Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

Ecological balance in the native population dynamics may cause the paradox of pest control with harvesting

We analyze a time-discrete mathematical model of host-parasite population dynamics with harvesting, in which the host can be regarded as a pest. We harvest a portion of the host population at a moment in each parasitism season. The principal target of the harvesting is the host; however, the parasite population may also be affected and reduced by a portion. Our model involves the Beverton-Holt type density effect on the host population. We investigate the condition in which the harvesting of the host results in an eventual increase of its equilibrium population size, analytically proving that the paradoxical increase could occur even when the harvesting does not directly affect the parasite population at all. We show that the paradox of pest control could be caused essentially by the interspecific relationship and the intraspecific density effect.     

The Feeding Behaviour of a Sit-and-Wait Predator,Ranatra dispar (Heteroptera: Nepidae): The Combined Effect of Food Deprivation and Prey Size on the Behavioural Components of Prey Capture

Previous work has shown a significant effect of hunger on the predatory behaviour in a sit-and-wait predator Ranatra dispar, the water stick insect (Bailey 1986 a). The experiments reported here were designed to investigate the combined effect of prey size and hunger on the predatory behaviour in order to identify which behavioural components are effected. It was found that the hunger level determines whether R. dispar will initially be aroused or not but the distance at which the arousal takes place is influenced by the size of the prey. This is believed to reflect the capacity and interrelation between visual and mechanoreceptor, sensory organs. The decision to strike at a prey is, although again influenced by hunger, significantly affected by prey size. The distance of the prey when the strike takes place is affected by hunger not the size of the prey. The outcome of the strike is determined by the size of the prey, not the hunger level of the predator. This is believed to reflect the relationship between strike trajectory, leg morphology and prey size. Food deprivation affects all components of predatory behaviour of R. dispar leading up to prey capture, by increasing not only distance of response but also the number of strikes, hits, and captures per unit presentation of prey. It does not affect capture efficiency which remains at about 70 to 80 %. Food deprivation also increases the range of prey sizes that R. dispar responds to and attempts to capture. The effect of food deprivation is considered to reflect a motivational change in responsiveness to particular prey stimuli usually described as a sensitization of particular stimulus-response relations rather than the food deprivation affecting the sensory mechanisms. The predatory success in relation to size of model prey suggested a hypothetical size that could be captured, irrespective of predator motivational level, which is based primarily on the relationship between the shape of the grasping leg and size of prey.     

Further characterization of the long-run population distribution of a deleterious gene

In a population of constant size, there is an equilibrium distribution for every deleterious autosomal dominant gene. This equilibrium represents the balance between selection and mutation. The purpose of this paper is to describe an approximate method of computing the equilibrium distribution and an exact method of computing its cumulants. If the surrounding population has experienced prolonged growth or decline, then an equilibrium does not develop. However, one can show that the variance of the number of carriers divided by the current population size does stabilize; this quantity is an increasing function of the growth rate.     

The Invisible Cliff: Abrupt Imposition of Malthusian Equilibrium in a Natural-Fertility,Agrarian Society

Analysis of a natural fertility agrarian society with a multi-variate model of population ecology isolates three distinct phases of population growth following settlement of a new habitat: (1) a sometimes lengthy copial phase of surplus food production and constant vital rates; (2) a brief transition phase in which food shortages rapidly cause increased mortality and lessened fertility; and (3) a Malthusian phase of indefinite length in which vital rates and quality of life are depressed, sometimes strikingly so. Copial phase duration declines with increases in the size of the founding group, maximum life expectancy and fertility; it increases with habitat area and yield per hectare; and, it is unaffected by the sensitivity of vital rates to hunger. Transition phase duration is unaffected by size of founding population and area of settlement; it declines with yield, life expectancy, fertility and the sensitivity of vital rates to hunger. We characterize the transition phase as the Malthusian transition interval (MTI), in order to highlight how little time populations generally have to adjust. Under food-limited density dependence, the copial phase passes quickly to an equilibrium of grim Malthusian constraints, in the manner of a runner dashing over an invisible cliff. The three-phase pattern diverges from widely held intuitions based on standard Lotka-Verhulst approaches to population regulation, with implications for the analysis of socio-cultural evolution, agricultural intensification, bioarchaeological interpretation of food stress in prehistoric societies, and state-level collapse.     

