A symbiotic self-cross diffusion model

In this work, we show existence and non-existence results of coexistence states for a Lotka-Volterra symbiotic model with self and cross-diffusion in one species. We study the behavior of the set of positive solutions when the cross-diffusion or the self-diffusion parameter is large.In this work we show existence and non-existence results of coexistence states for a Lotka-Volterra symbiotic model with self and cross-diffusion in one species. We study the behavior of the set of positive solutions when the cross-diffusion or the self-d199-10833856CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOMTM2012-31304Dancer, E.N., On the indices of fixed points of mappings in cones and applications (1983) J. Math. Anal. Appl., 91, pp. 131-151Delgado, M., Montenegro, M., Suárez, A., A Lotka-Volterra symbiotic model with cross-diffusion (2009) J. Differential Equations, 246, pp. 2131-2149Delgado, M., López-Gómez, J., Suárez, A., On the symbiotic Lotka-Volterra model with diffusion and transport effects (2000) J. Differential Equations, 160, pp. 175-262Gidas, B., Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations (1981) Comm. Partial Differential Equations, 6, pp. 883-901Korman, P., Leung, A., On the existence and uniqueness of positive steady-states in the Volterra-Lotka ecological models with diffusion (1987) Appl. Anal., 26, pp. 145-160Kuto, K., A strongly copuled diffusion effect on the stationary solution set of a prey-predator model (2007) Adv. Differential Equations, 12, pp. 145-172Kuto, K., Yamada, Y., Positive solutions for Lotka-Volterra competition systems with large cross-diffusion (2010) Appl. Anal., 89, pp. 1037-1066Kuto, K., Yamada, Y., Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type (2009) Differential Integral Equa-tions, 22, p. 725752Li, L., Coexistence theorems of steady states for predator-prey interacting systems (1988) Trans. Amer. Math. Soc., 305, pp. 143-166Ling, Z., Pedersen, M., Coexistence of two species in a strongly coupled cooperative model (2007) Math. Comput. Modelling, 45, pp. 371-377López-Gómez, J., The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems (1996) J. Differential Equations, 127, pp. 263-294López-Gómez, J., Pardo, R., Coexistence regions in Lotka-Volterra models with diffusion (1992) Nonlinear Anal, 19, pp. 11-28Lou, Y., Necessary and sufficient condition for the existence of positive solutions of certain cooperative system (1996) Nonlinear Anal, 26, pp. 1079-1095Lou, Y., Ni, W.M., Diffusion vs cross-diffusion: an elliptic approach (1999) J. Differential Equations, 154, pp. 157-190Pao, C.V., Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion (2005) Nonlinear Anal, 60, pp. 1197-1217Ruan, W.H., Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients (1996) J. Math. Anal. Appl., 197, pp. 558-578Ryu, K., Ahn, I., Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics (2003) J. Math. Anal. Appl., 283, pp. 46-65Ryu, K., Ahn, I., Positive steady-states for two interacting species models with linear self-cross diffusions (2003) Discrete Contin. Dyn. Syst., 9, pp. 1049-1061Yamada, Y., Positive solutions for Lotka-Volterra systems with cross-diffusion (2008) Hand-book of differential equations: stationary partial differential equations, 6, pp. 411-501. , Handb. Differ. Equ., Elsevier/North-Holland, Amsterda