zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Effect of time delay on a detritus-based ecosystem. (English) Zbl 1127.92042
Summary: Models of detritus-based ecosystems with delay have received a great deal of attention for the last few decades. This paper deals with the dynamical analysis of a nonlinear model of a detritus-based ecosystem involving detritivores and predators of detritivores. We have obtained criteria for local stability of various equilibrium points and persistence of the model system. Next, we have introduced discrete time delays due to recycling of dead organic matters and gestation of nutrients to the growth equations of various trophic levels. With the delay differential equations model system we have studied the effect of time delays on the stability behaviour. Next, we have obtained an estimate for the length of the time delay to preserve the stability of the model system. Finally, the existence of Hopf-bifurcating small amplitude periodic solutions is derived by considering the time delay as a bifurcation parameter.

MSC:
92D40Ecology
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
WorldCat.org
Full Text: DOI EuDML
References:
[1] M. Bandyopadhyay, “Global stability and bifurcation in a delayed nonlinear autotroph-herbivore model,” Nonlinear Phenomena in Complex Systems, vol. 7, no. 3, pp. 238-249, 2004.
[2] M. Bandyopadhyay and R. Bhattacharya, “Non-linear bifurcation analysis of a detritus based ecosystem,” Nonlinear Studies, vol. 10, no. 4, pp. 357-372, 2003. · Zbl 1040.92041
[3] E. Beretta and Y. Kuang, “Convergence results in a well-known delayed predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 204, no. 3, pp. 840-853, 1996. · Zbl 0876.92021 · doi:10.1006/jmaa.1996.0471
[4] G. Birkhoff and G. C. Rota, Ordinary Differential Equation, Ginn, Massachusetts, 1982. · Zbl 0183.35601
[5] F. Charles, “Utilisation of fresh detritus derived from Cystoseira mediterranea and Posidonia oceania by deposit-feeding bivalve Abra ovata,” Journal of Experimental Marine Biology and Ecology, vol. 174, no. 1, pp. 43-64, 1993. · doi:10.1016/0022-0981(93)90250-R
[6] J. M. Cushing, Integrodifferential Equations and Delay Model in Population Dynamics, Springer, Heidelberg, 1977. · Zbl 0363.92014
[7] P. Das and A. B. Roy, “Oscillations in delay differential equation model of reproductive hormones in men with computer simulations,” Journal of Biological Systems, vol. 2, no. 1, pp. 73-90, 1994. · doi:10.1142/S0218339094000076
[8] M. C. Dash, Fundamentals of Ecology, Tata McGraw Hill, New Delhi, 2001.
[9] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhüuser Boston, Massachusetts, 2005. · Zbl 1069.35001
[10] M. A. Faust and R. A. Gulledge, “Associations of microalgae and meiofauna in floating detritus at a mangrove Island, Twin Cays, Belize,” Journal of Experimental Marine Biology and Ecology, vol. 197, no. 2, pp. 159-175, 1996. · doi:10.1016/0022-0981(95)00159-X
[11] H. I. Freedman and V. S. H. Rao, “The trade-off between mutual interference and time lags in predator-prey systems,” Bulletin of Mathematical Biology, vol. 45, no. 6, pp. 991-1004, 1983. · Zbl 0535.92024 · doi:10.1007/BF02458826
[12] H. I. Freedman and P. Waltman, “Persistence in models of three interacting predator-prey populations,” Mathematical Biosciences, vol. 68, no. 2, pp. 213-231, 1984. · Zbl 0534.92026 · doi:10.1016/0025-5564(84)90032-4
[13] K. Gopalsamy, “Harmless delays in model systems,” Bulletin of Mathematical Biology, vol. 45, no. 3, pp. 295-309, 1983. · Zbl 0514.34060 · doi:10.1007/BF02459394
[14] K. Gopalsamy, “Delayed responses and stability in two-species systems,” Journal of the Australian Mathematical Society. Series B. Applied Mathematics, vol. 25, no. 4, pp. 473-500, 1984. · Zbl 0552.92016 · doi:10.1017/S0334270000004227
[15] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1992. · Zbl 0752.34039
[16] B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1981. · Zbl 0474.34002
[17] A. Hastings, “Delays in recruitment at different trophic levels: effects on stability,” Journal of Mathematical Biology, vol. 21, no. 1, pp. 35-44, 1984. · Zbl 0547.92014 · doi:10.1007/BF00275221
[18] I. R. Joint, “Microbial production of an estuarine mudflat,” Estuarine Coastal and Marine Science, vol. 7, no. 2, pp. 185-195, 1978. · doi:10.1016/0302-3524(78)90074-9
[19] S. E. Jorgensen, “Energy and ecological system analysis,” in Complex Ecosystems, B. C. Pattern and S. E. Jorgensen, Eds., Prentice Hall, New York, 1994.
