On a predator-prey system with cross diffusion representing the tendency of predators in the presence of prey species. (English) Zbl 1160.35021

The present paper is concerned with the existence of positive solution of the following Lotka-Volterra predator-prey system with cross-diffusion \[ -\Delta u=u(a_1-u-b_{1,2}v), \quad -D\Delta u-\Delta v=v (a_2+b_{2,1}u-v) \quad \text{in } \Omega, \] \((u,v)=(0,0) \text{on}\; \partial \Omega\). The system represents the tendency of predators to avoid the group defense by a large number of prey or diffuse in the direction of higher concentration of the prey species. The approach uses the method of upper and lower solutions. Sufficient conditions for the existence of positive solutions are provided and the non-existence of positive solutions is also investigated.


35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)
Full Text: DOI


[1] Dubey, B.; Das, B.; Hussain, J., A predator-prey interaction model with self and cross-diffusion, Ecol. Model., 141, 67-76 (2001)
[2] Ghoreishi, A.; Logan, R., Positive solutions of a class of biological models in a heterogeneous environment, Bull. Austral. Math. Soc., 44, 1, 79-94 (1991) · Zbl 0735.35051
[3] Gurtin, M. E., Some mathematical models for population dynamics that lead to segregation, Quart. Appl. Math., 32, 1-9 (1974/75) · Zbl 0298.92006
[4] Jorné, J., Negative ionic cross diffusion coefficients in electrolytic solutions, J. Theoret. Biol., 55, 2, 529-532 (1975)
[5] Kuto, K.; Yamada, Y., Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197, 315-348 (2004) · Zbl 1205.35116
[6] Li, L., Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305, 1, 143-166 (1988) · Zbl 0655.35021
[7] Lou, Y.; Ni, W. M., Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131, 1, 79-131 (1996) · Zbl 0867.35032
[8] Lou, Y.; Ni, W. M., Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154, 1, 157-190 (1999) · Zbl 0934.35040
[9] Okubo, A.; Levin, S. A., Diffusion and Ecological Problems: Modern Perspectives, Interdiscip. Appl. Math., vol. 14 (2001), Springer-Verlag: Springer-Verlag New York · Zbl 1027.92022
[10] Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0780.35044
[11] Pao, C. V., Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Anal., 60, 1197-1217 (2005) · Zbl 1074.35034
[12] Ryu, K.; Ahn, I., Coexistence theorem of steady states for nonlinear self-cross-diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283, 1, 46-65 (2003) · Zbl 1115.35321
[13] Shigesada, N.; Kawasaki, K.; Teramoto, E., Spatial segregation of interacting species, J. Theoret. Biol., 79, 1, 83-99 (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.