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Discrete & Continuous Dynamical Systems - B

2016 , Volume 21 , Issue 2

Special issue dedicated to the memory of Paul Waltman

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Special issue dedicated to the memory of Paul Waltman
Sze-Bi Hsu , Hal L. Smith and  Xiaoqiang Zhao
2016, 21(2): i-ii doi: 10.3934/dcdsb.2016.21.2i +[Abstract](45) +[PDF](142.2KB)
This volume is dedicated to the memory of Paul Waltman. Many of the authors of articles contained here were participants at the NCTS International Conference on Nonlinear Dynamics with Applications to Biology held May 28-30, 2014 at National Tsing-Hua University, Hsinchu, Taiwan. The purpose of the conference was to survey new developments in nonlinear dynamics and its applications to biology and to honor the memory of Professor Paul Waltman for his influence on the development of Mathematical Biology and Dynamical Systems. Attendees at the conference included Paul's sons Fred and Dennis, many of Paul's former doctoral and post-doctoral students, many others who, although not students of Paul, nevertheless were recipients of Paul's valuable advice and council, and many colleagues from all over the world who were influenced by Paul's mathematics and by his personality. We thank the NCTS for its financial support of the conference and Dr. J.S.W. Wong for supporting the conference banquet.

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Nutrient limitations as an explanation of Gompertzian tumor growth
Ebraheem O. Alzahrani and  Yang Kuang
2016, 21(2): 357-372 doi: 10.3934/dcdsb.2016.21.357 +[Abstract](37) +[PDF](436.5KB)
An intuitive and influential two-compartment model of cancer cell growth proposed by Gyllenberg and Webb in 1989 [18] with transition rates between proliferating and quiescent cells reproduces some important features of the well known Gompertzian growth model. However, other plausible mechanisms may also be capable of producing similar dynamics. In this paper, we formulate a resource limited three-compartment model of avascular spherical solid tumor growth and study its dynamics. The resource, such as oxygen, is assumed to enter the tumor proportional to its surface area and the dead cells form the necrotic core inside the tumor. We show the tumor growth of our model mimics that of Gompertzian model, and the solutions of our model are naturally bounded. We also identify general and explicit expressions of the tumor final sizes and study the stability of the tumor at steady states. In contrast to the Gyllenberg-Webb model, our model confirms that tumor size at the positive steady state is strictly decreasing function of the dead cell removal rate. We also present two intriguing mathematical open questions.
Microbial disease in coral reefs: An ecosystem in transition
Joydeb Bhattacharyya and  Samares Pal
2016, 21(2): 373-398 doi: 10.3934/dcdsb.2016.21.373 +[Abstract](31) +[PDF](2311.0KB)
Infectious disease outbreaks are considered an important factor for the degradation of coral reefs. Reef-building coral species are susceptible to the influences of black band disease (BBD), characterized by cyanobacteria-dominated microbial mat that migrates rapidly across infected corals, leaving empty coral skeletons behind. We investigate coral-macroalgal phase shift in presence of BBD infection by means of an eco-epidemiological model under the assumption that the transmission of BBD occurs through both contagious and non-contagious pathways. It is observed that in presence of low coral-recruitment rate on algal turf, reduced herbivory and high macroalgal immigration, the system exhibits hysteresis through a saddle-node bifurcation and a transcritical bifurcation. Also, the system undergoes a supercritical Hopf bifurcation followed by a saddle-node bifurcation if BBD-transmission rate crosses certain critical value. We examine the effects of incubation time lag of infectious agents develop in susceptible corals after coming in contact with infected corals and a time lag in the recovery of algal turf in response to grazing of herbivores by performing equilibrium and stability analyses of delay-differential forms of the ODE model. Computer simulations have been carried out to illustrate different analytical results.
Computation of $\mathcal R $ in age-structured epidemiological models with maternal and temporary immunity
Zhilan Feng , Qing Han , Zhipeng Qiu , Andrew N. Hill and  John W. Glasser
2016, 21(2): 399-415 doi: 10.3934/dcdsb.2016.21.399 +[Abstract](49) +[PDF](1207.0KB)
For infectious diseases such as pertussis, susceptibility is determined by immunity, which is chronological age-dependent. We consider an age-structured epidemiological model that accounts for both passively acquired maternal antibodies that decay and active immunity that wanes, permitting re-infection. The model is a 6-dimensional system of partial differential equations (PDE). By assuming constant rates within each age-group, the PDE system can be reduced to an ordinary differential equation (ODE) system with aging from one age-group to the next. We derive formulae for the effective reproduction number ${\mathcal R}$ and provide their biological interpretation in some special cases. We show that the disease-free equilibrium is stable when ${\mathcal R}<1$ and unstable if ${\mathcal R}>1$.
