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Simple food chain in a chemostat with distinct removal rates. (English) Zbl 0943.92034
We consider a model describing predator-prey interactions in a chemostat that incorporates both general response functions and distinct removal rates. In this case, the conservation law fails. To overcome this difficulty, we make use of a novel way of constructing a Lyapunov function in the study of the global stability of a predator-free steady state. Local and global stability of other steady states, persistence analysis, as well as numerical simulations are also presented. Our findings are largely in line with those of an identical removal rate case.

MSC:
92D40 Ecology
34D20 Stability of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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[1] 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
[2] Drake, J.F.; Tsuchiya, H.M., Predation of Escherichia coli by copoda steuii, Appl. environ. microbiol., 31, 870-874, (1976)
[3] Hale, J.K., Ordinary differential equations, (1980), Krieger Malabar · Zbl 0186.40901
[4] Jost, J.L.; Drake, S.F.; Fredrickson, A.G.; Tsuchiya, M., Interaction of tetrahymena pyriformis, Escherichia coli, azotobacter vinelandii and glucose in a minimal medium, J. bacteriol., 113, 834-840, (1976)
[5] Kuang, Y., Limit cycles in a chemostat-related models, SIAM J. appl. math., 49, 1759-1767, (1989) · Zbl 0683.34021
[6] LaSalle, J., Some extensions of Lyapunov’s second method, IRE trans. circuit, CT-7, 520-527, (1960)
[7] Li, B., Global asymptotic behavior of the chemostat: general response functions and different removal rates, SIAM J. appl. math., 59, 411-422, (1999) · Zbl 0916.92025
[8] B. Li, Analysis of Chemostat-Related Models with Distinct Removal Rates, Ph.D. thesis, Arizona State University, 1998.
[9] Sell, G., What is a dynamical system?, () · Zbl 0379.34035
[10] Smith, H.L.; Waltman, P., The theory of the chemostat, (1995), Cambridge Univ. Press Cambridge · Zbl 0860.92031
[11] Thieme, H.R., Persistence under relaxed point-dissipativity with an application to an epidemic model, SIAM J. math. anal., 24, 407-435, (1993) · Zbl 0774.34030
[12] Tsuchiya, H.M.; Drake, S.F.; Jost, J.L.; Fredrickson, A.G., Predator – prey interactions of dictyostelium discordeum and Escherichia coli in continuous culture, J. bacteriol., 110, 1147-1153, (1972)
[13] Wolkowicz, G.S.K.; Lu, Z., Global dynamics of a mathematical model of competition in the chemostat: general response function and differential death rates, SIAM J. appl. math., 52, 222-233, (1992) · Zbl 0739.92025
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