# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

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## Mathematical Biosciences & Engineering

2008 , Volume 5 , Issue 4

A special issue on
Tribute to Thomas G. Hallam's Contributions to Mathematical Ecology,

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2008, 5(4): i-iii doi: 10.3934/mbe.2008.5.4i +[Abstract](806) +[PDF](713.7KB)
Abstract:
Thomas Guy Hallam began his career as a faculty member in the Department of Mathematics at Florida State University, working in the area of comparison theorems for ordinary differential equations. While at Florida State he organized a mathematical modeling course and thus became interested in mathematical biology. He began to wonder how he, as a mathematician, might address the mounting environmental problems. He took courses in oceanography and ecology and delved deeply into the literature. During the summer of 1974, he gave a full series of lectures on mathematical biology at the University of São Carlos in São Paulo, Brazil. In 1976, he took a year's leave at the University of Georgia, Athens, in the Departments of Mathematics, Zoology, and the Institute of Ecology, where he met Tom Gard and Ray Lassiter, with whom he has had career-long interactions. In Athens, he became interested in ecotoxicology and partial differential equation models of physiologically structured populations.

2008, 5(4): 601-616 doi: 10.3934/mbe.2008.5.601 +[Abstract](1054) +[PDF](253.3KB)
Abstract:
We develop a numerical method for estimating parameters in a structured erythropoiesis model consisting of a nonlinear system of partial differential equations. Convergence theory for the computed parameters is provided. Numerical results for estimating the growth rate of precursor cells as a function of the erythropoietin concentration and the decay rate of erythropoietin as a function of the total number of precursor cells from computationally generated data are provided. Standard errors for such parameters are also given.
2008, 5(4): 617-645 doi: 10.3934/mbe.2008.5.617 +[Abstract](1170) +[PDF](559.7KB)
Abstract:
Arenaviruses are associated with rodent-transmitted diseases in humans. Five arenaviruses are known to cause human illness: Lassa virus, Junin virus, Machupo virus, Guanarito virus and Sabia virus. In this investigation, we model the spread of Machupo virus in its rodent host Calomys callosus. Machupo virus infection in humans is known as Bolivian hemorrhagic fever (BHF) which has a mortality rate of approximately 5-30% [31].
Machupo virus is transmitted among rodents through horizontal (direct contact), vertical (infected mother to offspring) and sexual transmission. The immune response differs among rodents infected with Machupo virus. Either rodents develop immunity and recover (immunocompetent) or they do not develop immunity and remain infected (immunotolerant). We formulate a general deterministic model for male and female rodents consisting of eight differential equations, four for females and four for males. The four states represent susceptible, immunocompetent, immunotolerant and recovered rodents, denoted as $S$, $I^t$, $I^c$ and $R$, respectively. A unique disease-free equilibrium (DFE) is shown to exist and a basic reproduction number $\mathcal R_0$ is computed using the next generation matrix approach. The DFE is shown to be locally asymptotically stable if $\mathcal R_0<1$ and unstable if $\mathcal R_0>1$.
Special cases of the general model are studied, where there is only one immune stage, either $I^t$ or $I^c$. In the first model, $SI^cR^c$, it is assumed that all infected rodents are immunocompetent and recover. In the second model, $SI^t$, it is assumed that all infected rodents are immunotolerant. For each of these models, the basic reproduction numbers are computed and their relationship to the basic reproduction number of the general model determined. For the $SI^t$ model, it is shown that bistability may occur, the DFE and an enzootic equilibrium, with all rodents infectious, are locally asymptotically stable for the same set of parameter values. A simplification of the $SI^t$ model yields a third model, where the sexes are not differentiated, and therefore, there is no sexual transmission. For this third simplified model, the dynamics are completely analyzed. It is shown that there exists a DFE and possibly two additional equilibria, one of which is globally asymptotically stable for any given set of parameter values; bistability does not occur. Numerical examples illustrate the dynamics of the models. The biological implications of the results and future research goals are discussed in the conclusion.
2008, 5(4): 647-667 doi: 10.3934/mbe.2008.5.647 +[Abstract](1049) +[PDF](380.0KB)
Abstract:
We consider ordinary least squares parameter estimation problems where the unknown parameters to be estimated are probability distributions. A computational framework for quantification of uncertainty (e.g., standard errors) associated with the estimated parameters is given and sample numerical findings are presented.
2008, 5(4): 669-680 doi: 10.3934/mbe.2008.5.669 +[Abstract](869) +[PDF](283.3KB)
Abstract:
Species augmentation is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. In this paper, we develop a differential equations model and optimal control formulation for a continuous time augmentation of a general declining population. We find a characterization for the optimal control and show numerical results for scenarios of different illustrative parameter sets. The numerical results provide considerably more detail about the exact dynamics of optimal augmentation than can be readily intuited. The work and results presented in this paper are a first step toward building a general theory of population augmentation, which accounts for the complexities inherent in many conservation biology applications.