[20] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Massachusetts, 1993. · Zbl 0777.34002
[21] J. R. Linley, “Studies on larval biology of Leptoconops (Kieff.)(Diptera: Ceratopogonidae),” Bulletin of Entomological Research, vol. 58, no. 1, pp. 1-24, 1968.
[22] J. R. Linley and G. M. Adams, “Ecology and behaviour of immature Culicoides melleus(Coq.) (Diptera: Ceratogonidae),” Bulletin of Entomological Research, vol. 62, no. 1, pp. 113-127, 1972.
[23] N. MacDonald, Time Lags in Biological Models, vol. 27 of Lecture Notes in Biomathematics, Springer, Berlin, 1978. · Zbl 0403.92020
[24] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York, 1976. · Zbl 0346.58007
[25] A. Martin and S. Ruan, “Predator-prey models with delay and prey harvesting,” Journal of Mathematical Biology, vol. 43, no. 3, pp. 247-267, 2001. · Zbl 1008.34066 · doi:10.1007/s002850100095
[26] R. M. May, “Time delay versus stability in population models with two and three trophic levels,” Ecology, vol. 54, no. 2, pp. 315-325, 1973. · doi:10.2307/1934339
[27] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 2001. · Zbl 1044.92047
[28] J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, Berlin, 1993. · Zbl 0779.92001
[29] S. Ray and A. Choudhury, “Salinity tolerance of Culex sitiens (Weid.) (Diptera: Culicidae) larvae in laboratory condition,” Current Science, vol. 57, no. 3, pp. 159-160, 1988.
[30] S. Ruan, “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,” Quarterly of Applied Mathematics, vol. 59, no. 1, pp. 159-173, 2001. · Zbl 1035.34084
[31] A. K. Sarkar and D. Ghosh, “Role of detritus in a general prey-predator model of Sunderban Estuary, India,” Biosystems, vol. 44, no. 2, pp. 153-160, 1997. · doi:10.1016/S0303-2647(97)00053-1
[32] A. K. Sarkar, D. Mitra, and A. B. Roy, “Stability of partially closed producer-consumer system via decomposer,” Ganit. Journal of Bangladesh Mathematical Society, vol. 10, no. 1-2, pp. 21-27, 1990. · Zbl 0825.92120
[33] T. F. Thingstad and T. I. Langeland, “Dynamics of a chemostat culture: the effect of a delay in cell response,” Journal of Theoretical Biology, vol. 48, no. 1, pp. 149-159, 1974. · doi:10.1016/0022-5193(74)90186-6
[34] E. W. Vetter, “Population dynamics of a dense assemblage of marine detritivores,” Journal of Experimental Marine Biology and Ecology, vol. 226, no. 1, pp. 131-161, 1998. · doi:10.1016/S0022-0981(97)00246-3
[35] V. Volterra, Lecons sur la Théorie Mathématique de la Lutte Pour la Vie, Gauthier-Villars, Paris, 1931. · Zbl 0002.04202