Modeling and Specification of the Aquatic Ecological Emergence using Genetic Programming Nelson Fernández, Gustavo Marcano-Valero, José Aguilar, Oswaldo Terán, Carlos Gershenson Abstract—A major endeavor of ecology is to understand the emergence of complexity. This task requires the integration of knowledge and theories, moving from physical to social sciences. We use genetic programming to develop mathematical relationships between ecological emergence and variables such as self-organization, homeostasis, autopoiesis and complexity. These variables were initially formalized on the basis of in- formation theory. The emergence models found were applied and tested with a case study involving an arctic lake and a tropical lake. In these lakes, the variables of limiting nutrients, biomass and physico-chemical components were taken into account for the automated generation of the model equations. The results show that the model follows in the dynamics of the aquatic ecological components selected accurately. In this context, ecological emergence can be calculated and studied. I. Introduction T He validation of generality, realism and accuracy of the models developed by Genetic programming (GP) and others machine learning techniques, is a permanent challenge to ecological modelling [1]. Also, in ecology a major endeavor is to understand the complex behaviors expressed by their self-organized and emergent dynamics at different scales [2]. The advantage of GP models consisted in that they mimic natural adaptation and evolution by computers. Based on biological information processing, GP has been observed as an appropriated evolutionary algorithm to find computational solutions for ecological problems. In this sense, GP has emerged as a promising intelligent paradigm for complex aspects in ecology [3]. Particularly, GP has been used like identification method [4], [5], [6], [7]. For the identification problems, GP has advantages compared with other machines learning techniques, because it needs little a priori knowl- edge, which can be obtained from the data set of the system [6]. GP works very well for flexible selection of the models structure and parameters. The literature reveals the benefits of the GP within the framework of system identification [8]. In Ecology, formal measures of complexity and other emergent properties have been proposed in order to compare Nelson Fernández is with Laboratorio de Hidroinformática, Uni- versidad de Pamplona,Colombia. (email:nfernandez@unipamplona.edu.co, http://unipamplona.academia.edu/NelsonFernandez) Gustavo Marcano and José Aguilar are with Centro de Mi- croelectrónica y Sistemas Distribuidos, Universidad de los Andes, Veô snezuela(gustavexx@gmail.com, aguilar@ula.ve) Oswaldo Terà ˛ an is with Centro de Simulación y Modelos, Universidad de los Andes, Venezuela(email:oteran@ula.ve) Carlos Gershenson is with Instituto de Matemáticas Aplicadas y en Sistemas and Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México (email:cgg@unam.mx) different components and/or ecosystems at different scales [9], [10], [11], [12].The information theory has been useful for the development of several models of complexity and it has been used in different ways as it can see in [13]. Recently, results obtained in discrete dynamical systems [14], [15] show that a formal description of the complex- ity behavior of different systems, in terms of properties as emergence (change-adaptability), self-organization (order- regularity), autopoiesis (autonomy) and homeostasis (regular- ity), is suitable. With the application of GP, this paper is focused in the identification of mathematical relationships for emergence, self-organization and autopoiesis properties, in order to com- plement the understanding of the complex aspects of the ecological systems. Particularly, we would like to define, using GP, the mathematical relationship that might exist between these three variables. As a study case, we consider an Arctic Lake and a Tropical Lake. The data of lakes were obtained using The Aquatic Ecosystem Simulator [16]. The paper is organized as follows: Section II presents the measures for the properties of emergence, self-organization and the reformulated measures of complexity, homeostasis and autopoiesis. All measures developed were based on previous works. Section III presents the ecosystems studied. Section IV describes the procedure to obtain the models and the models obtained with GP. The model results are analyzed in Section V. Section VI concludes the paper. II. Measures for Emergence,Self-Organization, Complexity ,Homeostasis and Autopoiesis The measures applied in this paper have as the basis the re- cent developments proposed by Gershenson and Fern’andez [14], and refined measures, based on axioms, presented in [17]. In particular, emergence and self-organization measures are the same of the propose in these works. The remain measures of Complexity, Homeostasis and Autopoiesis were reformulated, in order to avoid strong correlations among them. It is important to highlight that there are no agreed definitions—or even notions—of complexity, emergence, or self-organization. In that sense, the measures we propose here aim at being general and simple, but useful to describe these processes from an information theory perspective. A. Emergence In general, Emergence refers to properties of a phe- nomenon that are present now and were not before. If we suppose these properties as non-trivial, we could say