Polymorphism in sexual versus non-sexual disease transmission

Pathogens causing sexually transmitted diseases (STDs) often consist of related strains that cause non-sexually transmitted, or ''ordinary infectious'', diseases (OIDs). We use differential equation models of single populations to derive conditions under which a genetic variant with one (e.g. sexual) transmission mode can invade and successfully displace a genetic variant with a different (e.g. non-sexual) transmission mode. Invasion by an STD is easier if the equilibrium population size in the presence of an OID is smaller; conversely an OID can invade more easily if the equilibrium size of the population with the STD is larger. Invasion of an STD does not depend on the degree of sterility caused by the infection, but does depend on the added mortality caused by a resident OID. In contrast, the ability of an OID to invade a population at equilibrium with an STD decreases as the degree of sterility caused by the STD increases. When equilibrium population sizes for a population infected with an STD are above the point at which non-sexual contacts exceed sexual contacts (the sexual–social crossover point) and when equilibrium population sizes for an OID are below this point, there can be a stable genetic polymorphism for transmission mode. This is most likely when the STD is mildly sterilizing, and the OID causes low or intermediate levels of added mortality. Because we assume the strains are competitively equivalent and there are no heterogeneities associated with the transmission process, the polymorphism is maintained by density-dependent selection brought about by pathogen effects on population size.     

Frequency- and density-dependent selection on a quantitative character

The equilibrium distribution of a quantitative character subject to frequency- and density-dependent selection is found under different assumptions about the genetical basis of the character that lead to a normal distribution in a population. Three types of models are considered: (1) one-locus models, in which a single locus has an additive effect on the character, (2) continuous genotype models, in which one locus or several loci contribute additively to a character, and there is an effectively infinite range of values of the genotypic contributions from each locus, and (3) correlation models, in which the mean and variance of the character can change only through selection at modifier loci. It is shown that the second and third models lead to the same equilibrium values of the total population size and the mean and variance of the character. One-locus models lead to different equilibrium values because of constraints on the relationship between the mean and variance imposed by the assumptions of those models.——The main conclusion is that, at the equilibrium reached under frequency- and density-dependent selection, the distribution of a normally distributed quantitative character does not depend on the underlying genetic model as long as the model imposes no constraints on the mean and variance.     

Prey persistence and abundance in systems with intraguild predation and type-2 functional responses

The apparent prevalence of intraguild predation in productive environments has been regarded as puzzling because some simple models suggest that the intraguild prey species is often either reduced in abundance or driven extinct at high resource productivity. While various theoretical mechanisms that avoid this prediction have been uncovered, they have often been viewed as being narrowly applicable. This article examines the fate of the intraguild prey in models in which consumer species may have type-2 functional responses; these are usually characterized by sustained fluctuations in population density at high enough resource productivities. The models also include adaptive, but imperfect diet choice by the top predator. We concentrate on two situations: (1) the prey exhibits less saturation in its functional response to the resource than does the predator and (2) the predator is unable to persist on the basal resource alone. The reasons given by previous studies for discounting these cases are re-examined. The present analysis shows that prey abundance often increases with increasing productivity in both cases, as does the range of prey parameters that allows prey persistence. It is also possible for the prey to coexist with the predator in spite of having a larger equilibrium requirement for the resource. Different assumptions about the dynamics of diet choice can have a large impact on population responses to enrichment. We argue that the persistence and/or increase in abundance of intraguild prey at higher productivity should not be regarded as puzzling because observations are consistent with a range of theoretical models that reflect commonly observed mechanisms.     

The role of size-specific predation in the evolution and diversification of prey life histories

Some of the best empirical examples of life-history evolution involve responses to predation. Nevertheless, most life-history theory dealing with responses to predation has not been formulated within an explicit dynamic food-web context. In particular, most previous theory does not explicitly consider the coupled population dynamics of the focal species and its predators and resources. Here we present a model of life-history evolution that explores the evolutionary consequences of size-specific predation on small individuals when there is a trade-off between growth and reproduction. The model explicitly describes the population dynamics of a predator, the prey of interest, and its resource. The selective forces that cause life-history evolution in the prey species emerge from the ecological interactions embodied by this model and can involve important elements of frequency dependence. Our results demonstrate that the strength of the coupling between predator and prey in the community determines many aspects of life-history evolution. If the coupling is weak (as is implicitly assumed in many previous models), differences in resource productivity have no effect on the nature of life-history evolution. A single life-history strategy is favored that minimizes the equilibrium resource density (if possible). If the coupling is strong, then higher resource productivities select for faster growth into the predation size refuge. Moreover, under strong coupling it is also possible for natural selection to favor an evolutionary diversification of life histories, possibly resulting in two coexisting species with divergent life-history strategies.     

