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Discrete & Continuous Dynamical Systems - B

2017 , Volume 22 , Issue 3

Special issue in honor of Stephen Cantrell on the occasion of his 60th birthday

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2017, 22(3): ⅰ-ⅱ doi: 10.3934/dcdsb.201703i +[Abstract](21) +[HTML](10) +[PDF](77.4KB)
Optimal control and cost-effectiveness analysis of a three age-structured transmission dynamics of chikungunya virus
Folashade B. Agusto
2017, 22(3): 687-715 doi: 10.3934/dcdsb.2017034 +[Abstract](177) +[HTML](3) +[PDF](1294.4KB)

Chikungunya is an RNA viral disease, transmitted to humans by infected Aedes aegypti or Aedes albopictus mosquitoes. In this paper, an age-structured deterministic model for the transmission dynamics of Chikungunya virus is presented. The model is locally and globally asymptotically stable when the reproduction number is less than unity. A global sensitivity analysis using the reproduction number indicates that the mosquito biting rate, the transmission probability per contact of mosquitoes and of humans, mosquito recruitment rate and the death rate of the mosquitoes are the parameters with the most influence on Chikungunya transmission dynamics. Optimal control theory was then applied, using the results from the sensitivity analysis, to minimize the number infected humans, with time dependent control variables (impacting mosquito biting rate, transmission probability, death rate and recovery rates in humans).

The numerical simulations indicate that Chikungunya can be reduced by the application of these controls. The benefits associated with these health interventions are evaluated using cost-effectiveness analysis and these shows that using mono-control strategy involving treatment of infected individuals is the most cost-effective strategy of this category. With pairs of control, the pairs involving treatment of infected individuals and mosquitoes adulticiding, is the most cost-effective strategy of this category and is more cost-effective than using the triple control strategy involving personal protection, treatment of infected humans and mosquitoes adulticiding.

Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response
Eric Avila-Vales , Gerardo García-Almeida and  Erika Rivero-Esquivel
2017, 22(3): 717-740 doi: 10.3934/dcdsb.2017035 +[Abstract](75) +[HTML](0) +[PDF](1303.5KB)

In this article, the clasical Bazykin model is modifed with Bedding–ton–DeAngelis functional response, subject to self and cross-diffusion, in order to study the spatial dynamics of the model.We perform a detailed stability and Hopf bifurcation analysis of the spatial model system and determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. We present some numerical simulations of time evolution of patterns to show the important role played by self and cross-diffusion as well as other parameters leading to complex patterns in the plane.

Macroalgal allelopathy in the emergence of coral diseases
Joydeb Bhattacharyya and  Samares Pal
2017, 22(3): 741-762 doi: 10.3934/dcdsb.2017036 +[Abstract](42) +[HTML](6) +[PDF](3327.2KB)

Microbial disease in corals associated with the proliferation of benthic macroalgae are the major contributors to the decline of coral reefs over the past few decades. Several benthic macroalgae species produce allelopathic chemical compounds that negatively affect corals. The emergence of microbial diseases in corals occurs simultaneously with the elevated abundance of benthic macroalgae. The release of allelochemicals by toxic-macroalgae enhances microbial activity on coral surfaces via the release of dissolved compounds. Proliferation of benthic macroalgae in coral reefs results in increased physical contacts between corals and macroalgae, triggering the susceptibility of coral disease. The abundance of macroalgae changes the community structure towards macroalgae dominated reef ecosystem. We investigate coral-macroalgal phase shift in presence of macroalgal allelopathy and microbial infection on corals by means of an eco-epidemiological model under the assumption that the transmission of infection is mediated by the pathogens shed by infectious corals and under the influence of macroalgae in the environment. We perform equilibrium and stability analysis on our non-linear ODE model and found that the system is capable of exhibiting the existence of two stable configurations of the community under the same environmental conditions by allowing saddle-node bifurcations that involves in creation and destruction of fixed points and associated hysteresis effect. It is shown that the system undergoes a sudden change of transition when the transmission rate of the infection crosses some certain critical thresholds. Computer simulations have been carried out to illustrate different analytical results.

