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Loops and branches of coexistence states in a Lotka–Volterra competition model. (English) Zbl 1154.35011
Summary: A two-species Lotka–Volterra competition–diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate $$\mu$$ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of $$\mu$$. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers $$l$$ and $$b$$, the interspecific competition coefficients can be chosen such that there exist at least $$l$$ bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least $$b$$ other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K57 Reaction-diffusion equations 92D10 Genetics and epigenetics 92D25 Population dynamics (general)
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