This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space ∂u1∂t=Δu1+u1(x,y,t)r1(t)-a1(t)u1(x,y,t)-b1(t)u2(x,y,t),∂u2∂t=dΔu2+u2(x,y,t)r2(t)-a2(t)u1(x,y,t)-b2(t)u2(x,y,t),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$\end{document}where Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} denotes ∂2∂x2+∂2∂y2\documentclass[12pt]{minimal}
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\begin{document}$$\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }$$\end{document}, x,y∈R\documentclass[12pt]{minimal}
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\begin{document}$$x,y\in {\mathbb {R}}$$\end{document} and d>0\documentclass[12pt]{minimal}
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\begin{document}$$d>0$$\end{document} is a constant, the functions ri(t),ai(t)\documentclass[12pt]{minimal}
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\begin{document}$$r_i(t),a_i(t)$$\end{document} and bi(t)\documentclass[12pt]{minimal}
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\begin{document}$$b_i(t)$$\end{document} are T-periodic and Hölder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution Φ1(x+ct,t),Φ2(x+ct,t)\documentclass[12pt]{minimal}
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\begin{document}$$\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) $$\end{document} connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.
