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Resumen En este trabajo estudiamos la estabilidad epistemológica de profesores debutantes. Para ello, hemos hecho un estudio cualitativo de su ETM-idóneo en el momento en que desarrollan un dominio matemático con sus estudiantes, de modo de clarificar los elementos que ponen en juego cuando despliegan una tarea en el aula. Mostramos resultados relativos al favorecimiento eventual que hacen esos profesores de la concreción de las génesis semiótica, instrumental y discursiva, y de la circulación entre los distintos polos de los planos epistemológico y cognitivo. Del estudio se desprende que la epistemología del profesor debutante no es estable, debido a una tensión entre su ETM-personal y su ETM-idóneo.

In this research, we propose a genetic decomposition for the solution set of a system of linear equations with two unknowns, by means of a transit from a homogeneous to a non-homogeneous linear system, in a Cartesian geometric context. To validate our genetic decomposition, we designed instruments that we applied to students from a secondary school mathematics teacher training program. Thus, and by using implicative statistics, we were able to confirm the mental constructions and mechanisms considered in our genetic decomposition. The results show lack of understanding of what a solution for a system is, difficulties in articulating the geometrical and algebraic aspects, and the convenience of using an alternative strategy in the case of systems of three or more linear equations., En esta investigación proponemos una descomposición genética para el conjunto solución de un sistema de ecuaciones lineales con dos incógnitas, mediante un tránsito desde sistemas homogéneos a no homogéneos, en un contexto geométrico-cartesiano. Para validar nuestra descomposición genética diseñamos instrumentos que aplicamos a estudiantes de formación inicial del profesorado de matemáticas para educación secundaria. Con ello, y haciendo uso de la estadística implicativa, logramos confirmar las estructuras mentales dispuestas en nuestra descomposición genética. Los resultados evidencian cierta incomprensión de lo que es una solución para un sistema, dificultades para articular los aspectos geométricos con los algebraicos y la conveniencia de utilizar una estrategia alternativa para el caso de un sistema de tres o más ecuaciones lineales.

Los obstáculos epistemológicos suelen tener raíces profundas en la propia Matemática, que pueden pesquisarse en la historia de la disciplina, y se caracterizan a la vez por la persistencia con la cual reaparecen en diversas situaciones y lo determinante que son para el logro de los aprendizajes. Estos obstáculos con frecuencia no son advertidos por el docente, bien sea porque ha reemplazado oportunamente sus propias concepciones (semánticas) por otras de carácter teórico –superando así el obstáculo, sin reparar explícitamente en ello–, o porque no ha logrado aun hacer esa substitución. En este trabajo presentamos una ilustración particularmente relevante de lo anterior, que se refiere a la persistencia de un obstáculo ligado al concepto de infinito en personas en distinto estadio de formación. Luego mostramos una característica adicional del obstáculo que llamamos resistencia. Posteriormente, utilizamos diversas perspectivas teóricas propiamente didácticas para adentrarnos en la cuestión. Finalmente, proponemos algunas reflexiones que se pueden derivar de nuestro estudio.

Presentamos los resultados de un estudio comparativo entre las trayectorias y estrategias de modelización movilizadas por matemáticos y profesores de matemáticas al abordar una situación hipotética que implica la construcción de un modelo matemático. La investigación se desarrolló dentro de un enfoque cualitativo con un grupo de profesores de matemáticas y estudio de casos de investigadores matemáticos. Los resultados muestran que el proceso de construcción de un Modelo Matemático difiere principalmente por el tipo de procesos cognitivos desarrollados en ambos grupos de estudio y las trayectorias de modelización son dependientes de las representaciones gráficas que los individuos utilizan para abordar el problema.

