On hybrid competitive Lotka-Volterra ecosystems

Nonlinear Analysis, Theory, Methods and Applications

Volumn 71, Issue 12, 2009, Pages

  Zhu, C ^a   Yin, G ^b  

^a UNIVERSITY OF WISCONSIN MILWAUKEE   (United States)

^b WAYNE STATE UNIVERSITY   (United States)

Author keywords

Asymptotic behavior; Lotka Volterra model; Random environment; Regime switching diffusion

Indexed keywords

ASYMPTOTIC BEHAVIORS; EXTERNAL ENVIRONMENTS; FINITE MOMENTS; FINITE TIME; LARGE TIME BEHAVIOR; LOTKA-VOLTERRA; LOTKA-VOLTERRA MODELS; NUMERICAL EXPERIMENTAL; POPULATION SIZES; POSITIVE SOLUTION; RANDOM ENVIRONMENT; REGIME SWITCHING; STOCHASTIC DIFFERENTIAL EQUATIONS;

DIFFERENTIAL EQUATIONS; DIFFUSION; ECOSYSTEMS; POPULATION STATISTICS; SWITCHING SYSTEMS;

ASYMPTOTIC ANALYSIS;

EID: 72149109277     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2009.01.166     Document Type: Article

References (30)

1. Lotka A.J. Elements of Physical Biology (1925), William and Wilkins, Baltimore
2. Volterra V. Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi. Mem. Acad. Lincei 2 (1926) 31-113
3. Takeuchi Y., and Adachi N. The existence of globally stable equilibria of ecosystems of the generalized Volterra type. J. Math. Biol. 10 (1980) 401-415
4. Li X.Z., Tang C.L., and Ji X.H. The creteria for globally stable equilibrium in n-dimensional Lotka-Volterra Systems. J. Math. Anal. Appl. 240 (1999) 600-606
5. Xiao D., and Li W. Limit cycles for the competitive three dimensional Lotka-Volterra system. J. Differential Equations 164 (2000) 1-15
6. Arnold L., Horsthemke W., and Stucki J.W. The influence of external real and white noise on the Lotka-Volterra model. Biomedical J. 21 (1979) 451-471
7. Goel N.S., Maitra S.C., and Montroll E.W. Nonlinear Models of Interacting Populations (1971), Academic Press, New York
8. Khasminskii R.Z., and Klebaner F.C. Long term behavior of solutions of the Lotka-Volterra systems under small random perturbations. Ann. Appl. Probab. 11 (2001) 952-963
9. Imhof L.A. The long-run behavior of stochastic replicator dynamics. Ann. Appl. Probab. 15 (2005) 1019-1045
10. Khasminskii R.Z., and Potsepun N. On the replicator dynamics behavior under Stratonovich type random perturbations. Stoch. Dyn. 6 (2006) 197-211
11. Mao X., Sabanis S., and Renshaw R. Asymptotic behavior of the stochastic Lotka-Volterra model. J. Math. Anal. Appl. 287 (2003) 141-156
12. Du N.H., and Sam V.H. Dynamics of a stochastic Lotka-Volterra model perturbed by white noise. J. Math. Anal. Appl. 324 (2006) 82-97
13. Mao X., Marion G., and Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Process. Appl. 97 (2002) 95-110
14. Cohen J.E., Łuczak T., Newman C.M., and Zhou Z.-M. Stochastic structure and nonlinear dynamics of food webs: Qualitative stability in a Lotka-Volterra cascade model. Proc. R. Soc. Lond. B 240 (1990) 607-627
15. Jeffries C. Stability of predation ecosystem models. Ecology 57 (1976) 1321-1325
16. Stratonovich R.L. A new representation for stochastic integrals and equations. SIAM J. Control 4 (1966) 362-371
17. Khasminskii R.Z. Stochastic Stability of Differential Equations (1980), Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands
18. Øksendal B. Stochastic Differential Equations. 6th ed. (2005), Springer, Berlin
19. Du N.H., Kon R., Sato K., and Takeuchi Y. Dynamical behavior of Lotka-Volterra competition systems: Non-autonomous bistable case and the effect of telegraph noise. J. Comput. Appl. Math. 170 (2004) 399-422
20. Luo Q., and Mao X. Stochastic population dynamics under regime switching. J. Math. Anal. Appl. 334 (2007) 69-84
21. Slatkin M. The dynamics of a population in a Markovian environment. Ecology 59 (1978) 249-256
22. Takeuchi Y., Du N.H., Hieu N.T., and Sato K. Evolution of predator-prey systems decribed by a Lotka-Volterra equation under random environment. J. Math. Anal. Appl. 323 (2006) 938-957
23. Skorohod A.V. Asymptotic Methods in the Theory of Stochastic Differential Equations (1989), Amer. Math. Soc., Providence, RI
24. Friedman A. Stochastic Differential Equations and Applications, Vols. I and II (1975), Academic Press, New York
25. Zhu C., and Yin G. Asymptotic properties of hybrid diffusion systems. SIAM J. Control Optim. 46 (2007) 1155-1179
26. Liptser R. A strong law of large numbers for local martingales. Stochastics 3 (1980) 217-228
27. Ladde G.S., and Šiljak D.D. Stability of multispecies communities in randomly varying environment. J. Math. Biol. 2 (1975) 165-178
28. Alvarez L.H.R., and Shepp L.A. Optimal harvesting of stochastically fluctuating populations. J. Math. Biol. 37 (1998) 155-177
29. Lungu E.M., and Øksendal B. Optimal harvesting from a population in a stochastic crowded environment. Math. Biosci. 145 (1997) 47-75
30. Edelstein-Keshet L. Mathematical Models in Biology (1988), Random House, New York