Competitive Lotka-Volterra population dynamics with jumps.(English)Zbl 1228.93112

Summary: This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a Stochastic Differential Equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the $$p$$th moment with $$p>0$$ and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our $$n$$-dimensional model.

MSC:

 93E03 Stochastic systems in control theory (general) 60J60 Diffusion processes 60J05 Discrete-time Markov processes on general state spaces 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text:

References:

 [1] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039 [2] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002 [3] Li, X.; Tang, C.; Ji, X., The criteria for globally stable equilibrium in $$n$$-dimensional lotka – volterra systems, J. math. anal. appl., 240, 600-606, (1999) · Zbl 0947.34044 [4] Takeuchi, Y.; Adachi, N., The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. math. biol., 10, 401-415, (1980) · Zbl 0458.92019 [5] Takeuchi, Y.; Adachi, N., The stability of generalized Volterra equations, J. math. anal. appl., 62, 453-473, (1978) · Zbl 0388.45011 [6] Xiao, D.; Li, W., Limit cycles for the competitive three dimensional lotka – volterra system, J. differential equations, 164, 1-15, (2000) · Zbl 0960.34022 [7] Gard, T., Persistence in stochastic food web models, Bull. math. biol., 46, 357-370, (1984) · Zbl 0533.92028 [8] Gard, T., Stability for multispecies population models in random environments, Nonlinear anal., 10, 1411-1419, (1986) · Zbl 0598.92017 [9] Mao, X.; Marion, G.; Renshaw, E., Environmental noise suppresses explosion in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046 [10] Mao, X.; Yuan, C.; Zou, J., Stochastic differential delay equations of population dynamics, J. math. anal. appl., 304, 296-320, (2005) · Zbl 1062.92055 [11] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press · Zbl 1126.60002 [12] G. Hu, K. Wang, On stochastic logistic equation with Markovian switching and white noise, Osaka J. Math. (2010) (preprint). · Zbl 1239.60045 [13] Jiang, D.; Shi, N., A note on nonautonomous logistic equation with random perturbation, J. math. anal. appl., 303, 164-172, (2005) · Zbl 1076.34062 [14] Liu, M.; Wang, K., Persistence and extinction in stochastic non-autonomous logistic systems, J. math. anal. appl., 375, 443-457, (2011) · Zbl 1214.34045 [15] Zhu, C.; Yin, G., On hybrid competitive lotka – volterra ecosystems, Nonlinear anal., 71, (2009), e1370-e1379 · Zbl 1238.34059 [16] Zhu, C.; Yin, G., On competitive lotka – volterra model in random environments, J. math. anal. appl., 357, 154-170, (2009) · Zbl 1182.34078 [17] Roubik, D., Experimental community studies: time-series tests of competition between african and neotropical bees, Ecology, 64, 971-978, (1983) [18] Roughgarden, J., Theory of population genetics and evolutionary ecology: an introduction, (1979), Macmillan New York [19] Peng, S.; Zhu, X., Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic process. appl., 116, 370-380, (2006) · Zbl 1096.60026 [20] Prato, D.; Zabczyk, J., Ergodicity for infinite dimensional systems, (1996), Cambridge University Press · Zbl 0849.60052 [21] Lipster, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228, (1980) · Zbl 0435.60037 [22] Kunita, H., Itô’s stochastic calculus: its surprising power for applications, Stochastic process. appl., 120, 622-652, (2010) · Zbl 1202.60079 [23] Øksendal, B.; Sulem, A., Applied stochastic control of jump diffusions, (2007), Springer Berlin · Zbl 1116.93004 [24] Applebaum, D., Lévy processes and stochastics calculus, (2009), Cambridge University Press
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.