Dirac-concentrations in an integro-pde model from evolutionary game theory

King-Yeung Lam

doi:10.3934/dcdsb.2018205

Article Contents

2019, Volume 24, Issue 2: 737-754. Doi: 10.3934/dcdsb.2018205

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Dirac-concentrations in an integro-pde model from evolutionary game theory

King-Yeung Lam

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

Received: July 31, 2017

Revised: January 31, 2018

Early access: June 2018

Published: January 2019

The first author is supported by NSF grant DMS-1411476

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Abstract

Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Motivated by the existence of moving Dirac-concentrations in the time-dependent problem, we study the qualitative properties of steady states in the limit of small diffusion. Under different conditions on the growth rate and interaction kernel as motivated by the framework of adaptive dynamics, we will show that as the diffusion rate tends to zero the steady state concentrates (ⅰ) at a single location; (ⅱ) at two locations simultaneously; or (ⅲ) at one of two alternative locations. The third result in particular shows that solutions need not be unique. This marks an important difference of the non-local equation with its local counterpart.

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Mathematics Subject Classification: Primary: 35K55, 35F21, 92D15; Secondary: 47G20, 49L25.

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Figure 1. The left, center and right panels illustrate the sign of $\frac{K(x,y)}{r(x)} - \frac{K(y,y)}{r(y)}$ as a function of $x$ and $y$, under the assumptions of Theorems 1, 2 and 3 respectively. Here $x$ and $y$ are the strategy of the invader and resident species respectively. $\frac{K(x,y)}{r(x)} - \frac{K(y,y)}{r(y)} < 0$ (resp. $>0$) means invasion of resident with strategy ''$y$" by invader with strategy ''$x$" is a success (resp. failure)

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