Global Analysis of Beddington-DeAngelis Type Chemostat Model with Nutrient Recycling and Impulsive Input

ABSTRACT

In this paper, a Beddington-DeAngelis type chemostat model with nutrient recycling and impulsive input is considered. Except using Floquet theorem, introducing a new method combining with comparison theorem of impulse differential equation and by using the Liapunov function method, the sufficient and necessary conditions on the permanence and extinction of the microorganism are obtained. Two examples are given in the last section to verify our mathematical results. The numerical analysis shows that if only the system is permanent, then it also is globally attractive.

Cite this paper

M. Rehim, L. Sun and A. Muhammadhaji, "Global Analysis of Beddington-DeAngelis Type Chemostat Model with Nutrient Recycling and Impulsive Input,"*Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 1097-1105. doi: 10.4236/am.2013.48148.

M. Rehim, L. Sun and A. Muhammadhaji, "Global Analysis of Beddington-DeAngelis Type Chemostat Model with Nutrient Recycling and Impulsive Input,"

References

[1] H. L. Smith, “Competitive Coexistence in an Oscillating Chemostat,” SIAM Journal on Applied Mathematics, Vol. 40, No. 3, 1981, pp. 498-522. doi:10.1137/0140042

[2] S. B. Hsu, “A Competition Model for a Seasonally Fluc tuation Nutrient,” Journal of Mathematical Biology, Vol. 9, No. 2, 1980, pp. 115-132.

[3] G. J. Butler, S. B. Hsu and P. Waltman, “A Mathematical Model of the Chemostat with Periodic Washout Rate,” SIAM Journal on Applied Mathematics, Vol. 45, No. 3, 1985, pp. 435-449. doi:10.1137/0145025

[4] P. Lemas, “Coexistence of Three Competing Microbial Populations in a Chemostat with Periodic Input and Dilu tion Rate,” Mathematical Bioscience, Vol. 129, No. 2, 1995, pp. 111-142. doi:10.1016/0025-5564(94)00056-6

[5] S. S. Pilyugin and P. Waltman, “Competition in Unstirred Chemostat with Periodic Input and Washout,” SIAM Journal on Applied Mathematics, Vol. 59, No. 2, 1999, pp. 1157-1177.

[6] X. He and S. Ruan, “Global Stability in Chemostat-Type Plankton Models with Delayed Nutrient Recycling,” Re search Report, The School of Mathematical and Statistics, University of Sydney, 1996, pp. 96-98.

[7] S. Ruan, “The Effect of Delays on Stability and Persis tence in Plankton Models,” Nonlinear Analysis: Real Word Applications, Vol. 24, No. 4, 1995, pp. 575-585.

[8] S. Ruan and X. He, “Global Stability in Chemostat-Type Competition Models with Nutrient Recycling,” SIAM Journal on Applied Mathematics, Vol. 58, No. 1, 1998, pp. 170-192. doi:10.1137/S0036139996299248

[9] V. S. H. Rao and P. R. S. Rao, “Global Stability in Che mostat Models Involving Time Delays and Wall Growth,” Nonlinear Analysis: Real Word Applications, Vol. 5, No. 1, 2005, pp. 141-158.

[10] S. R. J. Jang, “Dynamics of Variable-Yield Nutrient Phytoplankton-Zooplankton Models with Nutrient Recy cling and Self-Shading,” Journal of Mathematical Biol ogy, Vol. 40, No. 3, 2000, pp. 229-250. doi:10.1007/s002850050179

[11] S. Sun and L. Chen, “Complex Dynamics of a Chemotat with Variable Yield and Periodically Impulsive Perturba tion on the Substrate,” Journal of Mathematical Chemis try, Vol. 43, No. 1, 2008, pp. 338-348. doi:10.1007/s10910-006-9200-z

[12] V. Lakshmikantham, D. D. Bainov and P. S. Simeoonov, “Theory of Impulsive Differential Equations,” World Sci ence, Singapore City, 1989. doi:10.1142/0906

[13] D. D. Bainov and P. S. Simeoonov, “Impulsive Differen tial Equations: Periodic Solutions and Applications,” Longman Scientific and Thechnical, London, 1993.

