Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

Abstract

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.

Codimension-Two Bifurcations in Animal Aggregation Models with Symmetry

“…We now use X as the phase space to (12). Since we consider an equilibrium with isotropy subgroup O(2), there is an action of O(2) on X, given by (5) and (6).…”

“…Kernels K r,a,al indicate whether the interactions take place inside the repulsion range (K r ), attraction range (K a ) or alignment range (K al ). Although these kernels can be described by a variety of continuous and discontinuous functions [12], in this article we consider only translated Gaussian kernels:…”

“…The first approach can be found mainly in individualbased models, for which studies into the mathematical mechanisms behind the patterns are still a difficult task. The second approach can be found in continuum models, for which numerical simulations can sometimes become too complicated (see the review in [12]). There are, however, models that combine analytical results with numerical simulations.…”

“…These mathematical models are used to suggest plausible biological mechanisms regarding the interactions at the micro-scale level (e.g., changes in individuals' speed or turning, or changes of inter-individual social interaction ranges) that could explain the observed macro-scale level dynamics (e.g., the shape, size and dynamics of the group) [10,21,20,24,29,23,28,4,47,2,8,34,38,39]. Generally, these models either (i) focus on numerical simulations with the purpose of comparing the simulated aggregation patterns with the available data (and thus ascertain the correctness of the micro-scale level assumptions incorporated into the models), or (ii) focus on the analytical results, e.g., show the existence of particular solutions exhibited by these models, or try to connect biological interactions at the micro-scale and macro-scale levels [12]. The first approach can be found mainly in individualbased models, for which studies into the mathematical mechanisms behind the patterns are still a difficult task.…”

Fronts under arrest: Nonlocal boundary dynamics in biology

“…A variety of natural mechanisms lead to nonlocal effects: An animal's ability to see and hear its surroundings leads to convolutional averages in swarming, aggregation, and alignment models of collective behavior [1][2][3][4][5][6][7]; The propensity of a species to broadly forage according to super diffusive processes, such as Lévy flights, motivates the use of fractional Laplacians in partial differential equation (PDE) models of albatrosses, sharks, and criminals [8][9][10][11]; The physical coupling between neurons also leads to nonlocal operators in continuum models for nerve signal propagation and hallucinations [12][13][14][15]. Even local PDEs used for understanding phase transitions and optics, such as the Allen-Cahn equation and singularly perturbed reaction-diffusion systems, exhibit an effective nonlocal coupling at the interface between different domain boundaries [16][17][18].…”