Tian, Rongrong; Wei, Jinlong; Wu, Jiang-Lun On a generalized population dynamics equation with environmental noise. (English) Zbl 1466.60122 Stat. Probab. Lett. 168, Article ID 108944, 7 p. (2021). Summary: We establish the existence and uniqueness of global (in time) positive strong solutions for a generalized population dynamics equation with environmental noise, while the global existence fails for the deterministic equation. Particularly, we prove the global existence of positive strong solutions for the following stochastic differential equation \[d X_t = (\theta X_t^{m_0} + k X_t^m) d t + \varepsilon X_t^{\frac{m + 1}{2}} \varphi (X_t) d W_t,\ t > 0,\ X_t > 0,\ m > m_0 \geqslant 1,\ X_0 = x > 0,\] with \(\theta, k, \varepsilon \in \mathbb{R}\) being constants and \(\varphi (r) = r^\vartheta\) or \(| \log (r) |^\vartheta\) (\( \vartheta > 0 \)), and we also show that the index \(\vartheta > 0\) is sharp in the sense that if \(\vartheta = 0\), one can choose certain proper constants \(\theta, k\) and \(\varepsilon\) such that the solution \(X_t\) will explode in a finite time almost surely. Cited in 2 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 92D25 Population dynamics (general) Keywords:stochastic differential equation; environmental noise; explosion; positive strong solution PDFBibTeX XMLCite \textit{R. Tian} et al., Stat. Probab. Lett. 168, Article ID 108944, 7 p. (2021; Zbl 1466.60122) Full Text: DOI Link References: [1] Agudov, N. V.; Spagnolo, B., Noise-enhanced stability of periodically driven metastable states, Phys. Rev. E, 64, 3, 1-4 (2001) [2] Butler, G.; Freedman, H. I.; Waltman, P., Uniformly persistent systems, Proc. Amer. Math. Soc., 96, 3, 425 (1986) · Zbl 0603.34043 [3] Chow, P. L., Explosive solutions of stochastic reaction-diffusion equations in mean \(L^p\)-norm, J. Differential Equations, 250, 2567-2580 (2011) · Zbl 1213.35410 [4] Dozzi, M.; López-Mimbela, J. A., Finite-time blowup and existence of global positive solutions of a semi-linear spde, Stochastic Process. Appl., 120, 6, 767-776 (2010) · Zbl 1193.35258 [5] Gammaitoni, L.; Hänggi, P.; Jung, P., Stochastic resonance, Rev. Modern Phys., 70, 1, 223-287 (1998) [6] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1989), North-Holland · Zbl 0684.60040 [7] Kifer, Y., Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel J. Math., 70, 1, 1-47 (1990) · Zbl 0732.58037 [8] Li, P. S.; Yang, X.; Zhou, X., A general continuous-state nonlinear branching process, Ann. Appl. Probab., 29, 4, 2523-2555 (2019) · Zbl 1466.60179 [9] Lv, G.; Duan, J., Impacts of noise on a class of partial differential equations, J. Differential Equations, 258, 6, 2196-2220 (2015) · Zbl 1319.60143 [10] Lv, G.; Duan, J.; Wang, L.; Wu, J.-L., Impacts of noise on ordinary differential equations, Dynam. Systems Appl., 27, 2, 25-36 (2018) [11] Mantegna, R. N.; Spagnolo, B., Noise enhanced stability in an unstable system, Phys. Rev. Lett., 76, 4, 563-566 (1996) [12] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97, 1, 95-110 (2002) · Zbl 1058.60046 [13] Niu, M.; Xie, B., Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations, Nonlinear Anal. RWA, 13, 3, 1346-1352 (2012) · Zbl 1239.60065 [14] Ramanan, K.; Zeitouni, O., The quasi-stationary distribution for small random perturbations of certain one-dimensional maps, Stochastic Process. Appl., 84, 1, 25-51 (1999) · Zbl 0997.60074 [15] Rozenfeld, A. F.; Tessone, C. J.; Albano, E.; Wio, H. S., On the influence of noise on the critical and oscillatory behavior of a predator-prey model: coherent stochastic resonance at the proper frequency of the system, Phys. Lett. A, 280, 1-2, 45-52 (2001) · Zbl 0972.37005 [16] Turchin, P., Complex Population Dynamics: A Theoretical Empirical Synthesis (2003), Princeton University Press: Princeton University Press Princeton and Oxford · Zbl 1062.92077 [17] Valenti, D.; Fiasconaro, A.; Spagnolo, B., Stochastic resonance and noise delayed extinction in a model of two competing species, Physica A, 331, 3, 477-486 (2004) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.