Description of an ecological niche for a mixed local/nonlocal dispersal: An evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes

By Serena Dipierro, Enrico Valdinoci. Physica A: Statistical Mechanics and its Applications, 2021

149 citations recorded by the source corpus.

Abstract We propose here a motivation for a mixed local/nonlocal problem with a new type of Neumann condition. Our description is based on formal expansions and approximations. In a nutshell, a biological species is supposed to diffuse either by a random walk or by a jump process, according to prescribed probabilities. If the process makes an individual exit the niche, it must come to the niche right away, by selecting the return point according to the underlying stochastic process. More precisely, if the random particle exits the domain, it is forced to immediately reenter the domain, and the new point in the domain is chosen randomly by following a bouncing process with the same distribution as the original one. By a suitable definition outside the niche, the density of the population ends up solving a mixed local/nonlocal equation, in which the dispersion is given by the superposition of the classical and the fractional Laplacian. This density function satisfies two types of Neumann conditions, namely the classical Neumann condition on the boundary of the niche, and a nonlocal Neumann condition in the exterior of the niche.

Subfields: Modeling and Simulation, Molecular Biology, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/53285fd07bc337b4e4eb1921da6f2ec31ba760c2

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Citations in the corpus

On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, background): “This kind of equation appears in image processing, Lévy processes etc, see Dipierro-Valdinoci [25] and the references therein.”
Regularity results for solutions of mixed local and nonlocal elliptic equations (2022, background): “For instance, as observed in [21], these operators describe a biological species whose individuals diffuse either by a random walk or by a jump process, according to prescribed probabilities.”
A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, unknown): “…subject of investigation, arising naturally in several ﬁelds, for instance as the superposition of diﬀerent types of stochastic processes such as a classical random walk and a L´evy ﬂight, which has also interesting application in the study of optimal animal foraging strategies, see [20,21,26,28].”
A Faber-Krahn inequality for mixed local and nonlocal operators (2021, background): “Further applications arise in the theory of optimal searching strategies, biomathematics and animal foraging, see [30, 31] and the references therein.”
A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators (2021, background): “The study of mixed local/nonlocal operators has been recently received an increasing level of attention, both in view of their intriguing mathematical structure, which combines the classical setting and the features typical of nonlocal operators in a framework that is not scale-invariant [40, 45, 46”
Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations (2022, background): “We also refer to Biagi et al. [4, 8], Dipierro et al. [22], Dipierro et al. [21], Dipierro and Valdinoci [23] and the references therein.”
Linear theory for a mixed operator with Neumann conditions (2020, unknown): “To this end, see [37], we set a variational formulation related to the operator in (1) which endows the equation in the set with a suitable Neumann condition (see [39] for a thorough description of the biological motivation).”
Mixed local and nonlocal elliptic operators: regularity and maximum principles (2020, background): “the appendix in [28] for a thorough discussion of this phenomenon and [30] for the description of a biological population in an ecological niche modeled by a mixed operator.”