Type II functional response for continuous, physiologically structured models

The goal of this work is to formulate a general Holling-type functional, or behavioral, response for continuous physiologically structured populations, where both the predator and the prey have physiological densities and certain rules apply to their interactions. The physiological variable can be, for example, a development stage, weight, age, or a characteristic length. The model leads to a Fredholm integral equation for the functional response, and, when inserted into population balance laws, it produces a coupled system of partial differential-integral equations for the two species, with a nonlocal integral term that arises from rules of interaction in the functional response. The general model is, typically, analytically intractable, but specialization to a structured prey-unstructured predator model leads to some analytic results that reveal interesting and unexpected dynamics caused by the presence of size-dependent handling times in the functional response. In this case, steady-states are shown to exist over long times, similar to the stable age-structure solutions for the McKendick-von Foerster model with exponential growth rates determined by the Euler-Lotka equation. But, for type II responses, there are early transient oscillations in the number of births that bifurcate in a few generations into either the decaying or growing steady-state. The bifurcation parameter is the initial level of prey. This special case is applied to a problem of the biological control of a structured pest population (e.g., aphids) by a predator (e.g., lady beetles).     

Strain replacement in an epidemic model with super-infection and perfect vaccination

Several articles in the recent literature discuss the complexities of the impact of vaccination on competing subtypes of one micro-organism. Both with competing virus strains and competing serotypes of bacteria, it has been established that vaccination has the potential to switch the competitive advantage from one of the pathogen subtypes to the other resulting in pathogen replacement. The main mechanism behind this process of substitution is thought to be the differential effectiveness of the vaccine with respect to the two competing micro-organisms. In this article, we show that, if the disease dynamics is regulated by super-infection, strain substitution may indeed occur even with perfect vaccination. In fact we discuss a two-strain epidemic model in which the first strain can infect individuals already infected by the second and, as far as vaccination is concerned, we consider a best-case scenario in which the vaccine provides perfect protection against both strains. We find out that if the reproduction number of the first strain is smaller than the reproduction number of the second strain and the first strain dominates in the absence of vaccination then increasing vaccination levels promotes coexistence which allows the first strain to persist in the population even if its vaccine-dependent reproduction number is below one. Further increase of vaccination levels induces the domination of the second strain in the population. Thus the second strain replaces the first strain. Large enough vaccination levels lead to the eradication of the disease.     

A theoretical study on mathematical modelling of an infectious disease with application of optimal control

In this paper, we propose and analyze an epidemic problem which can be controlled by vaccination as well as treatment. In the first part of our analysis we study the dynamical behavior of the system with fixed control for both vaccination and treatment. Basic reproduction number is obtained in all possible cases and it is observed that the simultaneous use of vaccination and treatment control is the most favorable case to prevent the disease from being epidemic. In the second part, we take the controls as time dependent and obtain the optimal control strategy to minimize both the infected populations and the associated costs. All the analytical results are verified by simulation works. Some important conclusions are given at the end of the paper.     

Oscillations in a size-structured prey-predator model

This article introduces a predator–prey model with the prey structured by body size, based on reports in the literature that predation rates are prey-size specific. The model is built on the foundation of the one-species physiologically structured models studied earlier. Three types of equilibria are found: extinction, multiple prey-only equilibria and possibly multiple predator–prey coexistence equilibria. The stabilities of the equilibria are investigated. Comparison is made with the underlying ODE Lotka–Volterra model. It turns out that the ODE model can exhibit sustain oscillations if there is an Allee effect in the net reproduction rate, that is the net reproduction rate grows for some range of the prey’s population size. In contrast, it is shown that the structured PDE model can exhibit sustain oscillations even if the net reproductive rate is strictly declining with prey population size. We find that predation, even size-non-specific linear predation can destabilize a stable prey-only equilibrium, if reproduction is size specific and limited to individuals of large enough size. Furthermore, we show that size-specific predation can also destabilize the predator–prey equilibrium in the PDE model. We surmise that size-specific predation allows for temporary prey escape which is responsible for destabilization in the predator–prey dynamics.     

