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Discrete & Continuous Dynamical Systems - S

2011 , Volume 4 , Issue 6

Issue on biomathematics: Newly developed applied mathematics
and new mathematics arising from biosciences

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Preface
Jianjun Paul Tian, Murray R. Bremner, Reinhard Laubenbacher and Banghe Li
2011, 4(6): i-ii doi: 10.3934/dcdss.2011.4.6i +[Abstract](286) +[PDF](80.5KB)
Abstract:
The First Joint Meeting of the American Mathematical Society and the Chinese Mathematical Society took place in Shanghai, China, December 17-21, 2008. It was organized by the Shanghai Mathematical Society, and hosted by Fudan University in Shanghai. Leading researchers from China and the United States participated in the conference. The conference was a major event for advancing mathematical research, and especially for developing international communication and cooperation among mathematicians from China and the United States. The conference program consisted of seven plenary talks, invited talks of eighteen special sessions, and many contributed talks. The topics in the special sessions covered a wide range of mathematics, applied mathematics and mathematical biology.

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Derivations in power-associative algebras
Joseph Bayara, André Conseibo, Artibano Micali and Moussa Ouattara
2011, 4(6): 1359-1370 doi: 10.3934/dcdss.2011.4.1359 +[Abstract](287) +[PDF](357.7KB)
Abstract:
In this paper we investigate derivations of a commutative power-associative algebra. Particular cases of stable and partially stable algebras are inspected. Some attention is paid to the Jordan case. Further results are given. Especially, we show that the core of a $n^{th}$-order Bernstein algebra which is power-associative is a Jordan algebra.
Train algebras of degree 2 and exponent 3
Joseph Bayara, André Conseibo, Moussa Ouattara and Artibano Micali
2011, 4(6): 1371-1386 doi: 10.3934/dcdss.2011.4.1371 +[Abstract](320) +[PDF](209.5KB)
Abstract:
In this paper we investigate the structure of weighted algebras satisfying the equation $(x^3)^2 = \omega(x)^3x^3$, a class of algebras properly containing the class of Bernstein algebras. We give the classification of these algebras in dimension three. Some results about the structure of algebras satisfying the more general equation $(x^n)^2 = \omega(x)^nx^n$, for $n\geq 2$, are also obtained.
Polynomial identities for ternary intermolecular recombination
Murray R. Bremner
2011, 4(6): 1387-1399 doi: 10.3934/dcdss.2011.4.1387 +[Abstract](265) +[PDF](327.8KB)
Abstract:
The operation of binary intermolecular recombination, originating in the theory of DNA computing, permits a natural generalization to $n$-ary operations which perform simultaneous recombination of $n$ molecules. In the case $n = 3$, we use computer algebra to determine the polynomial identities of degree $\le 9$ satisfied by this trilinear nonassociative operation. Our approach requires computing a basis for the nullspace of a large integer matrix, and for this we compare two methods: the row canonical form, and the Hermite normal form with lattice basis reduction. In the conclusion, we formulate some conjectures for the general case of $n$-ary intermolecular recombination.
Topological symmetry groups of $K_{4r+3}$
Dwayne Chambers, Erica Flapan and John D. O'Brien
2011, 4(6): 1401-1411 doi: 10.3934/dcdss.2011.4.1401 +[Abstract](350) +[PDF](343.5KB)
Abstract:
We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of a complete graph of the form $K_{4r+3}$ in $S^3$.
Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning
Wenrui Hao, Jonathan D. Hauenstein, Bei Hu, Yuan Liu, Andrew J. Sommese and Yong-Tao Zhang
2011, 4(6): 1413-1428 doi: 10.3934/dcdss.2011.4.1413 +[Abstract](299) +[PDF](547.6KB)
Abstract:
The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D. Lander, Q. Nie, Journal of Theoretical Biology, 248 (2007), 579--589] has multiple steady state solutions. In this paper, we describe the computation of seven steady state solutions found by discretizing the boundary value problem using a finite difference scheme and solving the resulting polynomial system using algorithms from numerical algebraic geometry. The stability of each of these steady state solutions is studied by mathematical analysis and numerical simulations via a time marching approach. The results of this paper show that three of the seven steady state solutions are stable and the location of the organizer of a zebrafish embryo determines which stable steady state pattern the multi-stability system converges to. Numerical simulations also show that the system is robust with respect to the change of the organizer size.
