Papers by J.Ignacio Tello
We consider a limit case of a system of two equations arising in magnetic recording for a one-dim... more We consider a limit case of a system of two equations arising in magnetic recording for a one-dimensional domain. The system models the tape deflection when it is driven over a magnetic head profile, which is a known function. The system is reduced to a second order nonlinear equation, where the unknown u appears evaluated in a finite set of distinguished points {x i } n i=1 of the domain.
Siam Journal on Mathematical Analysis, Nov 13, 2014
In this paper we consider a system of three parabolic equations modeling the behavior of two biol... more In this paper we consider a system of three parabolic equations modeling the behavior of two biological species moving attracted by a chemical factor. The chemical substance verifies a parabolic equation with slow diffusion. The system contains second order terms in the first two equations modeling the chemotactic effects. We apply an iterative method to obtain the global existence of solutions using that the total mass of the biological species is conserved. The stability of the homogeneous steady states is studied by using an energy method. A final example is presented to illustrate the theoretical results.
Summary In this paper we study a nonlinear system of difierential equations which arises from a s... more Summary In this paper we study a nonlinear system of difierential equations which arises from a stationary 2-dimensional Ice-Sheet Model describing the ice-streaming phenomenon. The system consists of a multivalued nonlinear PDE of parabolic type coupled with a flrst order PDE and an ODE involving a nonlocal term. We study the uniqueness of weak solution under suitable assumptions (physically reasonable).
Mathematical Biosciences and Engineering, 2015
This work studies a general reaction-diffusion model for acid-mediated tumor invasion, where tumo... more This work studies a general reaction-diffusion model for acid-mediated tumor invasion, where tumor cells produce excess acid that primarily kills healthy cells, and thereby invade the microenvironment. The acid diffuses and could be cleared by vasculature, and the healthy and tumor cells are viewed as two species following logistic growth with mutual competition. A key feature of this model is the density-limited diffusion for tumor cells, reflecting that a healthy tissue will spatially constrain a tumor unless shrunk. Under appropriate assumptions on model parameters and on initial data, it is shown that the unique heterogeneous state is nonlinearly stable, which implies a long-term coexistence of the healthy and tumor cells in certain parameter space. Our theoretical result suggests that acidity may play a significant role in heterogeneous tumor progression.
Quarterly of Applied Mathematics, 2015
Theoretical Biology and Medical Modelling, 2015
The immunotherapy using dendritic cells (DCs) against different varieties of cancer is an approac... more The immunotherapy using dendritic cells (DCs) against different varieties of cancer is an approach that has been previously explored which induces a specific immune response. This work presents a mathematical model of DCs immunotherapy for melanoma in mice based on work by Experimental Immunotherapy Laboratory of the Medicine Faculty in the Universidad Autonoma de Mexico (UNAM). The model is a five delay differential equation (DDEs) which represents a simplified view of the immunotherapy mechanisms. The mathematical model takes into account the interactions between tumor cells, dendritic cells, naive cytotoxic T lymphocytes cells (inactivated cytotoxic cells), effector cells (cytotoxic T activated cytotoxic cells) and transforming growth factor β cytokine (T G F-β). The model is validated comparing the computer simulation results with biological trial results of the immunotherapy developed by the research group of UNAM. The results of the growth of tumor cells obtained by the control immunotherapy simulation show a similar amount of tumor cell population than the biological data of the control immunotherapy. Moreover, comparing the increase of tumor cells obtained from the immunotherapy simulation and the biological data of the immunotherapy applied by the UNAM researchers obtained errors of approximately 10 %. This allowed us to use the model as a framework to test hypothetical treatments. The numerical simulations suggest that by using more doses of DCs and changing the infusion time, the tumor growth decays compared with the current immunotherapy. In addition, a local sensitivity analysis is performed; the results show that the delay in time " τ", the maximal growth rate of tumor "r" and the maximal efficiency of tumor cytotoxic cells rate "aT" are the most sensitive model parameters. By using this mathematical model it is possible to simulate the growth of the tumor cells with or without immunotherapy using the infusion protocol of the UNAM researchers, to obtain a good approximation of the biological trials data. It is worth mentioning that by manipulating the different parameters of the model the effectiveness of the immunotherapy may increase. This last suggests that different protocols could be implemented by the Immunotherapy Laboratory of UNAM in order to improve their results.
Quarterly of Applied Mathematics, 2011
We consider the problem of a rigid surface moving over a flat plane. The surfaces are separated b... more We consider the problem of a rigid surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The relative position of the surfaces is unknown except for the initial time t = 0. The total load applied over the upper surface is a know constant for t > 0. The mathematical model consists in a coupled system formed by Reynolds variational inequality for incompressible fluids and Newton ′ s second Law. In this paper we study the global existence and uniqueness of solutions of the evolution problem when the position of the surface presents only one degree of freedom, under extra assumptions on its geometry. The existence of steady states is also studied.
