AIMS Mathematics, 4(6): 1721–1744. DOI:10.3934/math.2019.6.1721 Received: 17 October 2018 Accepted: 27 November 2019 http://www.aimspress.com/journal/Math Published: 23 December 2019 Research article Biological advection and cross-diffusion with parameter regimes Jaywan Chung1, Yong-Jung Kim2;∗, Ohsang Kwon3 and Chang-Wook Yoon4 1 Thermoelectric Conversion Research Center, Korea Electrotechnology Research Institute, 12, Bulmosan-ro 10 beon-gil, Changwon-si, Gyeongsangnam-do, 51543, Korea 2 Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong, Daejeon 305-701, Korea 3 Department of Mathematics, Chungbuk National University, Chungdae-ro 1, Seowon-gu, Cheongju, Chungbuk 362-763, Korea 4 College of Science & Technology, Korea University, Sejong 30019, Republic of Korea * Correspondence: Email:
[email protected]; Tel: +82423502739; Fax: +82423505710. Abstract: Advection and cross-diffusion terms are obtained as dispersal strategies of biological species. The focus of the paper is their connection to a given population dynamics. In particular, meaningful parameter regimes as biological dispersal are obtained. Eventually, we obtain a systematic approach to construct an advection or a cross-diffusion term from a given population dynamics and find meaningful parameter regimes as biological advection and cross-diffusion. Keywords: cross diffusion; chemotaxis; biological dispersal; Lotka-Volterra equations; evolution of dispersal Mathematics Subject Classification: 35K57, 35Q92 1. Introduction A realistic dispersal of biological organisms should take account the interaction between individuals and the response to the environmental variations. The importance of having such a diffusion model has been emphasized by many researchers (see Skellam [25,26] and Okubo & Levin [24, Chapter 5]). In a reaction-diffusion equation, such interactions and responses are given in the reaction term and hence it is important to appropriately design the dispersal part depending on the reaction term.