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. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 1048 – 1060 ¨ ITAK ˙ c TUB ⃝ doi: Colombeau solutions of a nonlinear stochastic predator–prey equation Ulu˘ gC ¸ APAR∗ ¨ ˙ MDBF, Sabancı Universitesi, Orhanlı, Tuzla, Istanbul Turkey Received: • Accepted: • Published Online: • Printed: Abstract: 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. Key words: Stochastic nonlinear predator–prey equation, generalized solutions, random Colombeau distributions 1. Introduction This article is concerned basically with the stochastic version of the following deterministic semilinear hyperbolic system (Lotka–Volterra) in 2 variables [2]: D1 u1 (x, t) = (∂t + c1 ∂x )u1 (x, t) = λ1 u1 (x, t)u2 (x, t) D2 u2 (x, t) = (∂t + c2 ∂x )u2 (x, t) = λ2 u1 (x, t)u2 (x, t) uj (x, 0) = γj (x) ; j = 1, 2, λ1 λ2 0 small enough depending on ϕ are significant. Therefore, in order to assert that a property holds for T ∈ G(ℜn ) it is sufficient to verify it on some representative fT for large q and small ϵ. The following definition utilizes this feature. A generalized function T ∈ G(ℜn ) is said to be of L1loc -type with respect to variable x1 if it has a representative fT with the following property: For every K × Kn−1 ⊂ ℜ × ℜn−1 compact, ∃q ∈ N such that ∀ϕ ∈ Aq (ℜn ), ∃M > 0, η > 0 with ∫ supx′ ∈Kn−1 K |fT (ϕϵ , x′ , x1 )|dx1 ≤ M, 0 0 works out for every ϕ ∈ Aq (ℜn ). . Association with Schwartz distributions T ∈ G is said to be associated to V ∈ D′ , denoted T ≈ V if ∃ a representative fT , T = fT +N
