A mathematical model proposed to explain the horizontal structure of prey and predator populations is represented by a semilinear parabolic system of equations. In this paper some mixed problems for this kind of system, Lotka-Volterra system, are considered and asymptotic behaviors of the solutions are investigated by use of Energy Method. A remarkable fact is that, in opposition to Steele's conjecture (see ), the solution in the case of the Neumann boundary condition is asymptotically spatially homogeneous but does not tend to any constant steady state solution.
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Masayasu Mimura, Takaaki Nishida, On a Certain Semilinear Parabolic System Related to the Lotka-Volterra Ecological Model. Publ. Res. Inst. Math. Sci. 14 (1978), no. 2, pp. 269–282