Spatiotemporal patterns driven by cross-diffusion of a uni-directional
consumer-resource (C-R) system with Holling-II type functional response are
investigated in this paper. The existence of a unique positive steady state of the
considered system is studied first. The linear stability analysis shows that the
cross-diffusion is the key mechanism for the formation of spatiotemporal patterns
through Turing bifurcation. We choose the cross-diffusion coefficient as bifurcation
parameter and discuss three different types of Turing bifurcations, corresponding to
simple, double non-resonant and double resonant cases. Based on weakly nonlinear
analysis with the multiple scale method and the adjoint system theory, we derive the
amplitude equations of the Turing patterns near the Turing bifurcation point and
obtain the analytical approximation solutions of the patterns for each case. Specially,
some qualitative results of amplitude equations of the resonant case are given in
detail. Finally, numerical simulations are performed to illustrate the weakly nonlinear
theoretical predictions and through these simulations some patterns (single mode
pattern, mixed-mode pattern, super-squares pattern, roll pattern, hexagonal pattern)
are found. Simultaneously, numerical simulations show that the resource supplying
rate has an important impact on the direction of Turing bifurcation.

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• Cross-diffusion-driven Turing instability and weakly nonlinear analysis of Turing patterns in a uni-directional consumer-resource system