Enseñanza de las Ciencias. Revista de investigación y experiencias didácticas, 2017
La introducción de la modelación matemática en el currículo de enseñanza media chileno, hace ya u... more 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.
Climate-induced defences play a key role in the nonlinear dynamics of plankton evolution in a lak... more Climate-induced defences play a key role in the nonlinear dynamics of plankton evolution in a lake. The phenomenology thereof is linked to the existence of a depth range where organisms are optically protected from their predators, and also find a sufficient supply of oxygen for survival. A statistical parameterisation of a previously proposed dynamical model for plankton concentration is analysed
In this work we analyze the consequences of incorporating the phenomenon of depensation, also kno... more In this work we analyze the consequences of incorporating the phenomenon of depensation, also known as Allee effect, into the bioeconomic model proposed by Smith in [20]. The Smith's model is one of the simplest bioeconomic models used in the management of renewable resources, which related the biomass of the exploited resources and the nominal fishery's effort of open-access. The mathematical prop-erties of the original model experiment significant changes, due to the fact that in the new proposed model the equilibrium point (0, 0) is an attractor point (local or global) for all the values of the parameters, which indicates that the resources can extinct producing a collapse in fishery if the ratio biomass-effort is small, as it should be intuitively predictable by the fishery industry. We show that fishery can oscillate or cycle around a positive equilibrium point, if the ratio between biomass and fishing effort is sufficiently large.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright a b s t r a c t 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.
The main purpose of this work is to analyze a Gause type predator-prey model in which two ecologi... more The main purpose of this work is to analyze a Gause type predator-prey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation. We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions. Trajectories starting at one side of this separatrix curve have the equilibrium point (0,0) as their ω-limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point (K,0) in which the predators disappear and prey attains their carrying capacity. We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong influence on the system dynamics, resulting in multiple domains of attraction. Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.
Enseñanza de las Ciencias. Revista de investigación y experiencias didácticas, 2017
La introducción de la modelación matemática en el currículo de enseñanza media chileno, hace ya u... more 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.
Climate-induced defences play a key role in the nonlinear dynamics of plankton evolution in a lak... more Climate-induced defences play a key role in the nonlinear dynamics of plankton evolution in a lake. The phenomenology thereof is linked to the existence of a depth range where organisms are optically protected from their predators, and also find a sufficient supply of oxygen for survival. A statistical parameterisation of a previously proposed dynamical model for plankton concentration is analysed
In this work we analyze the consequences of incorporating the phenomenon of depensation, also kno... more In this work we analyze the consequences of incorporating the phenomenon of depensation, also known as Allee effect, into the bioeconomic model proposed by Smith in [20]. The Smith's model is one of the simplest bioeconomic models used in the management of renewable resources, which related the biomass of the exploited resources and the nominal fishery's effort of open-access. The mathematical prop-erties of the original model experiment significant changes, due to the fact that in the new proposed model the equilibrium point (0, 0) is an attractor point (local or global) for all the values of the parameters, which indicates that the resources can extinct producing a collapse in fishery if the ratio biomass-effort is small, as it should be intuitively predictable by the fishery industry. We show that fishery can oscillate or cycle around a positive equilibrium point, if the ratio between biomass and fishing effort is sufficiently large.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright a b s t r a c t 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.
The main purpose of this work is to analyze a Gause type predator-prey model in which two ecologi... more The main purpose of this work is to analyze a Gause type predator-prey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation. We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions. Trajectories starting at one side of this separatrix curve have the equilibrium point (0,0) as their ω-limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point (K,0) in which the predators disappear and prey attains their carrying capacity. We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong influence on the system dynamics, resulting in multiple domains of attraction. Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.
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