Applied Mathematics and Computation, 2015
In this paper we study the numerical resolution of a reinforced random walk model arising in hapt... more In this paper we study the numerical resolution of a reinforced random walk model arising in haptotaxis and the stabilization of solutions. The model consists of a system of two differential equations, one parabolic equation with a second order non-linear term (haptotaxis term) coupled to an ODE in a bounded two dimensional domain. We assume radial symmetry of the solutions. The scheme of resolution is based on the application of the characteristics method together with a finite element one. We present some numerical simulations which illustrate some features of the numerical stabilization of solutions.
In this paper we study non-negative radially symmetric solutions of the parabolic-elliptic Keller... more In this paper we study non-negative radially symmetric solutions of the parabolic-elliptic Keller-Segel system
Journal of Differential Equations, 2015
We study the behavior of two biological populations "u" and "v" attracted by the same chemical su... more We study the behavior of two biological populations "u" and "v" attracted by the same chemical substance whose behavior is described in terms of second order parabolic equations. The model considers a logistic growth of the species and the interactions between them are relegated to the chemoattractant production. The system is completed with a third equation modeling the evolution of chemical. We assume that the chemical "w" is a non-diffusive substance and satisfies an ODE, more precisely,
We consider a simple mathematical model of distribution of morphogens (signaling molecules respon... more We consider a simple mathematical model of distribution of morphogens (signaling molecules responsible for the differentiation of cells and the creation of tissue patterns) similar to the model proposed by Lander, Nie and Wan in 2002. The model consists of a system of two equations: a PDE of parabolic type modeling the distribution of free morphogens with a dynamic boundary
In this paper we consider a general system of reaction-diffusion equations and introduce a compar... more In this paper we consider a general system of reaction-diffusion equations and introduce a comparison method to obtain qualitative properties of its solutions. The comparison method is applied to study the stability of homogeneous steady states and the asymptotic behavior of the solutions of different systems with a chemotactic term. The theoretical results obtained are slightly modified to be applied to the problems where the systems are coupled in the differentiated terms and / or contain nonlocal terms. We obtain results concerning the global stability of the steady states by comparison with solutions of Ordinary Differential Equations.
SIAM, Journal on Mathematical Analysis, 2014
In this paper we consider a system of three parabolic equations modeling the behavior of two biol... more In this paper we consider a system of three parabolic equations modeling the behavior of two biological species moving attracted by a chemical factor. The chemical substance verifies a parabolic equation with slow diffusion. The system contains second order terms in the first two equations modeling the chemotactic effects. We apply an iterative method to obtain the global existence of solutions using that the total mass of the biological species is conserved. The stability of the homogeneous steady states is studied by using an energy method. A final example is presented to illustrate the theoretical results.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2013
In this paper we consider a numerical approach to reach the equilibrium position of a journal bea... more In this paper we consider a numerical approach to reach the equilibrium position of a journal bearing with radial loading. The system consists of an external cylinder surrounding a rotating shaft. The problem is modelled by the hydrodynamic Reynolds equation with a cavitation model of Elrod-Adams. Both equations are coupled to Newton's second law which describes the position of the shaft. The problem is considered as an inverse problem where the coefficient of the equation is unknown. The numerical approach to solve the inverse problem is based on a trust-region algorithm along with the finite element method. The Heaviside function in the Elrod-Adams equation is approximated by a third order Hermite polynomial. The trust-region algorithm for solving the inverse problem showed another way of solution, different from the ones that exist at this moment.
Nonlinearity, 2013
In this paper, we study a system of partial differential equations describing the evolution of a ... more In this paper, we study a system of partial differential equations describing the evolution of a population under chemotactic effects with non-local reaction terms. We consider an external application of chemoattractant in the system and study the cases of one and two populations in competition. By introducing global competitive/cooperative factors in terms of the total mass of the populations, we obtain, for a range of parameters, that any solution with positive and bounded initial data converges to a spatially homogeneous state with positive components. The proofs rely on the maximum principle for spatially homogeneous sub-and super-solutions.
Nonlinear Analysis: Real World Applications, 2002
Nonlinear Analysis: Real World Applications, 2010
This paper deals with a nonlinear system of parabolic-elliptic type with a logistic source term a... more This paper deals with a nonlinear system of parabolic-elliptic type with a logistic source term and coupled boundary conditions related to pattern formation. We prove the existence of a unique positive global in time classical solution. We also analyze the associated stationary problem. Moreover it is proved, under the assumption of sufficiently strong logistic damping, that there is only one nonzero homogeneous equilibrium, and all the solutions to the nonstationary problem tend to this steady state for large times.
Nonlinear Analysis: Theory, Methods & Applications, 2013
We study a parabolic-elliptic chemotactic system describing the evolution of a population's densi... more We study a parabolic-elliptic chemotactic system describing the evolution of a population's density ''u'' and a chemoattractant's concentration ''v''. The system considers a nonconstant chemotactic sensitivity given by ''χ (N − u)'', for N ≥ 0, and a source term of logistic type ''λu(1 − u)''. The existence of global bounded classical solutions is proved for any χ > 0, N ≥ 0 and λ ≥ 0. By using a comparison argument we analyze the stability of the constant steady state u = 1, v = 1, for a range of parameters.
Mathematical Models and Methods in Applied Sciences, 2014
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Papers by J.Ignacio Tello