Stefan problem, traveling fronts, and epidemic spread
Karl P. Hadeler
2016, 21(2): 417-436 doi: 10.3934/dcdsb.2016.21.417 +[Abstract](35) +[PDF](480.2KB)
The scalar reaction diffusion equation with a nonlinearity of logistic type has a minimal speed $c_0$ for standard traveling fronts. It is shown that also for speeds $0 < c < c_0$ there are traveling fronts but these are solutions to free boundary value (Stefan) problems. Furthermore, these speeds depend in a monotone way on the Stefan coefficient which links the loss of matter at the free boundary to the displacement per time. The results are extended to correlated random walks, Cattaneo systems and, in particular, to models for epidemic spread. In the epidemic problems a dichotomy phenomenon shows up: For small values of the Stefan coefficient there are no fronts indicating that for such values and certain data the free boundary stays bounded.
Classification of potential flows under renormalization group transformation
Sze-Bi Hsu , Bernold Fiedler and  Hsiu-Hau Lin
2016, 21(2): 437-446 doi: 10.3934/dcdsb.2016.21.437 +[Abstract](28) +[PDF](323.3KB)
Competitions between different interactions in strongly correlated electron systems often lead to exotic phases. Renormalization group is one of the powerful techniques to analyze the competing interactions without presumed bias. It was recently shown that the renormalization group transformations to the one-loop order in many correlated electron systems are described by potential flows. Here we prove several rigorous theorems in the presence of renormalization-group potential and find the complete classification for the potential flows. In addition, we show that the relevant interactions blow up at the maximal scaling exponent of unity, explaining the puzzling power-law Ansatz found in previous studies. The above findings are of great importance in building up the hierarchy for relevant couplings and the complete classification for correlated ground states in the presence of generic interactions.
An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius
Wen Jin and  Horst R. Thieme
2016, 21(2): 447-470 doi: 10.3934/dcdsb.2016.21.447 +[Abstract](32) +[PDF](479.8KB)
Persistence and local stability of the extinction state are studied for discrete-time population models $x_n = F(x_{n-1})$, $n \in \mathbb{N}$, with a map $F$ on the cone $X_+$ of an ordered normed vector space $X$. Since sexual reproduction is accounted for, the first order approximation of $F$ at 0 is an order-preserving homogeneous map $B$ on $X_+$ that is not additive. The cone spectral radius of $B$ acts as the threshold parameter that separates persistence from local stability of 0, the extinction state. An important ingredient of the persistence theory for the induced semiflow is a homogeneous order-preserving eigenfunctional $\theta:X_+ \to \mathbb{R}_+$ of $B$ that is associated with the cone spectral radius and interacts with an appropriate persistence function $\rho$. Applications are presented for spatially distributed or rank-structured populations that reproduce sexually.
Spread of phage infection of bacteria in a petri dish
Don A. Jones , Hal L. Smith and  Horst R. Thieme
2016, 21(2): 471-496 doi: 10.3934/dcdsb.2016.21.471 +[Abstract](42) +[PDF](494.9KB)
We extend our previous work on the spatial spread of phage infection of immobile bacteria on an agar coated plate by explicitly including loss of viruses by both adsorption to bacteria and by decay of free viruses and by including a distributed virus latent period and distributed burst size rather than fixed values of these key parameters. We extend earlier results on the spread of virus and on the existence of traveling wave solutions when the basic reproductive number for virus, $\mathcal{R}_0$, exceeds one and we compare the results with those obtained in earlier work. Finally, we formulate and analyze a model of multiple virus strains competing to infect a common bacterial host in a petri dish.
Cascade flocking with free-will
Le Li , Lihong Huang and  Jianhong Wu
2016, 21(2): 497-522 doi: 10.3934/dcdsb.2016.21.497 +[Abstract](28) +[PDF](860.4KB)
We consider a self-organized system with a hierarchy structure to allow multiple leaders in the highest rank, and with free-will. In the model, we use both Cucker-Smale and Motsch-Tadmor functions for the pair influence of agents, and we derive suffcient conditions for such a system to converge to a flock, where agents ultimately move in the same velocity. We provide examples to show our suffcient conditions are sharp, and we numerically observe that such a self-organized system may have agents moving in different (final) velocities but maintain finite distance from each other due to the free-will.
Competition of two phytoplankton species for light with wavelength
Chiu-Ju Lin
2016, 21(2): 523-536 doi: 10.3934/dcdsb.2016.21.523 +[Abstract](27) +[PDF](397.4KB)
We study the competition of two phytoplankton species for a single resource-light with wavelength in a well mixed water column. The model was proposed by Stomp et al. in 2007, in this model each species prefers different interval of light spectrum. Their experimental results show that colorful phytoplankton species coexist.
