Equivalence relationships between stage-structured population models

Matrix population models are widely applied in conservation ecology to help predict future population trends and guide conservation effort. Researchers must decide upon an appropriate level of model complexity, yet there is little theoretical work to guide such decisions. In this paper we present an analysis of a stage-structured model, and prove that the model's structure can be simplified and parameterised in such a way that the long-term growth rate, the stable-stage distribution and the generation time are all invariant to the simplification. We further show that for certain structures of model the simplified models require less effort in data collection. We also discuss features of the models which are not invariant to the simplification and the implications of our results for the selection of an appropriate model. We illustrate the ideas using a population model for short-tailed shearwaters (Puffinus tenuirostris). In this example, model simplification can increase parameter elasticity, indicating that an intermediate level of complexity is likely to be preferred.     

Effects of sterile insect releases on a population under predation or parasitism

A continuous-time differential equation model was constructed which describes the population dynamics of a predator prey system in which sterile prey are released in a program designed to eradicate or reduce the prey population. It was found that the dynamics of the system behave quite differently when predators are present. Two conditions were found which have differing implications for the control program. If the predators still exist when the wild prey population declines to extinction, then the SIRM is assisted by the predators, sometimes to a considereble extent. If the predators decline to extinction before the wild prey population goes extinct, then the predators may or may not assist the SIRM depending on the parameters of the system. If the predators do assist the SIRM, then a potentially dangerous situation exists in which an explosion of the prey population could occur after the predators go extinct. Predator polyphagy would probably minimize this danger of an explosion since it would stabilize the predator population.     

Density, climate and varying return points: an analysis of long-term population fluctuations in the threatened European tree frog

Experimental research has identified many putative agents of amphibian decline, yet the population-level consequences of these agents remain unknown, owing to lack of information on compensatory density dependence in natural populations. Here, we investigate the relative importance of intrinsic (density-dependent) and extrinsic (climatic) factors impacting the dynamics of a tree frog (Hyla arborea) population over 22 years. A combination of log-linear density dependence and rainfall (with a 2-year time lag corresponding to development time) explain 75% of the variance in the rate of increase. Such fluctuations around a variable return point might be responsible for the seemingly erratic demography and disequilibrium dynamics of many amphibian populations.     

Robustness: predicting the effects of life history perturbations on stage-structured population dynamics

Matrix-based models lie at the core of many applications across the physical, engineering and life sciences. In ecology, matrix models arise naturally via population projection matrices (PPM). The eigendata of PPMs provide detailed quantitative and qualitative information on the dynamic behaviour of model populations, especially their asymptotic rates of growth or decline. A fundamental task in modern ecology is to assess the effect that perturbations to life-cycle transition rates of individuals have on such eigendata. The prevailing assessment tools in ecological applications of PPMs are direct matrix simulations of eigendata and linearised extrapolations to the typically non-linear relationship between perturbation magnitude and the resulting matrix eigenvalues. In recent years, mathematical systems theory has developed an analytical framework, called 'Robustness Analysis and Robust Control', encompassing also algorithms and numerical tools. This framework provides a systematic and precise approach to studying perturbations and uncertainty in systems represented by matrices. Here we lay down the foundations and concepts for a 'robustness' inspired approach to predictive analyses in population ecology. We treat a number of application-specific perturbation problems and show how they can be formulated and analysed using these robustness methodologies.     

An interpretation of aphid population development by statistical model

Shiyomi (1967a and b) proposed two models which describe the reproduction and the plant-to-plant movement of aphids. For the explanation of the whole process of development of population of aphids, the above two models were incorporated into a new model (called Model C). This model is superior in the following points to the negative binomial model:

1. Model C has 7 parameters and gives a fuller explanation compared with the negative binomial model which has 2 parameters.
2. Model C describes the structure of population at any stage of its development, while the negative binomial model describes that of a well developed stage of population.

     

Evidence of population structuring following population genetic analyses of Fasciola hepatica from Argentina

Fasciola hepatica, the liver fluke, is a trematode parasite that causes disease of economic importance in livestock. As a zoonosis this parasite also poses a risk to human health in areas where it is endemic. Population genetic studies can reveal the mechanisms responsible for genetic structuring (non-panmixia) within parasite populations and provide valuable insights into population dynamics, which in turn enables theoretical predictions of evolutionary dynamics such as the evolution of drug resistance. Here we genotyped 320 F. hepatica collected from 14 definitive hosts from four provinces in Argentina. STRUCTURE analysis indicated three population clusters, and principal coordinate analysis confirmed this, showing population clustering across provinces. Similarly, pairwise F_ST values amongst all four provinces were significant, with standardised pairwise F_ST (F′_ST) ranging from 0.0754 to 0.6327. Therefore, population genetic structure was evident across these four provinces in Argentina. However, there was no evidence of deviation from Hardy–Weinberg equilibrium, so it appears that within these sub-populations there is largely random mating. We identified 263 unique genotypes, which gave a clonal diversity of 82%. Parasites with identical genotypes, clones, accounted for 26.6% of the parasites studied and were found in 12 of the 14 hosts studied, suggesting some clonemate transmission.     

