Advances in Differential Equations

Positive steady states for prey-predator models with cross-diffusion

Kimie Nakashima and Yoshio Yamada

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This paper is concerned with the existence of positive solutions for boundary value problems of nonlinear elliptic systems which arise in the study of the Lotka-Volterra prey-predator model with cross-diffusion. Making use of the theory of the fixed point index we can derive sufficient conditions for the coexistence of positive steady states. Moreover, when cross-diffusion effects are comparatively small, we can get a necessary and sufficient condition for the coexistence. The uniqueness result is also given in the special case when the spatial dimension is one.

Article information

Adv. Differential Equations, Volume 1, Number 6 (1996), 1099-1122.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q80: PDEs in connection with classical thermodynamics and heat transfer
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35K57: Reaction-diffusion equations 92D25: Population dynamics (general) 92D40: Ecology


Nakashima, Kimie; Yamada, Yoshio. Positive steady states for prey-predator models with cross-diffusion. Adv. Differential Equations 1 (1996), no. 6, 1099--1122.

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