Pattern Formation on Trees

Networks having the geometry and the connectivity of trees are considered as the spatial support of spatiotemporal dynamical processes. A tree is characterized by two parameters: its ramification and its depth. The local dynamics at the nodes of a tree is described by a nonlinear map, given rise to a coupled map lattice system. The coupling is expressed by a matrix whose eigenvectors constitute a basis on which spatial patterns on trees can be expressed by linear combination. The spectrum of eigenvalues of the coupling matrix exhibit a nonuniform distribution which manifest itself in the bifurcation structure of the spatially synchronized modes. These models may describe reaction-diffusion processes and several other phenomena occurring on heterogeneous media with hierarchical structure.Comment: Submitted to Phys. Rev. E, 15 pages, 9 fig

Generation, Ranking and Unranking of Ordered Trees with Degree Bounds

We study the problem of generating, ranking and unranking of unlabeled ordered trees whose nodes have maximum degree of $\Delta$. This class of trees represents a generalization of chemical trees. A chemical tree is an unlabeled tree in which no node has degree greater than 4. By allowing up to $\Delta$ children for each node of chemical tree instead of 4, we will have a generalization of chemical trees. Here, we introduce a new encoding over an alphabet of size 4 for representing unlabeled ordered trees with maximum degree of $\Delta$. We use this encoding for generating these trees in A-order with constant average time and O(n) worst case time. Due to the given encoding, with a precomputation of size and time O(n^2) (assuming $\Delta$ is constant), both ranking and unranking algorithms are also designed taking O(n) and O(nlogn) time complexities.Comment: In Proceedings DCM 2015, arXiv:1603.0053

Propagation failure of excitation waves on trees and random networks

Excitation waves are studied on trees and random networks of coupled active elements. Undamped propagation of such waves is observed in those networks. It represents an excursion from the resting state and a relaxation back to it for each node. However, the degrees of the nodes influence drastically the dynamics. Excitation propagates more slowly through nodes with larger degrees and beyond some critical degree waves lose their stability and disappear. For regular trees with a fixed branching ratio, the critical degree is determined with an approximate analytical theory which also holds locally for the early stage of excitation spreading in random networks.Comment: 7 pages, 7 figures, submitted to ep

Local Knowledge and Natural Resource Management in a Peasant Farming Community Facing Rapid Change: A Critical Examination

Environmental degradation is a major global problem, and addressing it is a key Millennium Development Goal. Its impacts are not just environmental (e.g., species loss), but also economic (e.g., reduced agricultural productivity), with degradation increasingly cited as a key cause of rural poverty in the developing world. The degradation literature typically emphasises common property or 'open access' natural resources, and how perverse incentives or missing institutions lead optimising private actors to degrade them. By contrast, the present paper considers degradation occurring on private farms in peasant communities. This is a critical yet delicate issue, given the poverty of such areas and questions about the role of farmers in either degrading or regenerating rural lands The paper examines natural resource management by peasant farmers in rural Tanzania. Its key concern is how the local knowledge informing farmers' management decisions adapts to challenges associated with environmental degradation and market liberalisation. Given their poverty, this question could have direct implications for the capacity of households to successfully meet their livelihood needs. Based on fresh empirical data, the paper finds that differential farmer knowledge helps explain the large differences in how households and communities respond to the degradation challenge. The implication is that some farmers adapt more effectively to emerging challenges than others, despite all being rational, optimising agents who follow the management strategies they deem best. The paper thus provides a critique of local knowledge, implying that some farmers experience adaptation slippages while others race ahead with effective adaptations. The paper speaks to the chronic poverty that plagues many rural communities in the developing world. Specifically, it helps explain the failure of proven 'sustainable agriculture' technologies to disseminate readily beyond an initial group of early innovators, and suggests a means to help 'scale up' local successes. Its key policy implication is to inform improved capacity building for peasant communities.