1
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Xie X. Steady solution and its stability of a mathematical model of diabetic atherosclerosis. J Biol Dyn 2023; 17:2257734. [PMID: 37711027 PMCID: PMC10576982 DOI: 10.1080/17513758.2023.2257734] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2023] [Accepted: 09/05/2023] [Indexed: 09/16/2023]
Abstract
Atherosclerosis is a leading cause of death worldwide. Making matters worse, nearly 463 million people have diabetes, which increases atherosclerosis-related inflammation. Diabetic patients are twice as likely to have a heart attack or stroke. In this paper, we consider a simplified mathematical model for diabetic atherosclerosis involving LDL, HDL, glucose, insulin, free radicals (ROS), β cells, macrophages and foam cells, which satisfy a system of partial differential equations with a free boundary, the interface between the blood flow and the plaque. We establish the existence of small radially symmetric stationary solutions to the model and study their stability. Our analysis shows that the plague will persist due to hyperglycemia even when LDL and HDL are in normal range, hence confirms that diabetes increase the risk of atherosclerosis.
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Affiliation(s)
- Xuming Xie
- Department of Mathematics, Morgan State University, Baltimore, MD, USA
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2
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Wang H, Salmaniw Y. Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping. J Math Biol 2023; 86:71. [PMID: 37029822 DOI: 10.1007/s00285-023-01905-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2022] [Revised: 03/12/2023] [Accepted: 03/16/2023] [Indexed: 04/09/2023]
Abstract
The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in chemistry or physics. Hence, it is essential to incorporate such knowledge-based processes into animal movement models. Here, we summarize popular deterministic mathematical models derived from first principles that begin to incorporate such influences on movement behaviour mechanisms. Most generally, these models take the form of nonlocal reaction-diffusion-advection equations, where the nonlocality may appear in the spatial domain, the temporal domain, or both. Mathematical rules of thumb are provided to judge the model rationality, to aid in model development or interpretation, and to streamline an understanding of the range of difficulty in possible model conceptions. To emphasize the importance of biological conclusions drawn from these models, we briefly present available mathematical techniques and introduce some existing "measures of success" to compare and contrast the possible predictions and outcomes. Throughout the review, we propose a large number of open problems relevant to this relatively new area, ranging from precise technical mathematical challenges, to more broad conceptual challenges at the cross-section between mathematics and ecology. This review paper is expected to act as a synthesis of existing efforts while also pushing the boundaries of current modelling perspectives to better understand the influence of cognitive movement mechanisms on movement behaviours and space use outcomes.
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Affiliation(s)
- Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
| | - Yurij Salmaniw
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
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3
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Foutel-Rodier F, Blanquart F, Courau P, Czuppon P, Duchamps JJ, Gamblin J, Kerdoncuff É, Kulathinal R, Régnier L, Vuduc L, Lambert A, Schertzer E. From individual-based epidemic models to McKendrick-von Foerster PDEs: a guide to modeling and inferring COVID-19 dynamics. J Math Biol 2022; 85:43. [PMID: 36169721 PMCID: PMC9517997 DOI: 10.1007/s00285-022-01794-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2021] [Revised: 02/21/2022] [Accepted: 05/11/2022] [Indexed: 11/25/2022]
Abstract
We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual’s infection age, i.e., the time elapsed since infection, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type i is simply obtained by integrating the probability of being in state i at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (ODEs) to account for disease complexity. We show that such models can always be rewritten in our framework, thus, providing a low-dimensional yet equivalent representation of these complex models. Third, beyond the simplicity of the approach, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.
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Affiliation(s)
- Félix Foutel-Rodier
- Département de Mathématiques, Université du Québec á Montréal, Montréal, QC, Canada.
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France.