Mathematical analysis of a cholera model with public health interventions

Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. We formulate a mathematical model that captures some essential dynamics of cholera transmission to study the impact of public health educational campaigns, vaccination and treatment as control strategies in curtailing the disease. The education-induced, vaccination-induced and treatment-induced reproductive numbers R(E), R(V), R(T) respectively and the combined reproductive number R(C) are compared with the basic reproduction number R(0) to assess the possible community benefits of these control measures. A Lyapunov functional approach is also used to analyse the stability of the equilibrium points. We perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Graphical representations are provided to qualitatively support the analytical results.     

Harvest-induced maturation evolution under different life-history trade-offs and harvesting regimes

The potential of harvesting to induce adaptive changes in exploited populations is now increasingly recognized. While early studies predicted that elevated mortalities among larger individuals select for reduced maturation size, recent theoretical studies have shown conditions under which other, more complex evolutionary responses to size-selective mortality are expected. These new predictions are based on the assumption that, owing to the trade-off between growth and reproduction, early maturation implies reduced growth. Here we extend these findings by analyzing a model of a harvested size-structured population in continuous time, and by systematically exploring maturation evolution under all three traditionally acknowledged costs of early maturation: reduced fecundity, reduced growth, and/or increased natural mortality. We further extend this analysis to the two main types of harvest selectivity, with an individual's chance of getting harvested depending on its size and/or maturity stage. Surprisingly, we find that harvesting mature individuals not only favors late maturation when the costs of early maturation are low, but promotes early maturation when the costs of early maturation are high. To our knowledge, this study therefore is the first to show that harvesting mature individuals can induce early maturation.     

Reproduction numbers for epidemics on networks using pair approximation

One way to describe the spread of an infection on a network is by using the method of pair approximation. This method is a deterministic pair-based variant of the usual methods used to describe the progress of an epidemic in randomly mixing populations. However, although the ideas of pair approximation are intuitively clear, it is not straightforward to make all assumptions used explicit. Furthermore, in literature problems arise in defining basic quantities like the basic reproduction number R(0) and the real-time epidemic growth rate parameter r. We formulate the pair approximations and the needed assumptions explicitly. We discuss problems inherent to this method. Furthermore, we define a new reproduction number, similar to R(0) and a new real-time growth rate parameter similar to r. We illustrate the methods of the paper by an example for which we can compare the approximation of the reproduction number with exact results.     

Analysis of SIR epidemic models with nonlinear incidence rate and treatment

This paper deals with the nonlinear dynamics of a susceptible-infectious-recovered (SIR) epidemic model with nonlinear incidence rate, vertical transmission, vaccination for the newborns of susceptible and recovered individuals, and the capacity of treatment. It is assumed that the treatment rate is proportional to the number of infectives when it is below the capacity and constant when the number of infectives reaches the capacity. Under some conditions, it is shown that there exists a backward bifurcation from an endemic equilibrium, which implies that the disease-free equilibrium coexists with an endemic equilibrium. In such a case, reducing the basic reproduction number less than unity is not enough to control and eradicate the disease, extra measures are needed to ensure that the solutions approach the disease-free equilibrium. When the basic reproduction number is greater than unity, the model can have multiple endemic equilibria due to the effect of treatment, vaccination and other parameters. The existence and stability of the endemic equilibria of the model are analyzed and sufficient conditions on the existence and stability of a limit cycle are obtained. Numerical simulations are presented to illustrate the analytical results.     