Equilibrium submanifold for a biological system
Hongyu He and Naohiro Kato
2011, 4(6): 1429-1441 doi: 10.3934/dcdss.2011.4.1429 +[Abstract](287) +[PDF](402.6KB)
Abstract:
The complexity in a biological system may be caused by both the number of variables involved and the number of system constants that can vary. A biological system in the subcellular level often stabilizes after a certain period of time. Its asymptote can then be described as an equilibrium under certain continuity assumptions. The biological quantities at the equilibrium can be detected by experiments and they observe some mathematical equations. The purpose of this paper is to study the equilibrium submanifold of vesicle trafficking in a two-compartment system. We compute the equilibrium submanifold under some fairly general assumption on the system constants. The disconnectedness of the equilibrium submanifold may have biological implications. We show that, unlike many other systems, the equilibrium is determined largely by system constants rather than the initial state. In particular, the equilibrium submanifold is locally a real algebraic variety, with small generic dimension and large degenerate dimension. Our result suggests that some biological system may be studied by algebraic or geometric methods.
Boolean models of bistable biological systems
Franziska Hinkelmann and Reinhard Laubenbacher
2011, 4(6): 1443-1456 doi: 10.3934/dcdss.2011.4.1443 +[Abstract](284) +[PDF](889.0KB)
Abstract:
This paper presents an algorithm for approximating certain types of dynamical systems given by a system of ordinary delay differential equations by a Boolean network model. Often Boolean models are much simpler to understand than complex differential equations models. The motivation for this work comes from mathematical systems biology. While Boolean mechanisms do not provide information about exact concentration rates or time scales, they are often sufficient to capture steady states and other key dynamics. Due to their intuitive nature, such models are very appealing to researchers in the life sciences. This paper is focused on dynamical systems that exhibit bistability and are described by delay equations. It is shown that if a certain motif including a feedback loop is present in the wiring diagram of the system, the Boolean model captures the bistability of molecular switches. The method is applied to two examples from biology, the lac operon and the phage $\lambda$ lysis/lysogeny switch.
The dynamics of zeroth-order ultrasensitivity: A critical phenomenon in cell biology
Qingdao Huang and Hong Qian
2011, 4(6): 1457-1464 doi: 10.3934/dcdss.2011.4.1457 +[Abstract](288) +[PDF](374.8KB)
Abstract:
It is well known since the pioneering work of Goldbeter and Koshland [Proc. Natl. Acad. Sci. USA, vol. 78, pp. 6840-6844 (1981)] that cellular phosphorylation- dephosphorylation cycle (PdPC), catalyzed by kinase and phosphatase under saturated condition with zeroth order enzyme kinetics, exhibits ultrasensitivity, sharp transition. We analyse the dynamics aspects of the zeroth order PdPC kinetics and show a critical slowdown akin to the phase transition in condensed matter physics. We demonstrate that an extremely simple, though somewhat mathematically "singular" model is a faithful representation of the ultrasentivity phenomenon. The simplified mathematical model will be valuable, as a component, in developing complex cellular signaling netowrk theory as well as having a pedagogic value.
An enzyme kinetics model of tumor dormancy, regulation of secondary metastases
Yangjin Kim and Khalid Boushaba
2011, 4(6): 1465-1498 doi: 10.3934/dcdss.2011.4.1465 +[Abstract](289) +[PDF](967.0KB)
Abstract:
In this paper we study 1 dimensional (1D) and 2D extended version of a two compartment model for tumor dormancy suggested by Boushaba et al. [3]. The model is based on the idea that the vascularization of a secondary tumor can be suppressed by inhibitor originating from a larger primary tumor. It has been observed emergence of a polypoid melanoma at a site remote from a primary polypoid melanoma after excision of the latter. The authors observed no recurrence of the melanoma at the primary site, but did observe secondary tumors at secondary sites five to seven centimeters from the primary site within a period of one month after the excision of the primary site. 1D and 2D simulations show that when the tumors are sufficiently remote, the primary tumor will not influence the secondary tumors while, if they are too close together, the primary tumor can effectively prevent the growth of the secondary tumors, even after it is removed. The sensitivity analysis was carried out for the 1D model. It has been long observed that surgery should be followed by other treatment options such as chemotherapy. 2D simulation suggests a possible treatment options with different dosage schedule after a surgery in order to achieve better clinical outcome.
