Stochastic one layer shallow water equations with Lévy noise

Abstract

Dedicated to Peter Eris Kloeden on the occasion of his 70 th birthday.

Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises

“…In the case one of section 4, we have shown that the jump noises with large positive coefficients can force the population for equation (1) to be extinct with probability one. Interestingly, we have found in the case two of section 4 that the jump noises with positive coefficients facilitate the permanence of the specie for equation (6).…”

“…From Theorem 2.7, we can obtain the following conclusions: (a) If h(t, z) > 0 for t ≥ 0 and z ∈ Z, the jump noises are favorable for the permanence of equation 6; (b) If −1 < h(t, z) < 0 for t ≥ 0 and z ∈ Z, the jump noises are unfavorable for the permanence of equation (6).…”

“…Thus, modeled with white noise cannot capture these phenomena. Introducing Lévy noises into the model may be a reasonable way to accommodate such phenomena; see [1,2,3,4,5,6,9,10,20,23,24,25,26,28,32,33,36,38]. In addition to stochastic disturbances, populations in ecology are often faced with the seasonal variations from environment and biological activity, which could be from the changes of weather, food supply, mating habits, hunting or harvesting seasons.…”

Nonlinear Stochastic parabolic partial differential equations with a monotone operator of the Ladyzenskaya-Smagorinsky type, driven by a Levy noise

“…Note that in Lemma 4.14 and in Lemma 4.15 we do not obtain convergence of the mentioned processes in L 2 ( Ω; L 2 ([0, T ]; H)) as in e.g. [15] or [19]. The lack of such a result could pose a problem while passing to the limit n → ∞ in Step 2.…”

“…Interest in SPDEs with Lévy noise has been growing in the recent years, see e.g., [12], [17], [15], [42], [43]. In [13] and [52] the authors use processes with jumps to model phase transitions in bursts of wind that contribute to the dynamics of El Niño.…”