# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

2016 , Volume 13 , Issue 4

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2016, 13(4): 631-652 doi: 10.3934/mbe.2016012 +[Abstract](338) +[PDF](691.9KB)
Abstract:
We study a model of the chemostat with several species in competition for a single resource. We take into account the intra-specific interactions between individuals of the same population of micro-organisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristics, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we provide necessary and sufficient conditions on the control parameters of the system to have a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the effect of the intra-specific competition on the coexistence region of the species.
2016, 13(4): 653-671 doi: 10.3934/mbe.2016013 +[Abstract](434) +[PDF](1124.3KB)
Abstract:
We develop statistical and mathematical based methodologies for determining (as the experiment progresses) the amount of information required to complete the estimation of stable population parameters with pre-specified levels of confidence. We do this in the context of life table models and data for growth/death for three species of Daphniids as investigated by J. Stark and J. Banks [17]. The ideas developed here also have wide application in the health and social sciences where experimental data are often expensive as well as difficult to obtain.
2016, 13(4): 673-695 doi: 10.3934/mbe.2016014 +[Abstract](407) +[PDF](539.0KB)
Abstract:
In this paper, we establish the exploitation of a single population modeled by the Beverton--Holt difference equation with periodic coefficients. We begin our investigation with the harvesting of a single autonomous population with logistic growth and show that the harvested logistic equation with periodic coefficients has a unique positive periodic solution which globally attracts all its solutions. Further, we approach the investigation of the optimal harvesting policy that maximizes the annual sustainable yield in a novel and powerful way; it serves as a foundation for the analysis of the exploitation of the discrete population model. In the second part of the paper, we formulate the harvested Beverton--Holt model and derive the unique periodic solution, which globally attracts all its solutions. We continue our investigation by optimizing the sustainable yield with respect to the harvest effort. The results differ from the optimal harvesting policy for the continuous logistic model, which suggests a separate strategy for populations modeled by the Beverton--Holt difference equation.
2016, 13(4): 697-722 doi: 10.3934/mbe.2016015 +[Abstract](443) +[PDF](595.1KB)
Abstract:
The question of the effects of environmental toxins on ecological communities is of great interest from both environmental and conservational points of view. Mathematical models have been applied increasingly to predict the effects of toxins on a variety of ecological processes. Motivated by the fact that individuals with different sizes may have different sensitivities to toxins, we develop a toxin-mediated size-structured model which is given by a system of first order fully nonlinear partial differential equations (PDEs). It is very possible that this work represents the first derivation of a PDE model in the area of ecotoxicology. To solve the model, an explicit finite difference approximation to this PDE system is developed. Existence-uniqueness of the weak solution to the model is established and convergence of the finite difference approximation to this unique solution is proved. Numerical examples are provided by numerically solving the PDE model using the finite difference scheme.
2016, 13(4): 723-739 doi: 10.3934/mbe.2016016 +[Abstract](492) +[PDF](765.1KB)
Abstract:
In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold $R_0$ which completely governs the disease dynamics: when $R_0<1$, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when $R_0>1$, the disease is permanent. It is interesting that the threshold value $R_0$ bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.
2016, 13(4): 741-785 doi: 10.3934/mbe.2016017 +[Abstract](311) +[PDF](1301.6KB)
Abstract:
A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving arbitrary'' distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence and maintenance of TB resistant strains are discussed. The possibility of exogenous re-infection is added and shown to be capable of supporting multiple equilibria; a situation that increases the challenges faced by public health experts. We show that exogenous re-infection may help established resilient communities of actively-TB infected individuals that cannot be eliminated using approaches based exclusively on the ability to bring the control reproductive number just below $1$.
2016, 13(4): 787-812 doi: 10.3934/mbe.2016018 +[Abstract](323) +[PDF](684.4KB)
Abstract:
We propose a model of two-species competition in the chemostat for a single growth-limiting, nonreproducing resource that extends that of Roy [38]. The response functions are specified to be Michaelis-Menten, and there is no predation in Roy's work. Our model generalizes Roy's model to general uptake functions. The competition is exploitative so that species compete by decreasing the common pool of resources. The model also allows allelopathic effects of one toxin-producing species, both on itself (autotoxicity) and on its nontoxic competitor (phytotoxicity). We show that a stable coexistence equilibrium exists as long as (a) there are allelopathic effects and (b) the input nutrient concentration is above a critical value. The model is reconsidered under instantaneous nutrient recycling. We further extend this work to include a zooplankton species as a fourth interacting component to study the impact of predation on the ecosystem. The zooplankton species is allowed to feed only on the two phytoplankton species which are its perfectly substitutable resources. Each of the models is analyzed for boundedness, equilibria, stability, and uniform persistence (or permanence). Each model structure fits very well with some harmful algal bloom observations where the phytoplankton assemblage can be envisioned in two compartments, toxin producing and non-toxic. The Prymnesium parvum literature, where the suppressing effects of allelochemicals are quite pronounced, is a classic example. This work advances knowledge in an area of research becoming ever more important, which is understanding the functioning of allelopathy in food webs.
2016, 13(4): 813-840 doi: 10.3934/mbe.2016019 +[Abstract](398) +[PDF](555.7KB)
Abstract:
In this paper, an epidemic model is investigated for infectious diseases that can be transmitted through both the infectious individuals and the asymptomatic carriers (i.e., infected individuals who are contagious but do not show any disease symptoms). We propose a dose-structured vaccination model with multiple transmission pathways. Based on the range of the explicitly computed basic reproduction number, we prove the global stability of the disease-free when this threshold number is less or equal to the unity. Moreover, whenever it is greater than one, the existence of the unique endemic equilibrium is shown and its global stability is established for the case where the changes of displaying the disease symptoms are independent of the vulnerable classes. Further, the model is shown to exhibit a transcritical bifurcation with the unit basic reproduction number being the bifurcation parameter. The impacts of the asymptomatic carriers and the effectiveness of vaccination on the disease transmission are discussed through through the local and the global sensitivity analyses of the basic reproduction number. Finally, a case study of hepatitis B virus disease (HBV) is considered, with the numerical simulations presented to support the analytical results. They further suggest that, in high HBV prevalence countries, the combination of effective vaccination (i.e. $\geq 3$ doses of HepB vaccine), the diagnosis of asymptomatic carriers and the treatment of symptomatic carriers may have a much greater positive impact on the disease control.
2016, 13(4): 841-855 doi: 10.3934/mbe.2016020 +[Abstract](407) +[PDF](495.2KB)
Abstract:
Eating behaviors among a large population of children are studied as a dynamic process driven by nonlinear interactions in the sociocultural school environment. The impact of food association learning on diet dynamics, inspired by a pilot study conducted among Arizona children in Pre-Kindergarten to 8th grades, is used to build simple population-level learning models. Qualitatively, mathematical studies are used to highlight the possible ramifications of instruction, learning in nutrition, and health at the community level. Model results suggest that nutrition education programs at the population-level have minimal impact on improving eating behaviors, findings that agree with prior field studies. Hence, the incorporation of food association learning may be a better strategy for creating resilient communities of healthy and non-healthy eaters. A Ratatouille effect can be observed when food association learners become food preference learners, a potential sustainable behavioral change, which in turn, may impact the overall distribution of healthy eaters. In short, this work evaluates the effectiveness of population-level intervention strategies and the importance of institutionalizing nutrition programs that factor in economical, social, cultural, and environmental elements that mesh well with the norms and values in the community.
2016, 13(4): 857-885 doi: 10.3934/mbe.2016021 +[Abstract](412) +[PDF](1627.8KB)
Abstract:
This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.

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