The Role of the Delay Time in the Modeling of Biological Range Expansions
- Ortega-Cejas, Vicente; Fort, Joaquim; Méndez López, Vicenç
The time interval between successive migrations of biological species causes a delay time in the reaction-diffusion equations describing their space-time dynamics. This lowers the predicted speed of the waves of advance, as compared to classical models. It has been shown that this delay-time effect improves the modeling of human range expansions. Here, we demonstrate that it can also be important for other species. We present two new examples where the predictions of the time-delayed and the classical (Fisher) approaches are compared to experimental data. No free or adjustable parameters are used. We show that the importance of the delay effect...
Linear stability analysis and metastable solutions for a phase-field model
- Jiménez Casas, Ángela; Rodríguez Bernal, Aníbal
We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable solutions that evolve very slowly in time.
Models of aggregation in Dictyostelium discoideum: On the track of spiral waves
- Herrero García, Miguel Ángel; Sastre, Leandro
This work is concerned with some aspects of the social life of the amoebae Dictyostelium discoideum (Dd). In particular, we shall focus on the early stages of the starvation-induced aggregation of Dd cells. Under such circumstances, amoebae are known to exchange a chemical messenger (cAMP) which acts as a signal to mediate their individual behaviour. This molecule is released from aggregation centres and advances through aggregation fields, first as circular waves and later on as spiral patterns. We shall recall below some of the basic features of this process, paying attention to the mathematical models that have been derived to...
Biological optimization of tumor radiosurgery
- Cappuccio, Antonio; Herrero García, Miguel Ángel; Nuñez, Luis
In tumor radiosurgery, a high dose of radiation is delivered in a single session. The question then naturally arises of selecting an irradiation strategy of high biological efficiency. In this study, the authors propose a mathematical framework to investigate the biological effects of heterogeneity and rate of dose delivery in radiosurgery. The authors simulate a target composed by proliferating and hypoxic tumor cells as well as by normal tissue. Treatment outcome is evaluated by a functional of the dose distribution that counts the LQ-surviving fractions of each cell type. Prescriptions on intensity, homogeneity, and duration of radiation delivery are incorporated...
Tumour radiotherapy and its mathematical modelling
- Cappuccio, Antonio; Herrero García, Miguel Ángel; Nuñez, Luis
Radiotherapy consists in the delivery of ionizing radiation with curative goals. It represents a major modality for treating solid malignancies in any anatomical site. It requires extremely precise dosimetry and delivery to control the lesion while sparing the surrounding normal tissue. This is currently achieved by means of a combination of techniques. Among these, mathematical modelling may contribute to design improved treatment plans. In the present work, we provide an overview of the basic features of radiotherapy. Thereafter, we describe concrete examples of how mathematical modelling has so far been employed to simulate radiotherapy and to theoretically explore alternative strategies....
Through a glass, darkly: Biology seen from mathematics Comment on "Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives" by N. Bellomo and B. Carbonaro
- Herrero García, Miguel Ángel
On the growth of filamentary structures in planar media
- Andreucci, D.; Herrero García, Miguel Ángel; Velázquez, J.J. L.
We analyse a mathematical model for the growth of thin filaments into a two dimensional medium. More exactly, we focus on a certain reaction/diffusion system, describing the interaction between three chemicals (an activator, an inhibitor and a growth factor), and including a fourth cell variable characterising irreversible incorporation to a filament. Such a model has been shown numerically to generate structures shaped like nets. We perform an asymptotical analysis of the behaviour of solutions, in the case when the system has parameters very large and very small, thereby allowing the onset of different time and space scales. In particular, we...
Singularity patterns in a chemotaxis model
- Herrero García, Miguel Ángel; Velázquez, J.J. L.
The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up of solutions implies chemotactic collapse, namely, concentration of species to form sporae. The model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity, which has the form ut=Δu−χ(uv), 0=Δv+(u−1), on ΩR2, where Ω is an open disc with no-flux (homogeneous Neumann) boundary conditions. The initial conditions are continuous functions u(x,0)=u0(x)≥0, v(x,0)=v0(x)≥0 for xΩ. Under these conditions, the...
