The effects of a single stage-structured population model with impulsive toxin input and time delays in a polluted environment

Uncontrolled contribution of pollutant to the environment has led many species to extinction and several others are at the verge of extinction. This article deals with the dynamics of a single stage-structured population model with impulsive toxin input and time delays (including constant individual maturation time delay and pollution time delay) in a polluted environment, in which we assume that only the mature individuals are affected by pollutants. We obtain conditions for the global attractivity of the population-extinction periodic solution and the permanence of the population. We show that maturation time delay and impulsive toxin input can bring great effects on the dynamics of the system, and pollution time delay is harmless. Numerical simulations confirm our theoretical results.     

Stability results for a size-structured population model with resources-dependence and inflow

It is undoubted that the survival of individuals of populations is dependent on resources (e.g., foods). We formulate a system of integro-differential equations to model the dynamics of a size-structured and resources-dependent population, a kind of inflow of newborn individuals from external environment is considered. The resource-dependence is incorporated through the size growth, mortality, fertility and feeding rates of the target population. The existence of the stationary size distributions are discussed, and the linear stability is investigated by means of the semigroup theory and the characteristic equation technique, some sufficient conditions for stability/instability of stationary states are obtained, and two examples and the corresponding simulations are presented.     

An Age-Structured Population Dynamics Model with Females' Pregnancy and Child Care

We present a deterministic model for an age-structured population dynamics taking into account females' pregnancy, maternal care of offsprings, and environmental pressure with or without spatial migration. The model is based on the age-density notion for a group formed by a mother and her offsprings under maternal care. A harmonic-mean-type mating function of sexes without formation of permanent pairs is used. It is assumed that each sex has the pre-reproductive, reproductive, and post-reproductive age intervals. All adult individuals are divided into males, single females, fertilized females, and females taking child care. Individuals of post-reproductive age belong to the group of singles. All individuals of pre-reproductive age are divided into the young and juvenile groups. Only young offsprings are assumed to be under maternal care. Juvenile individuals can live without maternal care. The model consists of integro-PDEs subject to the conditions of integral type. The existence and uniqueness theorem is proved in the case of unlimited population. Separable solutions and their long-time behavior are studied for the limited nondispersing population. In the case of random migration two types of separable solutions and their long-time behavior for the homogeneous Dirichlet and Neumann boundary conditions are studied. In the case of directed migration in one-dimensional domain with special initial and Dirichlet boundary conditions, the unlimited invasive population dynamics is studied. In particular, an explicit formula for the migration front is given.     

The dynamics of an age-structured TB transmission model with relapse

This paper deals with the global dynamics for a tuberculosis transmission model with age-structure and relapse. The time delay in the progression from the latent individuals to becoming the infectious individuals is also considered in our model. We perform some rigorous analyses for the model, including presenting an explicit formula for the basic reproduction number of the model, addressing the persistence of the solution semiflow and the existence of a global attractor. Based on these analyses, we establish some results about stability and instability of the solutions for our model. At end, the model is applied to describe tuberculosis transmission in China. The number of the total population and the number of the annual newly reported TB cases both match the statistical data well. The number of the total population, the latent individuals, the infectious individuals, the Purified Protein Derivative (PPD) positive rate, and the prevalence rate from 2020 to 2035 all are presented.     

Mathematical models of bipolar disorder

We use limit cycle oscillators to model bipolar II disorder, which is characterized by alternating hypomanic and depressive episodes and afflicts about 1% of the United States adult population. We consider two non-linear oscillator models of a single bipolar patient. In both frameworks, we begin with an untreated individual and examine the mathematical effects and resulting biological consequences of treatment. We also briefly consider the dynamics of interacting bipolar II individuals using weakly-coupled, weakly-damped harmonic oscillators. We discuss how the proposed models can be used as a framework for refined models that incorporate additional biological data. We conclude with a discussion of possible generalizations of our work, as there are several biologically-motivated extensions that can be readily incorporated into the series of models presented here.     

