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A plateau pika model with spatial cross-diffusion is investigated. By analyzing the corresponding characteristic equations, the local stability of an coexistence steady state is discussed when d21 is small enough. However, when d21 is large enough, the model shows Turing bifurcation if B2 -4AC > 0. Furthermore, it is proved that if, R > R0, βK > d and cross-diffusion rates are zero, the positive coexistence steady state is globally asymptotically stable. A nonconstant positive solution bifurcates from the coexistent steady state by the Leray-Schauder degree theory. Numerical simulations are carried out to illustrate the main results.