### Positive solutions of a diffusive predator-prey mutualist model with cross-diffusion

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H. Amann and J. Lopez-Gomez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), no. 2, 336-374.

S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal. 49 (2002), no. 3, 361-430.

W. Chen and R. Peng, Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model, J. Math. Anal. Appl. 291 (2004), no. 2, 550-564.

W. Chen and M. Wang, Non-constant positive steady-states of a predator-prey-mutualist model, Chinese Ann. Math. Ser. B 25 (2004), no. 2, 243-254.

W. Chen and M. Wang, Positive steady states of a competitor-competitor-mutualist model, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), no. 1, 53-57.

M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340.

M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161-180.

E.N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), no. 1, 131-151.

E.N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (1984), no. 2, 729-743.

E.N. Dancer and Y.H. Du, Positive solutions for a three-species competition system with diffusion. I. General existence results, Nonlinear Anal. 24 (1995), no. 3, 337-357.

E.N. Dancer and Y.H. Du, Positive solutions for a three-species competition system with diffusion. II. The case of equal birth rates, Nonlinear Anal. 24 (1995), no. 3, 359-373.

L. Hei, Global bifurcation of co-existence states for a predator-prey-mutualist model with diffusion, Nonlinear Anal. Real World Appl. 8 (2007), no. 2, 619-635.

T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988), no. 1, 143-166.

C.V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487-513.

B. Rai, H.I. Freedman and J.F. Addicott, Analysis of three-species models of mutualism in predator-prey and competitive systems, Math. Biosci. 65 (1983), no. 1, 13-50.

W. Ruan and W. Feng, On the fixed point index and multiple steady-state solutions of reaction-diffusion systems, Differential Integral Equations 8 (1995), no. 2, 371-391.

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal. 169 (1999), no. 2, 494-531.

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations 246 (2009), no. 7, 2788-2812.

J. Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York, 1983.

S. Xu, Global stability of a reaction-diffusion system of a competitor-competitor-mutualist model, Taiwanese J. Math. 15 (2011), no. 4, 1617-1627.

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, Handbook of differential equations: stationary partial differential equations. Vol. VI, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 411-501.

S.N. Zheng, A reaction-diffusion system of a competitor-competitor-mutualist model, J. Math. Anal. Appl. 124 (1987), no. 1, 254-280.

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