Mathematical Model Of Population Dynamics Research Articles

Change of dynamic regimes in the population of species with short life cycles: Results of an analytical and numerical study

There is a phenomenon of multiregimism found in the elementary mathematical model of population dynamics, meaning the possibility for different dynamic regimes to exist under the same conditions, with transition to these regimes dependent on the initial numerical values. The effect in question comes into existence in the model which has several different limiting regimes (attractors): equilibrium, regular fluctuations, and chaotic attractor. The revealed phenomenon of multiregimism lets us explain the initiation of fluctuations as well as disappearance of fluctuations. Adequacy of the model's dynamic regimes is depicted by their correlation with the actual dynamics of population size of bank vole (Myodes glareolus). It is shown that the impact of climatic factors on a reproductive process of a population noticeably extends the range of possible dynamic regimes and, in fact, leads to random migration over attraction basins of these regimes.

Mathematical modeling of population dynamics with Allee effect

Allee effect that refers to a positive relationship between individual fitness and population density provides an important conceptual framework in conservation biology. While declining Allee effect causes reduction in extinction risk in low-density population, it provides a benefit in limiting establishment success or spread of invading species. Population models that incorporated Allee effect confer the fundamental role which plays for shaping the population dynamics. In particular, non-spatial predator–prey and invasion models have shown the influence of Allee effects on population dynamics, and spatial models have illustrated its critical roles for pattern formation and the manifestation of traveling wave fronts. We highlight all such no-spatial and spatial population models and their contributions in deeper understanding of population dynamics. In addition, we briefly outline the trends for future research on Allee effect which we think are interesting and widely open.

Mathematical model of predator-prey interaction with accounting the areal factors and resistant factor of habitats of populations

The article gives coverage to refinement upon the mathematical model of predator-prey interaction taking into account the areal factors and resistant factor of habitats of populations. It is grounded the physical interpretation of proposed mathematical model and its connection with diffusion process, which could be described with the Ehrenfests diffusion model that lead to formation of Markov chain.. For further use of proposed mathematical model of population dynamics, it is made its identification and the estimation of adequacy by comparing the relative error of simulation results

The Approximate Bayesian Computation approach to reconstructing population dynamics and size from settlement data: demography of the Mesolithic-Neolithic transition at Lepenski Vir

Demographic aspects of prehistoric populations have an important role in current archaeological theory and empirical research. In this study, we develop a method to estimate population dynamics and population size and apply it to data on house remains at one of key European Mesolithic-Neolithic transitional sites - Lepenski Vir (Serbia). Lepenski Vir is a site located in the Danube Gorges, well-known for its trapezoidal house floors and stone sculpture. It was most intensively occupied between ~6200 and ~6000 cal BC, the so called Transitional phase, which corresponds to the beginning of the Neolithic in Central Balkans. We combine archaeological evidence and ethnographic information with mathematical models of population dynamics and house accumulation within a Bayesian framework (Approximate Bayesian Computation) to derive posterior distributions of growth rate and population size estimates for the Lepenski Vir population in this period.

A model for population dynamics of the mimetic butterfly Papilio polytes in the Sakishima Islands, Japan

We present a mathematical model for population dynamics of the mimetic swallowtail butterfly Papilio polytes in the Sakishima Islands, Japan. The model includes four major variables, that is, population densities of three kinds of butterflies (two female forms f. cyrus, f. polytes and the unpalatable butterfly Pachliopta aristolochiae) and their predator. It is well-known that the non-mimic f. cyrus resembles and attracts the male most, and the mimic f. polytes mimics the model butterfly P. aristolochiae. Based on experimental evidence, we assume that two forms f. cyrus and f. polytes interact under intraspecific competition for resources including the male, and the growth rate of f. cyrus is higher than that of f. polytes. We further assume that both the benefit of mimicry for the mimic f. polytes and the cost for the model are dependent on their relative frequencies, i.e. the motality of the mimic by predation decreases with increase in frequency of the model, while the motality of the model increases as the frequency of the mimic increases. Taking the density-dependent effect through carrying capacity into account, we set up a model system consisting of three ordinary differential equations (ODEs), analyze it mathematically and provide computer simulations that confirm the analytical results. Our results reproduce field records on population dynamics of P. polytes in the Miyako-jima Island. They also explain the positive dependence of the relative abundance (RA) of the mimic on the advantage index (AI) of the mimicry in the Sakishima Islands defined in Section 2.