Linking immunological and epidemiological dynamics of HIV: the case of super-infection

In this paper, a two-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the two strains eliminate each other with the strain with the larger immunological reproduction persisting. However, on the population scale superinfection is possible, with the strain with larger immunological reproduction number super-infecting the strain with the smaller immunological reproduction number. The two models are linked through the age-since-infection structure of the epidemiological variables. In addition, the between-host transmission and the disease-induced death rate depend on the within-host viral load. The immunological reproduction numbers, the epidemiological reproduction numbers and invasion reproduction numbers are computed. Besides the disease-free equilibrium, there are two population-level strain one and strain two isolated equilibria, as well as a population-level coexistence equilibrium when both invasion reproduction numbers are greater than one. The single-strain population-level equilibria are locally asymptotically stable suggesting that in the absence of superinfection oscillations do not occur, a result contrasting previous studies of HIV age-since-infection structured models. Simulations suggest that the epidemiological reproduction number and HIV population prevalence are monotone functions of the within-host parameters with reciprocal trends. In particular, HIV medications that decrease within-host viral load also increase overall population prevalence. The effect of the immunological parameters on the population reproduction number and prevalence is more pronounced when the initial viral load is lower.     

The structure of optimal time- and age-dependent harvesting in the Lotka-McKendrik population model

The paper analyzes optimal harvesting of age-structured populations described by the Lotka-McKendrik model. It is shown that the optimal time- and age-dependent harvesting control involves only one age at natural conditions. This result leads to a new optimization problem with the time-dependent harvesting age as an unknown control. The integral Lotka model is employed to explicitly describe the time-varying age of harvesting. It is proven that in the case of the exponential discounting and infinite horizon the optimal strategy is a stationary solution with a constant harvesting age. A numeric example on optimal forest management illustrates the theoretical findings. Discussion and interpretation of the results are provided.     

Analysis of a competitive prey–predator system with a prey refuge

Gauss's competitive exclusive principle states that two competing species having analogous environment cannot usually occupy the same space at a time but in order to exploit their common environment in a different manner, they can co-exist only when they are active in different times. On the other hand, several studies on predators in various natural and laboratory situations have shown that competitive coexistence can result from predation in a way by resisting any one prey species from becoming sufficiently abundant to outcompete other species such that the predator makes the coexistence possible. It has also been shown that the use of refuges by a fraction of the prey population exerts a stabilizing effect in the interacting population dynamics. Further, the field surveys in the Sundarban mangrove ecosystem reveal that two detritivorous fishes, viz. Liza parsia and Liza tade (prey population) coexist in nature with the presence of the predator fish population, viz. Lates calcarifer by using refuges.     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

The role of sexually abstained groups in two-sex demographic and epidemic logistic models with non-linear mortality

We describe several gender structured population models governed by logistic growth with non-linear death rate. We extend these models to include groups of people isolated from sexual activity and individuals exposed to a mild and long-lasting sexually transmitted disease, i.e. without disease-induced mortality and recovery. The transmission of the disease is modeled through formation/separation of heterosexual couples assuming that one infected individual automatically infects his/her partner. We are interested in how the non-reproductive class may change the demographic tendencies in the general population and whether they can curb the growth of the infected group while keeping the healthy one at acceptable levels. A comparison of the equilibrium total population size in the presence and the absence of the isolated class is also provided.     

Identifying the Relative Priorities of Subpopulations for Containing Infectious Disease Spread

In response to the outbreak of an emerging infectious disease, e.g., H1N1 influenza, public health authorities will take timely and effective intervention measures to contain disease spread. However, due to the scarcity of required resources and the consequent social-economic impacts, interventions may be suggested to cover only certain subpopulations, e.g., immunizing vulnerable children and the elderly as well as closing schools or workplaces for social distancing. Here we are interested in addressing the question of how to identify the relative priorities of subpopulations for two measures of disease intervention, namely vaccination and contact reduction, especially when these measures are implemented together at the same time. We consider the measure of vaccination that immunizes susceptible individuals in different age subpopulations and the measure of contact reduction that cuts down individuals’ effective contacts in different social settings, e.g., schools, households, workplaces, and general communities. In addition, we construct individuals’ cross-age contact frequency matrix by inferring basic contact patterns respectively for different social settings from the socio-demographical census data. By doing so, we present a prioritization approach to identifying the target subpopulations that will lead to the greatest reduction in the number of disease transmissions. We calculate the relative priorities of subpopulations by considering the marginal effects of reducing the reproduction number for the cases of vaccine allocation by age and contact reduction by social setting. We examine the proposed approach by revisiting the real-world scenario of the 2009 Hong Kong H1N1 influenza epidemic and determine the relative priorities of subpopulations for age-specific vaccination and setting-specific contact reduction. We simulate the influenza-like disease spread under different settings of intervention. The results have shown that the proposed approach can improve the effectiveness of disease control by containing disease transmissions in a host population.     