A computational study of avian influenza
Shu Liao, Jin Wang and Jianjun Paul Tian
2011, 4(6): 1499-1509 doi: 10.3934/dcdss.2011.4.1499 +[Abstract](271) +[PDF](343.3KB)
Abstract:
We propose a PDE model and conduct numerical simulation to study the temporal and spatial dynamics of the Avian Influenza, and investigate its epidemic and possibly pandemic effects in both the bird and human populations. We present several numerical examples to carefully study the population dynamics with small initial perturbations. Our results show that in the absence of external controls, any small amount of initial infection would lead to an outbreak of the influenza with considerably high death rates in both birds and human beings.
Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$
Dan Liu, Shigui Ruan and Deming Zhu
2011, 4(6): 1511-1532 doi: 10.3934/dcdss.2011.4.1511 +[Abstract](295) +[PDF](670.3KB)
Abstract:
Nongeneric bifurcation analysis near rough heterodimensional cycles associated to two saddles in $\mathbb{R}^4$ is presented under inclination flip. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincaré return map under the nongeneric conditions and further obtain the bifurcation equations. Coexistence of a heterodimensional cycle and a unique periodic orbit is proved after perturbations. New features produced by the inclination flip that heterodimensional cycles and homoclinic orbits coexist on the same bifurcation surface are shown. It is also conjectured that homoclinic orbits associated to different equilibria coexist.
Update sequence stability in graph dynamical systems
Matthew Macauley and Henning S. Mortveit
2011, 4(6): 1533-1541 doi: 10.3934/dcdss.2011.4.1533 +[Abstract](340) +[PDF](329.5KB)
Abstract:
In this article, we study finite dynamical systems defined over graphs, where the functions are applied asynchronously. Our goal is to quantify and understand stability of the dynamics with respect to the update sequence, and to relate this to structural properties of the graph. We introduce and analyze three different notions of update sequence stability, each capturing different aspects of the dynamics. When compared to each other, these stability concepts yield different conclusions regarding the relationship between stability and graph structure, painting a more complete picture of update sequence stability.
Conjectures for the existence of an idempotent in $\omega $-polynomial algebras
Michelle Nourigat and Richard Varro
2011, 4(6): 1543-1551 doi: 10.3934/dcdss.2011.4.1543 +[Abstract](236) +[PDF](321.8KB)
Abstract:
The existence of idempotent elements in baric algebras defined by $\omega$-polynomial identities ($\omega$-PI algebras) is an important problem for the study of genetic algebras. We conjecture here two criteria on the existence of an idempotent. These criteria are based on the existence of 1/2 as double root of a polynomial built from the identity defining a $\omega$-PI algebra. We show that these criteria are true in all the algebras studied until now and for which we have results concerning the existence of idempotent elements.
Backward problems of nonlinear dynamical equations on time scales
Yunfei Peng, X. Xiang and W. Wei
2011, 4(6): 1553-1564 doi: 10.3934/dcdss.2011.4.1553 +[Abstract](318) +[PDF](343.9KB)
Abstract:
In this paper, the backward problem of nonlinear dynamical equations on time scales is considered. Introducing the reasonable weak solution of the nonlinear backward problem, the existence of weak solution for nonlinear dynamical equation on time scales and its properties are presented.
Topology and dynamics of boolean networks with strong inhibition
Yongwu Rong, Chen Zeng, Christina Evans, Hao Chen and Guanyu Wang
2011, 4(6): 1565-1575 doi: 10.3934/dcdss.2011.4.1565 +[Abstract](273) +[PDF](190.9KB)
Abstract:
A major challenge in systems biology is to understand interactions within biological systems. Such a system often consists of units with various levels of activities that evolve over time, mathematically represented by the dynamics of the system. The interaction between units is mathematically represented by the topology of the system. We carry out some mathematical analysis on the connections between topology and dynamics of such networks. We focus on a specific Boolean network model - the Strong Inhibition Model. This model defines a natural map from the space of all possible topologies on the network to the space of all possible dynamics on the same network. We prove this map is neither surjective nor injective. We introduce the notions of "redundant edges" and "dormant vertices" which capture the non-injectiveness of the map. Using these, we determine exactly when two different topologies yield the same dynamics and we provide an algorithm that determines all possible network solutions given a dynamics.
Algebraic model of non-Mendelian inheritance
Jianjun Paul Tian
2011, 4(6): 1577-1586 doi: 10.3934/dcdss.2011.4.1577 +[Abstract](288) +[PDF](247.7KB)
Abstract:
Evolution algebra theory is used to study non-Mendelian inheritance, particularly organelle heredity and population genetics of Phytophthora infectans. We not only can explain a puzzling feature of establishment of homoplasmy from heteroplasmic cell population and the coexistence of mitochondrial triplasmy, but also can predict all mechanisms to form the homoplasmy of cell populations, which are hypothetical mechanisms in current mitochondrial disease research. The algebras also provide a way to easily find different genetically dynamic patterns from the complexity of the progenies of Phytophthora infectans which cause the late blight of potatoes and tomatoes. Certain suggestions to pathologists are made as well.