Density-dependent parameters and demographic equilibrium in open populations

Populations near their equilibrium are expected to show density-dependence through a negative feedback on at least one demographic parameter, e.g. survival and/or productivity. Nevertheless, it is not always clear which vital rate is affected the most, and even less whether this dependence holds in open populations in which immigration and emigration are also important. We assessed the relative importance of population density in the variation of local survival, recruitment, proportion of transients (emigrants) and productivity through the analysis of detailed life-histories of 4286  seabirds from a colony that reached an apparent demographic equilibrium after a period of exponential increase. We provide evidence that the role of population density and resource availability changes according to the demographic parameter considered. Estimates indicated that transients increased from 5% to 20% over the study period, suggesting an average turnover of about 1400 individuals per year. The parameters most influenced by population density alone were local survival and probability of transience. Recruitment was negatively associated with population density during the increasing phase but unexpected high values were also recorded at high population levels. These high values were explained by a combination of population size and food availability. Mean productivity varied with food availability, independently from population variations. The population density alone explained up to a third of the yearly variation of the vital rates considered, suggesting that open populations are equally influenced by stochastic and density-independent events (such as environmental perturbations) than by intrinsic (i.e. density-dependent) factors.     

Resource partitioning among competing species—A coevolutionary approach

A reasonably general theory for predicting the outcome of coevolution among interacting species is developed. It is applied to a model for resource partitioning among competing species.Current theory for resource partitioning is based on derivations of a “limiting similarity”—i.e., a limit to how similar competitors can be to one another consistent with coexistence. This theory presumes there is a mechanism, perhaps invasion and extinction, which causes competitors to attain the limiting similarity. The view taken in this paper is that partitioning is an evolutionary compromise between pressures for character displacement and disadvantages inherent in the shift to different resource types.A set of principles is offered for the evolution of the parameters in ecological models. (1) For single population models natural selection causes the parameters ultimately to assume those values which produce the highest equilibrium population size. (2) For models of interacting populations, but without interspecific frequency-dependence, natural selection causes the parameters to assume values which produce either the highest or lowest equilibrium population size for any species depending on the sign of the “feedback” in the community obtained by deleting that species. (3) For models of interacting populations with interspecific frequency dependence natural selection leads to parameter values which produce intermediate equilibrium population sizes. A function called the conditional equilibrium population size is introduced. Provided (a) the mean fitness is a maximum in each species at a stable coevolutionary equilibrium and (b) there is negative density-dependence in each species then natural selection causes the parameters to assume values which produce the highest conditional equilibrium population size for each species.These coevolutionary principles, applied to a model for resource partitioning, entail that the niche separation between species relative to given niche widths, increases with the variety of available resources and decreases with the number of competing populations. Also, the evolution of character displacement between two species does not proceed far enough to maximize the equilibrium population sizes of the species involved. These results imply that the relationship between the niche overlap (of nearest neighbors) and species diversity is qualitatively different depending on whether the variety of resources at any place covaries with the species diversity there. Without covariation niche overlap increases with species diversity; with covariation overlap may decrease with species diversity. This study provides the beginning of a theory for the convergent evolution of community structure.     

Population and prehistory III: Food-dependent demography in variable environments

The population dynamics of preindustrial societies depend intimately on their surroundings, and food is a primary means through which environment influences population size and individual well-being. Food production requires labor; thus, dependence of survival and fertility on food involves dependence of a population’s future on its current state. We use a perturbation approach to analyze the effects of random environmental variation on this nonlinear, age-structured system. We show that in expanding populations, direct environmental effects dominate induced population fluctuations, so environmental variability has little effect on mean hunger levels, although it does decrease population growth. The growth rate determines the time until population is limited by space. This limitation introduces a tradeoff between population density and well-being, so population effects become more important than the direct effects of the environment: environmental fluctuation increases mortality, releasing density dependence and raising average well-being for survivors. We discuss the social implications of these findings for the long-term fate of populations as they transition from expansion into limitation, given that conditions leading to high well-being during growth depress well-being during limitation.     