Determinants of age at first reproduction and lifetime breeding success revealed by full paternity assignment in a male ungulate

Age at first reproduction is an important determinant of individual variation in reproductive success in ungulates, but few studies have examined its relationship with later fitness‐related traits in males. We used a long‐term individual based study of a harvested moose population to quantify the individual reproductive performance and survival of males, as well as to examine the determinants of age at first reproduction and consequences of age at first reproduction on lifetime breeding success. The probability that a male successfully reproduced at the age of two was negatively related to the mean age of adult males in the population, but the relationship weakened with increasing population size. Large antlers and large body mass relative to other males in the population increased the number of calves sired at their first successful mating season. In addition, those that successfully reproduced as two year‐olds were more likely to sire calves the next year, making them more productive at a given age compared to those that first reproduced at the age of three or older. We emphasize the importance for males to start reproducing as soon as possible in a harvested population to gain lifetime fitness benefits, as surviving the hunt is a major determinant of reproductive success in this population. We found no costs of early reproduction in males, hence leading to high individual heterogeneity in male reproductive performance. The apparent lack of reproductive costs could partly be explained by the age distribution in the population, individual variation in early‐life body mass and antler size, and differences in probabilities of being hunted of successful and unsuccessful males.     

What shall I do now? State-dependent variations of life-history traits with aging in Wandering Albatrosses

Allocation decisions depend on an organism's condition which can change with age. Two opposite changes in life‐history traits are predicted in the presence of senescence: either an increase in breeding performance in late age associated with terminal investment or a decrease due to either life‐history trade‐offs between current breeding and future survival or decreased efficiency at old age. Age variation in several life‐history traits has been detected in a number of species, and demographic performances of individuals in a given year are influenced by their reproductive state the previous year. Few studies have, however, examined state‐dependent variation in life‐history traits with aging, and they focused mainly on a dichotomy of successful versus failed breeding and non‐breeding birds. Using a 50‐year dataset on the long‐lived quasi‐biennial breeding wandering albatross, we investigated variations in life‐history traits with aging according to a gradient of states corresponding to potential costs of reproduction the previous year (in ascending order): non‐breeding birds staying at sea or present at breeding grounds, breeding birds that failed early, late or were successful. We used multistate models to study survival and decompose reproduction into four components (probabilities of return, breeding, hatching, and fledging), while accounting for imperfect detection. Our results suggest the possible existence of two strategies in the population: strict biennial breeders that exhibited almost no reproductive senescence and quasi‐biennial breeders that showed an increased breeding frequency with a strong and moderate senescence on hatching and fledging probabilities, respectively. The patterns observed on survival were contrary to our predictions, suggesting an influence of individual quality rather than trade‐offs between reproduction and survival at late ages. This work represents a step further into understanding the evolutionary ecology of senescence and its relationship with costs of reproduction at the population level. It paves the way for individual‐based studies that could show the importance of intra‐population heterogeneity in those processes.     

Harvesting in a resource dependent age structured Leslie type population model

We analyse the effect of harvesting in a resource dependent age structured population model, deriving the conditions for the existence of a stable steady state as a function of fertility coefficients, harvesting mortality and carrying capacity of the resources. Under the effect of proportional harvest, we give a sufficient condition for a population to extinguish, and we show that the magnitude of proportional harvest depends on the resources available to the population. We show that the harvesting yield can be periodic, quasi-periodic or chaotic, depending on the dynamics of the harvested population. For populations with large fertility numbers, small harvesting mortality leads to abrupt extinction, but larger harvesting mortality leads to controlled population numbers by avoiding over consumption of resources. Harvesting can be a strategy in order to stabilise periodic or quasi-periodic oscillations in the number of individuals of a population.     