A HABITAT BASED MODEL FOR THE DISTRIBUTION OF FOREST INTERIOR NESTING BIRDS IN A FRAGMENTED LANDSCAPE

ABSTRACT. Increased awareness of the plight of many forest dwelling species has made necessary the development of methods for projecting the spatial distribution of these populations. This is particularly important for populations that currently occupy forest fragments and that are likely to be exposed to further disruption of their natural habitat. In this paper we develop a model for predicting the distribution of a bird population that evolved as forest interior dwellers. This model uses as its basis knowledge of the relationship between demographic characteristics of the population and the qualities of the habitat where individuals reside. We make the assumption that individuals will be naturally drawn to areas where they might expect greater reproductive success and repelled from areas where there is a high degree of intraspecific competition (high density). We apply the model to the ovenbird population in a large region of the Midwest. We use the model to examine the relative extent to which the surplus production from two major source areas supports extensive sink populations. The basic diffusion model parameterized by county forest cover data projects a population distribution which compares favorably with the results from the breeding bird count.     

Modeling the depletion of dissolved oxygen in a water body located near a city

Water bodies located nearby cities are much prone to pollution, especially in the developing countries, where effluents treatment facilities are generally lacking. The main reason for this phenomenon is the increasing population in the cities, and the large number of industries located near them. This leads to generation of huge amounts of domestic and industrial sewage that is discharged into the water bodies, increasing their organic pollutant load and resulting in the depletion of dissolved oxygen. In this paper, we propose a mathematical model for this situation, focusing especially on the resulting quality of the water, determined by the level of dissolved oxygen. The model also accounts for resources needed for the population survival and for the industrial operations. In addition, we describe also the decomposition of organic pollutants by bacteria in the aquatic medium. Feasibility conditions and stability criteria of the system's equilibria are determined analytically. The results show that human population and industries are relevant influential factors responsible for the increase in organic pollutants and the decrease in dissolved oxygen in the water body, in the sense that they may exert a destabilizing effect on the system. The numerical simulations confirm the analytical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Steady states in a structured epidemic model with Wentzell boundary condition

We introduce a non-linear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass; hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for e.g., Wolbachia in a mosquito population. Therefore, the (infinite dimensional) non-linearity arises in the recruitment term. First, we establish global existence of solutions and the principle of linearised stability for our model. Then, in our main result, we formulate simple conditions which guarantee the existence of non-trivial steady states of the model. Our method utilises an operator theoretic framework combined with a fixed-point approach. Finally in the last section, we establish a sufficient condition for the local asymptotic stability of the positive steady state.     

Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences

Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible and illustrate that while epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of “epigenetic load” and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a “proof of concept” of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end.     

Mathematical modeling and analysis of the depletion of dissolved oxygen in water bodies

In this paper, a nonlinear mathematical model is proposed to study the depletion of dissolved oxygen in a water body caused by industrial and household discharges of organic matters (pollutants). The problem is formulated as a food chain model by considering various interaction processes (biodegradation and biochemical) involving organic pollutants, bacteria, protozoa, an aquatic population and dissolved oxygen. Using stability theory, it is shown that as the rate of introduction of organic pollutants in a water body increases, the concentration of dissolved oxygen decreases due to various interaction processes. It is found that if the organic pollutants are continuously discharged into water body, the concentration of dissolved oxygen may become negligibly small, thus threatening the survival of aquatic populations. However, by using some effort to control the cumulative discharge of these pollutants into the water body, the concentration of dissolved oxygen can be maintained at a desired level.     

Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure

A periodic ratio-dependent predator-prey model with time delays and stage structure for both prey and predator is investigated. It is assumed that immature individuals and mature individuals of each species are divided by a fixed age, and that immature predators do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solutions of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

Fast game dynamics coupled to slow population dynamics: A single population with hawk-dove strategies

We study a model of a population subdivided into two subpopulations corresponding to hawk and dove tactics. It is assumed that the hawk and dove individuals compete for a resource every Day, I.e., at a fast time scale. This fast part of the model is coupled to a slow part which describes the growth of the subpopulations and the long term effects of the encounters between the individuals which must fight to have an access to the resource. We aggregate the model into a single equation for the total population. It is shown that in the case of a constant game matrix, the total population grows according to a logistic curve whose τ and K parameters are related to the coefficients of the hawk-dove game matrix. Our result shows that high equilibrium density populations are mainly doves, whereas low equilibrium density populations are mainly hawks. We also study the case of a density dependent game matrix for which the gain is linearly decreasing with the total density.     