La introducción de la modelación matemática en el currículo de enseñanza media chileno, hace ya una decena de años, ha relevado la necesidad de estudios relativos a las prácticas de su enseñanza en la formación inicial de profesores en el país. Este trabajo presenta los resultados de una investigación que utiliza el marco conocimiento especializado del profesor de matemáticas (MTSK,por sus siglas en inglés) para analizar los conocimientos y la reflexión sobre modelación puestos en juego por estudiantes en formación inicial durante un ciclo de 14 sesiones de 90 minutos cada una. Los resultados muestran un progreso en el conocimiento matemático y en el conocimiento pedagógico del contenido. El trabajo discute posibles maneras de establecer modelos de enseñanza de la modelación y un momento propicio para hacer uso de una propuesta sobre la materia., The introduction of mathematical modelling in the Chilean secondary education curriculum, some ten years ago, has highlighted the need for studies concerning the teaching practices on modelling in the initial training of teachers in the country. This paper presents the results of a research that uses the theoretical frame for the understanding of the Mathematics Teacher Specialized Knowledge, MTSK, to analyze the knowledge and reflection about modelling put into play by trainee teachers during a cycle of 14 sessions of 90 minutes each. The results show a progress in the mathematical knowledge and in the pedagogical content knowledge. The paper discusses possible ways to establish models for the teaching of modelling and anappropriate moment to use a proposal on the subject.

This study shows how, in the initial training of mathematics teachers, it is possible to promote processes of abstraction and mathematisation through modelling a real situation with the support of auxiliary material to mediate understanding. By adapting elements of the theoretical and methodological framework called Abstraction in Context (AiC), participants’ discussions while building a mathematical model—in a nested epistemic actions—are analysed. Two specific points are discussed in this paper. The first aims to identify how different types of knowledge emerge when an individual is faced with a modelling task. The second is regarding the use of auxiliary material as a means of metaphorising a situation. It was evidenced how the material favours the construction of a mathematical model through the simplification and idealisation that it brings. The meaning constructed for the model is supported in recognising a decreasing behaviour as a part of a whole.

This work deals with some typographical mistakes into the above-referenced paper. Although they do not affect the main results, it is necessary to make due corrections. We affirm that the results and conclusions obtained are correct and the errors have no further implications in the aforementioned paper.

A seminar was organised to promote interdisciplinary work between teachers from different departments at a university in order to design pedagogical situations to teach mathematical modelling in the first years of higher education. This chapter presents the views of teachers from different disciplines related to skills needed to work with models in physics, biology and mathematics. The participants discussed how models are present in their courses. They also highlight the students’ skills that are relevant to success in working with models in sciences. The main findings were that interpreting graphs and structuring a model with information from other models are considered relevant competences by the sample of science teachers. The discussion on designing teaching activities also revealed the university teachers’ conceptions regarding mathematical modelling and models.

In this work, a bidimensional differential equation system obtained by modifying the well-known predator–prey Rosenzweig–MacArthur model is analyzed by considering prey growth influenced by the Allee effect. One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence. The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predator–prey model, thereby increasing the risk of ecological extinction.

This work deals with the analysis of a predator–prey model derived from the Leslie–Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered. It is well-known that the Leslie–Gower model has a unique globally asymptotically stable equilibrium point. However, it is shown here the Allee effect significantly modifies the original system dynamics, as the studied model involves many non-topological equivalent behaviors. None, one or two equilibrium points can exist at the interior of the first quadrant of the modified Leslie–Gower model with strong Allee effect on prey. However, a collapse may be seen when two positive equilibrium points exist. Moreover, we proved the existence of parameter subsets for which the system can have: a cusp point (Bogdanov–Takens bifurcation), homoclinic curves (homoclinic bifurcation), Hopf bifurcation and the existence of two limit cycles, the innermost stable and the outermost unstable, in inverse stability as they usually appear in the Gause-type predator–prey models. In contrast, the system modelling an special of weak Allee effect, may include none or just one positive equilibrium point and no homoclinic curve; the latter implies a significant difference between the mathematical properties of these forms of the phenomenon, although both systems show some rich and interesting dynamics.

