Publicación: Análisis de estabilidad en sistemas impulsivos. Aplicación a la coexistencia de especies vegetales sometidas a incendios
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2022-07-08
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Atribución-NoComercial-SinDerivadas 4.0 Internacional
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Universidad Nacional de Educación a Distancia (España). Facultad de Ciencias. Departamento de Física Fundamental
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El objetivo fundamental de este trabajo es determinar las condiciones de supervivencia y coexistencia de las especies vegetales de un modelo que representa las poblaciones de un ecosistema subdividido jerarquizado. Asimismo, se analiza el efecto de la aparición de perturbaciones, incendios, tanto periódicos como estocásticos, haciendo uso de sistemas de ecuaciones diferenciales ordinarias impulsivas. Para ello, se introducen las bases matemáticas que fundamentarán el estudio, definiendo en primer lugar los modelos de competencia entre especies, a continuación los sistemas impulsivos y finalmente los conceptos de equilibrio. Se continúa con el estudio concreto para el caso de una especie, donde se hace un desarrollo analítico para el caso sin incendios y con incendios periódicos, mientras que para el caso de incendios estocásticos se hace un análisis numérico fundamentado en algoritmos escritos en código Python. Además, se desarrolla el análisis para el caso de dos especies, haciendo uso de métodos analíticos típicos como el Principio de Estabilidad Lineal, las desigualdades impulsivas y las funciones de Lyapunov. Para el caso de dos especies en presencia de incendios estocásticos se desarrolla el estudio numérico pertinente aplicando algoritmos en base Python. Los resultados muestran que en los casos sin incendios y con incendios periódicos es posible determinar, en base a criterios generales y analíticos, las condiciones de supervivencia y coexistencia de las especies, mientras que es posible abordar, a través del análisis numérico, el caso de incendios estocásticos a pesar de la complejidad extra que este modelo aporta a la dinámica del sistema. Las simulaciones numéricas confirman que, frente al caso de incendios periódicos, el modelo de incendios estocásticos dependientes de biomasa es un modelo más realista que, en el caso límite estudiado, muestra unas condiciones de supervivencia de la segunda especie que coinciden con el caso de fuegos periódicos.
The main objective of this work is to determine the survival and coexistence conditions of plant species in a model representing the populations of a hierarchical subdivided ecosystem. Likewise, the effect of the occurrence of disturbances, fires, both periodic and stochastic, is analyzed using systems of impulsive ordinary differential equations. For this purpose, the mathematical bases that will support the study are introduced, defining first the models of competition between species, then the impulsive systems and finally the concepts of equilibrium. The study continues with the concrete study for the case of a species, where an analytical development is made for the case without fires and with periodic fires, while for the case of stochastic fires a numerical analysis is made based on algorithms written in Python code. In addition, the analysis for the two-species case is developed, making use of typical analytical methods such as the Linear Stability Principle, impulsive inequalities and Lyapunov equations. For the two-species case in the presence of stochastic fires, the relevant numerical study is developed by applying Python-based algorithms. The results show that in the cases without fires and with periodic fires it is possible to determine, based on general and analytical criteria, the survival and coexistence conditions of the species, while it is possible to approach, through numerical analysis, the case of stochastic fires despite the extra complexity that this model brings to the dynamics of the system. Numerical simulations confirm that, compared to the case of periodic fires, the biomass-dependent stochastic fire model is a more realistic model that, in the limiting case studied, shows survival conditions of the second species that coincide with the case of periodic fires.
The main objective of this work is to determine the survival and coexistence conditions of plant species in a model representing the populations of a hierarchical subdivided ecosystem. Likewise, the effect of the occurrence of disturbances, fires, both periodic and stochastic, is analyzed using systems of impulsive ordinary differential equations. For this purpose, the mathematical bases that will support the study are introduced, defining first the models of competition between species, then the impulsive systems and finally the concepts of equilibrium. The study continues with the concrete study for the case of a species, where an analytical development is made for the case without fires and with periodic fires, while for the case of stochastic fires a numerical analysis is made based on algorithms written in Python code. In addition, the analysis for the two-species case is developed, making use of typical analytical methods such as the Linear Stability Principle, impulsive inequalities and Lyapunov equations. For the two-species case in the presence of stochastic fires, the relevant numerical study is developed by applying Python-based algorithms. The results show that in the cases without fires and with periodic fires it is possible to determine, based on general and analytical criteria, the survival and coexistence conditions of the species, while it is possible to approach, through numerical analysis, the case of stochastic fires despite the extra complexity that this model brings to the dynamics of the system. Numerical simulations confirm that, compared to the case of periodic fires, the biomass-dependent stochastic fire model is a more realistic model that, in the limiting case studied, shows survival conditions of the second species that coincide with the case of periodic fires.
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Facultad de Ciencias
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Física Fundamental