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autotroph; herbivore; nutrient recycling; global stability; Hopf-bifurcation; variable delay; two-timing expansion
We consider a mathematical model of nutrient-autotroph-herbivore interaction with nutrient recycling from both autotroph and herbivore. Local and global stability criteria of the model are studied in terms of system parameters. Next we incorporate the time required for recycling of nutrient from herbivore as a constant discrete time delay. The resulting DDE model is analyzed regarding stability and bifurcation aspects. Finally, we assume the recycling delay in the oscillatory form to model the daily variation in nutrient recycling and deduce the stability criteria of the variable delay model. A comparison of the variable delay model with the constant delay one is performed to unearth the biological relevance of oscillating delay in some real world ecological situations. Numerical simulations are done in support of analytical results.
[1] Bandyopadhyay, M., Bhattacharyya, R., Mukhopadhyay, B.: Dynamics of an autotroph-herbivore ecosystem with nutrient recycling. Ecol. Model. 176 (2004), 201-209. DOI 10.1016/j.ecolmodel.2003.10.030
[2] Beltrami, E., Carroll, T. O.: Modeling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol. 32 (1994), 857-863. DOI 10.1007/BF00168802 | Zbl 0825.92122
[3] Benson, D. L., Sherratt, J. A., Maini, P. K.: Diffusion driven instability in an inhomogeneous domain. Bull. Math. Biol. 55 (1993), 365-384. DOI 10.1007/BF02460888 | Zbl 0758.92003
[4] Benson, D. L., Maini, P. K., Sherratt, J. A.: Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients. Math. Comp. Model. 17 (1993), 29-34. DOI 10.1016/0895-7177(93)90025-T | MR 1236162 | Zbl 0784.92004
[5] Beretta, E., Bischi, G. I., Solimano, F.: Stability in chemostat equations with delayed recycling. J. Math. Biol. 28 (1990), 99-111. DOI 10.1007/BF00171521 | MR 1036414 | Zbl 0665.45006
[6] Bhattacharyya, R., Mukhopadhyay, B.: Oscillation and persistence in a mangrove ecosystem in presence of delays. J. Biol. Syst. 11 (2003), 351-364. DOI 10.1142/S021833900300097X | Zbl 1041.92038
[7] Bischi, G. I.: Effects of time lags on transient characteristics of a nutrient recycling model. Math. Biosci. 109 (1992), 151-175. DOI 10.1016/0025-5564(92)90043-V
[8] Colinvaux, P.: Ecology II. John Wiley & Sons Chichester (1993).
[9] Crawley, M. J.: Herbivory. The dynamics of animal plant interaction. Studies in Ecology Vol. 10. University of California Press Berkley (1983).
[10] Angelis, D. I. De, Bartell, S. M., Brenkert, A. L.: Effect of nutrient recycling and food chain length on the resilience. Amer. Naturalist 134 (1989), 778-805. DOI 10.1086/285011
[11] Angelis, D. L. De: Dynamics of Nutrient Recycling and Food Webs. Chapman & Hall London (1992).
[12] Farkas, M., Freedman, H. I.: The stable coexistence of competing species on a renewable resource. J. Math. Anal. Appl. 138 (1989), 461-472. DOI 10.1016/0022-247X(89)90303-X | MR 0991036 | Zbl 0661.92021
[13] Ghosh, D., Sarkar, A. K.: Oscillatory behaviour of an autotroph-herbivore system with a type-III uptake function. Int. J. Syst. Sci. 28 (1997), 259-264. DOI 10.1080/00207729708929385 | Zbl 0874.92033
[14] Ghosh, D., Sarkar, A. K.: Stability and oscillations in a resource-based model of two interacting species with nutrient cycling. Ecol. Model. 107 (1998), 25-33. DOI 10.1016/S0304-3800(97)00203-2
[15] Ghosh, D., Sarkar, A. K.: Qualitative analysis of autotroph-herbivore system with nutrient diffusion. Korean J. Comput. Appl. Math. 6 (1999), 589-599. MR 1710208 | Zbl 0938.92033
[16] Hallam, T. G.: Structural sensitivity of grazing formulations in nutrient controlled plankton models. J. Math. Biol. 5 (1978), 261-280. DOI 10.1007/BF00276122 | MR 0645277 | Zbl 0379.92015
[17] Hassard, B. D., Kazarinoff, N. D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press Cambridge (1981). MR 0603442 | Zbl 0474.34002
[18] Holling, C. S.: The functional response population regulation. Mem. Entomol. Soc. Can. 45 (1965), 1-60. DOI 10.4039/entm9745fv