Inducible defenses are a form of phenotypic plasticity that potentially modify direct interactions between various members of an ecological community, generating trait-mediated indirect effects. In this work, the hypothesis that inducible defenses increase the stability of tritrophic chains is tested, through the numerical analysis of a continuous-time model that discriminate between defenses affecting attack rate of predators, and defenses affecting predator handling time. In addition, discrimination between feeding costs of defenses affecting attack rate, and metabolic costs affecting feeding requirement for zero growth are considered. System stability was examined by computing dominant Lyapunov exponents, and through continuation routines of bifurcation points. Background parameter values were taken from two published studies. Our results show that a tritrophic system will generally be stabilized by the incorporation of inducible defenses and by their associated costs, but a number of new outcomes were obtained. Different long-term behavior is predicted if either one or two prey populations exhibit defenses. In the latter case, the defense of the basal prey dominates the dynamics. Handling time based inducible defenses exert a stronger stabilizing effect than attack rate based ones, but also impose a higher extinction risk for top predators. Inducible defenses in particular and trait-mediated indirect effects in general can be important sources of stability in natural systems.

Simple patch-occupancy models of competitive metacommunities have shown that coexistence is possible as long as there is a competition-colonization tradeoff such as that of superior competitors and dispersers. In this paper, we present a model of competition between three species in a dynamic landscape, where patches are being created and destroyed at a different rate. In our model, species interact according to a linear non-transitive hierarchy, such that species Y(3) outcompetes and can invade patches occupied by species Y(2) and this species in turn can outcompete and invade patches occupied by the inferior competitor Y(1). In this hierarchy, inferior competitors cannot invade patches of species with higher competitive ability. Analytical results show that there are regions in the parameter space where coexistence can occur, as well as regions where each of the species exists in isolation depending on species' life-history traits associated with their colonization abilities and extinction proneness as well as with the dynamics of habitat patches. In our model, the condition for coexistence depends explicitly on patch dynamics, which in turn modulate the limiting similarity for species coexistence. Coexistence in metacommunities inhabiting dynamic landscapes although possible is harder to attain than in static ones.

The aim of this paper is to correct two mistakes in [Appl. Math. Modell. (2011) 366–381], which are: the function defining the time rescaling given and the inclusion of a parameter outside of model. For a modified Leslie–Gower type predator–prey model considering the Allee effect on prey, a change of variables and a new time rescaling generating a diffeomorphism is proved; a topologically equivalent system to the original one is obtained, which is the same studied in the mentioned paper; we claim that the results and conclusions obtained are correct and the errors have not further implications.

Periodic solutions have been found for some infectious disease models of the SI and SEI types. Here four SEI models with either disease-reduced or uniform reproduction are examined to determine the model features that do and do not lead to periodic solutions. The two SEI models with the simple mass action incidence βXY can have periodic solutions for some parameter values, but the two SEI models with the standard mass action incidence λXY N do not have periodic solutions. For some intermediate values of λ in the SEI model with incidence λXY N and uniform reproduction, the interior equilibrium is a saddle whose stable manifold separates the attractive regions for the disease-free equilibrium and the susceptible-free equilibrium.

In this work we analyze a predator-prey model proposed by A. Kent et al. in Ecol. Model.162, 233 (2003), in which two aspect of the model are considered: an effect of emigration or immigration on prey population to constant rate and a prey threshold level for predators. We prove that the system when the immigration effect is introduced in the model has a dynamics that is similar to the Rosenzweig-MacArthur model. Also, when emigration is considered in the model, we show that the behavior of the system is strongly dependent on this phenomenon, this due to the fact that trajectories are highly sensitive to the initial conditions, in similar way as when Allee effect is assumed on prey. Furthermore, we determine constraints in the parameters space for which two stable attractor exist, indicating that the extinction of both population is possible in addition with the coexistence of oscillating of populations size in a unique stable limit cycle. We also show that the consideration of a threshold level of prey population for the predator is not essential in the dynamics of the model.

En este trabajo discutimos la significacion ecologica y matematica de la dimension del sistema en modelos de dinamica poblacional en tiempo continuo. Mostramos como la dimension del sistema refleja los supuestos ecologicos y afecta el espectro de resultados dinamicos asi como la tratabilidad matematica de los modelos. Acentuamos que la dimension de un modelo no es siempre equivalente al numero de variables de estado, y presentamos condiciones bajo las cuales la dimension del sistema es alterada.