Native intra- and inter-specific reactions may cause the paradox of pest control with harvesting

We analyse a general time-discrete mathematical model of host-parasite population dynamics with harvesting, in which the host can be regarded as a pest. We harvest a portion of the host population at a moment in each year. Our model involves the density effect on the host population. We investigate the condition in which the harvesting of the host results in a paradoxical increase of its equilibrium population size. Our results imply that for a family of pest-enemy systems, the paradox of pest control could be caused essentially by the interspecific relationship and the intraspecific density effect.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

Likelihoods and simulation methods for a class of nonneutral population genetics models

Methods for simulating samples and sample statistics, under mutation-selection-drift equilibrium for a class of nonneutral population genetics models, and for evaluating the likelihood surface, in selection and mutation parameters, are developed and applied for observed data. The methods apply to large populations in settings in which selection is weak, in the sense that selection intensities, like mutation rates, are of the order of the inverse of the population size. General diploid selection is allowed, but the approach is currently restricted to models, such as the infinite alleles model and certain K-models, in which the type of a mutant allele does not depend on the type of its progenitor allele. The simulation methods have considerable advantages over available alternatives. No other methods currently seem practicable for approximating likelihood surfaces.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Simplifying a physiologically structured population model to a stage-structured biomass model

We formulate and analyze an archetypal consumer-resource model in terms of ordinary differential equations that consistently translates individual life history processes, in particular food-dependent growth in body size and stage-specific differences between juveniles and adults in resource use and mortality, to the population level. This stage-structured model is derived as an approximation to a physiologically structured population model, which accounts for a complete size-distribution of the consumer population and which is based on assumptions about the energy budget and size-dependent life history of individual consumers. The approximation ensures that under equilibrium conditions predictions of both models are completely identical. In addition we find that under non-equilibrium conditions the stage-structured model gives rise to dynamics that closely approximate the dynamics exhibited by the size-structured model, as long as adult consumers are superior foragers than juveniles with a higher mass-specific ingestion rate. When the mass-specific intake rate of juvenile consumers is higher, the size-structured model exhibits single-generation cycles, in which a single cohort of consumers dominates population dynamics throughout its life time and the population composition varies over time between a dominance by juveniles and adults, respectively. The stage-structured model does not capture these dynamics because it incorporates a distributed time delay between the birth and maturation of an individual organism in contrast to the size-structured model, in which maturation is a discrete event in individual life history. We investigate model dynamics with both semi-chemostat and logistic resource growth.     

A population biological approach to the collective dynamics of countries undergoing demographic transition

Birth rates have been declining in higher-income countries since the middle of the 19th century. A growing number of other countries have entered this demographic transition to lower fertility, as socioeconomic development continues. Analyses of this demographic transition vary widely, but most analyze individual populations in isolation from others, and most come from fields outside the biological sciences. Here, we develop a population biological model of population dynamics in higher-income countries. Individual countries evolve through density-regulated growth, where gradual evolution toward higher population densities boosts productivity (and hence socioeconomic growth) through economics of agglomeration and scale, in turn reducing birth rates. The exchange of technology and capital between countries can further boost productivity gains in any given country, thus contributing to its demographic transition. As a result, countries can down-regulate one another's population growth through mutual improvements in productivity. The model is fitted to time series data on population size, GDP per capita, and birth rates for the United States, the United Kingdom, and France. The metapopulation dynamics are also characterized across a range of parameter values close to the fitted values. This work may help advance population biological approaches to understanding the implications of the fertility demographic transition for modern human populations. This is relevant to developing long-term predictions of the earth's total population size, which must be based upon a model that incorporates underlying mechanisms.     

Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution

The effect of dispersal on population size and stability is explored for a population that disperses passively between two discrete habitat patches. Two basic models are considered. In the first model, a single population experiences density-dependent growth in both patches. A graphical construction is presented which allows one to determine the spatial pattern of abundance at equilibrium for most reasonable growth models and rates of dispersal. It is shown under rather general conditions that this equilibrium is unique and globally stable. In the second model, the dispersing population is a food-limited predator that occurs in both a source habitat (which contains a prey population) and a sink habitat (which does not). Passive dispersal between source and sink habitats can stabilize an otherwise unstable predator-prey interaction. The conditions allowing this are explored in some detail. The theory of optimal habitat selection predicts the evolutionarily stable distribution of a population, given that individuals can freely move among habitats so as to maximize individual fitness. This theory is used to develop a heuristic argument for why passive dispersal should always be selectively disadvantageous (ignoring kin effects) in a spatially heterogeneous but temporally constant environment. For both the models considered here, passive dispersal may lead to a greater number of individuals in both habitats combined than if there were no dispersal. This implies that the evolution of an optimal habitat distribution may lead to a reduction in population size; in the case of the predator-prey model, it may have the additional effect of destabilizing the interaction. The paper concludes with a discussion of the disparate effects habitat selection might have on the geographical range occupied by a species.     

Natural selection and density-dependent population growth

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.