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.     

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.     

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.     

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.