Concentration phenomenon in some non-local equation
Olivier Bonnefon , Jérôme Coville and  Guillaume Legendre
2017, 22(3): 763-781 doi: 10.3934/dcdsb.2017037 +[Abstract](97) +[HTML](13) +[PDF](530.7KB)

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation

supplemented by the initial condition \begin{document}$u(0,\cdot)=u_0$\end{document} in \begin{document}$\Omega $\end{document}, where the domain \begin{document}$\Omega $\end{document} is a, the functions \begin{document}$k$\end{document} and \begin{document}$m$\end{document} are non-negative kernels satisfying integrability conditions and the function \begin{document}$a$\end{document} is continuous. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function \begin{document}$u$\end{document} represents the density of individuals characterized by the trait, the domain of trait values \begin{document}$\Omega $\end{document} is a bounded subset of \begin{document}$\mathbb{R}^N$\end{document}, the kernels \begin{document}$k$\end{document} and \begin{document}$m$\end{document} respectively account for the competition between individuals and the mutations occurring in every generation, and the function \begin{document}$a$\end{document} represents a growth rate. When the competition is independent of the trait, that is, the kernel \begin{document}$k$\end{document} is independent of \begin{document}$x$\end{document}, (\begin{document}$k(x,y)=k(y)$\end{document}), we construct a positive stationary solution which belongs to \begin{document}$d\mu$\end{document} inthe space of Radon measures on \begin{document}$\Omega $\end{document}. \begin{document}$\mathbb{M}(\Omega )$\end{document}.Moreover, in the case where this measure \begin{document}$d\mu$\end{document} is regular and bounded, we prove its uniqueness and show that, for any non-negative initial datum in \begin{document}$L^1(\Omega )\cap L^{\infty}(\Omega )$\end{document}, the solution of the Cauchy problem converges to this limit measure in \begin{document}$L^2(\Omega )$\end{document}. We also exhibit an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. The numerical simulations seem to reveal a dependence of the limit measure with respect to the initial datum.

A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions
Alfonso Castro and  Rosa Pardo
2017, 22(3): 783-790 doi: 10.3934/dcdsb.2017038 +[Abstract](40) +[HTML](16) +[PDF](724.6KB)

Building on the a priori estimates established in [3], we obtain a priori estimates for classical solutions to ellipticproblems with Dirichlet boundary conditions on regions with convex-starlike boundary. This includes ring-like regions. Arguments that go back to [4] are used to prove a priori bounds near the convex part of the boundary.Using that the boundary term in the Pohozaev identity on the boundary of a star-like region does not change sign, the proof isconcluded.

Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management
Renhao Cui , Haomiao Li , Linfeng Mei and  Junping Shi
2017, 22(3): 791-807 doi: 10.3934/dcdsb.2017039 +[Abstract](41) +[HTML](2) +[PDF](527.5KB)

A reaction-diffusion logistic population model with spatially nonhomogeneous harvesting is considered. It is shown that when the intrinsic growth rate is larger than the principal eigenvalue of the protection zone, then the population is always sustainable; while in the opposite case, there exists a maximum allowable catch to avoid the population extinction. The existence of steady state solutions is also studied for both cases. The existence of an optimal harvesting pattern is also shown, and theoretical results are complemented by some numerical simulations for one-dimensional domains.

Effects of superinfection and cost of immunity on host-parasite co-evolution
Liman Dai and  Xingfu Zou
2017, 22(3): 809-829 doi: 10.3934/dcdsb.2017040 +[Abstract](31) +[HTML](2) +[PDF](590.0KB)

In this paper, we investigate the cost of immunological up- regulation caused by infection in a between-host transmission dynamical model with superinfection. After introducing a mutant host to an existing model, we explore this problem in (A) monomorphic case and (B) dimorphic case. For (A), we assume that only strain 1 parasite can infect the mutant host. We identify an appropriate fitness for the invasion of the mutant host by analyzing the local stability of the mutant free equilibrium. After specifying a trade-off between the production and recovery rates of infected hosts, we employ the adaptive dynamical approach to analyze the evolutionary and convergence stabilities of the corresponding singular strategy, leading to some conditions for continuously stable strategy, evolutionary branching point and repeller. For (B), a new fitness is introduced to measure the invasion of mutant host under the assumption that both parasite strains can infect the mutant host. By considering two trade-off functions, we can study the conditions for evolutionary stability, isoclinic stability and absolute convergence stability of the singular strategy. Our results show that the host evolution would not favour high degree of immunological up-regulation; moreover, superinfection would help the parasite with weaker virulence persist in hosts.