Periodic solutions of a model for tumor virotherapy
Daniel Vasiliu and Jianjun Paul Tian
2011, 4(6): 1587-1597 doi: 10.3934/dcdss.2011.4.1587 +[Abstract](340) +[PDF](590.3KB)
Abstract:
In this article we study periodic solutions of a mathematical model for brain tumor virotherapy by finding Hopf bifurcations with respect to a biological significant parameter, the burst size of the oncolytic virus. The model is derived from a PDE free boundary problem. Our model is an ODE system with six variables, five of them represent different cell or virus populations, and one represents tumor radius. We prove the existence of Hopf bifurcations, and periodic solutions in a certain interval of the value of the burst size. The evolution of the tumor radius is much influenced by the value of the burst size. We also provide a numerical confirmation.
Novel dynamics of a simple Daphnia-microparasite model with dose-dependent infection
Kaifa Wang and Yang Kuang
2011, 4(6): 1599-1610 doi: 10.3934/dcdss.2011.4.1599 +[Abstract](329) +[PDF](348.7KB)
Abstract:
Many experiments reveal that Daphnia and its microparasite populations vary strongly in density and typically go through pronounced cycles. To better understand such dynamics, we formulate a simple two dimensional autonomous ordinary differential equation model for Daphnia magna-microparasite infection with dose-dependent infection. This model has a basic parasite production number $R_0=0$, yet its dynamics is much richer than that of the classical mathematical models for host-parasite interactions. In particular, Hopf bifurcation, stable limit cycle, homoclinic and heteroclinic orbit can be produced with suitable parameter values. The model indicates that intermediate levels of parasite virulence or host growth rate generate more complex infection dynamics.
On fuzzy filters of Heyting-algebras
Wei Wang and Xiao-Long Xin
2011, 4(6): 1611-1619 doi: 10.3934/dcdss.2011.4.1611 +[Abstract](364) +[PDF](287.4KB)
Abstract:
The concept of fuzzy filter of Heyting-algebras was introduced and some important properties were discussed. Some special kinds of fuzzy filters were defined and we prove that fuzzy Boolean filter is equivelent to fuzzy implicative filter in Heyting-algebras. And the relation among the fuzzy filters were proposed.
Turing instability in a coupled predator-prey model with different Holling type functional responses
Zhifu Xie
2011, 4(6): 1621-1628 doi: 10.3934/dcdss.2011.4.1621 +[Abstract](340) +[PDF](330.5KB)
Abstract:
In a reaction-diffusion system, diffusion can induce the instability of a positive equilibrium which is stable with respect to a constant perturbation, therefore, the diffusion may create new patterns when the corresponding system without diffusion fails, as shown by Turing in 1950s. In this paper we study a coupled predator-prey model with different Holling type functional responses, where cross-diffusions are included in such a way that the prey runs away from predator and the predator chase preys. We conduct the Turing instability analysis for each Holling functional response. We prove that if a positive equilibrium solution is linearly stable with respect to the ODE system of the predator-prey model, then it is also linearly stable with respect to the model. So diffusion and cross-diffusion in the predator-prey model with Holling type functional responses given in this paper can not drive Turing instability. However, diffusion and cross-diffusion can still create non-constant positive solutions for the model.
Dynamics of boolean networks
Yi Ming Zou
2011, 4(6): 1629-1640 doi: 10.3934/dcdss.2011.4.1629 +[Abstract](268) +[PDF](379.8KB)
Abstract:
Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an $n$-dimensional vector space over the field of two elements to itself. A fundamental problem in studying these dynamical systems is to link their long term behaviors to the structures of the functions that define them. In this paper, a method for deriving a Boolean network's dynamical information via its disjunctive normal form is explained. For a given Boolean network, a matrix with entries $0$ and $1$ is associated with the polynomial function that represents the network, then the information on the fixed points and the limit cycles is derived by analyzing the matrix. The described method provides an algorithm for the determination of the fixed points from the polynomial expression of a Boolean network. The method can also be used to construct Boolean networks with prescribed limit cycles and fixed points. Examples are provided to explain the algorithm.

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