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Mathematical Biosciences & Engineering

2014 , Volume 11 , Issue 6

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Spatiotemporal complexity in a predator--prey model with weak Allee effects
Yongli Cai, Malay Banerjee, Yun Kang and Weiming Wang
2014, 11(6): 1247-1274 doi: 10.3934/mbe.2014.11.1247 +[Abstract](97) +[PDF](1130.0KB)
In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.
Modeling colony collapse disorder in honeybees as a contagion
Christopher M. Kribs-Zaleta and Christopher Mitchell
2014, 11(6): 1275-1294 doi: 10.3934/mbe.2014.11.1275 +[Abstract](59) +[PDF](2651.8KB)
Honeybee pollination accounts annually for over $14 billion in United States agriculture alone. Within the past decade there has been a mysterious mass die-off of honeybees, an estimated 10 million beehives and sometimes as much as 90% of an apiary. There is still no consensus on what causes this phenomenon, called Colony Collapse Disorder, or CCD. Several mathematical models have studied CCD by only focusing on infection dynamics. We created a model to account for both healthy hive dynamics and hive extinction due to CCD, modeling CCD via a transmissible infection brought to the hive by foragers. The system of three ordinary differential equations accounts for multiple hive population behaviors including Allee effects and colony collapse. Numerical analysis leads to critical hive sizes for multiple scenarios and highlights the role of accelerated forager recruitment in emptying hives during colony collapse.
Epidemic models for complex networks with demographics
Zhen Jin, Guiquan Sun and Huaiping Zhu
2014, 11(6): 1295-1317 doi: 10.3934/mbe.2014.11.1295 +[Abstract](106) +[PDF](717.7KB)
In this paper, we propose and study network epidemic models with demographics for disease transmission. We obtain the formula of the basic reproduction number $R_{0}$ of infection for an SIS model with births or recruitment and death rate. We prove that if $R_{0}\leq1$, infection-free equilibrium of SIS model is globally asymptotically stable; if $R_{0}>1$, there exists a unique endemic equilibrium which is globally asymptotically stable. It is also found that demographics has great effect on basic reproduction number $R_{0}$. Furthermore, the degree distribution of population varies with time before it reaches the stationary state.
Dynamics of two phytoplankton populations under predation
Jean-Jacques Kengwoung-Keumo
2014, 11(6): 1319-1336 doi: 10.3934/mbe.2014.11.1319 +[Abstract](64) +[PDF](704.8KB)
The aim of this paper is to investigate the manner in which predation and single-nutrient competition affect the dynamics of a non-toxic and a toxic phytoplankton species in a homogeneous environment (such as a chemostat). We allow for the possibility that both species serve as prey for an herbivorous zooplankton species. We assume that the toxic phytoplankton species produces toxins that affect only its own growth (autotoxicity). The autotoxicity assumption is ecologically explained by the fact that the toxin-producing phytoplankton is not mature enough to produce toxins that will affect the growth of its nontoxic competitor. We show that, in the absence of phytotoxic interactions and nutrient recycling, our model exhibits uniform persistence. The removal rates are distinct and we use general response functions. Finally, numerical simulations are carried out to show consistency with theoretical analysis. Our model has similarities with other food-chain models. As such, our results may be relevant to a wider spectrum of population models, not just those focused on plankton. Some open problems are discussed at the end of this paper.
Modeling the impact of twitter on influenza epidemics
Kasia A. Pawelek, Anne Oeldorf-Hirsch and Libin Rong
2014, 11(6): 1337-1356 doi: 10.3934/mbe.2014.11.1337 +[Abstract](141) +[PDF](643.1KB)
Influenza remains a serious public-health problem worldwide. The rising popularity and scale of social networking sites such as Twitter may play an important role in detecting, affecting, and predicting influenza epidemics. In this paper, we develop a simple mathematical model including the dynamics of ``tweets'' --- short, 140-character Twitter messages that may enhance the awareness of disease, change individual's behavior, and reduce the transmission of disease among a population during an influenza season. We analyze the model by deriving the basic reproductive number and proving the stability of the steady states. A Hopf bifurcation occurs when a threshold curve is crossed, which suggests the possibility of multiple outbreaks of influenza. We also perform numerical simulations, conduct sensitivity test on a few parameters related to tweets, and compare modeling predictions with surveillance data of influenza-like illness reported cases and the percentage of tweets self-reporting flu during the 2009 H1N1 flu outbreak in England and Wales. These results show that social media programs like Twitter may serve as a good indicator of seasonal influenza epidemics and influence the emergence and spread of the disease.
A model for the nonlinear mechanism responsible for cochlear amplification
Kimberly Fessel and Mark H. Holmes
2014, 11(6): 1357-1373 doi: 10.3934/mbe.2014.11.1357 +[Abstract](55) +[PDF](2206.2KB)
A nonlinear model for the mechanism responsible for the amplification of the sound wave in the ear is derived using the geometric and material properties of the system. The result is a nonlinear beam equation, with the nonlinearity appearing in a coefficient of the equation. Once derived, the beam problem is analyzed for various loading conditions. Based on this analysis it is seen that the mechanism is capable of producing a spatially localized gain, as required by any amplification mechanism, but it is also capable of increasing the spatial contrast in the signal.
Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures
Toshikazu Kuniya, Yoshiaki Muroya and Yoichi Enatsu
2014, 11(6): 1375-1393 doi: 10.3934/mbe.2014.11.1375 +[Abstract](85) +[PDF](852.5KB)
In this paper, we formulate an SIR epidemic model with hybrid of multigroup and patch structures, which can be regarded as a model for the geographical spread of infectious diseases or a multi-group model with perturbation. We show that if a threshold value, which corresponds to the well-known basic reproduction number $R_0$, is less than or equal to unity, then the disease-free equilibrium of the model is globally asymptotically stable. We also show that if the threshold value is greater than unity, then the model is uniformly persistent and has an endemic equilibrium. Moreover, using a Lyapunov functional technique, we obtain a sufficient condition under which the endemic equilibrium is globally asymptotically stable. The sufficient condition is satisfied if the transmission coefficients in the same groups are large or the per capita recovery rates are small.
A new model with delay for mosquito population dynamics
Hui Wan and Huaiping Zhu
2014, 11(6): 1395-1410 doi: 10.3934/mbe.2014.11.1395 +[Abstract](131) +[PDF](767.9KB)
In this paper, we formulate a new model with maturation delay for mosquito population incorporating the impact of blood meal resource for mosquito reproduction. Our results suggest that except for the usual crowded effect for adult mosquitoes, the impact of blood meal resource in a given region determines the mosquito abundance, it is also important for the population dynamics of mosquito which may induce Hopf bifurcation. The existence of a stable periodic solution is proved both analytically and numerically. The new model for mosquito also suggests that the resources for mosquito reproduction should not be ignored or mixed with the impact of blood meal resources for mosquito survival and both impacts should be considered in the model of mosquito population. The impact of maturation delay is also analyzed.
A stochastic simulation model for Anelosimus studiosus during prey capture: A case study for determination of optimal spacing
Michele L. Joyner, Chelsea R. Ross, Colton Watts and Thomas C. Jones
2014, 11(6): 1411-1429 doi: 10.3934/mbe.2014.11.1411 +[Abstract](75) +[PDF](979.4KB)
In this paper, we develop a stochastic differential equation model to simulate the movement of a social/subsocial spider species, Anelosimus studiosus, during prey capture using experimental data collected in a structured environment. In a subsocial species, females and their maturing offspring share a web and cooperate in web maintenance and prey capture. Furthermore, observations indicate these colonies change their positioning throughout the day, clustered during certain times of the day while spaced out at other times. One key question was whether or not the spiders spaced out ``optimally'' to cooperate in prey capture. In this paper, we first show the derivation of the model where experimental data is used to determine key parameters within the model. We then use this model to test the success of prey capture under a variety of different spatial configurations for varying colony sizes to determine the best spatial configuration for prey capture.
Effects of elongation delay in transcription dynamics
Xuan Zhang, Huiqin Jin, Zhuoqin Yang and Jinzhi Lei
2014, 11(6): 1431-1448 doi: 10.3934/mbe.2014.11.1431 +[Abstract](79) +[PDF](1358.5KB)
In the transcription process, elongation delay is induced by the movement of RNA polymerases (RNAP) along the DNA sequence, and can result in changes in the transcription dynamics. This paper studies the transcription dynamics that involved the elongation delay and effects of cell division and DNA replication. The stochastic process of gene expression is modeled with delay chemical master equation with periodic coefficients, and is studied numerically through the stochastic simulation algorithm with delay. We show that the average transcription level approaches to a periodic dynamics over cell cycles at homeostasis, and the elongation delay can reduce the transcription level and increase the transcription noise. Moreover, the transcription elongation can induce bimodal distribution of mRNA levels that can be measured by the techniques of flow cytometry.
A mathematical model for antibiotic control of bacteria in peritoneal dialysis associated peritonitis
Colette Calmelet, John Hotchkiss and Philip Crooke
2014, 11(6): 1449-1464 doi: 10.3934/mbe.2014.11.1449 +[Abstract](48) +[PDF](440.1KB)
A study of the process of pharmacokinetics-pharmacodynamics (PKPD) of antibiotics and their interaction with bacteria during peritoneal dialysis associated peritonitis (PDAP) is presented. We propose a mathematical model describing the evolution of bacteria population in the presence of antibiotics for different peritoneal dialysis regimens. Using the model along with experimental data, clinical parameters, and physiological values, we compute variations in PD fluid distributions, drug concentrations, and number of bacteria in peritoneal and extra-peritoneal cavities. Scheduling algorithms for the PD exchanges that minimize bacteria count are investigated.

2016  Impact Factor: 1.035




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