Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case
The authors consider the boundary value problem (1) \(u'' = - \lambda u + u^ 2 + bu \nu\), \(\nu'' = - \mu \nu - cu \nu+\nu^ 2\), \(x \in (0,1)\), \(u(0) = u(1) = \nu (0) = \nu (1) = 0\). This problem usually arises in biology and chemistry in modeling the behaviour of two interacting species on (0,1). The authors show that the problem (1) has a coexistence state if and only if the relations \(\lambda > \lambda_ 1 (b \theta_ \mu)\) and \(\mu > \lambda_ 1 (-c \theta_ \lambda)\) hold. Moreover if these relations are satisfied then (1) has exactly one coexistence state, and this state is linearly stable, provided \(b\) (or \(c)\) is small enough.
scientific article; zbMATH DE number 459187