Permanence and partial extinction in an impulsive delay competitive system with the effect of toxic substances

Journal of Mathematical Chemistry

Volumn 46, Issue 4, 2009, Pages 1213-1231

  Liu, Zhijun ^a,b   Hui, Jing ^c   Wu, Jianhua ^a  

^a SHAANXI NORMAL UNIVERSITY   (China)

^b HUBEI UNIVERSITY FOR NATIONALITIES   (China)

^c GUANGXI UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

Author keywords

Competitive system; Delay; Impulse; Permanence and partial extinction; Toxicant

Indexed keywords

EID: 70350167115     PISSN: 02599791     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10910-008-9513-1     Document Type: Article

References (31)

1. Hallam T.G., Clark C.E.: Nonautonomous logistic equations as models of populations in a deteriorating environment. J. Theor. Biol 93, 303-311 (1982).
2. Freedman H.I., Schukla J.B.: Models for the effect of toxicant in single species and predator-prey systems. J. Math. Biol 30, 15-30 (1990).
3. Smith J.M.: Models in Ecology. Cambridge University, Cambridge (1974).
4. Chattopadhyay J.: Effect of toxic substances on a two-species competitive system. Ecol. Model 84, 287-289 (1996).
5. Mukhopadhyay A., Chattopadhyay J., Tapaswi P.K.: A delay differential model of plankton allelopathy. Math. Biosci 149, 167-189 (1998).
6. Liu B., Tao F.M.: The optimal pulse harvesting policy on a single-species population model with birth pulses in a polluted environment. Adv. Complex Syst 10, 173-196 (2007).
7. Liu B., Chen L.S., Zhang Y.J.: The effects of impulsive toxicant input on a population in a polluted environment. J. Bio. Syst 11, 265-274 (2003).
8. Bernatik A., Zimmerman W., Pitt M., Strizik M., Nevrly V., Zelinger Z.: Modelling accidental releases of dangerous gases into the lower troposphere from mobile sources. Process Safety Environ. Protect 86, 198-207 (2008).
9. Kassomenos P., Karayannis A., Panagopoulos I., Karakitsios S., Petrakis M.: Modelling the dispersion of a toxic substance at a workplace. Environ. Model. Software 23, 82-89 (2008).
10. Papadimitriou Ch., Palaska G., Lazaridou M., Samaras P., Sakellaropoulos G.P.: The effects of toxic substances on the activated sludge microfauna. Desalination 211, 177-191 (2007).
11. Sarkar R.R., Mukhopadhyay B., Bhattacharyya R., Banerjee S.: Time lags can control algal bloom in two harmful phytoplankton-zooplankton system. Appl. Math. Comput 186, 445-459 (2007).
12. Li Z., Chen F.D.: Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances. Appl. Math. Comput 182, 684-690 (2006).
13. Song X.Y., Chen L.S.: Periodic solution of a delay differential equation of plankton allelopathy. Acta. Math. Sci. Ser A 23, 8-13 (2003) (Chinese).
14. Liu Z.J., Chen L.S.: Periodic solution of a two-species competitive system with toxicant and birth pulse. Chaos Solitons Fract 32, 1703-1712 (2007).
15. Clark C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1976).
16. Goh B.S.: Management and Analysis of Biological Populations. Elsevier Scientific, Netherlands (1980).
17. Bainov D.D., Simeonov P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, Burnt Mill (1993).
18. Lakshmikantham V., Bainov D.D., Simeonov P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989).
19. D'onofrio A.: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci 179, 57-72 (2002).
20. Liu X.N., Chen L.S.: Global dynamics of the periodic logistic system with periodic impulsive perturbations. J. Math. Anal. Appl 289, 279-291 (2004).
21. W.B. Wang, J.H. Shen, Z.G. Luo, Partial survival and extinction in two competing species with impulses. Nonlinear Anal. RWA. doi:10.1016/j.nonrwa.2007.11.012.
22. Liu B., Chen L.S.: The periodic competing Lotka-Volterra model with impulsive effect. Math. Med. Biol 21, 129-145 (2004).
23. Lakmeche A., Arino O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treat. Dynam. Contin. Discrete Impuls. Syst 7, 165-187 (2000).
24. Wang F.Y., Pang G.P., Hui J.: Analysis of a Monod-Haldene type food chain chemostat with seasonally variably pulsed input and washout. J. Math. Chem 43, 601-619 (2008).
25. Sun S.L., Chen L.S.: Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration. J. Math. Chem 42, 837-847 (2007).
26. Sun L.S., Chen L.S.: Complex dynamics of a chemostat with variable yield and periodically impulsive perturbation on the substrate. J. Math. Chem 43, 338-349 (2008).
27. Hui J., Chen L.S.: Dynamic complexities in ratio-dependent predator-prey ecosystem models with birth pulse and pesticide pulse. Int. J. Bifur. Chaos Appl. Sci. Eng 14, 2893-2903 (2004).
28. Tang S.Y., Chen L.S.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol 44, 185-199 (2002).
29. Tang S.Y., Chen L.S.: Multiple attractors in stage-structured population models with birth pulses. Bull. Math. Biol 65, 479-495 (2003).
30. W.B. Wang, J.H. Shen, J.J. Nieto, Permanence and periodic solution of predator-prey system with Holling type functional response and impulses. Discrete Dyn. Nat. Soc. Art. ID 81756, 15 pp (2007).
31. Tang S.Y., Chen L.S.: The periodic predator-prey Lotka-Volterra model with impulsive effect. J. Mech. Med. Biol 3&4, 267-296 (2002).