Extinction and uniform strong persistence of a size-structured population model
Keng Deng and  Yixiang Wu
2017, 22(3): 831-840 doi: 10.3934/dcdsb.2017041 +[Abstract](32) +[HTML](0) +[PDF](364.5KB)

In this paper, we study the long-time behavior of a size-structured population model. We define a basic reproduction number \begin{document}$\mathcal{R}$\end{document} and show that the population dies out in the long run if \begin{document}$\mathcal{R}<1$\end{document}. If \begin{document}$\mathcal{R}>1$\end{document}, the model has a unique positive equilibrium, and the total population is uniformly strongly persistent. Most importantly, we show that there exists a subsequence of the total population converging to the positive equilibrium.

Malaria incidence and anopheles mosquito density in irrigated and adjacent non-irrigated villages of Niono in Mali
Moussa Doumbia and  Abdul-Aziz Yakubu
2017, 22(3): 841-857 doi: 10.3934/dcdsb.2017042 +[Abstract](29) +[HTML](2) +[PDF](1077.5KB)

In this paper, we extend the mathematical model framework of Dembele et al. (2009), and use it to study malaria disease transmission dynamics and control in irrigated and non-irrigated villages of Niono in Mali. In case studies, we use our "fitted" models to show that in support of the survey studies of Dolo et al., the female mosquito density in irrigated villages of Niono is much higher than that of the adjacent non-irrigated villages. Many parasitological surveys have observed higher incidence of malaria in non-irrigated villages than in adjacent irrigated areas. Our "fitted" models support these observations. That is, there are more malaria cases in non-irrigated areas than the adjacent irrigated villages of Niono. As in Chitnis et al., we use sensitivity analysis on the basic reproduction numbers in constant and periodic environments to study the impact of the model parameters on malaria control in both irrigated and non-irrigated villages of Niono.

Persistence in phage-bacteria communities with nested and one-to-one infection networks
Dan A. Korytowski and  Hal L. Smith
2017, 22(3): 859-875 doi: 10.3934/dcdsb.2017043 +[Abstract](28) +[HTML](0) +[PDF](512.4KB)

We show that a bacteria and bacteriophage system with either a perfectly nested or a one-to-one infection network is permanent, a.k.a uniformly persistent, provided that bacteria that are superior competitors for nutrient devote the least to defence against infection and the virus that are the most efficient at infectinghost have the smallest host range.By ensuring that the density-dependent reduction in bacterial growth rates are independent of bacterial strain, we are able to arrive at the permanence conclusion sought by Jover et al [3].The same permanence results hold for the one-to-one infection network considered by Thingstad [9] but without virus efficiency ordering.In some special cases, we show the global stability for the nested infection network, and obtain restrictions on the global dynamics for the one-to-one network.

Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods
Dung Le
2017, 22(3): 877-893 doi: 10.3934/dcdsb.2017044 +[Abstract](30) +[HTML](0) +[PDF](405.3KB)

The global existence of classical solutions to strongly coupled parabolic systems is shown to be equivalent to the availability of an iterative scheme producing a sequence of solutions with uniform continuity in the BMO norms. Amann's results on global existence of classical solutions still hold under much weaker condition that their BMO norms do not blow up in finite time. The proof makes use of some new global and local weighted Gagliardo-Nirenberg inequalities involving BMO norms.

Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model
Chengxia Lei and  Yihong Du
2017, 22(3): 895-911 doi: 10.3934/dcdsb.2017045 +[Abstract](41) +[HTML](1) +[PDF](380.0KB)

We give a complete description of the long-time asymptotic profile of the solution to a free boundary model considered recently in [10]. This model describes the spreading of an invasive species in an environment which shifts with a constant speed, and the research of [10] indicates that the species may vanish, or spread successfully, or fall in a borderline case.In the case of successful spreading, the long-time behavior of the population is not completely understood in [10].Here we show that the spreading of the species is governed by two traveling waves, one has the speed of the shifting environment, giving the profile of the retreating tail of the population, while the other has a faster speed determined by a semi-wave, representing the profile of the advancing front of the population.