[14] D. L. DeAngelis, R. A. Goldstein and R. V. Oneill, “A Model for Trophic Interaction,” Ecology, Vol. 56, No. 4, 1975, pp. 661-692. doi:10.2307/1936298

[15] J. R. Beddington, “Mutual Interference between Parasites and Its Effect on Searching Efficiency,” Journal of Ani mal Ecology, Vol. 44, No. 1, 1975, pp. 331-340. doi:10.2307/3866

[16] Z. D. Teng, R. Gao, R. Mehbuba and K. Wang, “Global Behaviors of Monod type Chemostat Model with Nutrient Recycling and Impusive Onput,” Journal of Mathemati cal Chemistry, Vol. 47, No. 1, 2010, pp. 276-294. doi:10.1007/s10910-009-9567-8

[1] H. L. Smith, “Competitive Coexistence in an Oscillating Chemostat,” SIAM Journal on Applied Mathematics, Vol. 40, No. 3, 1981, pp. 498-522. doi:10.1137/0140042

[2] S. B. Hsu, “A Competition Model for a Seasonally Fluc tuation Nutrient,” Journal of Mathematical Biology, Vol. 9, No. 2, 1980, pp. 115-132.

[3] G. J. Butler, S. B. Hsu and P. Waltman, “A Mathematical Model of the Chemostat with Periodic Washout Rate,” SIAM Journal on Applied Mathematics, Vol. 45, No. 3, 1985, pp. 435-449. doi:10.1137/0145025

[4] P. Lemas, “Coexistence of Three Competing Microbial Populations in a Chemostat with Periodic Input and Dilu tion Rate,” Mathematical Bioscience, Vol. 129, No. 2, 1995, pp. 111-142. doi:10.1016/0025-5564(94)00056-6

[5] S. S. Pilyugin and P. Waltman, “Competition in Unstirred Chemostat with Periodic Input and Washout,” SIAM Journal on Applied Mathematics, Vol. 59, No. 2, 1999, pp. 1157-1177.

[6] X. He and S. Ruan, “Global Stability in Chemostat-Type Plankton Models with Delayed Nutrient Recycling,” Re search Report, The School of Mathematical and Statistics, University of Sydney, 1996, pp. 96-98.

[7] S. Ruan, “The Effect of Delays on Stability and Persis tence in Plankton Models,” Nonlinear Analysis: Real Word Applications, Vol. 24, No. 4, 1995, pp. 575-585.

[8] S. Ruan and X. He, “Global Stability in Chemostat-Type Competition Models with Nutrient Recycling,” SIAM Journal on Applied Mathematics, Vol. 58, No. 1, 1998, pp. 170-192. doi:10.1137/S0036139996299248

[9] V. S. H. Rao and P. R. S. Rao, “Global Stability in Che mostat Models Involving Time Delays and Wall Growth,” Nonlinear Analysis: Real Word Applications, Vol. 5, No. 1, 2005, pp. 141-158.

[10] S. R. J. Jang, “Dynamics of Variable-Yield Nutrient Phytoplankton-Zooplankton Models with Nutrient Recy cling and Self-Shading,” Journal of Mathematical Biol ogy, Vol. 40, No. 3, 2000, pp. 229-250. doi:10.1007/s002850050179

[11] S. Sun and L. Chen, “Complex Dynamics of a Chemotat with Variable Yield and Periodically Impulsive Perturba tion on the Substrate,” Journal of Mathematical Chemis try, Vol. 43, No. 1, 2008, pp. 338-348. doi:10.1007/s10910-006-9200-z

[12] V. Lakshmikantham, D. D. Bainov and P. S. Simeoonov, “Theory of Impulsive Differential Equations,” World Sci ence, Singapore City, 1989. doi:10.1142/0906

[13] D. D. Bainov and P. S. Simeoonov, “Impulsive Differen tial Equations: Periodic Solutions and Applications,” Longman Scientific and Thechnical, London, 1993.

[14] D. L. DeAngelis, R. A. Goldstein and R. V. Oneill, “A Model for Trophic Interaction,” Ecology, Vol. 56, No. 4, 1975, pp. 661-692. doi:10.2307/1936298

[15] J. R. Beddington, “Mutual Interference between Parasites and Its Effect on Searching Efficiency,” Journal of Ani mal Ecology, Vol. 44, No. 1, 1975, pp. 331-340. doi:10.2307/3866

[16] Z. D. Teng, R. Gao, R. Mehbuba and K. Wang, “Global Behaviors of Monod type Chemostat Model with Nutrient Recycling and Impusive Onput,” Journal of Mathemati cal Chemistry, Vol. 47, No. 1, 2010, pp. 276-294. doi:10.1007/s10910-009-9567-8