    We classify the global behavior of the model by stability of equilibrium and the theory of monotone dynamical systems. The conclusion is that one of the following holds: one species competitively excludes the other, two species coexist, bistability of two species. We also analyze the special case when species have linear growth function, and the outcome is either competitive exclusion or two species coexistence; the results are consistent with the Stomp's experiments in 2004.
Oscillations in age-structured models of consumer-resource mutualisms
Zhihua Liu , Pierre Magal and  Shigui Ruan
2016, 21(2): 537-555 doi: 10.3934/dcdsb.2016.21.537 +[Abstract](30) +[PDF](616.6KB)
In consumer-resource interactions, a resource is regarded as a biotic population that helps to maintain the population growth of its consumer, whereas a consumer exploits a resource and then reduces its growth rate. Bi-directional consumer-resource interactions describe the cases where each species acts as both a consumer and a resource of the other, which is the basis of many mutualisms. In uni-directional consumer-resource interactions one species acts as a consumer and the other as a material and/or energy resource while neither acts as both. In this paper we consider an age-structured model for uni-directional consumer-resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer. Examples include a predator-prey system in which the prey is able to kill or consume predator eggs or larvae and the insect pollinator and the host plant relationship in which the plants provide food, seeds, nectar and other resources for the pollinators while the pollinators have both positive and negative effects on the plants. By carrying out local analysis and bifurcation analysis of the model, we discuss the stability of the positive equilibrium and show that under some conditions a non-trivial periodic solution through Hopf bifurcation appears when the maturation parameter passes through some critical values.
On the analytic integrability of the Liénard analytic differential systems
Jaume Llibre and  Claudia Valls
2016, 21(2): 557-573 doi: 10.3934/dcdsb.2016.21.557 +[Abstract](28) +[PDF](389.5KB)
We consider the Liénard analytic differential systems $\dot x = y$, $\dot y =-g(x) -f(x)y$, where $f,g: \mathbb{R}\to \mathbb{R}$ are analytic functions and the origin is an isolated singular point. Then for such systems we characterize the existence of local analytic first integrals in a neighborhood of the origin and the existence of global analytic first integrals.
Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise
Shujuan Lü , Hong Lu and  Zhaosheng Feng
2016, 21(2): 575-590 doi: 10.3934/dcdsb.2016.21.575 +[Abstract](28) +[PDF](419.0KB)
In this work, we analyze the stochastic fractional Ginzburg-Landau equation with multiplicative noise in two spatial dimensions with a particular interest in the asymptotic behavior of its solutions. To get started, we first transfer the stochastic fractional Ginzburg-Landau equation into a random equation whose solutions generate a random dynamical system. The existence of a random attractor for the resulting random dynamical system is explored, and the Hausdorff dimension of the random attractor is estimated.
Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession
Manjun Ma and  Xiao-Qiang Zhao
2016, 21(2): 591-606 doi: 10.3934/dcdsb.2016.21.591 +[Abstract](31) +[PDF](389.3KB)
This paper is devoted to the study of propagation phenomena for a two-species competitive reaction-diffusion model with seasonal succession in the monostable case. By appealing to theory of traveling waves and spreading speeds for monotone semiflows, we establish the existence of the minimal wave speed for rightward traveling waves and its coincidence with the rightward spreading speed. We also obtain a set of sufficient conditions for the spreading speed to be linearly determinate.
Growth of single phytoplankton species with internal storage in a water column
Linfeng Mei , Sze-Bi Hsu and  Feng-Bin Wang
2016, 21(2): 607-620 doi: 10.3934/dcdsb.2016.21.607 +[Abstract](50) +[PDF](372.7KB)
In this paper, we analyze a system modeling the growth of single phytoplankton populations in a water column, where population growth increases monotonically with the nutrient quota stored within individuals. We establish a threshold result on the global extinction and persistence of phytoplankton. Condition for persistence is shown to depend on the principal eigenvalue of a boundary value problem, which is related to the physical transport properties of the water column (i.e. the diffusivity and the sinking speed), nutrient uptake rate, and growth rate.
Competition between two similar species in the unstirred chemostat
Hua Nie , Yuan Lou and  Jianhua Wu
2016, 21(2): 621-639 doi: 10.3934/dcdsb.2016.21.621 +[Abstract](41) +[PDF](404.2KB)
This paper deals with the competition between two similar species in the unstirred chemostat. Due to the strict competition of the unstirred chemostat model, the global dynamics of the system is attained by analyzing the equilibria and their stability. It turns out that the dynamics of the system essentially depends upon certain function of the growth rate. Moreover, one of the semi-trivial stationary solutions or the unique coexistence steady state is a global attractor under certain conditions. Biologically, the results indicate that it is possible for the mutant to force the extinction of resident species or to coexist with it.
