential properties are more robust. In some cases, adding noise reveals deterministic structure which otherwise could not be observed. Simulations suggest that similar results hold for flows whose attractor is almost two-dimensional.     

A spatially explicit model for tropical tree diversity patterns

A simple two-parameter model resembling the classical voter model is introduced to describe macroecological properties of tropical tree communities. The parameters of the model characterize the speciation- and global-dispersion rates. Monte Carlo type computer simulations are performed on the model, investigating species abundances and the spatial distribution of individuals and species. Simulation results are critically compared with the experimental data obtained from a tree census on a 50 hectare area of the Barro Colorado Island (BCI), Panama. Fitting to only two observable quantities from the BCI data (total species number and the slope of the log-log species-area curve at the maximal area), it is possible to reproduce the full species-area curve, the relative species abundance distribution, and a more realistic spatial distribution of species.     

Autocatalysis in cultural ecology: model ecosystems and the dynamics of biocultural evolution

Using a well-known mathematical model frequently applied in theoretical population dynamics, certain ecological mechanisms are investigated that are inherent in the organic evolution of cultural capacities in man. Culture is argued to involve ecological interactions exhibiting analogies to the interaction of chemical species in autocatalytic biomolecular reactions. In the model, biocultural evolution proceeds by more and more broadening ecological niches and, thus, releasing competitive selection pressure on the populations involved. This, in turn, facilitates the maintenance of polymorphism in these populations as well as the individual acquisition of organic traits through learning and cultural transmission. The result is that the genetic variance in phenotypic expressions decreases at an accelerated rate.     

A dynamic model of water gain and loss in larval Mexican bean beetle

Abstract. Quantitative effects of temperature, vapour pressure deficit, host, and larval body size were experimentally determined. A simulation model for dynamic water balance in the Mexican bean beetle, Epilachna varivestis Mulsant, is presented and parameters are estimated from laboratory data for water gain/loss equations. The model is based on water loss through the cuticle, spiracles and frass, and water gain through ingestion.     

Trophic exploitation in grassland food chains: simple models and a field experiment

This study provides insight into the importance of top carnivores (top-down control) and nutrient inputs (bottom-up control) in structuring food chains in a terrestrial grassland system. Qualitative predictions about food chain structure are generated using 4 simple models, each differing in assumptions about some key component in the population dynamics of the herbivore trophic level. The four model systems can be classified broadly into two groups (1) those that assume plant resource intake by herbivores is limited by search rate and handling time as described by classic Lotka-Volterra models; and (2) those that assume plant resource intake by herbivores is limited externally by the supply rate of resources as described by alternatives to Lotka-Volterra formulations. The first class of models tends to ascribe greater importance to top-down control of food chain structure whereas the second class places greater weight on bottom-up control. I evaluated the model predictions using experimentally assembled grassland food chains in which I manipulated nutrient inputs and carnivore (wolf spider) abundance to determine the degree of top-down and bottom-up control of grassland plants and herbivores (grasshoppers). The experimental results were most consistent with predictions of the second class of models implying a predominance of bottom-up control of food chain structure.     

Mathematical models of microbial growth and competition in the chemostat regulated by cell-bound extracellular enzymes

A mathematical model of growth and competitive interaction of microorganisms in the chemostat is analyzed. The growth-limiting nutrient is not in a form that can be directly assimilated by the microorganisms, and must first be transformed into an intermediate product by cell-bound extracellular enzymes. General monotone functions, including Michaelis-Menten and sigmoidal response functions, are used to describe nutrient conversion and growth due to consumption of the intermediate product. It is shown that the initial concentration of the species is an important determining factor for survival or washout. When there are two species whose growth is limited by the same nutrient, three different modes of competition are described. Competitive coexistence steady states are shown to be possible in two of them, but they are always unstable. In all of our numerical simulations, the system approaches a steady state corresponding to the washout of one or both of the species from the chemostat.Research supported by NSF grant DMS-90-96279Research supported by NSERC grant A-9358     

Dynamical consequences of harvest in discrete age-structured population models

The role of harvest in discrete age-structured one-population models has been explored. Considering a few age classes only, together with the overcompensatory Ricker recruitment function, we show that harvest acts as a weak destabilizing effect in case of small values of the year-to-year survival probability P and as a strong stabilizing effect whenever the survival probability approaches unity. In the latter case, assuming n=2 age classes, we find that harvest may transfer a population from the chaotic regime to a state where the equilibrium point (x₁*, x₂*) becomes stable. However, as the number of age classes increases (which acts as a stabilizing effect in non-exploited models), we find that harvest acts more and more destabilizing, in fact, when the number of age classes has been increased to n=10, our finding is that in case of large values of the survival probabilities, harvest may transfer a population from a state where the equilibrium is stable to the chaotic regime, thus exactly the opposite of what was found in case of n=2. On the other hand, if we replace the Ricker relation with the generalized Beverton and Holt recruitment function with abruptness parameter larger than 2, several of the conclusions derived above are changed. For example, when n is large and the survival probabilities exceed a certain threshold, the equilibrium will always be stable.Revised version: 18 September 2003     

A stochastic modeling of early HIV-1 population dynamics

In a recent paper, Tuckwell and Le Corfec [J. Theor. Biol. 195 (1998) 450-463] applied the multi-dimensional diffusion process to model early human immunodeficiency virus type-1 (HIV-1) population dynamics. The purpose of this paper is to assess certain features and consequences of their model in the context of Tan and Wu's stochastic approach [Math. Biosci. 147 (1998) 173-205].     