[19] Jang, S. R.-J.: Dynamics of variable-yield nutrient-phytoplankton-zooplankton models with nutrient recycling and self-shading. J. Math. Biol. 40 (2000), 229-250. DOI 10.1007/s002850050179 | MR 1752127 | Zbl 0998.92039
[20] Kreysig, E.: Introductory Functional Analysis with Applications. John Wiley & Sons New York (1978). MR 0467220
[21] Ludwig, D., Jones, D. D., Holling, C. S.: Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J. Anim. Econ. 47 (1978), 315-332. DOI 10.2307/3939
[22] Maini, P. K., Benson, D. L., Sherratt, J. A.: Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients. IMA J. Math. Appl. Med. Biol. 9 (1992), 197-213. DOI 10.1093/imammb/9.3.197 | MR 1202777 | Zbl 0767.92004
[23] Mukherjee, D., Ray, S., Sinha, D. K.: Bifurcation analysis of a detritus-based ecosystem with time delay. J. Biol. Syst. 8 (2000), 255-261. DOI 10.1142/S0218339000000183 | Zbl 1009.92036
[24] Mukopadhyay, B., Bhattacharyya, R.: A delay-diffusion model of marine plankton ecosystem exhibiting cyclic nature of blooms. J. Biol. Phys. 31 (2005), 3-22. DOI 10.1007/s10867-005-2306-x
[25] Mukopadhyay, B., Bhattacharyya, R.: Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity. Ecol. Model. 198 (2006), 163-173. DOI 10.1016/j.ecolmodel.2006.04.005
[26] Mukhopadhyay, B., Bhattacharyya, R.: Modeling the role of diffusion coefficients on Turing instability in a reaction-diffusion prey-predator system. Bull. Math. Biol. 68 (2006), 293-313. DOI 10.1007/s11538-005-9007-2 | MR 2224770
[27] Muratori, S., Rinaldi, S.: Low- and high-frequency oscillations in three-dimensional food chain systems. SIAM J. Appl. Math. 52 (1992), 1688-1706. DOI 10.1137/0152097 | MR 1191357 | Zbl 0774.92024
[28] Nisbet, R. M., Mckinstry, J., Gurney, W. S. C.: A ``strategic'' model of material cycling in a closed ecosystem. Math. Biosci. 64 (1983), 99-113. DOI 10.1016/0025-5564(83)90030-5 | Zbl 0524.92027
[29] Pardo, O.: Global stability for a phytoplankton-nutrient system. J. Biol. Syst. 8 (2000), 195-209. DOI 10.1142/S0218339000000122
[30] Powell, T., Richardson, P. J.: Temporal variation, spatial heterogeneity and competition for resources in plankton systems: theoretical models. Am. Nat. 125 (1985), 431-463. DOI 10.1086/284352
[31] Ruan, S.: Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. J. Math. Biol. 31 (1993), 633-654. DOI 10.1007/BF00161202 | MR 1237092 | Zbl 0779.92021
[32] Ruan, S.: The effect of delays on stability and persistence in plankton models. Nonlinear Anal., Theory Methods Appl. 24 (1995), 575-585. DOI 10.1016/0362-546X(95)93092-I | MR 1315696 | Zbl 0830.34067
[33] Ruan, S.: Oscillations in plankton models with nutrient recycling. J. Theor. Biol. 208 (2001), 15-26. DOI 10.1006/jtbi.2000.2196
[34] Ruan, S., Wolkowicz, G.: Uniform persistence in plankton models with delayed nutrient recycling. Can. Appl. Math. Q. 3 (1995), 219-235. MR 1360033
[35] Sarkar, A. K., Mitra, D., Roy, S., Roy, A. B.: Permanence and oscillatory co-existence of a detritus-based prey-predator model. Ecol. Model. 53 (1991), 147-156. DOI 10.1016/0304-3800(91)90146-R
[36] Sarkar, A. K., Roy, A. B.: Oscillatory behavior in a resource-based plant-herbivore model with random herbivore attack. Ecol. Model. 68 (1993), 213-226. DOI 10.1016/0304-3800(93)90018-N
[37] Sikder, A., Roy, A. B.: Limit cycles in a prey-predator systems. Appl. Math. Lett. 6 (1993), 91-95. DOI 10.1016/0893-9659(93)90042-L | MR 1347294
[38] Schley, D., Gourley, S. A.: Linear stability criteria for population models with periodically perturbed delays. J. Math. Biol. 40 (2000), 500-524. DOI 10.1007/s002850000034 | MR 1770938 | Zbl 0961.92026
[39] Sherratt, J. A.: Turing bifurcations with a temporally varying diffusion coefficient. J. Math. Biol. 33 (1995), 295-308. DOI 10.1007/BF00169566 | MR 1331510 | Zbl 0812.92024
[40] Sherratt, J. A.: Diffusion-driven instability in oscillating environments. Eur. J. Appl. Math. 6 (1995), 355-372. DOI 10.1017/S0956792500001893 | MR 1343087 | Zbl 0847.35066
[41] Smith, O. L.: Food Webs. Chapman & Hall London (1982).
[42] Svirezhev, Y. M., Logofet, D. O.: Stability of Biological Communities. Mir Moscow (1983). MR 0723326
[43] Whittaker, R. H.: Communities and Ecosystems. Macmillan New York (1975).
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