The mathematical modeling and the stability analysis of multilane traffic in the macroscopic scale is considered. We propose a new first order model derived from microscopic dynamics with lane changing, leading to a coupled system of hyperbolic balance laws. The macroscopic limit is derived without assuming ad hoc space and time scalings. The analysis of the stability of the equilibria of the model is discussed. The proposed numerical tests confirm the theoretical findings between the macroscopic and microscopic modeling, and the results of the stability analysis.

Competitive systems can exhibit both hierarchical (transitive) and cyclic (intransitive) structures. Despite theoretical interest in cyclic competition, which offers richer dynamics and occupies a larger subset of the scope of possible competitive systems, most real-world systems are predominantly transitive. Why? Here, we introduce a generic mechanism that promotes transitivity, even when there is ample room for cyclicity. We demonstrate that, if competitive outcomes depend smoothly on competitor attributes, then similar competitors compete transitively. We quantify the rate of convergence to transitivity given the similarity of the competitors and the smoothness of the performance function. Thus, we prove the adage regarding apples and oranges. Similar objects admit well-ordered comparisons; diverse objects may not.

We analyze the searching strategies of a forager diffusing in the whole space via an equation of fractional type. Specifically, the diffusion of the forager is regulated by a Lévy flight whose exponent can be chosen in order to optimize a suitable foraging efficiency functional. Here, the dimension of the space is arbitrary. On the one hand, we show that the exponent corresponding to the limit case of heavy-tailed Lévy flights is a pessimizer for the efficiency functional. On the other hand, we prove that, in situations of biological interest, one finds the most rewarding strategies arbitrarily close to. The combination of these results gives that the most rewarding searching option may turn out to be unfeasible, or at least unreliable, in practice, since small perturbations of the optimal searching exponent lead to pessimal patterns. The cases analyzed specifically are those of a target located in the proximity of the forager and that of sparse prey modeled by a target infinitely far from the initial position of the seeker. The efficiency functionals taken into account are either of pointwise type (in which the predator and the prey are modeled by moving points) or of set-dependent type (in which the predator and the prey correspond to regions of space with uniform density, thus modeling also the case of a sight range of the biological individuals involved). To implement our analysis, we also provide a number of structural results about finiteness, continuity, and asymptotic behaviors of the efficiency functionals. It is suggestive to relate the adoption of the most rewarding searching pattern close to pessimizers to a “high-risk/high-gain” strategy, in which the forager aims at high-energy content prey to mitigate the risk of failure. This setting is also connected to foraging modes of “ambush” type.

.This work addresses the modelling of oxygenic photogranules (OPGs), by introducing a mathematical model which describes both the genesis and growth of photogranules and the related treatment process. The photogranule has been modelled as a free boundary domain with radial symmetry, which evolves over time as a result of microbial growth, attachment, and detachment processes. Hyperbolic and parabolic PDEs have been considered to model at mesoscale the transport and growth of sessile biomass and the diffusion and conversion of soluble substrates. The macroscale behavior of the system has been modelled through first order impulsive ordinary differential equations (IDEs), which reproduce a sequencing batch reactor (SBR) configuration. Phototrophic biomass has been considered for the first time in granular biofilms, and cyanobacteria and microalgae have been accounted separately, to model their metabolic differences. To describe the key role of cyanobacteria in the photogranulation process, the attachment velocity of all suspended microbial species has been modelled as a function of the cyanobacteria concentration in suspended form. The model takes into account the main biological processes involved in OPG-based systems: metabolic activity of cyanobacteria, microalgae, heterotrophic and nitrifying bacteria, microbial decay, extracellular polymeric substances (EPS) secretion, symbiotic and competitive interactions between different species, light-dark cycle, light attenuation across the granule, and photoinhibition phenomena. The model has been integrated numerically, and the results show its consistency in describing the photogranule evolution and ecology, and highlight the advantages of the OPG technology, analyzing the effects of the influent wastewater composition and light conditions on the process.Keywordsphotogranulefree boundary value problemscyanobacteriabiofilmmicrobial ecologyMSC codes65Z0592-0892D40

In a "structured system" of equations, each equation depends on a specified subset of the variables. In this article, we explore properties common to "almost every" system with a fixed structure and how the properties can be read from the corresponding connection graph. A solution p of a system F (p) = c is called robust if it persists despite small changes in F . We establish methods for determining robustness that depends on the structure, as expressed in the properties of the corresponding directed graph of the structured system. The keys to understanding linear and nonlinear structured systems are subsets of variables that we call forward and backward bottlenecks. In particular, when robustness fails in a structured system, it is due to the existence of a unique "backward bottleneck", that we call a "minimax bottleneck". We present a numerical method for locating the minimax bottleneck. We show how to remove it by adding edges to the graph.

Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide range of applications in the physical and life sciences, ranging from thermal conduction in composite media, gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, and intercellular gap junctions. Many of these systems involve three-dimensional (3D) diffusion in an array of parallel planes with homogeneity in the lateral directions, so that they can be reduced to effective one-dimensional (1D) models. In this paper we develop a probabilistic model of singleparticle diffusion in 1D multi-layered media by constructing a multi-layered version of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM, each of which is restricted to a single layer. Each round of reflected BM is killed when the local time at one end of the layer exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer on the other side of the boundary with equal probability, and the process is iterated We proceed by constructing a last renewal equation for multi-layered snapping out BM that relates the full probability density to the probability densities of partially reflected BM in each layer. We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that the renewal equation and corresponding multi-layer diffusion equation are equivalent. We illustrate the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter-based method for surface absorption; each round of reflected BM is now killed according to a non-exponential distribution for each local time threshold. This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM. We show that for certain configurations, non-exponential killing leads to an effective time-dependent permeability that is normalizable but heavy-tailed.

Spatial systems with heterogeneities are ubiquitous in nature, from precipitation, temperature, and soil gradients controlling vegetation growth to morphogen gradients controlling gene expression in embryos. Such systems, generally described by nonlinear dynamical systems, often display complex parameter dependence and exhibit bifurcations. The dynamics of heterogeneous spatially extended systems passing through bifurcations are still relatively poorly understood, yet recent theoretical studies and experimental data highlight the resulting complex behaviors and their relevance to real-world applications. We explore the consequences of spatial heterogeneities passing through bifurcations via two examples strongly motivated by applications. These model systems illustrate that studying heterogeneity-induced behaviors in spatial systems is crucial for a better understanding of ecological transitions and functional organization in brain development.

We introduce the first two-way coupled model for the thermo-viscous damping of a mechanical structure (such as quartz tuning fork) that is forced by the weak acoustic and thermal waves generated when a laser source periodically interacts with a trace gas. The model is based on a Helmholtz system of thermo-visco-acoustic equations in the fluid, together with a system of equations for the temperature and the displacement of the structure. These two subsystems are coupled across the fluid-structure interface via several conditions. With this model, the user specifies the geometry of the structure and the viscous and thermal parameters of the fluid, and the model outputs an effective damping parameter and a signal strength that is proportional to the concentration of the trace gas. This new model is a significant improvement over existing one-way coupled models in which damping effects are incorporated via a priori laboratory measurements. Analytical solutions derived for an annular structure show reasonable agreement between the one-way and two-way coupled models at higher ambient pressures. However, at low ambient pressure the one-way coupled model does not adequately capture thermo-viscous effects.

Long-term behaviors of biochemical systems are described by steady states in deterministic models and stationary distributions in stochastic models. Obtaining their analytic solutions can be done for limited cases, such as linear or finite-state systems, as it generally requires solving many coupled equations. Interestingly, analytic solutions can be easily obtained when underlying networks have special topologies, called weak reversibility (WR) and zero deficiency (ZD), and the kinetic law follows a generalized form of mass-action kinetics. However, such desired topological conditions do not hold for the majority of cases. Thus, translating networks to have WR and ZD while preserving the original dynamics was proposed. Yet, this approach is limited because manually obtaining the desired network translation among the large number of candidates is challenging. Here, we prove necessary conditions for having WR and ZD after translation, and based on these conditions, we develop a user-friendly computational package, TOWARDZ, that automatically and efficiently identifies translated networks with WR and ZD. This allows us to quantitatively examine how likely it is to obtain WR and ZD after translation depending on the number of species and reactions. Importantly, we also describe how our package can be used to analytically derive steady states of deterministic models and stationary distributions of stochastic models. TOWARDZ provides an effective tool to analyze biochemical systems.

.Optimal packing is applied in physics for investigation of macroscopic properties of composites and porous media. The present paper demonstrates that the optimal packing of perfectly conducting disks on the plane corresponds to the minimal effective conductivity for macroscopically isotropic composites. Such an optimal location is studied and compared to the pure geometric problem for \(N\leq 8\) disks packing on the flat torus. The hexagonal (triangular) array of disks solves the problem for the triangle lattice numbers \(N=1,3,4,7, \ldots\). The rest of the cases \(N=2,5,6,8\) are investigated separately by minimization of structural sums constructed by means of the Eisenstein functions.Keywordseffective conductivity of disksoptimal packing of disks on torusstructural sumsEisenstein functionperiodic packingMSC codes74Q1552C15