北京东灵山地区辽东栎、大叶白蜡和五角枫种群分布格局与动态

对北京东灵山地区三块样地中的辽东栎(Quercus liaotungensis)、大叶白蜡(Fraxinus rhynchophylla)和五角枫(Acer mono)种群的分布格局和种群动态进行了考察.调查三块样地中每个树种的全部个体,按高度将它们分为4个不同龄级,用相邻格子样方法进行取样.在研究分布格局时,计算了扩散系数C、扩散型指数Iδ与IA、CA指数和聚块指数m/m等各项指标,测定它们的分布格局,并通过计算拟合它们的分布类型.结果表明,不同地点各种群的幼苗(<0.4 m)、小幼树(0.4～1 m)和大幼树(1～2 m)绝大多数呈聚集分布,少量呈随机分布;成林辽东栎成熟个体(≥2 m)是一种界于均匀分布和随机分布之间的状态,幼林辽东栎成体和灌丛大叶白蜡属于随机分布,成林大叶白蜡和五角枫、灌丛辽东栎呈聚集分布.在研究种群动态时,分别作出每个种群的龄级结构图,再对同一样地不同种群以及同一种群不同龄级个体的数量差异进行最小显著差法(LSD)检验和比较,结果表明检验结果和龄级结构图显示出来的趋势是一致的,即成林辽东栎种群具有较稳定的结构,成林大叶白蜡和五角枫表现为增长型结构,幼林辽东栎、大叶白蜡和五角枫都表现为衰退型结构,灌丛辽东栎具有稳定型结构,灌丛大叶白蜡具有增长型结构.研究表明,作为优势种的辽东栎种群个体之间的竞争很激烈,在乔木层中占有明显优势;作为伴生种的大叶白蜡的幼苗对环境的适应能力很强,在较低位层中占有明显优势.     

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.     

Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology

A single-species population dynamics with dispersal in a spatially heterogeneous environment is modeled by a nonlinear reaction-diffusion equation with a potential term. To each nonlinear kinetics there corresponds a bifurcation curve that describes the relation between the growth rate and the central density of a steady-state population distribution. Our main concern is an inverse problem for this correspondence. The existence of nonlinear kinetics realizing a prescribed bifurcation curve is established. It is shown that the freedom of such kinetics is of degree finite and even, depending only on the heterogeneity of the environment, and conversely that any nonnegative even integer occurs as the degree of freedom in some environments. A discussion is also made on under what kind of environment the degree is equal to zero or is positive. The mathematical analysis involves the development of a general theory for singular multiplicative Wiener-Hopf integral equations.     

Spatial population structure in the banner-tailed kangaroo rat,Dipodomys spectabilis

I attempted to characterize spatial units of local dynamics and dispersal in banner-tailed kangaroo rats (Dipodomys spectabilis), to determine if spatial structure influenced population dynamics in the way predicted by current metapopulation models. D. spectabilis exhibited a hierarchical spatial structure. Local populations that appeared as discrete entities on a scale of kilometers were subdivided into clusters of mounds on a scale of meters. This structure, however, cannot be characyerized in terms of the discrete habitat patches envisioned by the metapopulation models. Occupied areas were statistically distinguishable from the surrounding matrix, but this difference was only quantitative. There were no discrete boundaries between occupied areas and the matrix. Habitat within occupied areas was heterogeneous, and occupied areas in different locations were statistically distinguishable from each other. High heterogeneity within occupied areas, and high contrast among them, make it difficult to define what is a suitable habitat patch for D. spectabilis. On a smaller spatial scale, there was significant aggregation of resident mounds within occupied areas. These aggregations, however, do not correspond to discrete habitat patches. Rather, they appear to result from an interaction between fine-scale habitat heterogeneity and limited dispersal due to natal philopatry and low adult vagility. These complications make it difficult to identify habitat patches independent of the species' distribution. For species like D. spectabilis that are patchily distributed but do not occupy discrete habitat patches, a patch occupancy approach does not seem appropriate for describing spatial structure. Hierarchical spatial structure underscores the need for a framework that incorporates multiple scales of spatial structure, rather than one that pre-imposes a single spatial scale as being important for population dynamics. A framework that (i) considers patchiness as a combination of both habitat heterogeneity, and life-history and behavioral characteristics, and (ii) incorporates hierarchical spatial structure, appears to be the most suitable for conceptualizing spatial dynamics of behaviorally complex vertebrates such as D. spectabilis.