zbMATH — the first resource for mathematics

Analysis of a Lotka-Volterra food chain chemostat with converting time delays. (English) Zbl 1198.34167
Summary: A model of the food chain chemostat involving predator, prey and growth-limiting nutrients is considered. The model incorporates two discrete time delays in order to describe the time involved in converting processes. The Lotka-Volterra type increasing functions are used to describe the species uptakes. In addition to showing that solutions with positive initial conditions are positive and bounded, we establish sufficient conditions for the (i) local stability and instability of the positive equilibrium and (ii) global stability of the non-negative equilibria. Numerical simulation suggests that the delays have both destabilizing and stabilizing effects, and the system can produce stable periodic solutions, quasi-periodic solutions and strange attractors.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

34K20 Stability theory of functional-differential equations
92D40 Ecology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text: DOI
[1] Herbert, D.; Elsworth, R.; Telling, R.C., The continuous culture of bacteria: a theoretical and experimental study, J gen microbiol, 14, 1403-1418, (1956)
[2] Novick, A.; Sziliand, L., Description of the chemostat, Science, 112, 715-716, (1950)
[3] Smith, H.; Waltman, P., The theory of the chemostat, (1995), Cambridge University Press Cambridge, UK
[4] Waltman P. Competition models in population biology, CBMS-NSF region conf ser appl math, vol. 45, SIAM, Philadelphia, PA; 1983.
[5] Taylor, P.A.; Williams, J.L., Theoretical studies on the coexistence of competing species under continuous-flow conditions, Can J microbiol, 21, 90-98, (1975)
[6] Cannale, R.P., An analysis of models describing predator – prey interaction, Biotechnol bioeng, 12, 353-378, (1970)
[7] Butler, G.J.; Hsu, S.B.; Waltman, P., Coexistence of competing predators in a chemostat, J math biol, 17, 133-151, (1983) · Zbl 0508.92019
[8] Kuang, Y., Limit cycles in a chemostat-related model, SIAM J appl math, 49, 1759-1767, (1989) · Zbl 0683.34021
[9] Wang, F.; Hao, C.; Chen, L., Bifurcation and chaos in a tessiet type food chain chemostat with pulsed input and washout, Chaos, solitons & fractals, 32, 1547-1561, (2007) · Zbl 1126.92059
[10] Pang, G.; Wang, F.; Chen, L., Analysis of a monod – haldene type food chain chemostat with periodically varying substrate, Chaos, solitons & fractals, 38, 731-742, (2008) · Zbl 1146.34317
[11] Wang, F.; Pang, G.; Lu, Z., Analysis of a beddington – deangelis food chain chemostat with periodically varying dilution rate, Chaos, solitons & fractals, 40, 1609-1615, (2009) · Zbl 1198.34062
[12] Wang, F.; Pang, G., The global stability of a delayed predator – prey system with two stage-structure, Chaos, solitons & fractals, 40, 778-785, (2009) · Zbl 1197.34100
[13] MacDonald, N., Time delays in chemostat models, (), 33-53
[14] Wolkowicz, G.S.K.; Xia, H., Global asymptotic behavior of a chemostat model with discrete delays, SIAM J appl math, 57, 4, 1019-1043, (1997) · Zbl 0902.92023
[15] Ellermyer, S.F., Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth, SIAM J appl math, 54, 456-465, (1994) · Zbl 0794.92023
[16] Caperson, J., Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50, 182-192, (1969)
[17] Caswell, H.A., A simulation study of a time lag population model, J theor biol, 34, 419-439, (1972)
[18] Kuang, Y., Delay differential equations with applications in popular dynamics, (1993), Academic Press New York
[19] Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay-dependent parameters, SIAM J math anal, 33, 1144-1165, (2002) · Zbl 1013.92034
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.