The Conceprts of Delay Differential Equations and IT’S Application

All processes take time to complete. While physical processes such as acceleration and deceleration take little time compared to the times need to travel most distances, the times involved in biological processes such as gestation and maturation can be substantial when compared to the data-collection times in most population studies. Therefore, it is often imperative to explicitly incorporate these process times in mathematical models of population dynamics. These process times are often called delay times, and the model that incorporate such delay times are referred as delay differential equation (DDE) models. The models will examine some theoretical concepts and their applications to real life situation. The application examines measles and the time it takes to manifest or to its removal or treatment from the system. The solutions of the models will be displayed in graphical forms using MATLAB method. The analysis of the models indicate the times delay and its characteristics.

Prediction of artiodactyles population dynamics in hunting sector of Ukraine

Проведено аналіз можливості прогнозування динаміки популяцій за допомогою універсальної моделі, отриманої теоретичним шляхом на основі положень системології. Запропоновано ряд методик визначення робочих параметрів моделей, а також проведено оцінювання якості прогнозування кількості парнокопитних з їх (моделей) застосуванням.

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.

Continuous dependence of the solutions of impulsive differential equations on the initial conditions and barrier curves

The basic objects of investigation in this article are nonlinear impulsive differential equations. The impulsive moments coincide with the moments of meeting of the integral curve and some of the so-called barrier curves. For such type of equations, sufficient conditions are found under which the solutions are continuously dependent on the perturbations with respect to the initial conditions and barrier curves. The results are applied to a mathematical model of population dynamics.

A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.

Differentiability of solutions of impulsive differential equations with respect to the impulsive perturbations

The nonlinear impulsive differential equations with fixed moments of impulsive perturbation are the main object of investigation in this paper. Sufficient conditions for these types of equations are obtained, under which their solutions are continuously dependent and differentiable with respect to the initial conditions and the impulsive perturbations. The results are applied to a mathematical model of population dynamics.

Complex dynamics of the population with a simple age structure

The effects of the following modes of density-dependent control of population growth: density-dependent birth rate, adult survival rate, juvenile survival rate are compared based on the mathematical model of population dynamics. It is shown that the most efficient mechanisms limiting population size are decreasing with the growth of the adult population birth rate and/or the decreasing survival rate of the offspring with the increase in their number. However, these same mechanisms are responsible for oscillations of the population size and its chaotic change. The density-dependence of the adult survival rate is not efficient in constraining the population growth, but it can substantially limit the magnitude of oscillations of the population size.

Bridging gaps between statistical and mathematical modeling in ecology

Congratulations to Heisey and colleagues (2010) for an analysis of complex ecological data within a process-oriented framework and an equally thoughtful discussion of the ongoing challenges of linking process and pattern. Their work joins a growing literature linking modern statistical methods for describing patterns in data with expanded sets of ecologically-motivated mathematical models of population dynamics. My comments build on the authors’ framework and discussion to describe a continuum that (I feel) encompasses a conceptual path between process and pattern and illustrates how hierarchical models provide a convenient area within which to explore this continuum.

Mathematical model of population dynamics with the world population size stabilizing about a stationary level

Mathematical model of population dynamics with the world population size stabilizing about a stationary level

Population dynamics of a pathogen: the conundrum of vivax malaria

Building a mathematical model of population dynamics of pathogens within their host involves considerations of factors similar to those in ecology, as pathogens can prey on cells in the host. But within the multicellular host, attacked cell types are integrated with other cellular systems, which in turn intervene in the infection. For example, immune responses attempt to sense and then eliminate or contain pathogens, and homeostatic mechanisms try to compensate for cell loss. This review focuses on modeling applied to malarias, diseases caused by single-cell eukaryote parasites that infect red blood cells, with special concern given to vivax malaria, a disease often thought to be benign (if sometimes incapacitating) because the parasite only attacks a small proportion of red blood cells, the very youngest ones. However, I will use mathematical modeling to argue that depletion of this pool of red blood cells can be disastrous to the host if growth of the parasite is not vigorously check by host immune responses. Also, modeling can elucidate aspects of new field observations that indicate that vivax malaria is more dangerous than previously thought.Electronic supplementary materialThe online version of this article (doi:10.1007/s12551-010-0034-3) contains supplementary material, which is available to authorized users.