| | - François Blanquart
- Infection, Antimicrobials, Modeling, Evolution UMR 1137, Université de Paris, INSERM, Paris, France
| | - Philibert Courau
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
| | - Peter Czuppon
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Institute for Evolution and Biodiversity, University of Münster, 48149, Münster, Germany
| | - Jean-Jil Duchamps
- Laboratoire de mathématiques de Besançon UMR 6623, Université Bourgogne Franche-Comté, CNRS, F-25000, Besançon, France
| | - Jasmine Gamblin
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
| | - Élise Kerdoncuff
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Institut de Systématique, Biodiversité, Évolution UMR 7205, Muséum National d'Histoire Naturelle, CNRS, Paris, France
- Department of Molecular and Cell Biology, University of California, Berkeley, California, USA
| | - Rob Kulathinal
- Department of Biology, Temple University, Philadelphia, PA, USA
| | - Léo Régnier
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Laboratoire de Physique Théorique de la Matiére Condensée, CNRS/Sorbonne University, Paris, France
| | - Laura Vuduc
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Université Paris-Saclay, Centrale Supélec, MICS Lab Gif-sur-Yvette, Berkeley, France
| | - Amaury Lambert
- SMILE Group, Center for Interdisciplinary Research in Biology UMR 7241, Collège de France, CNRS, INSERM U 1050, PSL Research University, Paris, France
- Institut de Biologie de l'ENS, École Normale Supérieure, CNRS UMR 8197 INSERM U 1024, Paris, France
| | - Emmanuel Schertzer
- Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria
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Dimarco G, Perthame B, Toscani G, Zanella M. Kinetic models for epidemic dynamics with social heterogeneity. J Math Biol 2021; 83:4. [PMID: 34173890 PMCID: PMC8233611 DOI: 10.1007/s00285-021-01630-1] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2020] [Revised: 05/26/2021] [Accepted: 06/13/2021] [Indexed: 01/01/2023]
Abstract
We introduce a mathematical description of the impact of the number of daily contacts in the spread of infectious diseases by integrating an epidemiological dynamics with a kinetic modeling of population-based contacts. The kinetic description leads to study the evolution over time of Boltzmann-type equations describing the number densities of social contacts of susceptible, infected and recovered individuals, whose proportions are driven by a classical SIR-type compartmental model in epidemiology. Explicit calculations show that the spread of the disease is closely related to moments of the contact distribution. Furthermore, the kinetic model allows to clarify how a selective control can be assumed to achieve a minimal lockdown strategy by only reducing individuals undergoing a very large number of daily contacts. We conduct numerical simulations which confirm the ability of the model to describe different phenomena characteristic of the rapid spread of an epidemic. Motivated by the COVID-19 pandemic, a last part is dedicated to fit numerical solutions of the proposed model with infection data coming from different European countries.
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Affiliation(s)
- G Dimarco
- Mathematics and Computer Science Department, University of Ferrara, Ferrara, Italy
| | - B Perthame
- Sorbonne Université, CNRS, Université de Paris, Inria Laboratoire Jacques-Louis Lions, 75005, Paris, France
| | - G Toscani
- Mathematics Department, University of Pavia, Pavia, Italy
| | - M Zanella
- Mathematics Department, University of Pavia, Pavia, Italy.
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5
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Tricot A, Sokolov IM, Holcman D. Modeling the voltage distribution in a non-locally but globally electroneutral confined electrolyte medium: applications for nanophysiology. J Math Biol 2021; 82:65. [PMID: 34057627 DOI: 10.1007/s00285-021-01618-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2020] [Revised: 03/14/2021] [Accepted: 05/17/2021] [Indexed: 11/25/2022]
Abstract
The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson-Nernst-Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface membrane.
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Affiliation(s)
- A Tricot
- Data Modeling, Computational Biology and Predictive Medicine, Ecole Normale Supérieure PSL, 46 rue d'Ulm, 75005, Paris, France
| | - I M Sokolov
- Institute of Physics and IRIS Adlershof, Humboldt University Berlin, Newtonstr. 15, 12489, Berlin, Germany
| | - D Holcman
- Data Modeling, Computational Biology and Predictive Medicine, Ecole Normale Supérieure PSL, 46 rue d'Ulm, 75005, Paris, France.
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Cai L, Li Z, Yang C, Wang J. Global analysis of an environmental disease transmission model linking within-host and between-host dynamics. Appl Math Model 2020; 86:404-423. [PMID: 34219864 PMCID: PMC8248274 DOI: 10.1016/j.apm.2020.05.022] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R 0 for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R 0 < 1, and the endemic equilibrium is globally asymptotically stable when R 0 > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.