Transboundary capital and pollution flows and the emergence of regional inequalities
Simon Levin and  Anastasios Xepapadeas
2017, 22(3): 913-922 doi: 10.3934/dcdsb.2017046 +[Abstract](27) +[HTML](0) +[PDF](366.3KB)

We seek to explain the emergence of spatial heterogeneity regarding development and pollution on the basis of interactions associated with the movement of capital and polluting activities from one economy to another. We use a simple dynamical model describing capital accumulation along the lines of a fixed-savings-ratio Solow-type model capable of producing endogenous growth and convergence behavior, and pollution accumulation in each country with pollution diffusion between countries or regions. The basic mechanism underlying the movements of capital across space is the quest for locations where the marginal productivity of capital is relatively higher than the productivity at the location of origin. The notion that capital moves to locations of relatively higher productivity but not necessarily from locations of high concentration to locations of low concentration, does not face difficulties associated with the Lucas paradox. We show that, for a wide range of capital and pollution rates of flow, spatial heterogeneity emerges even between two economies with identical fundamental structures. These results can be interpreted as suggesting that the neoclassical convergence hypothesis might not hold under differential rates of flow of capital and polluting activities among countries of the same fundamental structure.

Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems
Julián López-Gómez , Marcela Molina-Meyer and  Paul H. Rabinowitz
2017, 22(3): 923-946 doi: 10.3934/dcdsb.2017047 +[Abstract](41) +[HTML](2) +[PDF](1861.0KB)

In [12], the structure of the set of possible solutions of a degenerate boundary value problem was studied. For solutions with one interior zero, there were two possibilities for the solution set. In this paper, numerical examples are given showing each of these possibilities can occur.

A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects
Komi Messan and  Yun Kang
2017, 22(3): 947-976 doi: 10.3934/dcdsb.2017048 +[Abstract](194) +[HTML](0) +[PDF](2277.8KB)

We propose and study a two patch Rosenzweig-MacArthur prey-predator model with immobile prey and predator using two dispersal strategies. The first dispersal strategy is driven by the prey-predator interaction strength, and the second dispersal is prompted by the local population density of predators which is referred as the passive dispersal. The dispersal strategies using by predator are measured by the proportion of the predator population using the passive dispersal strategy which is a parameter ranging from 0 to 1. We focus on how the dispersal strategies and the related dispersal strengths affect population dynamics of prey and predator, hence generate different spatial dynamical patterns in heterogeneous environment. We provide local and global dynamics of the proposed model. Based on our analytical and numerical analysis, interesting findings could be summarized as follow: (1) If there is no prey in one patch, then the large value of dispersal strength and the large predator population using the passive dispersal in the other patch could drive predator extinct at least locally. However, the intermediate predator population using the passive dispersal could lead to multiple interior equilibria and potentially stabilize the dynamics; (2) The large dispersal strength in one patch may stabilize the boundary equilibrium and lead to the extinction of predator in two patches locally when predators use two dispersal strategies; (3) For symmetric patches (i.e., all the life history parameters are the same except the dispersal strengths), the large predator population using the passive dispersal can generate multiple interior attractors; (4) The dispersal strategies can stabilize the system, or destabilize the system through generating multiple interior equilibria that lead to multiple attractors; and (5) The large predator population using the passive dispersal could lead to no interior equilibrium but both prey and predator can coexist through fluctuating dynamics for almost all initial conditions.

Management strategies in a malaria model combining human and transmission-blocking vaccines
Jemal Mohammed-Awel , Ruijun Zhao , Eric Numfor and  Suzanne Lenhart
2017, 22(3): 977-1000 doi: 10.3934/dcdsb.2017049 +[Abstract](150) +[HTML](0) +[PDF](1164.8KB)

We propose a new mathematical model studying control strategies of malaria transmission. The control is a combination of human and transmission-blocking vaccines and vector control (larvacide). When the disease induced death rate is large enough, we show the existence of a backward bifurcation analytically if vaccination control is not used, and numerically if vaccination is used. The basic reproduction number is a decreasing function of the vaccination controls as well as the vector control parameters, which means that any effort on these controls will reduce the burden of the disease. Numerical simulation suggests that the combination of the vaccinations and vector control may help to eradicate the disease. We investigate optimal strategies using the vaccinations and vector controls to gain qualitative understanding on how the combinations of these controls should be used to reduce disease prevalence in malaria endemic setting. Our results show that the combination of the two vaccination controls integrated with vector control has the highest impact on reducing the number of infected humans and mosquitoes.

Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer
Erica M. Rutter and  Yang Kuang
2017, 22(3): 1001-1021 doi: 10.3934/dcdsb.2017050 +[Abstract](77) +[HTML](0) +[PDF](1272.0KB)

Advanced prostate cancer is often treated by androgen deprivation therapy, which is initially effective but gives rise to fatal treatment-resistant cancer. Intermittent androgen deprivation therapy improves the quality of life of patients and may delay resistance towards treatment. Immunotherapy alters the bodies immune system to help fight cancer and has proven effective in certain types of cancer. We propose a model incorporating androgen deprivation therapy (intermittent and continual) in conjunction with dendritic cell vaccine immunotherapy. Simulations are run to determine the sensitivity of cancer growth to dendritic cell vaccine therapy administration schedule. We consider the limiting case where dendritic cells are administered continuously and perform analysis on the full model and the limiting cases of the model to determine necessary conditions for global stability of cancer eradication.

Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications
Wenxian Shen and  Xiaoxia Xie
2017, 22(3): 1023-1047 doi: 10.3934/dcdsb.2017051 +[Abstract](30) +[HTML](0) +[PDF](509.3KB)

In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.

Advection control in parabolic PDE systems for competitive populations
Kokum R. De Silva , Tuoc V. Phan and  Suzanne Lenhart
2017, 22(3): 1049-1072 doi: 10.3934/dcdsb.2017052 +[Abstract](43) +[HTML](1) +[PDF](1311.0KB)

This paper investigates the response of two competing species to a given resource using optimal control techniques. We explore the choices of directed movement through controlling the advective coefficients in a system of parabolic partial differential equations. The objective is to maximize the abundance represented by a weighted combination of the two populations while minimizing the 'risk' costs caused by the movements.

Eigenvectors of homogeneous order-bounded order-preserving maps
Horst R. Thieme
2017, 22(3): 1073-1097 doi: 10.3934/dcdsb.2017053 +[Abstract](33) +[HTML](1) +[PDF](469.5KB)

The existence of eigenvectors associated with the cone spectral radius is shown for homogenous, order-preserving, continuous maps that have compact and order-bounded powers (iterates). The order-boundedness makes it possible to show the existence of eigenvectors for perturbations of the maps using Hilbert's projective metric, while the power compactness or similar compactness properties together with a uniform continuity condition let the eigenvectors of the perturbations converge to an eigenvector of the original map.

On carrying-capacity construction, metapopulations and density-dependent mortality
J. X. Velasco-Hernández , M. Núñez-López , G. Ramírez-Santiago and  M. Hernández-Rosales
2017, 22(3): 1099-1110 doi: 10.3934/dcdsb.2017054 +[Abstract](34) +[HTML](3) +[PDF](994.5KB)

We present a mathematical model for competition between species that includes variable carrying capacity within the framework of niche construction. We make use the classical Lotka-Volterra system for species competition and introduce a new variable which contains the dynamics of the constructed niche. The paper illustrates that the total available patches at equilibrium always exceeds the constructedniche at equilibrium in the absence of species.

On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3
Zhi-Cheng Wang , Hui-Ling Niu and  Shigui Ruan
2017, 22(3): 1111-1144 doi: 10.3934/dcdsb.2017055 +[Abstract](93) +[HTML](2) +[PDF](626.9KB)

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space

where \begin{document} $\mathbf{x}∈ \mathbb{R}^3$ \end{document} and \begin{document} $t>0$ \end{document}. For the bistable case, namely \begin{document} $k_1,k_2>1$ \end{document}, it is well known that the system admits a one-dimensional monotone traveling front \begin{document} $\mathbf{\Phi}(x+ct)=\left(\Phi_1(x+ct),\Phi_2(x+ct)\right)$ \end{document} connecting two stable equilibria \begin{document} $\mathbf{E}_u=(1,0)$ \end{document} and \begin{document} $\mathbf{E}_v=(0,1)$ \end{document}, where \begin{document} $c∈\mathbb{R}$ \end{document} is the unique wave speed. Recently, two-dimensional Ⅴ-shaped fronts and high-dimensional pyramidal traveling fronts have been studied under the assumption that \begin{document} $c>0$ \end{document}. In this paper it is shown that for any \begin{document} $s>c>0$ \end{document}, the system admits axisymmetric traveling fronts

