A simple food chain with a growth inhibiting nutrient

Applied Mathematics and Computation

Volumn 104, Issue 2-3, 1999, Pages 277-298

  Jang, Sophia R J ^a   Allen, Linda J S ^a  

^a TEXAS TECH UNIVERSITY   (United States)

Author keywords

Chemostat; Food chain; Inhibition; Persistence; Threshold

Indexed keywords

EID: 0342993144     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0096-3003(98)10076-0     Document Type: Article

References (34)

1. J.F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng. 10 (1968) 707-723.
2. S. Busenberg, S.K. Kumar, P. Austin, G. Wake, The dynamics of a model of a plankton-nutrient interaction, Bull. Math. Biol. 52 (1990) 667-696.
3. A.W. Bush, A.E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theoret. Biol. 63 (1976) 385-395.
4. G.J. Butler, G.S.K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math. 45 (1985) 138-151.
5. G.J. Butler, S.B. Hsu, P. Waltman, Coexistence of competing predators in a chemostat, J. Math. Biol. 17 (1983) 133-151.
6. A.P.S.F. Chamon, The dynamics of a plankton model with diffusion, Master's Thesis, Texas Tech University, Lubbock, TX, 1995.
7. C.D. Collins, Formulation and validation of a mathematical models of phytoplankton growth, Ecology 61 (1980) 639-649.
8. G.T. Evans, J.S. Parslow, A model of annual plankton cycles, Biol. Oceanogr. 3 (1985) 327-347.
9. H.I. Freedman, P. Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci. 68 (1984) 213-231.
10. H.I. Freedman, J.W.-H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985) 69-86.
11. T.C. Gard, Persistence in food webs: Holling-type food chains, Math. Biosci. 49 (1980) 61-67.
12. T.C. Gard, T.G. Hallam, Persistence in food webs - I Lotka-Volterra food chains, Bull. Math. Biol. 41 (1979) 877-891.
13. T.G. Hallam, On persistence of aquatic ecosystems, in: N.R. Andersen, B.J. Zahurance (Eds.), Oceanic Sound Scattering Prediction, Plenum Press, New York, 1977.
14. T.G. Hallam, Structural sensibility of grazing formulation in nutrient controlled plankton models, J. Math. Biol. 5 (1978) 261-280.
15. G.W. Harrison, Global stability of predator-prey interactions, J. Math. Biol. 8 (1979) 159-171.
16. A. Hastings, T. Powell, Chaos in a three-species food chain, Ecology 72 (1991) 896-903.
17. H.W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci. 28 (1976) 335-356.
18. A. Klebanoff, A. Hastings, Chaos in three species food chains, J. Math. Biol. 32 (1994) 427-451.
19. S.K. Kumar, W.F. Vincent, P.C. Austin, G.C. Wake, Picoplankton and marine food chain dynamics in a variable mixed-layer: a reaction-diffusion model, Ecol. Modelling 57 (1991) 193-219.
20. J. LaSalle, S. Lefschetz, Stability by Liapunov's Direct Method, Academic Press, New York, 1961.
21. J.E. Marsden, M. McCracken, The Hopf Bifurcation and its Applications, Springer, New York, 1976.
22. R.V. O'Neill, D.L. DeAngelis, J.J. Pastor, B.J. Jackson, W.M. Post, Multiple nutrient limitations in ecological models, Ecol. Modelling. 46 (1989) 147-163.
23. S. Ruan, Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, J. Math. Biol. 31 (1994) 633-654.
24. H.L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, New York, 1995.
25. J.H. Steele, The Structure of Marine Ecosystems, Blackwell Scientific, Oxford, 1974.
26. J.H. Steele, E.W. Henderson, A simple plankton model, Am. Nat. 117 (1981) 676-691.
27. A.J. Taylor, Characteristic properties of model for the vertical distribution of phytoplankton under stratification, Ecol. Modelling 40 (1988) 175-199.
28. H.R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol 30 (1992) 755-763.
29. J.E. Truscott, Environmental forcing of simple plankton models, J. Plankton Res. 17 (1995) 2207-2232.
30. J.E. Truscott, J. Brindley, Ocean plankton population as excitable media, Bull. Math. Biol. 56 (1994) 981-998.
31. J.E. Truscott, J. Brindley, Equilibria, stability and excitability in a general class of plankton population models, Philos. Trans. R. Soc. London A 347 (1994) 703-718.
32. G.S.K. Wolkowicz, Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM J. Appl. Math. 52 (1992) 222-233.
33. J.S. Wroblewski, J.L. Sarmiento, G.R. Flierl, An ocean basin scale model of plankton dynamics in the North Atlantic. Solutions for the climatological oceanographic conditions in May, Global Biogeochem. Cycles 2 (1988) 199-218.
34. D.R. Yang, A.E. Humphrey, Dynamic and steady-state studies of phenol biodegradation in pure and mixed cultures, Biotechnol. Bioeng. 17 (1975) 1211-1235.