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In this paper, we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics [19] or nonlocal diffusion models [1]. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure, we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory.
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ON EQUILIBRIUM SOLUTIONS TO NONLOCAL MECHANISTIC MODELS IN ECOLOGY
Understanding the factors that drive species to move and develop territorial patterns is at the heart of spatial ecology. In many cases, mechanistic models, where the movement of species is based on local information, have been proposed to study such territorial patterns. In this work, we introduce a nonlocal system of reaction-advection-diffusion equations that incorporate the use of nonlocal information to influence the movement of species. One benefit of this model is that groups are able to maintain coherence without having a home-center. As incorporating nonlocal mechanisms comes with analytical and computational costs, we explore the potential of using long-wave approximations of the nonlocal model to determine if they are suitable alternatives that are more computationally efficient. We use the gradient flow-structure of the both local and nonlocal models to compute the equilibrium solutions of the mechanistic models via energy minimizers. Generally, the minimizers of the local models match the minimizers of the nonlocal model reasonably well, but in some cases, the differences in segregation strength between groups is highlighted. In some cases, as we scale the number of groups, we observe an increased savings in computational time when using the local model versus the nonlocal counterpart.
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- Award ID(s):
- 1909638
- PAR ID:
- 10324176
- Date Published:
- Journal Name:
- Journal of applied analysis and computation
- Volume:
- 11
- Issue:
- 6
- ISSN:
- 2156-907X
- Page Range / eLocation ID:
- 2664-2686
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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