NDLI: Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal

Content Provider	Springer Nature Link
Author	Lam, King Yeung
Copyright Year	2017
Abstract	We consider an integro-PDE model from evolutionary biology. The solution $$u_\epsilon (x,\alpha )$$ is structured by two variables $$x\in D \subset {\mathbb {R}}^k$$ and $$\alpha \in (\underline{\alpha },{\overline{\alpha }})\subset \subset {\mathbb {R}}_+$$ . The diffusion coefficient in the x direction depends on $$\alpha $$ and the diffusion coefficient in the $$\alpha $$ direction is a constant $$\epsilon ^2$$ . A special feature of this model is the appearance of the integral $${\hat{u}}_\epsilon (x)$$ of the solution in the $$\alpha $$ variable, which can be viewed as an infinite dimensional parameter of the problem. In a previous work, the existence of a steady state that exhibits Dirac-concentration in one of the variables yet remains regular in the other variables was proved independently by Lam and Lou (J Funct Anal 272:1755–1790, 2017) and by Perthame and Souganidis (Math Model Nat Phenom 11:154–166, 2016). In this paper, we tackle the long-time dynamics of solutions. When the environment function is non-constant, we show that the steady state is linearly stable by considering the corresponding nonlocal eigenvalue problem. Uniqueness of steady state is obtained from the stability result via a degree argument. When the environment function is a constant, the global asymptotic stability result is obtained. This problem can be regarded as a competition of infinitely many species parameterized by $$\alpha $$ . As with the competition model for three or more species, the integro-PDE model does not generate a monotone dynamical system so that it is necessary to consider all (real or complex) eigenvalues in determining its linear stability.
Ending Page	32
Page Count	32
Starting Page	1
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3
Volume Number	56
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2017-05-03
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Semilinear elliptic equations Problems related to evolution Stability problems
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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