Modeling And Analysis Of A Marine Bacteriophage Infection With Latency Period
- Edoardo Beretta; Istituto Di Biomatematica; Yang Kuang
A mathematical model for the marine bacteriophage infection with explicit latency period is proposed as a system of discrete time delay differential equations and its important mathematical features are analyzed. Let i represent the death constant rate in the class of infected bacteria during latency period "T ." Only the fraction "exp(; i T )" of the bacteria infected at "t ;T " will release by lysis at time "t""b" new phages, where b 2 (1# +1)iscalled "virus replication factor." Hence, if i ? 0, the main parameter on whichthe dynamics of the model depends on is "b exp(; i...
Anharmonicity and its significance to non-Ohmic electric conduction
- Makarov , Valeri A.; Velarde, Manuel G.; Chetverikov, Alexander; Ebeling, Werner
We provide here a thorough analysis of the interplay between anharmonic lattice dynamics (with exponential repulsion between units) and electric conduction in a driven-dissipative electrically charged one-dimensional system. First, we delineate the ranges of parameter values where, respectively, subsonic and supersonic wave solitons are possible along the lattice. Then, we study the consequences of the soliton-mediated coupling of light negative to heavy positive charges (lattice units). In the presence of an external electric field we obtain the current-field characteristics for a wide range of values of all parameters defining the system. Finally, we discuss the conditions for an Ohmic-non-Ohmic transition...
Tactile information processing in the trigeminal complex of the rat
- Makarov , Valeri A.; Pavlov, Alexey N.; Tupitsyn, Anatoly N.; Panetsos, Fivos; Moreno, Ángel; García-González, , Víctor; Sánchez-Jiménez,, Abel
We study mechanisms of information processing in the principalis (Pr5), oralis (Sp5o) and interpolaris (Sp5i) nuclei of the trigeminal sensory complex of the rat under whisker stimulation by short air puffs. After the standard electrophysiological description of the neural spiking activity we apply a novel wavelet based method quantifying the structural stability of Bring patterns evoked by a periodic whisker stimulation. We show that the response stability depends on the puff duration delivered to the vibrissae and differs among the analyzed nuclei. Pr5 and Sp5i exhibit the maximal stability to an intermediate stimulus duration, whereas Sp5o shows "preference" for short...
El control de mosquitos (Diptera: Culicidae) utilizando métodos biomatemáticos en la provincia Villa Clara
- Fimia Duarte, Rigoberto; Centro Provincial de Higiene, Epidemiología
y Microbiología de Villa Clara. Ave. Libertadores No. 99/C y D. Rpto.
Santa Catalina, Santa Clara. CP 20100. Villa Clara. Cuba.; Osés Rodríguez, Ricardo; Centro Meteorológico Provincial. Prolongación de Marta Abreu # 59 altos.
Santa Clara. Villa Clara. Cuba.; Otero Martín, Meylín; Centro Meteorológico Provincial. Prolongación de Marta Abreu # 59 altos.
Santa Clara. Villa Clara. Cuba.; Diéguez Fernández, Lorenzo; Unidad Municipal de Vigilancia y Lucha Antivectorial de Camagüey.
Cuba.; Cepero Rodriguez, Omelio; Facultad de Ciencias
Agropecuarias. Universidad entral “Marta Abreu” de Las Villas. Carretera a Camajuaní Km. 5 ½. Santa Clara. CP 54830. Villa Clara. Cuba.; Gonzalez Gonzalez, Ramon; Centro Provincial de Higiene,
Epidemiología y Microbiología de Villa Clara. Ave. Libertadores No. 99/C
y D. Rpto. Santa Catalina, Santa Clara. CP 20100. Villa Clara. Cuba.; Silveira Prado, Enrique A; Centro de Bioactivos Químicos.
Universidad Central “Marta Abreu” de Las Villas. Carretera a Camajuaní
Km. 5 ½. Santa Clara. CP 54830. Villa Clara. Cuba.; Corona Santander, Edgar; Centro Provincial de Higiene, Epidemiología y Microbiología de Villa Clara.
Ave. Libertadores No. 99/C y D. Rpto. Santa Catalina, Santa Clara. CP
20100. Villa Clara. Cuba.
Objetivo:Métodos: en el caso de los peces se realizó un estudio de cohorte en los Consejos Populares de Báez y Guaracabulla, pertenecientes al municipio Placetas, iniciándose la investigación en abril del 2006 hasta abril de 2007, para lo cual se seleccionó una muestra de 1740 depósitos, dividida en dos grupos: uno con 870 evaluar la eficacia de dos especies de peces fluviales conjuntamente con la modelación matemática en función del control de las larvas de mosquitos en la provincia Villa Clara. depósitos a los cuales se les sembraron peces(a razón de 3 ejemplares por depósito, siempre tratando de buscar el...