Towards a model of a floodplain fish population and its fishery

Synopsis A model is developed which describes the way in which the fish populations of African rivers and their fisheries are influenced by the different types of flood Regime. Ichthyomass and fish catch are dependent on both the extent of flooding during high water and the amount of water remaining in the system during the dry season. The relative number and individual weight of fish are determined by the intensity of flooding, whereas the total number surviving to the next year depends more on the low water regime. Catch per unit effort falls with increasing difference between areas flooded at high and low water. A negative log-log relationship exists between catch and the ratio of maximum area flooded to minimum area of water remaining in the system. This relationship of catch to flood ratio may form the basis for a general index for the evaluation of both year-to-year variations within a floodplain, and differences between floodplains. Lines of equal catch are also derived for various combinations of high and low water areas; these might be used as guidelines for the hydrological management of tropical flood-plains.     

Ecological–genetic approach in modeling the natural evolution of a population: Prospects and special aspects of verification

This study presents a complex approach for modeling the natural evolution of a population in terms of population number and dynamics of the genetic structure. A set of dynamic models that consider various types of natural selection was applied to describe possible mechanisms underlying the formation of existing genetic variations in litter sizes in coastal, inland, and farmed arctic fox populations (Alopex lagopus, family Canidae, order Carnivora). The r–K selection model for uniform population and the models with natural selection were assessed on various life cycle stages in a two-age population. The life cycle of arctic fox was fitted to the population model with two age stages. The different reproductive potentials and survivability of progeny on the early stage of life cycle were genetically determined using the model with a single diallelic gene. A monomorphism was obtained for a considered characteristic in a population of coastal arctic fox with constant food supply. Meanwhile, a polymorphism with cyclic fluctuations in population number and gene frequency was obtained in inland arctic fox populations, which could be due to cyclic fluctuations of prey. In farmed fox populations, the considered gene becomes pleiotropic (defines the survival rate of individuals on early and late stages of the life cycle) because of artificial selection performed by farmers to increase the reproductive success of breeders. The application of an appropriate model (with selection by pleiotropic gene) can be used to determine the elimination rate of low litter size alleles from the farmed populations. The possible applications of the proposed models for formulating and solving optimal control tasks in arctic fox populations are discussed too.     

A general measure of human population growth regulation

A general measure relating the relative effects of mortality and fertility in damping population growth has been derived from stable population theory. This measure, called the Index of Growth Regulation, can be calculated from a life table and a fertility schedule. It is a single number which integrates the fertility and mortality aspects of a population. It has intuitive meaning, and can be related to social aspects of the population. It can be used to compare any two populations, and for this purpose it has advantages over traditional comparative statistics such as life expectancy, which consider only mortality. By selecting life tables representing general stages of human cultural evolution, it has been shown that hunting-gathering populations were regulated more by mortality than by fertility factors. That is, more growth which could have occurred did not occur due to the incidence of mortality than to the incidence of infecundity. The two forces were of about equal weight before the industrial revolution, and fertility has had a predominant role in population regulation since the beginning of industrialization.     

Consistent micro, macro and state-based population modelling

A population system can be modelled using a micro model focusing on the individual entities, a macro model where the entities are aggregated into compartments, or a state-based model where each possible discrete state in which the system can exist is represented. However, the concepts, building blocks, procedural mechanisms and the time handling for these approaches are very different. For the results and conclusions from studies based on micro, macro and state-based models to be consistent (contradiction-free), a number of modelling issues must be understood and appropriate modelling procedures be applied. This paper presents a uniform approach to micro, macro and state-based population modelling so that these different types of models produce consistent results and conclusions. In particular, we demonstrate the procedures (distribution, attribute and combinatorial expansions) necessary to keep these three types of models consistent. We also show that the different time handling methods usually used in micro, macro and state-based models can be regarded as different integration methods that can be applied to any of these modelling categories. The result is free choice in selecting the modelling approach and the time handling method most appropriate for the study without distorting the results and conclusions.     

The effect of population density on the growth of an animal population

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Contribution from the Department of Fisheries, Kyoto University.

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Contribution from the Entomological Laboratory, Kyoto University, No. 201.     

Generation carryover of a fraction of population members as an animal adaptation to unstable environmental conditions

Summary A heterogeneous life cycle of individuals in a population was examined on its adaptive significance to an unstable environmental condition. The trend of population growth was simulated by a simple mathematical model in which a part of population in a certain generation was carried over to the next generation without participating in the reproduction. With the increase of the rate of carryover of individuals to the next generation the population fluctuation tended to be stabilized. A minute fraction of population is carried over, the effect is very large to prevent the population decline at a sequence of adverse environmental conditions. The population level increased greatly depending upon the extent of environmental change as far as the rate of carryover took an intermediate value. The optimum proportion of members to be carried over to the next generation was determined by the extent of environmental change and its frequency of occurrence. Contribution from the Entomological Laboratory, College of Agriculture, Kyoto University, No. 451.     