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Affiliation(s)
- Liming Cai
- School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P.R. China
| | - Zhaoqing Li
- School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P.R. China
| | - Chayu Yang
- Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
| | - Jin Wang
- Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
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7
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Seirin-Lee S, Sukekawa T, Nakahara T, Ishii H, Ei SI. Transitions to slow or fast diffusions provide a general property for in-phase or anti-phase polarity in a cell. J Math Biol 2020; 80:1885-917. [PMID: 32198524 DOI: 10.1007/s00285-020-01484-z] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2019] [Revised: 12/30/2019] [Indexed: 11/24/2022]
Abstract
Cell polarity is an important cellular process that cells use for various cellular functions such as asymmetric division, cell migration, and directionality determination. In asymmetric cell division, a mother cell creates multiple polarities of various proteins simultaneously within her membrane and cytosol to generate two different daughter cells. The formation of multiple polarities in asymmetric cell division has been found to be controlled via the regulatory system by upstream polarity of the membrane to downstream polarity of the cytosol, which is involved in not only polarity establishment but also polarity positioning. However, the mechanism for polarity positioning remains unclear. In this study, we found a general mechanism and mathematical structure for the multiple streams of polarities to determine their relative position via conceptional models based on the biological example of the asymmetric cell division process of C. elegans embryo. Using conceptional modeling and model reductions, we show that the positional relation of polarities is determined by a contrasting role of regulation by upstream polarity proteins on the transition process of diffusion dynamics of downstream proteins. We analytically prove that our findings hold under the general mathematical conditions, suggesting that the mechanism of relative position between upstream and downstream dynamics could be understood without depending on a specific type of bio-chemical reaction, and it could be the universal mechanism in multiple streams of polarity dynamics of the cell.
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8
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Abstract
We investigate an SIRS epidemic model with spatial diffusion and nonlinear incidence rates. We show that for small diffusion rate of the infected class [Formula: see text], the infected population tends to be highly localized at certain points inside the domain, forming K spikes. We then study three distinct destabilization mechanisms, as well as a transition from localized spikes to plateau solutions. Two of the instabilities are due to coarsening (spike death) and self-replication (spike birth), and have well-known analogues in other reaction-diffusion systems such as the Schnakenberg model. The third transition is when a single spike becomes unstable and moves to the boundary. This happens when the diffusion of the recovered class, [Formula: see text] becomes sufficiently small. In all cases, the stability thresholds are computed asymptotically and are verified by numerical experiments. We also show that the spike solution can transit into an plateau-type solution when the diffusion rates of recovered and susceptible class are sufficiently small. Implications for disease spread and control through quarantine are discussed.
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Affiliation(s)
- Chunyi Gai
- Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada
| | - David Iron
- Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada
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Abstract
Mathematical models of pathogen transmission in age-structured host populations, can be used to design or evaluate vaccination programs. For reliable results, their forces or hazard rates of infection (FOI) must be formulated correctly and the requisite contact rates and probabilities of infection on contact estimated from suitable observations. Elsewhere, we have described methods for calculating the probabilities of infection on contact from the contact rates and FOI. Here, we present methods for estimating the FOI from cross-sectional serological surveys or disease surveillance in populations with or without concurrent vaccination. We consider both continuous and discrete age, and present estimates of the FOI for vaccine-preventable diseases that confer temporary or permanent immunity.
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Affiliation(s)
- Zhilan Feng
- Purdue University, Department of Mathematics, West Lafayette IN, United States
| | - John W Glasser
- National Center for Immunization and Respiratory Diseases, CDC, Atlanta GA, United States
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10
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Bai M, Xu S. Asynchronous exponential growth for a two-phase size-structured population model and comparison with the corresponding one-phase model. J Biol Dyn 2018; 12:683-699. [PMID: 30067132 DOI: 10.1080/17513758.2018.1501104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2017] [Accepted: 07/05/2018] [Indexed: 06/08/2023]
Abstract
In this paper we study a two-phase size-structured population model with distributed delay in the birth process. This model distinguishes individuals by 'active' or 'resting' status. The individuals in the two life-stages have different growth rates. Only individuals in the 'active' stage give birth to the individuals in the 'active' stage or the 'resting' stage. The size of all the newborns is 0. By using the method of semigroups, we obtain that the model is globally well-posed and its solution possesses the property of asynchronous exponential growth. Moreover, we give a comparison between this two-phase model with the corresponding one-phase model and show that the asymptotic behaviours of the sum of the densities of individuals in the 'active' stage and the 'resting' stage and the solution of the corresponding one-phase model are different.