2008, 5(4): 681-690 doi: 10.3934/mbe.2008.5.681 +[Abstract](1071) +[PDF](124.2KB)
Abstract:
We establish the final size equation for a general age-of-infection epidemic model in a new simpler form if there are no disease deaths (total population size remains constant). If there are disease deaths, the final size relation is an inequality but we obtain an estimate for the final epidemic size.
2008, 5(4): 691-711 doi: 10.3934/mbe.2008.5.691 +[Abstract](994) +[PDF](2927.5KB)
Abstract:
Suspended organic and inorganic particles, resulting from the interactions among biological, physical, and chemical variables, modify the optical properties of water bodies and condition the trophic chain. The analysis of their optic properties through the spectral signatures obtained from satellite images allows us to infer the trophic state of the shallow lakes and generate a real time tool for studying the dynamics of shallow lakes. Field data (chlorophyll-a, total solids, and Secchi disk depth) allow us to define levels of turbidity and to characterize the shallow lakes under study. Using bands 2 and 4 of LandSat 5 TM and LandSat 7 ETM+ images and constructing adequate artificial neural network models (ANN), a classification of shallow lakes according to their turbidity is obtained. ANN models are also used to determine chlorophyll-a and total suspended solids concentrations from satellite image data. The results are statistically significant. The integration of field and remote sensors data makes it possible to retrieve information on shallow lake systems at broad spatial and temporal scales. This is necessary to understanding the mechanisms that affect the trophic structure of these ecosystems.
2008, 5(4): 713-727 doi: 10.3934/mbe.2008.5.713 +[Abstract](1340) +[PDF](206.0KB)
Abstract:
In this study, we expand on the susceptible-infected-susceptible (SIS) heterosexual mixing setting by including the movement of individuals of both genders in a spatial domain in order to more comprehensively address the transmission dynamics of competing strains of sexually-transmitted pathogens. In prior models, these transmission dynamics have only been studied in the context of nonexplicitly mobile heterosexually active populations at the demographic steady state, or, explicitly in the simplest context of SIS frameworks whose limiting systems are order preserving. We introduce reaction-diffusion equations to study the dynamics of sexually-transmitted diseases (STDs) in spatially mobile heterosexually active populations. To accomplish this, we study a single-strain STD model, and discuss in what forms and at what speed the disease spreads to noninfected regions as it expands its spatial range. The dynamics of two competing distinct strains of the same pathogen on this population are then considered. The focus is on the investigation of the spatial transition dynamics between the two endemic equilibria supported by the nonspatial corresponding model. We establish conditions for the successful invasion of a population living in endemic conditions by introducing a strain with higher fitness. It is shown that there exists a unique spreading speed (where the spreading speed is characterized as the slowest speed of a class of traveling waves connecting two endemic equilibria) at which the infectious population carrying the invading stronger strain spreads into the space where an equilibrium distribution has been established by the population with the weaker strain. Finally, we give sufficient conditions under which an explicit formula for the spreading speed can be found.
2008, 5(4): 729-741 doi: 10.3934/mbe.2008.5.729 +[Abstract](887) +[PDF](370.7KB)
Abstract:
Viral infections are one of the leading source of mortality worldwide. The great majority of them circulate and persist in wild reservoirs and periodically spill over into humans or domestic animals. In the wild reservoirs, the progression of disease is frequently quite different from that in spillover hosts. We propose a mathematical treatment of the dynamics of viral infections in wild mammals using models with alternative outcomes. We develop and analyze compartmental epizootic models assuming permanent or temporary immunity of the individuals surviving infections and apply them to rabies in bats. We identify parameter relations that support the existing patterns in the viral ecology and estimate those parameters that are unattainable through direct measurement. We also investigate how the duration of the acquired immunity affects the disease and population dynamics.
2008, 5(4): 743-756 doi: 10.3934/mbe.2008.5.743 +[Abstract](963) +[PDF](1268.5KB)
Abstract:
Temperate-zone bats are subject to serious energetic constraints due to their high surface area to volume relations, the cost of temperature regulation, the high metabolic cost of flight, and the seasonality of their resources. We present a novel, multilevel theoretical approach that integrates information on bat biology collected at a lower level of organization, the individual with its physiological characteristics, into a modeling framework at a higher level, the population. Our individual component describes the growth of an individual female bat by modeling the dynamics of the main body compartments (lipids, proteins, and carbohydrates). A structured population model based on extended McKendrick-von Foerster partial differential equations integrates those individual dynamics and provides insight into possible regulatory mechanisms of population size as well as conditions of population survival and extinction. Though parameterized for a specific bat species, all modeling components can be modified to investigate other bats with similar life histories. A better understanding of population dynamics in bats can assist in the development of management techniques and conservation strategies, and to investigate stress effects. Studying population dynamics of bats presents particular challenges, but bats are essential in some areas of concern in conservation and disease ecology that demand immediate investigation.