Generation carryover of a fraction of population members as an animal adaptation to unstable environmental conditions

A heterogeneous life cycle of individuals in a population was examined on its adaptive significance to an unstable environmental condition. The trend of population growth was simulated by a simple mathematical model in which a part of population in a certain generation was carried over to the next generation without participating in the reproduction. With the increase of the rate of carryover of individuals to the next generation the population fluctuation tended to be stabilized. A minute fraction of population is carried over, the effect is very large to prevent the population decline at a sequence of adverse environmental conditions. The population level increased greatly depending upon the extent of environmental change as far as the rate of carryover took an intermediate value. The optimum proportion of members to be carried over to the next generation was determined by the extent of environmental change and its frequency of occurrence.     

A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis

A dynamic population model of Ixodes scapularis, the vector of a number of tick-borne zoonoses in North America, was developed to simulate effects of temperature on tick survival and seasonality. Tick development rates were modelled as temperature-dependent time delays, calculated using mean monthly normal temperature data from specific meteorological stations. Temperature also influenced host-finding success in the model. Using data from stations near endemic populations of I. scapularis, the model reached repeatable, stable, cyclical equilibria with seasonal activity of different instars being very close to that observed in the field. In simulations run using data from meteorological stations in central and eastern Canada, the maximum equilibrium numbers of ticks declined the further north was the station location, and simulated populations died out at more northerly stations. Tick die-out at northern latitudes was due to a steady increase in mortality of all life stages with decreasing temperature rather than a specific threshold event in phenology of one life stage. By linear regression we investigated mean annual numbers of degree-days >0 degrees C (DD>0 degrees C) as a readily mapped index of the temperature conditions at the meteorological stations providing temperature data for the model. Maximum numbers of ticks at equilibrium were strongly associated with the mean DD>0 degrees C (r2>0.96, P<0.001), when the Province of origin of the meteorological station was accounted for (Quebec>Ontario, beta=103, P<0.001). The intercepts of the regression models provided theoretical limits for the establishment of I. scapularis in Canada. Maps of these limits suggested that the range of southeast Canada where temperature conditions are currently suitable for the tick, is much wider than the existing distribution of I. scapularis, implying that there is potential for spread. Future applications of the model in investigating climate change effects on I. scapularis are discussed.     

Non-linear age-dependent population diffusion

Summary A non-linear problem arising in the study of an age-dependent population diffusion is considered. Existence and uniqueness results together with a priori bounds for the growth of the population are obtained. Moreover the solutions are shown to depend continuously on the initial data.This work was done under the auspices of the G.N.A.F.A. of the National Research Council.     

The Galilean turn in population ecology

The standard mathematical models in population ecology assume that a population's growth rate is a function of its environment. In this paper we investigate an alternative proposal according to which the rate of change of the growth rate is a function of the environment and of environmental change. We focus on the philosophical issues involved in such a fundamental shift in theoretical assumptions, as well as on the explanations the two theories offer for some of the key data such as cyclic populations. We also discuss the relationship between this move in population ecology and a similar move from first-order to second-order differential equations championed by Galileo and Newton in celestial mechanics.     

Long-term population trends in widespread British moths

The Rothamsted Insect Survey has operated a Great Britain-wide network of light-traps since 1968. From these data we estimated the first ever national abundance indices and 35-year population trends for 338 species of common macro-moths. Although the number of trap sites which run each year is not constant, there is a representative, well-distributed core of traps that have run for 15 years. The proportion of operating sites catching a species and the annual geometric mean catch of successful traps were used to provide estimates of species range and absolute abundance. T, an index of long-term population trends, was used to compare trends among species. T was not biased by trap site turnover. The percentage of species displaying significant decreases (54%) was more than double that displaying increases (22%). Species found throughout Great Britain are decreasing most rapidly in the south and especially the southeast but species with a southerly distribution are increasing. Results of a preliminary overview suggest habitat and climate change may both play a role in changing species dynamics. The existence of estimates of abundances and trends for such a large species pool opens the way for much further research, linking trends with land-use changes, climate change and inter-specific dynamics.     

Optimal harvesting of an age-structured population

Here we investigate the optimal harvesting of an age-structured population. We use the McKendrick model of population dynamics, and optimize a discounted yield on an infinite time horizon. The harvesting function is allowed to depend arbitrarily on age and time and its magnitude is unconstrained. We obtain, in addition to existence, the qualitative result that an optimal harvesting policy consists of harvesting at no more than three distinct ages.