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Spatio-temporal coexistence in the cross-diffusion competition system. (English) Zbl 1468.37065

The authors consider the coupled PDEs \[ u_t = \{(d+\gamma v)u\}_{xx}+(r_1-a_1u-b_1v)u \] \[ v_t = dv_{xx}+(r_2-a_2u-b_2v)v \] on \(x\in(0,L)\) and \(t>0\) with \(u_x(0,t)=u_x(L,t)=0\) and \(v_x(0,t)=v_x(L,t)=0\). They expand the solution in Fourier series in space, resulting in an infinite set of ODEs for the coefficients of the series. A numerical investigation of these ODEs shows instabilities of the spatially-uniform steady state as either \(d\) or \(\gamma\) are varied. This motivates a center manifold and normal form calculation about a doubly degenerate Hopf bifurcation. The result is a rigorous proof of the existence of a small amplitude (in space) time-periodic solution of the PDE which bifurcates from the spatially-uniform steady state through a Hopf bifurcation.

MSC:

37N25 Dynamical systems in biology
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35B10 Periodic solutions to PDEs
35B36 Pattern formations in context of PDEs
35K59 Quasilinear parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)

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AUTO-07P; AUTO
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References:

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