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Abstract In this work, we investigate the existence of nonmonotone traveling wave solutions to a reaction‐diffusion system modeling social outbursts, such as rioting activity, originally proposed in Berestycki et al (Netw Heterog Media. 2015;10(3):443–475). The model consists of two scalar values, the level of unrestand a tension field. A key component of the model is a bandwagon effect in the unrest, provided the tension is sufficiently high. We focus on the so‐called tension‐inhibitive regime, characterized by the fact that the level of unrest has a negative feedback on the tension. This regime has been shown to be physically relevant for the spatiotemporal spread of the 2005 French riots. We use Geometric Singular Perturbation Theory to study the existence of such solutions in two situations. The first is when bothanddiffuse at a very small rate. Here, the time scale over which the bandwagon effect is observed plays a key role. The second case we consider is when the tension diffuses at a much slower rate than the level of unrest. In this case, we are able to deduce that the driving dynamics are modeled by the well‐known Fisher–Kolmogorov‐Petrovsky‐Piskunov (KPP) equation.
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Stability of fronts in the diffusive Rosenzweig–MacArthur model
Abstract We consider a diffusive Rosenzweig–MacArthur predator–prey model in the situation when the prey diffuses at a rate much smaller than that of the predator. In a certain parameter regime, the existence of fronts in the system is known: the underlying dynamical system in a singular limit is reduced to a scalar Fisher–KPP (Kolmogorov–Petrovski–Piskunov) equation and the fronts supported by the full system are small perturbations of the Fisher–KPP fronts. The existence proof is based on the application of the Geometric Singular Perturbation Theory with respect to two small parameters. This paper is focused on the stability of the fronts. We show that, for some parameter regime, the fronts are spectrally and asymptotically stable using energy estimates, exponential dichotomies, the Evans function calculation, and a technique that involves constructing unstable augmented bundles. The energy estimates provide bounds on the unstable spectrum which depend on the small parameters of the system; the bounds are inversely proportional to these parameters. We further improve these estimates by showing that the eigenvalue problem is a small perturbation of some limiting (as the modulus of the eigenvalue parameter goes to infinity) system and that the limiting system has exponential dichotomies. Persistence of the exponential dichotomies then leads to bounds uniform in the small parameters. The main novelty of this approach is related to the fact that the limit of the eigenvalue problem is not autonomous. We then use the concept of the unstable augmented bundles and by treating these as multiscale topological structures with respect to the same two small parameters consequently as in the existence proof, we show that the stability of the fronts is also governed by the scalar Fisher–KPP equation. Furthermore, we perform numerical computations of the Evans function to explicitly identify regions in the parameter space where the fronts are spectrally stable.
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- Award ID(s):
- 2106157
- PAR ID:
- 10535208
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- ISSN:
- 0022-2526
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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