A simple model of cellular automata with inertia and applications in pattern formation and ecotones phenomena

We propose a Cellular Automata (CA) model in which three ubiquitous and relevant processes in nature are present, namely, spatial competition, distinction between dynamically stronger and weaker agents and the existence of an inner resistance to changes in the actual state (=−1,0,+1) of each CA lattice cell n (which we call inertia) [1]. Considering ensembles of initial lattices, we study the average properties of the CA final stationary configuration structures resulting from the system time evolution. Assuming the inertia a (proper) control parameter, we identify qualitative changes in the CA spatial patterns resembling usual phase transitions. Interestingly, some of the observed features may be associated with continuous transitions (critical phenomena). However, certain quantities seem to present jumps, typical of discontinuous transitions. We argue that these apparent contradictory findings can be attributed to the inertia parameter’s discrete character. Along the presentation, we discuss few potential applications for the present CA formulation in different fields, like pattern formation in chemical reactions and the disorganized spread of housing in poor neighborhoods of big citys. But one specific problem is more closely analyzed under the present framework.

We show that our CA with inertia is a minimal [2], not dedicated, model capable of describing the basic aspects of ecotones: zones between geographic regions of distinct biomes. In such transition areas, there is the coexistence of groups of species coming from different ecosystems. In many concrete situations, it is still not completely understood how less fitted (e.g., to the nearby biomes conditions) exogenous animals and plants can survive. They eventually would perish in a more homogeneous environment if competing with the same stronger (better adapted) local endogenic species. We show that depending on our inertia parameter I (which in the ecotones case can be interpreted as an enhanced fitness of the weaker species, consequence of the competition between the stronger species), it becomes more difficult for the interactions to change the state of each cell n (representing individuals of a certain species living in a given geographic region). By playing with the initial conditions and the distribution of stronger and weaker species, it is possible to generate from our model spatial patterns of meta-communities qualitatively similar to those in real ecotones. In special, ecotones’ boundary extensions and shapes usually are driven by climate conditions and species competition. Such boundaries may or may not arise depending on the strength and interplay of these factors (say, quantified by a parameter λ). The variation of λ can modify or even destroy ecotones. The corresponding transition, as λ varies, appears to be akin to continuous phase transitions. Qualitatively, this is exactly what we observe from our CA lattices’ time evolution. Therefore, our CA with inertia can partially explain the already conjectured in the literature relation between continuous phase transitions and the emergence of ecological treats [3,4], in particular ecotones.
 

Authors: 
Marcos Da Luz
Room: 
2
Date: 
Monday, September 24, 2018 - 18:45 to 19:00

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