# American Institute of Mathematical Sciences

March  2021, 14(3): 919-933. doi: 10.3934/dcdss.2020228

## Spatio-temporal coexistence in the cross-diffusion competition system

 1 Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadainishi Miyazaki, 889-2192, Japan 2 Graduate School of Science and Technology, Meiji University, Kanagawa 214-8571, Japan

* Corresponding author: Shunsuke Kobayashi

Received  January 2019 Revised  September 2019 Published  March 2021 Early access  December 2019

We study a two component cross-diffusion competition system which describes the population dynamics between two biological species. Since the cross-diffusion competition system possesses the so-called population pressure effects, a variety of solution behaviors can be exhibited compared with the classical diffusion competition system. In particular, we discuss on the existence of spatially non-constant time periodic solutions. Applying the center manifold theory and the standard normal form theory, the cross-diffusion competition system is reduced to a two dimensional dynamical system around a doubly degenerate point. As a result, we show the existence of stable time periodic solutions in the system. This means spatio-temporal coexistence between two biological species.

Citation: Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
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Neutral stability curves in $(d,\gamma)$-plane. The horizontal axis and the vertical axis mean the value $d$ and the value $\gamma$, respectively. The parameter values are $r_1 = 5$, $r_2 = 2$, $a_1 = 3$, $a_2 = 1$, $b_1 = 1$, $b_2 = 3$ and $L = 1$
">Figure 2.  Global bifurcation diagrams for (3) with (4) when the value of $\gamma$ varies. The horizontal axis and the vertical axis mean the value $d$ and the boundary value $u(0)$, respectively. Solid curves and dashed curves mean stable branches and unstable ones, respectively. The marks $\square$ and $\blacksquare$ indicate a pitchfork bifurcation point and a Hopf bifurcation point, respectively. The parameter values are the same as the ones in Figure 1
in the neighborhood of the Hopf bifurcation points. The marks $\bullet$ indicate stable periodic solution branch. The parameter values are the same as the ones in Figure 2. (b) A periodic solution at $d = 0.01528$">Figure 3.  (a) Enlarged view of the bifurcation diagram for $\gamma = 1.7$ in Figure 2 in the neighborhood of the Hopf bifurcation points. The marks $\bullet$ indicate stable periodic solution branch. The parameter values are the same as the ones in Figure 2. (b) A periodic solution at $d = 0.01528$
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