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Affiliation(s)
- Meng Bai
- a School of Mathematics and Statistics Sciences , Zhaoqing University , Zhaoqing , 526061 Guangdong , People's Republic of China
| | - Shihe Xu
- a School of Mathematics and Statistics Sciences , Zhaoqing University , Zhaoqing , 526061 Guangdong , People's Republic of China
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Abstract
This work presents a mathematical model for the localization of multiple species of diffusion molecules on membrane surfaces. Morphological change of bilayer membrane in vivo is generally modulated by proteins. Most of these modulations are associated with the localization of related proteins in the crowded lipid environments. We start with the energetic description of the distributions of molecules on curved membrane surface, and define the spontaneous curvature of bilayer membrane as a function of the molecule concentrations on membrane surfaces. A drift-diffusion equation governs the gradient flow of the surface molecule concentrations. We recast the energetic formulation and the related governing equations by using an Eulerian phase field description to define membrane morphology. Computational simulations with the proposed mathematical model and related numerical techniques predict (i) the molecular localization on static membrane surfaces at locations with preferred mean curvatures, and (ii) the generation of preferred mean curvature which in turn drives the molecular localization.
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Affiliation(s)
- Michael Mikucki
- Department of Applied Mathematics & Statistics, Colorado School of Mines, Golden, Colorado, 80401-1887
| | - Y C Zhou
- Department of Mathematics, Colorado State University, Fort Collins, Colorado, 80523-1874
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Abstract
Microtubules (MTs) are protein filaments that provide structure to the cytoskeleton of cells and a platform for the movement of intracellular substances. The spatial organization of MTs is crucial for a cell's form and function. MTs interact with a class of proteins called motor proteins that can transport and position individual filaments, thus contributing to overall organization. In this paper, we study the mathematical properties of a coupled partial differential equation (PDE) model, introduced by White et al. in 2015, that describes the motor-induced organization of MTs. The model consists of a nonlinear coupling of a hyperbolic PDE for bound motor proteins, a parabolic PDE for unbound motor proteins, and a transport equation for MT dynamics. We locally smooth the motor drift velocity in the equation for bound motor proteins. The mollification is not only critical for the analysis of the model, but also adds biological realism. We then use a Banach Fixed Point argument to show local existence and uniqueness of mild solutions. We highlight the applicability of the model by showing numerical simulations that are consistent with in vitro experiments.
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Affiliation(s)
- Thomas Hillen
- a Department of Mathematical and Statistical Sciences , Centre for Mathematical Biology, University of Alberta , Edmonton , AB , Canada
| | - Diana White
- b Aix-Marseille University, Institute of Mathematics , Marseille , France
| | - Gerda de Vries
- a Department of Mathematical and Statistical Sciences , Centre for Mathematical Biology, University of Alberta , Edmonton , AB , Canada
| | - Adriana Dawes
- c Department of Mathematics , Ohio State University , Columbus , OH , USA
- d Department of Molecular Genetics , Ohio State University , Columbus , OH , USA
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Abstract
This paper deals with a plant-pollinator model with diffusion and time delay effects. By considering the distribution of eigenvalues of the corresponding linearized equation, we first study stability of the positive constant steady-state and existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. We then derive an explicit formula for determining the direction and stability of the Hopf bifurcation by applying the normal form theory and the centre manifold reduction for partial functional differential equations. Finally, we present an example and numerical simulations to illustrate the obtained theoretical results.
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Affiliation(s)
- Jirong Huang
- a School of Mathematical Sciences , Beijing Normal University , Beijing , People's Republic of China
| | - Zhihua Liu
- a School of Mathematical Sciences , Beijing Normal University , Beijing , People's Republic of China
| | - Shigui Ruan
- b Department of Mathematics , University of Miami , Coral Gables , FL , USA
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Mikucki M, Zhou Y. Fast Simulation of Lipid Vesicle Deformation Using Spherical Harmonic Approximation. Commun Comput Phys 2017; 21:40-64. [PMID: 28804520 PMCID: PMC5552105 DOI: 10.4208/cicp.oa-2015-0029] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate the membrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.
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Affiliation(s)
- Michael Mikucki
- Department of Applied Mathematics & Statistics, Colorado
School of Mines, Golden, Colorado, 80401, USA
| | - Yongcheng Zhou
- Department of Mathematics, Colorado State University, Fort Collins,
Colorado, 80523, USA
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15
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Abstract
Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction-diffusion model. In the case that the redistribution kernels are highly concentrated, we show that the integro-differential equation can be approximated by a reaction-diffusion equation, in which the proliferation rate contributes to both the diffusion term and the reaction term. We completely solve the corresponding critical domain size problem and the minimal wave speed problem. Birth-jump models can be applied in many areas in mathematical biology. We highlight an application of our results in the context of forest fire spread through spotting. We show that spotting increases the invasion speed of a forest fire front.
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Affiliation(s)
- T Hillen
- a Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology , University of Alberta , Edmonton , Canada
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