Ecoepidemic models with disease incubation and selective hunting

In this paper we consider ecoepidemic models in which the basic demographics is represented by predator-prey interactions, with the disease modeled by an SEI system. At first we consider a basic Lotka-Volterra type of interaction. Then we also introduce competition for resources among individuals of the prey population. Several variations of the model are presented, in which the prey intra-specific population pressure assumes different forms, depending on the virulence of the disease. Indeed, the latter may affect the exposed and infected individuals so much that they may not be able to compete with the sound ones for resources. A further distinguishing feature of this investigation lies in the way in which the predator actively selects the prey for hunting. For instance in some cases predators may discard the diseased ones, as less palatable, while in other situations they would instead search expressly for the infected, since these are weaker individuals and thus easier to hunt. The equilibria of the systems are analyzed, showing that in some cases bifurcations arise, contrary to what happens to similar classical Holling type I ecoepidemic models. These persistent oscillations seem to be triggered by the number of subpopulations present in the system, which is larger than those introduced in the former models, counting also the latent class. Furthermore, adding predation to an SEI epidemic model has profound effects on the stability of its equilibria. In particular, once the predators are introduced into an SEI epidemic at a stable endemic equilibrium, their presence destabilizes this equilibrium making the previous stable conditions unrecoverable.     

On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains

We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.     

Optimal harvesting for periodic age-dependent population dynamics with logistic term

We prove an asymptotic behavior result for an age-dependent population dynamics with logistic term and periodic vital rates. We investigate next an optimal harvesting problem related to a periodic age-structured model with logistic term. Existence of an optimal control and necessary optimality conditions are established. A conceptual algorithm to approximate the optimal pair is derived and some numerical experiments are presented.     

Evolution of dispersal in a spatially periodic integrodifference model

An integrodifference model describing the reproduction and dispersal of a population is introduced to investigate the evolution of dispersal in a spatially periodic habitat. The dispersal is determined by a kernel function, and the dispersal strategy is defined as the probability of population individuals’ moving to a different habitat. Both conditional and unconditional dispersal strategies are investigated, the distinction being whether dispersal depends on local environmental conditions. For competing unconditional dispersers, we prove that the population with the smaller dispersal probability always prevails. Alternatively, for conditional dispersers, it is shown that the strategy known as ideal free dispersal is both sufficient and necessary for evolutionary stability. These results extend those in the literature for discrete diffusion models in finite patchy landscapes and from reaction–diffusion models.     

Dynamic equilibria of group vaccination strategies in a heterogeneous population

In this paper we present an evolutionary variational inequality model of vaccination strategies games in a population with a known vaccine coverage profile over a certain time interval. The population is considered to be heterogeneous, namely its individuals are divided into a finite number of distinct population groups, where each group has different perceptions of vaccine and disease risks. Previous game theoretical analyses of vaccinating behaviour have studied the strategic interaction between individuals attempting to maximize their health states, in situations where an individual’s health state depends upon the vaccination decisions of others due to the presence of herd immunity. Here we extend such analyses by applying the theory of evolutionary variational inequalities (EVI) to a (one parameter) family of generalized vaccination games. An EVI is used to provide conditions for existence of solutions (generalized Nash equilibria) for the family of vaccination games, while a projected dynamical system is used to compute approximate solutions of the EVI problem. In particular we study a population model with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). The smaller group is considered much less vaccination inclined than the larger group. Under these hypotheses, considering that the vaccine coverage of the entire population is measured during a vaccine scare period, we find that our model reproduces a feature of real populations: the vaccine averse minority will react immediately to a vaccine scare by dropping their strategy to a nonvaccinator one; the vaccine inclined majority does not follow a nonvaccinator strategy during the scare, although vaccination in this group decreases as well. Moreover we find that there is a delay in the majority’s reaction to the scare. This is the first time EVI problems are used in the context of mathematical epidemiology. The results presented emphasize the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by EVI in this area of research.      

Application of Tikhonov’s regularization method to solve inverse problems for two population models

Tikhonov’s regularization method is applied to numerical solution of inverse problems for two population models. For the first model we solve the inverse problem that involves simultaneous determination of the mortality rate and the initial distribution of individuals given supplementary information on population density. For the second model we determine the growth rate of the individuals given additional information about their density. Examples of numerical solution are presented for both inverse problems. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 5–14, 2006.     

PERSISTENCE IN AN AGE-STRUCTURED POPULATION FOR A PATCH-TYPE ENVIRONMENT

A discrete-time model for an age-structured population in a patch-type environment is presented and analyzed. Comparison techniques for difference equations are used to find sufficient conditions for population persistence or extinction. The persistence and extinction theorem is used to define the critical patch number, the threshold for population persistence. Several examples are presented which illustrate the results of the theorems. The model is applied to a watersnake population.