Francisco J. Solis

Nonstandard discrete approximationspreserving stability properties of continuous mathematical models

Most numerical methods for differential equations introduce spurious solutions. Westudy the method presented by Mickens to obtain exact nonstandard methods for some ordinary differential equations. We show how to generalize his method to equations with no known exact analytical solution, and show that the new scheme has better stability properties than Runge-Kutta methods. We apply the method to several examples.

Modelling the effects of human papillomavirus in cervical cells

Worldwide, cervical cancer is the second most common cancer in women, after breast cancer. The prevalence of this malignant disease is estimated at 1.4 million cases worldwide, causing about 290,000 deaths and 500,000 new cases per year, of which 80% correspond to women living in developing countries. In this work we propose a family of ordered models for basal cells of the cervix corresponding to different stages ranging from normal cells to the formation of precancerous lesions. We analyse the first member of the family analytically and for the second member we developed a non-standard numerical method in order to extract some biological information.

Self-limitation in a discrete predator-prey model

We study the ecological and mathematical significance of a nonlinear discrete predator-prey model that includes several types of self-limitation on the prey. The model is derived for the dynamics of two interacting populations where predators feed only on prey of a certain age. We show how the introduction of different limitation factors can account for several important phenomena that affect the dynamic output of the models. We show why some of these factors contribute to a viable interaction between the two populations and some other factors originate unstable behavior with unbounded oscillations.

Chaos in the one-dimensional wave equation

This paper deals with the chaotic behavior of the solutions of a mixed problem for the one-dimensional wave equation with a quadratic boundary condition. This behavior is studied through the connection between the energy function and quadratic discrete dynamical systems.

Nonlinear juvenile predation population dynamics

A general nonlinear age-structured predator-prey model is analyzed to obtain the dynamics of two interacting populations that includes self-limitation on the prey and juvenile predation. Our aim is to identify mechanisms of newborn survival that allow us to explain viable interactions between the two populations in circumstances when their absence would otherwise result in unstable behavior with unbounded oscillations. To achieve our goal we apply some standard methods in the analysis of dynamical systems such as the Painleve property and bifurcation analysis.

Modeling the pursuit in natural systems: A relaxed oscillation approach

We provide families of dynamical systems that model a wide range of cyclic pursuit mechanisms present in many organizations, which include the immune system, food chains and financial markets. A basic characteristic under study is the asymptotic behavior of the individual pursue of a system particle, cell or individual, to neighborhood particles. Relaxation oscillations are included in the systems in order to analyze different asymptotic scenarios. Several simulations will be presented and discussed that fit within the pursuit feature.

Discrete mathematical models of an aggressive heterogeneous tumor growth with chemotherapy treatment

Discrete mathematical models are proposed to study the dynamics of interacting cells of an organism that is affected by an aggressive heterogeneous tumor. The models include the application of a chemotherapy treatment. Different combinations of two drug applications are considered in order to identify the best one to eliminate the tumor.

A classification of slow convergence near parametric periodic points of discrete dynamical systems

We study the phenomenon of slow convergence in families of discrete dynamical systems where the iteration function has a Puiseux series representation. Such occurrence consists in the slow convergence of orbits near non-hyperbolic parametric periodic points. We provide a precise new definition of the slowness of convergence which is based on literature results for the critical exponents associated with parametric periodic points. Such exponents establish a general classification for slow systems and provide a measure of rates of convergence. For dynamical systems whose iteration functions have Taylor series expansions, the new definition is natural with wider applicability. However, it can be also used for iteration functions where a more sophisticated approach, such as a Lagrange expansion, is needed. In addition, we show that even for such iteration functions, the critical exponent can be easily computed. The presented theoretical results are illustrated by numerical examples having different rates of convergence.

Evolution of a mathematical model of an aggressive-invasive cancer under chemotherapy

In this work, we first propose continuous models of an aggressive-invasive cancer which is characterized by its heterogeneity, by its high proliferation rates and by the potential invasion of cancer cells into surrounding tissue. An acidification environment mechanism is included in the models to analyze the level of invasion. Second, we introduce and analyze their discrete versions obtained by an appropriate discretization. Afterwards a chemotherapy treatment with a gradual effect is added, which includes important aspects as resistance and drug toxicity. Finally we make comparisons among different chemotherapy schemes to identify a suitable combination in order to eradicate or to reduce the growth of the tumor.

Discrete modeling of aggressive tumor growth with gradual effect of chemotherapy

Abstract Discrete mathematical models are proposed to study the dynamics of interacting cells of an organism that is affected by an aggressive heterogeneous tumor. The models include the application of a chemotherapy treatment with a gradual effect. Another factor included in the models is the competence among the different tumor cells. An effective treatment index is introduced in order to analyze the evolution of the tumor and to compare different treatments.