Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal

By Peter W. Bates, Guangyu Zhao. 2007

219 citations recorded by the source corpus.

Subfields: Applied Mathematics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/6bf514f7db350fc4d635b6309149607b6a6461d3

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Citations in the corpus

Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations (2012, background): “See, for example, [2], [6], [10], [12], [14], [19], [20], [25], [26], [30], [31], [32], [34], [53], for the study of spectral theory for nonlocal dispersal operators and the existence, uniqueness, and stability of nontrivial positive stationary solutions.”
Two species nonlocal diffusion systems with free boundaries (2019, background): “Therefore there exists L ≥ L0 such that λp ( L(−l,l) + al ) > 0 for l > L, and by [3, 7] and Proposition 3.”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, background): “his context, u(t,x) is the density of the considered population, Jis a dispersal kernel and f(x,s) is a KPP type non-linearity describing the growth rate of the population. When Ω is a bounded domain [3, 25, 28, 36, 41, 59], an optimal persistence criteria has been obtained using the sign of λp(M Ω ”
Pulsating fronts for nonlocal dispersion and KPP nonlinearity (2013, unknown): “ environment, [32, 33, 42]. Regarding equation (1.1) we quote [1, 2, 18, 20, 21, 23, 25] for the existence and characterisation of travelling fronts for this equation with homogenous nonlinearity and [3, 22, 24, 36, 42] for the study of the stationary problem. In what follows, we assume that J: RN →”
Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations (2014, background): “rm ( ∂tu = ν R D∪Dc k(y −x)[u(t,y) −u(t,x)]dy +F(t,x,u), x ∈ D,¯ Bn,bu = 0, x ∈ Dc (x ∈ RN if D = RN), (1.2) are often used to model diﬀusive systems which exhibit nonlocal internal interactions (see [4, 7, 10, 11, 12, 16, 19, 24, 26], etc.). In (1.2), D is a smooth bounded domain of RN or D = RN; D”
The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries (2018, unknown): “atial domains. We ﬁrstconsider the problem (3.7) ut= d Zℓ 2 ℓ1 J(x−y)u(t,y)dy−u(t,x) +f(u), t>0, x∈ [ℓ1,ℓ2], u(0,x) = u0(x), x∈ [ℓ1,ℓ2], where −∞ <ℓ1 <ℓ2 <+∞. Proposition 3.5 ( [3,11]). Denote Ω = (ℓ1,ℓ2) and suppose (J) and (f1)–(f4) hold. Then (3.7) admits a unique positive steady ”
On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications (2013, unknown): “The reader is referred to [1], [2], [4], [5], [6], [7], [8], [10], [11], [16], [22], [23], [24], [26], [28], [30], [32], etc.”
Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats (2010, unknown): “As the movements and interactions of many species in biology and ecology can occur between non-adjacent spatial locations, nonlocal dispersal has also been used to model the population spreading dynamics of species by many people (see [3], [15], [16], [17], [18], [31], [32], [35], and references the”
Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications (2012, background): “In [33], the authors proved that time independent KPP equations with nonlocal dispersal also exhibit monostable feature (see also [2,9] for the study of positive stationary solutions of time independent KPP equations with nonlocal dispersal).”