Demographic stochasticity and temporal autocorrelation in the dynamics of structured populations

Populations can show temporal autocorrelation in the dynamics arising from different mechanisms, including fluctuations in the demographic structure. This autocorrelation is often treated as a complicating factor in the analyses of stochastic population growth and extinction risk. However, it also reflects important information about the demographic structure. Here, we consider how temporal autocorrelation is related to demographic stochasticity in structured populations. Demographic stochasticity arises from inherent randomness in the demographic processes of individuals, like survival and reproduction, and the resulting impact on population growth is measured by the demographic variance. Earlier studies have shown that population structure have positive or negative effects on the demographic variance compared to a model where the structure is ignored. Here, we derive a new expression for the demographic variance of a structured population, using the temporal autocorrelation function of the population growth rate. We show that the relative difference in demographic variance when the structure is included or ignored (the effect of structure on demographic variance) is approximately twice the sum of the autocorrelations. We demonstrate the result for a simple hypothetical example, as well as a set of empirical examples using age‐structured models of 24 mammals from the demographic database COMADRE. In the empirical examples, the sum of the autocorrelation function was negative in all cases, indicating that age structure generally has a negative effect on the demographic variance (i.e. the demographic variance is lower compared to that of a model where the structure is ignored). Other kinds of structure, such as spatial heterogeneity affecting fecundity, can have positive effects on the demographic variance, and the sum of the autocorrelations will then be positive. These results yield new insights into the complex interplay between population structure, demographic variance, and temporal autocorrelation, that shapes the population dynamics and extinction risk of populations.     

Linked selected and neutral loci in heterogeneous environments

We analyze a system of ordinary differential equations modeling haplotype frequencies at a physically linked pair of loci, one selected and one neutral, in a population consisting of two demes with divergent selection regimes. The system is singularly perturbed, with the migration rate m between the demes serving as a small parameter. We use geometric singular perturbation theory to show that when m is sufficiently small, each solution not initially fixed for the same selected allele in both demes approaches one of a 1-dimensional continuum of equilibria. We then obtain asymptotic expansions of the solutions and show their validity on arbitrarily long finite time intervals. From these expansions we obtain formulas for the transient dynamics of F _ST (a measure of population structure) at both loci, as well as for the rate of genotyping error if the allelic state at the selected locus is inferred from that at the neutral (marker) locus. We examine two cases in detail, one modeling two populations in secondary contact after a period of evolution in allopatry, and the other modeling the origination and spread of a resistance allele.Electronic supplementary material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00285-006-0038-6 and is accessible for authorized users.     

Evolutionary responses to harvesting in ungulates

1. We investigate the evolutionary responses to harvesting in ungulates using a state-dependent, stochastic, density-dependent individual-based model of red deer Cervus elaphus (L.) females subject to different harvesting regimes. 2. The population's mean weight at first reproduction shifts towards light weights as harvesting increases, and its distribution changes from a single peak distribution under very low or high harvest rates, to a bimodal distribution under intermediate harvest rates. 3. These results suggest that, consistent with previous studies on aquatic species, harvesting-induced mortality may drive adaptive responses in ungulates by reducing the fitness benefits from adult survival and growth in favour of early and lightweight reproduction. 4. Selective harvesting for heavy animals has no additional effect on the evolutionarily stable strategy, suggesting that harvest rate is more important than the degree of selectivity in driving adaptive responses. However, selective harvesting of light females is positively associated with maturation weights even higher than those of a nonharvested population, probably due to the reduction in the fitness value of the offspring. 5. The average number of weight at maturation strategies in the population declines but the total number of strategies across all simulations increases with harvest rate, suggesting that harvesting-induced selection on weight at maturity overcomes the increase in strategy diversity expected from density-dependent release. 6. Yield initially increases with harvesting due to enhanced productivity of light females experiencing density-dependent release. However, it crashes under intense harvesting resulting in a population skewed to light, young and, therefore, less reproductive animals.