Bifurcation branch of stationary solutions for a Lotka–Volterra cross-diffusion system in a spatially heterogeneous environment

Fish–hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross-diffusion

“…Compared with the results in [21], we find that the results are quite different. For the case μ > 0, a spatial segregation can yield an unbounded fish-hook shaped bifurcation branch; for the case μ < 0, strong cooperation for u and weak for v can also yield an unbounded fish-hook shaped bifurcation branch; while in [21], a spatial segregation asserts a bounded fish-hook shaped bifurcation branch.…”

“…For example, Du et al [8,9,[11][12][13] have mainly studied degenerate effects of intraspecific pressures in some predator-prey or competitive models; Hutson et al [16][17][18][19] have mainly studied the spatial effects of birth rates in some diffusive competitive models. Recently, Kuto [21] studied a Lotka-Volterra predator-prey system with cross-diffusion in a spatially heterogeneous environment. By the methods of the bifurcation theory and Lyapunov-Schmidt reduction, Kuto obtained the global bifurcation branch of positive stationary solutions and found that the spatial segregation of ρ(x) and d(x) could cause the bifurcation branch to form a bounded fish-hook curve.…”

A Lotka–Volterra symbiotic model with cross-diffusion

“…In the second part of the paper, we have studied the profiles of the solutions when the crossdiffusion parameter β tends to +∞, this type of study is made in a slight different problem in [7] and [8], see also [14]. We show the following result.…”

“…(1) Fix μ > λ 1 and consider λ as a bifurcation parameter. We apply the Crandall-Rabinowitz theorem [1] (see also Section 2 of [9] and [20]) to conclude that λ = F (μ) is a simple bifurcation point from the semi-trivial solution (0, θ μ ), in fact it is the unique bifurcation point of positive solutions of (3.2) from (0, θ μ ). Moreover, from Theorem 4.1 in [12] there exists a continuum C of coexistence states of (3.2) emanating from (0, θ μ ) at λ = F (μ) which verifies at least one of the following alternatives:…”

“…When β > 0, that is, when the cross-diffusion is present, the competition (b < 0 and c < 0) and prey-predator (bc < 0) cases have been studied in more detail than the symbiotic case (b > 0 and c > 0), see for instance [4,6,9,[14][15][16][19][20][21]23]. Basically, in these papers the authors study existence, non-existence, uniqueness or multiplicity of positive solutions using fixed point index in positive cones, global and local bifurcation techniques; and also sub-supersolution methods in [18].…”