Colombeau solutions of a nonlinear stochastic predator--prey equation

The solution of a random semilinear hyperbolic system with singular initial data is sought as a random Colombeau distribution. The product of 2 additive white noises is well tackled within the theory of random Colombeau distributions. In the special case of a random predator--prey system, the exact Colombeau solution is obtained under some assumptions when the process is driven by doubly reflected Brownian motions.

Anahtar Kelimeler:

Stochastic nonlinear predator--prey equation, generalized solutions, random Colombeau distributions

Colombeau solutions of a nonlinear stochastic predator--prey equation

Keywords:

Stochastic nonlinear predator--prey equation, generalized solutions, random Colombeau distributions,

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Biagioni, H.A. A non-linear theory of generalized functions. Lecture Notes in Maths. No. 1421, Springer-Verlag (1990).
Colombeau, J.F., Oberguggenberger, M.: On a hyperbolic system with a compatible quadratic term: Generalized solutions, delta waves and multiplication of distributions. J. Commun. Partial Differ. Equations 15, 905–938 (1990). C ¸ apar, U., Aktu˘ glu, H.: A new construction of random Colombeau distributions. Statistics and Probability Letters 54, 291–299 (2001).
C ¸ apar, U. ; Aktu˘ glu, H.: An overall view of stochastics in Colombeau related algebras. Stochastic Analysis and Related Topics VIII, Progress in Probability Vol. 53, U.C ¸ apar, A.S. Ustunel, eds., 67–90, Birkhauser (2003).
Hasimoto, H.: Exact solution of a certain semi-linear system of partial differential equations related to a migrating predation problem. Proc. Japan Acad. 50 623–627 (1974).
Oberguggenberger, M.: Generalized solutions to semi-linear hyperbolic systems. Monatshefte Math, 3 599–607 (1987).
Oberguggenberger, M., Russo, F.: Nonlinear SPDEs: Colombeau solutions and pathwise limits. Proc. of the 6th Oslo-Silivri Worksop on Stochastic Analysis and Related Topics, 325–339, Birkhauser (1998).
Rosinger, E.E.: Generalized Solutions of Non-linear Partial Differential Equations. Lecture Notes in Maths, 684, Springer-Verlag (1978).