Jan M. Baetens

Phenomenological study of irregular cellular automata based on Lyapunov exponents and Jacobians

Originally, cellular automata (CA) have been defined upon regular tessellations of the n-dimensional Euclidean space, while CA on irregular tessellations have received only little attention from the scientific community, notwithstanding serious shortcomings are associated with the former manner of subdividing Rn. In this paper we present a profound phenomenological study of two-state, two-dimensional irregular CA from a dynamical systems viewpoint. We opted to exploit properly defined quantitative measures instead of resorting to qualitative methods for discriminating between behavioral classes. As such, we employ Lyapunov exponents, measuring the divergence rate of close trajectories in phase space, and Jacobians, formulated using Boolean derivatives and expressing the sensitivity of a cellular automaton to its inputs. Both are stated for two-state CA on irregular tessellations, enabling us to characterize these discrete dynamical systems, and advancing us to propose a classification scheme for this CA family. In addition, a relationship between these quantitative measures is established in extension of the insights already developed for the classical CA paradigm. Finally, we discuss the repercussions on the CA dynamics that arise when the geometric variability of the spatial entities is taken into account during the CA simulation.

Design and parameterization of a stochastic cellular automaton describing a chemical reaction

Although most of the work concerned with reaction kinetics concentrates on empirical findings, stochastic models, and differential equations, a growing number of researchers is exploring other methods to elucidate reaction kinetics. In this work, the parameterization of an utter discrete spatio‐temporal model, more specifically, a cellular automaton (CA), describing the reaction of HCl with CaCO3, is suggested. Furthermore, a system of partial differential equations (PDE), deduced from a set of CA rules, is implemented to compare both modeling paradigms. In this article, the experimental setup to acquire time series of data is explained, a stochastic CA‐based model and a continuous PDE‐based model capable of describing the reaction are proposed, the models are parameterized using the experimental data and, finally, the relationship between a discrete time step of the CA‐based model and the physical time is studied. Essentially, the parameterization of both models can be traced back to the quest for a solution of the inverse problem in which a (set of) rule(s), respectively a system of PDE, is deduced starting from the observed data. It is demonstrated that the proposed CA‐ and PDE‐based models are capable of describing the considered chemical reaction with a high accuracy, which is confirmed by a root mean squared error between the simulated and observed data of 0.388 and 0.869 g CO2, respectively. Further, it is shown that an exponential or linear relationship can be used to link the physical time to a discrete time step of the CA‐based model.

Automated image-based analysis of spatio-temporal fungal dynamics.

Due to their ability to grow in complex environments, fungi play an important role in most ecosystems and have for that reason been the subject of numerous studies. Some of the main obstacles to the study of fungal growth are the heterogeneity of growth environments and the limited scope of laboratory experiments. Given the increasing availability of image capturing techniques, a new approach lies in image analysis. Most previous image analysis studies involve manual labelling of the fungal network, tracking of individual hyphae, or invasive techniques that do not allow for tracking the evolution of the entire fungal network. In response, this work presents a highly versatile tool combining image analysis and graph theory to monitor fungal growth through time and space for different fungal species and image resolutions. In addition, a new experimental set-up is presented that allows for a functional description of fungal growth dynamics and a quantitative mutual comparison of different growth behaviors. The presented method is completely automated and facilitates the extraction of the most studied fungal growth features such as the total length of the mycelium, the area of the mycelium and the fractal dimension. The compactness of the fungal network can also be monitored over time by computing measures such as the number of tips, the node degree and the number of nodes. Finally, the average growth angle and the internodal length can be extracted to study the morphology of the fungi. In summary, the introduced method offers an updated and broader alternative to classical and narrowly focused approaches, thus opening new avenues of investigation in the field of mycology.

The impact of initial evenness on biodiversity maintenance for a four-species in silico bacterial community

Initial community evenness has been shown to be a key factor in preserving the functional stability of an ecosystem, but has not been accounted for in previous modelling studies. We formulate a model that allows the initial evenness of the community to be varied in order to investigate the consequent impact on system diversity. We consider a community of four interacting bacterial species, and present a stochastic, spatial individual-based model simulating the ecosystem dynamics. Interactions take place on a two-dimensional lattice. The model incorporates three processes: reproduction, competition and mobility. In addition to variable initial evenness, multiple competition schemes are implemented, modelling various possible communities, which results in diverse coexistence and extinction scenarios. Simulations show that long-term system behaviour is strongly dependent on initial evenness and competition structure. The system is generally unstable; higher initial evenness has a small stabilizing effect on ecosystem dynamics by extending the time until the first extinction.

On the topological sensitivity of cellular automata.

