In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involved on the inertial times, reveals some intriguing consequences. To show in detail the richness of such a scenario, we present, as an illustrative example, the pattern dynamics occurring in the hyperbolic generalization of the extended Klausmeier model. This is a simple two-species model used to describe the migration of vegetation stripes along the hillslope of semiarid environments. By means of a thorough comparison between analytical predictions and numerical simulations, we show that inertia, apart from enlarging the region of the parameter plane where wave instability occurs, may also modulate the key features of the coherent structures, solution of the CCGL equation. In particular, it is proven that inertial effects play a role, not only during transient regime from the spatially-homogeneous steady state toward the patterned state, but also in altering the amplitude, the wavelength, the angular frequency, and even the stability of the phase-winding solutions. 

Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models

We study in this work the global existence of solutions to a system of reaction cross diffusion equations appearing in the modeling of multiple sclerosis, in the one-dimensional case. Weak solutions are obtained for general initial data, and existence, uniqueness, stability and smoothness are proven when initial data are smooth.

Existence and regularity for a chemotaxis model involved in the modeling of multiple sclerosis

We present a theoretical and numerical study of the bifurcations of the stationary patterns supported by a chemotactic model of Multiple Sclerosis (MS). We derive the normal forms of the dynamics which allows to predict the appearance and stabilization of the emerging branches describing the concentric patterns typical of Balo's sclerosis, a very aggressive variant of MS. Spatial modulation of the Turing-type structures through a zigzag instability is also addressed. The nonlinear stage of the Eckhaus and zigzag instability is investigated numerically: defect-mediated wavenumber adjustments are recovered and the time of occurrence of phase-slips is studied as the system parameters are varied.

https://cab.unime.it/journals/index.php/AAPP/article/download/AAPP.96S3A9/AAPP96S3A9

Eckhaus and zigzag instability in a chemotaxis model of multiple sclerosis

Uncovering the mechanisms behind animal space use patterns is of vital importance for predictive ecology, thus conservation and management of ecosystems. Movement is a core driver of those patterns so understanding how movement mechanisms give rise to space use patterns has become an increasingly active area of research. This study focuses on a particular strand of research in this area, based around step selection analysis (SSA). SSA is a popular way of inferring drivers of movement decisions, but, perhaps less well-appreciated, it also parametrises a model of animal movement. Of key interest is that this model can be propogated forwards in time to predict the space use patterns over broader spatial and temporal scales than those that pertain to the proximate movement decisions of animals. Here, we provide a guide for understanding and using the various existing techniques for scaling-up step selection models to predict broad scale space use patterns. We give practical guidance on when to use which technique, as well as specific examples together with code in R and Python. By pulling together various disparate techniques into one place, and providing code and instructions in simple examples, we hope to highlight the importance of these techniques and make them accessible to a wider range of ecologists, ultimately helping expand the usefulness of step selection analysis.

/pdf/how-to-scale-up-from-animal-movement-decisions-to-gs3uou4f.pdf

How to scale up from animal movement decisions to spatiotemporal patterns: An approach via step selection

Global asymptotic behavior of solutions for a parabolic-parabolic-ODE chemotaxis system modeling multiple sclerosis

In this paper we study radially symmetric solutions for our recently proposed reaction–diffusion–chemotaxis model of Multiple Sclerosis. Through a weakly nonlinear expansion we classify the bifurcation at the onset and derive the amplitude equations ruling the formation of concentric demyelinating patterns which reproduce the concentric layers observed in Balo sclerosis and in the early phase of Multiple Sclerosis. We present numerical simulations which illustrate and fit the analytical results.

Axisymmetric solutions for a chemotaxis model of Multiple Sclerosis

A principal concern of ecological research is to unveil the causes behind observed spatio-temporal distributions of species. A key tactic is to correlate observed locations with environmental features, in the form of resource selection functions or other correlative species distribution models. In reality, however, the distribution of any population both affects and is affected by those surrounding it, creating a complex network of feedbacks causing emergent spatio-temporal features that may not correlate with any particular aspect of the underlying environment. Here, we study the way in which the movements of populations in response to one another can affect the spatio-temporal distributions of ecosystems. We construct a stochastic individual-based modelling (IBM) framework, based on stigmergent interactions (i.e. organisms leave marks which cause others to alter their movements) between and within populations. We show how to gain insight into this IBM via mathematical analysis of a partial differential equation (PDE) system given by a continuum limit. We show how the combination of stochastic simulations of the IBM and mathematical analysis of PDEs can be used to categorise emergent patterns into homogeneous vs. heterogeneous, stationary vs. perpetually-fluctuating, and aggregation vs. segregation. In doing so, we develop techniques for understanding spatial bifurcations in stochastic IBMs, grounded in mathematical analysis. Finally, we demonstrate through a simple example how the interplay between environmental features and between-population stigmergent interactions can give rise to predicted spatial distributions that are quite different to those predicted purely by accounting for environmental covariates.

/pdf/beyond-resource-selection-emergent-spatio-temporal-d2me1zj3.pdf

Beyond resource selection: emergent spatio-temporal distributions from animal movements and stigmergent interactions

Non-local advection is a key process in a range of biological systems, from cells within individuals to the movement of whole organisms. Consequently, in recent years, there has been increasing attention on modelling non-local advection mathematically. These often take the form of partial differential equations, with integral terms modelling the non-locality. One common formalism is the aggregation-diffusion equation, a class of advection diffusion models with non-local advection. This was originally used to model a single population, but has recently been extended to the multi-species case to model the way organisms may alter their movement in the presence of coexistent species. Here we prove existence theorems for a class of non-local multi-species advection-diffusion models, with an arbitrary number of co-existent species. We prove global existence for models in n=1 spatial dimension and local existence for n>1. We describe an efficient spectral method for numerically solving these models and provide example simulation output. Overall, this helps provide a solid mathematical foundation for studying the effect of inter-species interactions on movement and space use.

Valeria Giunta

Papers

Axisymmetric solutions for a chemotaxis model of Multiple Sclerosis

Existence and regularity for a chemotaxis model involved in the modeling of multiple sclerosis

Eckhaus and zigzag instability in a chemotaxis model of multiple sclerosis

Beyond resource selection: emergent spatio-temporal distributions from animal movements and stigmergent interactions

Local and Global Existence for Non-local Multi-Species Advection-Diffusion Models