# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021166
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## Analysis of stationary patterns arising from a time-discrete metapopulation model with nonlocal competition

 1 Department of Economics, Social Sciences University of Ankara, Ulus-Ankara, Turkey 2 Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

* Corresponding author

Received  October 2020 Revised  May 2021 Early access June 2021

Fund Project: This research of YK is partially funded by the NSF-DMS (Award Number 1716802); the NSFIOS/DMS (Award Number 1558127); DARPA-SBIR 2016.2 SB162-005; and the James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472)

The paper studies the pattern formation dynamics of a discrete in time and space model with nonlocal resource competition and dispersal. Our model is generalized from the metapopulation model proposed by Doebeli and Killingback [2003. Theor. Popul. Biol. 64, 397-416] in which competition for resources occurs only between neighboring populations. Our study uses symmetric discrete probability kernels to model nonlocal interaction and dispersal. A linear stability analysis of the model shows that solutions to this equation exhibits pattern formation when the dispersal rate is sufficiently small and the discrete interaction kernel satisfies certain conditions. Moreover, a weakly nonlinear analysis is used to approximate stationary patterns arising from the model. Numerical solutions to the model and the approximations obtained through the weakly nonlinear analysis are compared.

Citation: Ozgur Aydogmus, Yun Kang. Analysis of stationary patterns arising from a time-discrete metapopulation model with nonlocal competition. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021166
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The patchy environment $S$ and neighborhoods of a patch $q = (q_1, q_2).$ In Panels (a) and (b), von Neumann and Moore neighborhoods of patch $(q_1, q_2)$ (colored in red) are determined by patches colored in gray, respectively
Dispersion relations for the examples E1-4
Comparison between numerical solutions to the CML (on the left) for examples E1 and E3, and the weakly nonlinear first-order approximations of these solutions (on the right). Panel (a) illustrates stationary waves in a 1-dimensional habitat for E1. Similarly, panel (c) shows stationary patterns in a 2-dimensional habitat for E3. In panels (c) and (d), colors represent the population size and approximated population size, respectively
Comparison between numerical solutions to CML (on the left) and the weakly nonlinear first order approximation of these solutions (on the right). Panel (a) illustrates stationary waves in a 1-dimensional habitat for the parameters given in example E2
Comparison between numerical solution to CML (on the left) and the weakly nonlinear first order approximation of this solutions (on the right). Panel (a) illustrates stationary waves in a 2-dimensional habitat for E4. In panels (a) and (b), colors represent the population size and approximated population size, respectively
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