A delay model for viral infection in toxin producing phytoplankton and zooplankton system. (English) Zbl 1222.37098
Summary: An eco-epidemiological delay model is proposed and analysed for virally infected, toxin producing phytoplankton (TPP) and zooplankton system. It is shown that time delay can destabilize the otherwise stable non-zero equilibrium state. The coexistence of all species is possible through periodic solutions due to Hopf bifurcation. In the absence of infection the delay model may have a complex dynamical behavior which can be controlled by infection. Numerical simulation suggests that the proposed model displays a wide range of dynamical behaviors. Different parameters are identified that are responsible for chaos.
MSC:
| 37N25 | Dynamical systems in biology |
| 92D40 | Ecology |
| 34K13 | Periodic solutions to functional-differential equations |
| 34K18 | Bifurcation theory of functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
References:
| [1] | Rice, E. L., Allelopathy (1984), Academic Press: Academic Press London |
| [2] | Kirk, K.; Gilbert, J., Variations in herbivore response to chemical defences: zooplankton foraging on toxic cyanobacteria, Ecology, 73, 220-228 (1992) |
| [3] | Odum, E. P., Fundamentals of ecology (1971), W.B. Saunders Company: W.B. Saunders Company London |
| [4] | Sarkar, R. R.; Chattopadhayay, J., Occurrence of planktonic blooms under environmental fluctuations and its possible control mechanism-mathematical models and experimental observations, J Theor Biol, 224, 501-516 (2003) · Zbl 1465.92098 |
| [5] | Chattopadhayay, J.; Sarkar, R. R., A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA J Math Appl Med Biol, 19, 137-161 (2002) · Zbl 1013.92046 |
| [6] | Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-preys system, J Math Biol, 36, 389-406 (1998) · Zbl 0895.92032 |
| [7] | Gakkhar, S.; Sahani, S., Effects of seasonal growth on delayed prey-predator model, Chaos Solitons Fractals, 39, 230-239 (2009) · Zbl 1197.34134 |
| [8] | Gakkhar, S.; Sahani, S., Effect of seasonal growth on ratio-dependent delayed prey-predator model, Commun Nonlinear Sci Numer Simul, 14, 850-862 (2009) · Zbl 1221.34187 |
| [9] | Sarkar, Time lags can control algal blooms in two harmful phytoplankton-zooplankton systems, Appl Math Comput, 186, 445-459 (2007) · Zbl 1111.92065 |
| [10] | Kuang, Y., Delay differential equations with applications in population dynamics (1993), Academic Press: Academic Press San Diego · Zbl 0777.34002 |
| [11] | Fuhrman, J. A., Marine viruses and their biogeochemical and ecological effects, Nature, 399, 541-548 (1999) |
| [12] | Wommack, K. E.; Colwell, R. R., Virioplankton: viruses in aquatic ecosystems, Microbiol Mol Biol Rev, 64, 69-114 (2000) |
| [13] | Suttle, C. A.; Chan, A. M., Marine cyanophages infecting oceanic and coastal strains of Synechococcus: abundance, morphology, cross infectivity and growth characteristics, Mar Ecol Prog Ser, 92, 99-109 (1993) |
| [14] | Beltrami, E.; Carroll, T. O., Modelling the role of viral disease in recurrent phytoplankton blooms, J Math Biol, 32, 857-863 (1994) · Zbl 0825.92122 |
| [15] | Singh, B. K.; Chattopadhyay, J.; Sinha, S., A role of viral infection in a simple phytoplankton zooplankton system, J Theor Bio, 231, 153-166 (2004) · Zbl 1447.92475 |
| [16] | Xiao, Y.; Chen, L., Modelling and analysis of a predator-prey model with disease in the prey, Math Biosci, 171, 59-82 (2001) · Zbl 0978.92031 |
| [17] | Freedman, H. I., Deterministic mathematical models in population biology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023 |
| [18] | Hall, S. R.; Duffy, M. A.; Caceres, C. E., Selective predation and productivity jointly drive complex behaviour in host-parasite systems, Am Nat, 165, 70-81 (2005) |
| [19] | Gakkhar, S.; Negi, K., A mathematical model for viral infection in toxin producing phytoplankton and zooplankton system, Appl Math Comput, 179, 301-313 (2006) · Zbl 1096.92042 |
| [20] | Schmidt, G.; Tondl, A., Nonlinear vibrations (1986), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0589.73027 |
| [21] | Moore, J., Parasites and the behaviour of animals (2002), Oxford Univ. Press: Oxford Univ. Press Oxford |
| [22] | Murray, D. L.; Carry, J. R.; Keith, L. B., Interactive effects of sub lethal nematodes status on snowshoe hare vulnerability to predation?, J Anim Ecol, 66, 250-264 (1997) |
| [23] | Hudson, P. J.; Dobson, A. P.; Newborn, D., Do parasites make pray vulnerable to predation? Red grouse and parasites, J Anim Ecol, 61, 681-692 (1992) |
| [24] | Hale, J. K., Theory of functional differential equations (1977), Springer: Springer New York · Zbl 0352.34001 |
| [25] | Freedman, H. I.; Rao, V. S.H., The trade-off between mutual interference and time lags in predator-prey-systems, Bull Math Bio, 45, 991-1003 (1983) · Zbl 0535.92024 |
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