Destabilization threshold curves for diffusion systems with equal diffusivity under non-diagonal flux boundary conditions
Kunimochi Sakamoto
2016, 21(2): 641-654 doi: 10.3934/dcdsb.2016.21.641 +[Abstract](20) +[PDF](496.3KB)
This article deals with destabilizations of Turing type for diffusive systems with equal diffusivity under non-diagonal flux boundary conditions. Stability-instability threshold curves in the complex plane are described as the graph of a piecewise analytic function for simple $m$-dimensional domains $(m\geq 1)$. Also analyzed are effects caused by imposing homogeneous boundary conditions of Dirichlet or Neumann type on appropriate portions of the domain boundary.
Evolutionary dynamics of a multi-trait semelparous model
Amy Veprauskas and  J. M. Cushing
2016, 21(2): 655-676 doi: 10.3934/dcdsb.2016.21.655 +[Abstract](37) +[PDF](938.7KB)
We consider a multi-trait evolutionary (game theoretic) version of a two class (juvenile-adult) semelparous Leslie model. We prove the existence of both a continuum of positive equilibria and a continuum of synchronous 2-cycles as the result of a bifurcation that occurs from the extinction equilibrium when the net reproductive number $R_{0}$ increases through $1$. We give criteria for the direction of bifurcation and for the stability or instability of each bifurcating branch. Semelparous Leslie models have imprimitive projection matrices. As a result (unlike matrix models with primitive projection matrices) the direction of bifurcation does not solely determine the stability of a bifurcating continuum. Only forward bifurcating branches can be stable and which of the two is stable depends on the intensity of between-class competitive interactions. These results generalize earlier results for single trait models. We give an example that illustrates how the dynamic alternative can change when the number of evolving traits changes from one to two.
On a two-patch predator-prey model with adaptive habitancy of predators
Xiaoying Wang and  Xingfu Zou
2016, 21(2): 677-697 doi: 10.3934/dcdsb.2016.21.677 +[Abstract](41) +[PDF](946.7KB)
A two-patch predator-prey model with the Holling type II functional response is studied, in which predators are assumed to adopt adaptive dispersal to inhabit the better patch in order to gain more fitness. Analytical conditions for the persistence and extinction of predators are obtained under different scenarios of the model. Numerical simulations are conducted which show that adaptive dispersal can stabilize the system with either weak or strong adaptation, when prey and predators tend to a globally stable equilibrium in one isolated patch and tend to limit cycles in the other. Furthermore, it is observed that the adaptive dispersal may cause torus bifurcation for the model when the prey and predators population tend to limit cycles in each isolated patch.
Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting
Dongmei Xiao
2016, 21(2): 699-719 doi: 10.3934/dcdsb.2016.21.699 +[Abstract](41) +[PDF](561.2KB)
In this paper, we investigate a class of population model with seasonal constant-yield harvesting and discuss the effect of the seasonal harvesting on the survival of the population. It is shown that the population can be survival if and only if the model has at least a periodic solution and the initial amount of population is not lower than the minimum periodic solution. And if the population goes to extinction then it must be in the finite time. As an application of the conclusion, we systemically study the global dynamics of a logstic equation with seasonal constant-yield harvesting, and prove that there exists a threshold value $h_{MSY}$ of the intensity of harvesting, called the maximum sustainable yield, which is strictly greater than the maximum sustainable yield of logstic equation with constant-yield harvesting, such that the model has exactly two periodic solutions: one is attracting and the other is repelling if $0 < h < h_{MSY}$, a unique periodic solution which is semi-stable if $h=h_{MSY}$ and all solutions which go down to zero in the finite time if $h>h_{MSY}$. Hence, the logstic equation with seasonal constant-yield harvesting undergoes saddle-node bifurcation of the periodic solution as $h$ passes through $h_{MSY}$. Biologically, these theoretic results reveal that the seasonal constant-yield harvesting can increase the maximum sustainable yield such that the ecological system persists comparing to the constant-yield harvesting.
Stochastic epidemic models driven by stochastic algorithms with constant step
Lifeng Chen and  Jifa Jiang
2016, 21(2): 721-736 doi: 10.3934/dcdsb.2016.21.721 +[Abstract](30) +[PDF](565.9KB)
In this paper we propose the stochastic epidemic model that relates directly to the deterministic counterpart and reveal close connections between these two models. Under the classic assumptions, the sample path of process eventually converges to the disease-free equilibrium, even though the corresponding deterministic flow converges to an endemic equilibrium. From the fact that disease can occur sporadically, we adjust the stochastic model slightly by introducing a stochastic incidence and establish precise connections between the long-run behavior of the discrete stochastic process and its deterministic flow approximation for large populations.

2016  Impact Factor: 0.994




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