2008, 5(4): 757-770 doi: 10.3934/mbe.2008.5.757 +[Abstract](1601) +[PDF](866.0KB)
Abstract:
The increasing prevalence of HIV/AIDS in Africa over the past twenty-five years continues to erode the continent's health care and overall welfare. There have been various responses to the pandemic, led by Uganda, which has had the greatest success in combating the disease. Part of Uganda's success has been attributed to a formalized information, education, and communication (IEC) strategy, lowering estimated HIV/AIDS infection rates from 18.5% in 1995 to 4.1% in 2003. We formulate a model to investigate the effects of information and education campaigns on the HIV epidemic in Uganda. These campaigns affect people's behavior and can divide the susceptibles class into subclasses with different infectivity rates. Our model is a system of ordinary differential equations and we use data about the epidemics and the number of organizations involved in the campaigns to estimate the model parameters. We compare our model with three types of susceptibles to a standard SIR model.
2008, 5(4): 771-787 doi: 10.3934/mbe.2008.5.771 +[Abstract](853) +[PDF](308.6KB)
Abstract:
The dynamic behaviour of simple aquatic ecosystems with nutrient recycling in a chemostat, stressed by limited food availability and a toxicant, is analysed. The aim is to find effects of toxicants on the structure and functioning of the ecosystem. The starting point is an unstressed ecosystem model for nutrients, populations, detritus and their intra- and interspecific interactions, as well as the interaction with the physical environment. The fate of the toxicant includes transport and exchange between the water and the populations via two routes, directly from water via diffusion over the outer membrane of the organism and via consumption of contaminated food. These processes are modelled using mass-balance formulations and diffusion equations. At the population level the toxicant affects different biotic processes such as assimilation, growth, maintenance, reproduction, and survival, thereby changing their biological functioning. This is modelled by taking the parameters that described these processes to be dependent on the internal toxicant concentration. As a consequence, the structure of the ecosystem, that is its species composition, persistence, extinction or invasion of species and dynamics behaviour, steady state oscillatory and chaotic, can change. To analyse the long-term dynamics we use the bifurcation analysis approach. In ecotoxicological studies the concentration of the toxicant in the environment can be taken as the bifurcation parameter. The value of the concentration at a bifurcation point marks a structural change of the ecosystem. This indicates that chemical stressors are analysed mathematically in the same way as environmental (e.g. temperature) and ecological (e.g. predation) stressors. Hence, this allows an integrated approach where different type of stressors are analysed simultaneously. Environmental regimes and toxic stress levels at which no toxic effects occur and where the ecosystem is resistant will be derived. A numerical continuation technique to calculate the boundaries of these regions will be given.
2008, 5(4): 789-801 doi: 10.3934/mbe.2008.5.789 +[Abstract](1157) +[PDF](159.2KB)
Abstract:
In this paper, we formulate a mathematical model for malaria transmission that includes incubation periods for both infected human hosts and mosquitoes. We assume humans gain partial immunity after infection and divide the infected human population into subgroups based on their infection history. We derive an explicit formula for the reproductive number of infection, $R_0$, to determine threshold conditions whether the disease spreads or dies out. We show that there exists an endemic equilibrium if $R_0>1$. Using an numerical example, we demonstrate that models having the same reproductive number but different numbers of progression stages can exhibit different transient transmission dynamics.
2008, 5(4): 803-812 doi: 10.3934/mbe.2008.5.803 +[Abstract](938) +[PDF](137.3KB)
Abstract:
In this paper, the existence of positive periodic solutions of a class of periodic $n$-species Gilpin-Ayala impulsive competition systems is studied. By using the continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions is obtained. Our results are general enough to include some known results in this area.
2008, 5(4): 813-830 doi: 10.3934/mbe.2008.5.813 +[Abstract](858) +[PDF](1282.0KB)
Abstract:
Most animal populations are characterized by balanced sex ratios, but there exist several exceptions in which the sex ratio at birth is skewed. An interesting hypothesis proposed by Clark (1978) to explain male-biased sex ratios is the local resource competition theory: the bias may be expected in those species in which males disperse more than females, which are thus more prone to local competition for resources. Here we discuss some of the ideas underlying Clark's theory using a spatially explicit approach. In particular, we focus on the role of spatiotemporal heterogeneity as a possible determinant of biased sex ratios. We model spatially structured semelparous populations where either Ricker density dependence or environmental stochasticity can generate irregular spatiotemporal patterns. The proposed discrete-time model describes both genetic and complex population dynamics assuming that (1) sex ratio is genetically determined, (2) only young males can disperse, and (3) individuals locally compete for resources. The analysis of the model shows that no skewed sex ratios can arise in homogeneous habitats. Temporal asynchronized fluctuations between two distinct patches coupled with dispersal of young males is the minimum requirement for obtaining skewed sex ratios of demographic nature in local adult populations. However, the establishment of a male-biased sex ratio at birth in the long run is possible if dispersal is genetically determined and there is genetic linkage between sex ratio determination and dispersal.