Ever since the conceptualization of cellular automata (CA), much attention has been paid to the dynamical properties of these discrete dynamical systems, and, more in particular, to their sensitivity to the initial condition from which they are evolved. Yet, the sensitivity of CA to the topology upon which they are based has received only minor attention, such that a clear insight in this dependence is still lacking and, furthermore, a quantification of this so-called topological sensitivity has not yet been proposed. The lack of attention for this issue is rather surprising since CA are spatially explicit, which means that their dynamics is directly affected by their topology. To overcome these shortcomings, we propose topological Lyapunov exponents that measure the divergence of two close trajectories in phase space originating from a topological perturbation, and we relate them to a measure grasping the sensitivity of CA to their topology that relies on the concept of topological derivatives, which is introduced in this paper. The validity of the proposed methodology is illustrated for the 256 elementary CA and for a family of two-state irregular totalistic CA.

Cellular automata on irregular tessellations

To this day, most articles on cellular automata (CA) employ regular tessellations of ℝ n , notwithstanding this kind of tessellation suffers from a few serious drawbacks limiting the appraisal of regular CA as full-fledged modelling tools. In this article, we propose an extension to the classical CA paradigm, by allowing irregular tessellations of ℝ n , which, in essence, encloses the regular tessellations as a special case. After a short review on the major shortcomings of CA on regular tessellations, we establish a definition for CA on irregular tessellations of ℝ n that is followed by a profound elaboration of its constituents. In addition, we propose a graph representation that is applicable to an important family of irregular CA, and, in accordance with the framework developed for CA on regular tessellations, we present an enumeration scheme for the k-state, θ-sum irregular (outer-)totalistic CA. Illustrative examples conclude this article.

Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles

We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analog of Floquet theory for cellular automata.

On the decomposition of stochastic cellular automata

In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over the state set of a stochastic cellular automaton, i.e. images that show the average state of each cell during the evolution of the stochastic cellular automaton. The second property shows that stochastic cellular automata are equivalent to so-called stochastic mixtures of deterministic cellular automata. Based on this property, any stochastic cellular automaton can be decomposed into a set of deterministic cellular automata, each of which contributes to the behavior of the stochastic cellular automaton.

Predicting Metapopulation Responses to Conservation in Human-Dominated Landscapes

Loss of habitat to urbanization is a primary cause of population declines as human-dominated landscapes expand at increasing rates. Understanding how the relative effects of different conservation strategies is important to slow population declines for species in urban landscapes. We studied the wood thrush Hylocichla mustelina, a declining forest-breeding Neotropical migratory species, and umbrella species for forest-breeding songbirds, within the urbanized mid-Atlantic United States. We integrated 40 years of demographic data with contemporary metapopulation model simulations of breeding wood thrushes to predict population responses to differing conservation scenarios. We compared four conservation scenarios over a 30-year time period (2014–2044) representing A) current observed state (Null), B) replacing impervious surface with forest (Reforest), C) reducing brown-headed cowbird Molothrus ater parasitism pressure (Cowbird removal), and D) simultaneous reforesting and cowbird removal. Compared to the Null scenario, the Reforest scenario increased mean annual population trends by 54 % , the Remove cowbirds scenario increased mean annual population trends by 38 %, and the scenario combining reforestation and cowbird removal increased mean annual population trends by 98 %. Mean annual growth rates (λ) per site were greater in the Reforest (λ = 0.94) and Remove cowbirds (λ = 0.92) compared to the Null (λ = 0.88) model scenarios. However, only by combining the positive effects of reforestation and cowbird removal did wood thrush populations stop declining (λ = 1.00). Our results suggest that independently replacing impervious surface with forest habitat around forest patches and removing cowbirds may slow current negative population trends. Furthermore, conservation efforts that combine reforestation and cowbird removal may potentially benefit populations of wood thrushes and other similarly forest-breeding songbird species within urbanized fragmented landscapes that typify the mid-Atlantic United States.

Modelling three-dimensional fungal growth in response to environmental stimuli.

Most fungi grow by developing complex networks that enable the translocation of nutrients over large distances. Spatially explicit mathematical models are able to capture both the complexity of the fungal network and the biomass evolution, as such providing a powerful alternative to classical modelling paradigms. Unfortunately, most of these models restrict growth to two dimensions or confine it to a lattice, thereby resulting in unrealistic representations of fungal networks. In addition, interactions between fungi and their environment are often neglected. In response, this work presents a lattice-free three-dimensional fungal growth model that accounts for the interactions between the in silico fungus and different substrates and media. A sensitivity analysis was carried out to identify the key model parameters for future calibration. Finally, a scenario analysis covering a variety of growth conditions was conducted to illustrate the broad scope of the model and its ability to replicate in situ growth scenarios.