Abstract
In this work, we study coexistence states for a Lotka‐Volterra symbiotic system with cross‐diffusion under homogeneous Dirichlet boundary conditions. By using topological degree theory and bifurcation theory, we prove the existence and multiplicity of positive solutions under certain conditions on the parameters. Asymptotic behaviors of positive solutions are respectively studied as the cross‐diffusion coefficient tends to infinity and the interaction rate tends to zero. Finally, we compare our results with those of the Lotka‐Volterra predator and competition systems.
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