2008, 5(4): 831-842 doi: 10.3934/mbe.2008.5.831 +[Abstract](774) +[PDF](347.1KB)
Abstract:
We constructed differential equation models for the diurnal abundance and distribution of breeding glaucous-winged gulls (Larus glaucescens) as they moved among nesting and non-nesting habitat patches. We used time scale techniques to reduce the differential equations to algebraic equations and connected the models to field data. The models explained the data as a function of abiotic environmental variables with $R^{2}=0.57$. A primary goal of this study is to demonstrate the utility of a methodology that can be used by ecologists and wildlife managers to understand and predict daily activity patterns in breeding seabirds.
2008, 5(4): 843-857 doi: 10.3934/mbe.2008.5.843 +[Abstract](997) +[PDF](332.6KB)
Abstract:
A multiple species metapopulations model with density-dependent dispersal is presented. Assuming the network configuration matrix to be diagonizable we obtain a decoupling of the associated perturbed system from the homogeneous state. It was possible to analyze in detail the instability induced by the density-dependent dispersal in two classes of $k$-species interaction models: a hierarchically organized competitive system and an age-structured model.
2008, 5(4): 859-875 doi: 10.3934/mbe.2008.5.859 +[Abstract](925) +[PDF](478.8KB)
Abstract:
Simple, discrete-time, population models typically exhibit complex dynamics, like cyclic oscillations and chaos, when the net reproductive rate, $R$, is large. These traditional models generally do not incorporate variability in juvenile "risk,'' defined to be a measure of a juvenile's vulnerability to density-dependent mortality. For a broad class of discrete-time models we show that variability in risk across juveniles tends to stabilize the equilibrium. We consider both density-independent and density-dependent risk, and for each, we identify appropriate shapes of the distribution of risk that will stabilize the equilibrium for all values of $R$. In both cases, it is the shape of the distribution of risk and not the amount of variation in risk that is crucial for stability.
2008, 5(4): 877-887 doi: 10.3934/mbe.2008.5.877 +[Abstract](943) +[PDF](1129.6KB)
Abstract:
A spatially explicit model is developed to simulate the small fish community and its underlying food web, in the freshwater marshes of the Everglades. The community is simplified to a few small fish species feeding on periphyton and invertebrates. Other compartments are detritus, crayfish, and a piscivorous fish species. This unit food web model is applied to each of the 10,000 spatial cells on a 100 x 100 pixel landscape. Seasonal variation in water level is assumed and rules are assigned for fish movement in response to rising and falling water levels, which can cause many spatial cells to alternate between flooded and dry conditions. It is shown that temporal variations of water level on a spatially heterogeneous landscape can maintain at least three competing fish species. In addition, these environmental factors can strongly affect the temporal variation of the food web caused by top-down control from the piscivorous fish.
2008, 5(4): 889-903 doi: 10.3934/mbe.2008.5.889 +[Abstract](772) +[PDF](228.2KB)
Abstract:
Negative frequency-dependent selection is a well known microevolutionary process that has been documented in a population of Perissodus microlepis, a species of cichlid fish endemic to Lake Tanganyika (Africa). Adult P. microlepis are lepidophages, feeding on the scales of other living fish. As an adaptation for this feeding behavior P. microlepis exhibit lateral asymmetry with respect to jaw morphology: the mouth either opens to the right or left side of the body. Field data illustrate a temporal phenotypic oscillation in the mouth-handedness, and this oscillation is maintained by frequency-dependent selection. Since both genetic and population dynamics occur on the same time scale in this case, we develop a (discrete time) model for P. microlepis populations that accounts for both dynamic processes. We establish conditions on model parameters under which the model predicts extinction and conditions under which there exists a unique positive (survival) equilibrium. We show that at the positive equilibrium there is a 1:1 phenotypic ratio. Using a local stability and bifurcation analysis, we give further conditions under which the positive equilibrium is stable and conditions under which it is unstable. Destabilization results in a bifurcation to a periodic oscillation and occurs when frequency-dependent selection is sufficiently strong. This bifurcation is offered as an explanation of the phenotypic frequency oscillations observed in P. microlepis. An analysis of the bifurcating periodic cycle results in some interesting and unexpected predictions.

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