in \begin{document} $\mathbb{R}^3$ \end{document} connecting \begin{document} $\mathbf{E}_u=(1,0)$ \end{document} and \begin{document} $\mathbf{E}_v=(0,1)$ \end{document}, where \begin{document} $\mathbf{x}^\prime∈\mathbb{R}^2$ \end{document}. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the \begin{document} $x_3$ \end{document}-axis. Moreover, some important qualitative properties of the axisymmetric traveling fronts are given. When \begin{document} $s$ \end{document} tends to \begin{document} $c$ \end{document}, it is proven that the axisymmetric traveling fronts converge locally uniformly to planar traveling wave fronts in \begin{document} $\mathbb{R}^3$ \end{document}. The existence of axisymmetric traveling fronts is obtained by constructing a sequence of pyramidal traveling fronts and taking its limit. The qualitative properties are established by using the comparison principle and appealing to the asymptotic speed of propagation for the resulting system. Finally, the nonexistence of axisymmetric traveling fronts with concave/convex level sets is discussed.

Individual based models and differential equations models of nosocomial epidemics in hospital intensive care units
Glenn F. Webb
2017, 22(3): 1145-1166 doi: 10.3934/dcdsb.2017056 +[Abstract](38) +[HTML](2) +[PDF](2326.4KB)

Mathematical models of antibiotic resistant infection epidemics in hospital intensive care units are developed with two modeling methods, individual based models and differential equations based models. Both models dynamically track uninfected patients, patients infected with a nonresistant bacterial strain not on antibiotics, patients infected with a nonresistant bacterial strain on antibiotics, and patients infected with a resistant bacterial strain. The outputs of the two modeling methods are shown to be complementary with respect to a common parameterization, which justifies the differential equations modeling approach for very small patient populations present in an intensive care unit. The model outputs are classified with respect to parameters to distinguish the extinction or endemicity of the bacterial strains. The role of stewardship of antibiotic use is analyzed for mitigation of these nosocomial epidemics.

Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion
Yixiang Wu , Necibe Tuncer and  Maia Martcheva
2017, 22(3): 1167-1187 doi: 10.3934/dcdsb.2017057 +[Abstract](36) +[HTML](5) +[PDF](793.3KB)

This paper investigates a two strain SIS model with diffusion, spatially heterogeneous coefficients of the reaction part and distinct diffusion rates of the separate epidemiological classes. First, it is shown that the model has bounded classical solutions. Next, it is established that the model with spatially homogeneous coefficients leads to competitive exclusion and no coexistence is possible in this case. Furthermore, it is proved that if the invasion number of strain $j$ is larger than one, then the equilibrium of strain $i$ is unstable; if, on the other hand, the invasion number of strain $j$ is smaller than one, then the equilibrium of strain $i$ is neutrally stable. In the case when all diffusion rates are equal, global results on competitive exclusion and coexistence of the strains are established. Finally, evolution of dispersal scenario is considered and it is shown that the equilibrium of the strain with the larger diffusion rate is unstable. Simulations suggest that in this case the equilibrium of the strain with the smaller diffusion rate is stable.

The Filippov equilibrium and sliding motion in an internet congestion control model
Shu Zhang and  Yuan Yuan
2017, 22(3): 1189-1206 doi: 10.3934/dcdsb.2017058 +[Abstract](72) +[HTML](1) +[PDF](579.2KB)

We consider an Internet congestion control system which is presented as a group of differential equations with time delay, modeling the random early detection (RED) algorithm. Although this model achieves success in many aspects, some basic problems are not clear. We provide the result on the existence of the equilibrium and the positivity and boundedness of the solution. Also, we implement the model by route switch mechanism, based on the minimum delay principle, to model the dynamic routing. For the simple network topology, we show that the Filippov solution exists under some restrictions on parameters. For the case with a single user group and two alternative links, we prove that the discontinuous boundary, or equivalently the sliding region, always exists and is locally attractive. This result implies that for some cases this type of routing may deviate from the purpose of the original design.

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