Current Status of Mathematical Models for Population Dynamics of Prymnesium parvum in a Texas Reservoir1

Grover, James P., Jason W. Baker, Daniel L. Roelke, and Bryan W. Brooks, 2010. Current Status of Mathematical Models for Population Dynamics of Prymnesium parvum in a Texas Reservoir. Journal of the American Water Resources Association (JAWRA) 46(1):92‐107. DOI: 10.1111/j.1752‐1688.2009.00393.xAbstract: Blooms of the harmful alga Prymnesium parvum have apparently increased in frequency in inland waters of the United States, especially in western Texas. A suite of mathematical models was developed based on a chemostat (or continuously stirred tank reactor) framework, and calibrated with data from Lake Granbury, Texas. Inputs included data on flows, salinity, irradiance, temperature, zooplankton grazing, and nutrients. Parameterization incorporated recent laboratory studies relating the specific growth rate of P. parvum to such factors. Models differed in the number of algal populations competing with P. parvum, and whether competition occurred only by consumption of shared nutrients, or additionally through production of an allelopathic chemical by one of the populations, parameterized as cyanobacteria. Uncalibrated models did not reproduce the observed seasonal dynamics of P. parvum in Lake Granbury, which displayed a maximum population in late February during a prolonged bloom in cooler weather, and reduced abundance in summer. Sensitivity analyses suggested two modifications leading to predictions that better resembled observations. The first modification greatly reduces the optimal temperature for growth of P. parvum, an approach that disagrees with laboratory experiments indicating a strong potential for growth at temperatures above 20°C. The second modification increases the growth rate of P. parvum at all temperatures, in models including cyanobacterial allelopathy. Despite these adjustments, calibrated models did not faithfully simulate all features of the seasonal dynamics of P. parvum.

Persistence and fluctuation of lateral dimorphism in fishes.

Two morphological types ("righty" and "lefty") have been discovered in several fish species and are referred to as a typical example of antisymmetry. It has been suggested, first, that this dimorphism (called laterality) is inheritable; second, that the frequencies of laterality in each species fluctuate around 0.5; and third, that predators mainly exploit prey of the opposite laterality; that is, lefty and righty predators prey on righties and lefties, respectively. The latter is defined as "cross predation"; the antonym "parallel predation" means predation within the same laterality. We hypothesized that cross predation drives alternation of the survival and reproductive advantages between two morphological types, leading to frequency-dependent selection that maintains the dimorphism. To investigate this, we constructed mathematical models of population dynamics of one prey/one predator systems and three-trophic-level systems with omnivory. Mathematical analysis and computer simulations explained the behavior of the laterality frequency in nature well, insofar as cross predation dominated over parallel predation. Furthermore, the simulations showed that when only one of the morphological types exists in a species, the other type can invade. This suggests that dimorphism is maintained in all interacting species.

Single-species models for many-species food webs.

Most species live in species-rich food webs; yet, for a century, most mathematical models for population dynamics have included only one or two species. We ask whether such models are relevant to the real world. Two-species population models of an interacting consumer and resource collapse to one-species dynamics when recruitment to the resource population is unrelated to resource abundance, thereby weakening the coupling between consumer and resource. We predict that, in nature, generalist consumers that feed on many species should similarly show one-species dynamics. We test this prediction using cyclic populations, in which it is easier to infer underlying mechanisms, and which are widespread in nature. Here we show that one-species cycles can be distinguished from consumer resource cycles by their periods. We then analyse a large number of time series from cyclic populations in nature and show that almost all cycling, generalist consumers examined have periods that are consistent with one-species dynamics. Thus generalist consumers indeed behave as if they were one-species populations, and a one-species model is a valid representation for generalist population dynamics in many-species food webs.

Dynamics of a two sex population with gestation period

We investigate a mathematical model of population dynamics for a population of two sexes (male and female) in which new individuals are conceived in a process of mating between individuals of opposed sexes and their appearance is postponed by a period of

Population dynamics of biocontrol agents and pathogens in soils and rhizospheres

Understanding the dynamics between a pathogen and a biocontrol agent (BCA) in soil or in an infection court such as the rhizosphere is crucial for predicting the success of biological control. This is especially true for biological control using the strategy of reduction of initial inoculum prior to infection. By studying the population density fluxes over time, one can observe how the BCA and pathogen influence each other's population and life cycles, and how the biological and physical environment influence this relationship. Methods for quantifying fungi and bacteria in soil, including classical (dilution plating, baiting, bioassays), immunological (ELISA, IFC, immunomagnetic) and molecular (DNA hybridization, PCR, marker and reporter genes) methods, are discussed and critiqued. Finally, the value of mathematical modeling of population dynamics as a means of providing important information about the tripartite relationship between the pathogen, BCA, and host plant is reviewed.