Modelling vascular morphogenesis: current views on blood vessels development
- Herrero García, Miguel Ángel; Köhn Luque, Álvaro; Pérez-Pomares, José M.
In this work we present a comprehensive account of our current knowledge on vascular morphogenesis, both from a biological and a mathematical point of view. To this end, we first describe the basic steps in the known mechanisms of blood vessel morphogenesis, whose structure, function and unfolding properties are examined. We then provide a wide, although by no means exhaustive, account of mathematical models which are used to describe and discuss particular aspects of the overall biological process considered. We finally summarize the approaches presented, and suggest possible directions for future research. Details about some of the major signalling molecules...
From microscopic to macroscopic description of multicellular systems and biological growing tissues
- Bellomo, Nicola; Bellouquid, Abdelghani; Herrero García, Miguel Ángel
This paper presents an asymptotic theory for a large class of Boltzmann-type equations suitable to model the evolution of multicellular systems in biology. The mathematical approach described herein shows how various types of diffusion phenomena, linear and nonlinear, can be obtained in suitable asymptotic limits. Time scaling related to cell movement and biological activity are shown to play a crucial role in determining the macroscopic equations corresponding to each case.
On the role of mathematics in biology
- Herrero García, Miguel Ángel
Slow and fast invasion waves in a model of acid-mediated tumour growth
- Fasano, A.; Herrero García, Miguel Ángel; Rodrigo, Marianito R.
This work is concerned with a reaction-diffusion system that has been proposed as a model to describe acid-mediated cancer invasion. More precisely, we consider the properties of travelling waves that can be supported by such a system, and show that a rich variety of wave propagation dynamics, both fast and slow, is compatible with the model. In particular, asymptotic formulae for admissible wave profiles and bounds on their wave speeds are provided.
Mathematical models of vascular development
- Köhn Luque, Álvaro
El uso de las Matemáticas para comprender problemas relevantes en Biología se remonta al menos a los trabajos pioneros de D. Bernoulli sobre la eficacia de la vacuna contra la viruela (1760) y L. Euler sobre el flujo sanguíneo en arterias (1775). Desde entonces, el campo de la Biomatemática ha experimentado un extraordinario desarrollo, a lo largo del cual han surgido una gran cantidad de desafíos fascinantes. Uno de ellos es el de esclarecer la morfogénesis (la emergencia de las formas biológicas) haciendo uso de métodos matemáticos, una cuesti´on que ha sido abordada, entre otros, en los influyentes trabajos de...
Extinction times and size of the surviving species in a two-species competition process
- Gómez Corral, Antonio; López García, M.
We investigate a stochastic model for the competition between two species. Based on percentiles of the maximum number of individuals in the ecosystem, we present an approximating model for which the extinction time can be thought of as a phase-type random variable. We determine formulae for the probabilities of extinction and the moments of the extinction time. We discuss the use of several quasi-stationary assumptions. We include a comparative study between existing asymptotic results, results obtained from a simulation of the process, and our solution.
Asymptotic construction of pulses in the discrete Hodgkin-Huxley model for myelinated nerves
- Carpio Rodríguez, Ana María
A quantitative description of pulses and wave trains in the spatially discrete Hodgkin-Huxley model for myelinated nerves is given. Predictions of the shape and speed of the waves and the thresholds for propagation failure are obtained. Our asymptotic predictions agree quite well with numerical solutions of the model and describe wave patterns generated by repeated firing at a boundary.
Equilibrio vs. colapso de especies marinas comerciales en modelos estructurados por edades
- Morán Cabré, Manuel; Maroto Fernández, José María
En este artículo analizamos la existencia y estabilidad del equilibrio, como contrapartida al colapso, en modelos poblaciones de especies marinas estructurados por edades(análisis por cohortes). En particular, se establecen las condiciones necesarias y suficientes sobre la función de reclutamiento para que el equilibrio sea estable. En el caso de función de reclutamiento lineal con techo se muestra que, una disminución de las tasas de supervivencia de las cohortes puede provocar que se pase de una situación de estabilidad del equilibrio a otra en donde no existe equilibrio.