Multiple travelling waves in evolutionary game dynamics

Hutson, V.; Mischaikow, Konstantin; Vickers, G.T.

doi:10.1007/bf03167371

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…• Evolutionary game theory: In this field, one seeks to understand how players change their strategies depending on those of the other players [54,91,56,118,22]. When considering the possible spatial diffusion of the players, reaction-diffusion equations arise naturally, [53,55]. • Neuroscience: traveling wave phenomena also arise when modeling nerve impulses [28].…”

Section: A Model Problemmentioning

confidence: 99%

“…More precisely, we will minimize various functionals J, depending on the controls objetive and we will also consider different classes of admissible controls A. All the experiments will take the abstract form of min a∈A J(u a ; u 0 ) (55) where u a solves:…”

Section: 2mentioning

confidence: 99%

“…Reaction-diffusion systems deserve a special mention. For instance, in the context of evolutionary game theory, [53,54,55], one deals with more than two interaction strategies. Earlier in the paper we motivated the problem with a two strategy game that was reduced to a single evolution equation.…”

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confidence: 99%

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Control of reaction-diffusion models in biology and social sciences

Ruiz-Balet

Zuazua

2022

MCRF

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These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities.The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle.We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics.The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome.We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems' dynamics and, in particular, the use of traveling waves and steady-states' paths, can be employed to build controls driving the system to the desired target.We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction.Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and results.

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Section: A Model Problemmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Control of reaction-diffusion models in biology and social sciences

Ruiz-Balet

Zuazua

2022

MCRF

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“…For example, in evolutionary game theory, a two-strategy game (see [16,17,18]) , here u stands for the proportion of players using one strategy. The replicator dynamics (see [19]) with spatial diffusion of coordination game leads to bistable nonlinearities [4,5]:…”

Section: Motivationmentioning

confidence: 99%

Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

Ruiz-Balet¹,

Zuazua²

2019

Preprint

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Dynamic phenomena in social and biological sciences can often be modeled by employing reactiondiffusion equations. Frequently in applications, their control plays an important role when avoiding population extinction or propagation of infectious diseases, enhancing multicultural features, etc.When addressing these issues from a mathematical viewpoint one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [0, 1])). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible.In the present work, we analyze the controllability to constant steady-states of spatially homogeneous semilinear heat equations, with constraints in the state, and using boundary controls, which is indeed a natural way of acting on the system in the present context. The nonlinearities considered are among the most frequent: monostable and bistable ones. We prove that controlling the system to a constant steady-state may become impossible when the diffusivity is too small (or when the domain is large), due to the existence of barrier functions. When such an obstruction does not arise, we build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves and some connectivity of the set of steady-states. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady-states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain. The results are illustrated by numerical simulations of these models that find several applications, such as the extinction of minority languages or the survival of rare species in sufficiently large reserved areas.

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Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

Ruiz-Balet

Zuazua

2020

Journal de Mathématiques Pures et Appliquées

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Multiple travelling waves in evolutionary game dynamics

Cited by 3 publications

References 12 publications

Control of reaction-diffusion models in biology and social sciences

Control of reaction-diffusion models in biology and social sciences

Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations

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