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G. Nakamura

Publications and source records attributed to G. Nakamura.

6 recordsLinked to original sources
arXiv2022

Levy geometric graphs

We present a new family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's size but only on the considered scale. It follows a Gamma distribution and thus presents an exponential decay. Levy flights are particular random walks with some power-law increments of infinite variance. When building the geometric graphs from them, we show from dimensional arguments, that the number of connected components (clusters) follows an inverse power of the scale. The distribution of the size of their components, properly normalized, is scale-invariant, which reflects the self-similar nature of the underlying process. This allows to test if a graph (including non-spatial ones) could possibly result from an underlying Levy process. When the scale increases, these graphs never tend towards a single cluster, the giant component. In other words, while the autocorrelation of the process scales as a power of the distance, they never undergo a phase transition of percolation type. The Levy graphs may find applications in community detection and in the analysis of collective behaviors as in face-to-face interaction networks.

cond-mat.stat-mech↗
arXiv2021

Dispersal and organization of polarized cells: non-linear diffusion and cluster formation without adhesion

Experimental studies of cell motility in culture have shown that under adequate conditions these living organisms possess the ability to organize themselves into complex structures. Such structures may exhibit a synergy that greatly increases their survival rate and facilitate growth or spreading to different tissues. These properties are even more significant for cancer cells and related pathologies. Theoretical studies supported by experimental evidence have also shown that adhesion plays a significant role in cellular organization. Here we show that the directional persistence observed in polarized displacements permits the formation of stable cell aggregates in the absence of adhesion, even in low-density regimes. We introduce a discrete stochastic model for the dispersal of polarized cells with exclusion and derive the hydrodynamic limit. We demonstrate that the persistence coupled with the cell-cell exclusion hinders the cellular motility around other cells, leading to a non-linear diffusion which facilitates their capture into larger aggregates.

cond-mat.stat-mech↗
arXiv2021

Modeling cooperation and competition in biological communities

The far-reaching consequences of ecological interactions in the dynamics of biological communities remain an intriguing subject. For decades, competition has been a cornerstone in ecological processes, but mounting evidence shows that cooperation does also contribute to the structure of biological communities. Here, we propose a simple deterministic model for the study of the effects of facilitation and competition in the dynamics of such systems. The simultaneous inclusion of both effects produces rich dynamics and captures the context-dependence observed in the formation of ecological communities. The approach reproduces relevant aspects of primary and secondary plant succession, the effect invasive species, and the survival of rare species. The model also takes into account the role of the ecological priority effect and stress the crucial role of facilitation in conservation efforts and species coexistence.

q-bio.PE↗
arXiv2020

Confinement strategies in a simple SIR model

We propose a simple SIR model in order to investigate the impact of various confinement strategies on a most virulent epidemic. Our approach is motivated by the current COVID-19 pandemic. The main hypothesis is the existence of two populations of susceptible persons, one which obeys confinement and for which the infection rate does not exceed 1, and a population which, being non confined for various imperatives, can be substantially more infective. The model, initially formulated as a differential system, is discretised following a specific procedure, the discrete system serving as an integrator for the differential one. Our model is calibrated so as to correspond to what is observed in the COVID-19 epidemic. Several conclusions can be reached, despite the very simple structure of our model. First, it is not possible to pinpoint the genesis of the epidemic by just analysing data from when the epidemic is in full swing. It may well turn out that the epidemic has reached a sizeable part of the world months before it became noticeable. Concerning the confinement scenarios, a universal feature of all our simulations is that relaxing the lockdown constraints leads to a rekindling of the epidemic. Thus we sought the conditions for the second epidemic peak to be lower than the first one. This is possible in all the scenarios considered (abrupt, progressive or stepwise exit) but typically a progressive exit can start earlier than an abrupt one. However, by the time the progressive exit is complete, the overall confinement times are not too different. From our results, the most promising strategy is that of a stepwise exit. And in fact its implementation could be quite feasible, with the major part of the population (minus the fragile groups) exiting simultaneously but obeying rigorous distancing constraints.

q-bio.PE↗
arXiv2015

Unique continuation property of solutions to general second order elliptic systems

This paper concerns about the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumption which called {\sl basic assumptions}, but also some technical assumptions which we called {\sl further assumptions}. It is shown as usual by first applying the Holmgren transform to this inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate given via a partition of unity and Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.

math.AP↗