# Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model

Mathematical Modelling of Natural Phenomena (2012)

- Volume: 7, Issue: 3, page 99-116
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topMandal, P.S., and Banerjee, M.. "Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model." Mathematical Modelling of Natural Phenomena 7.3 (2012): 99-116. <http://eudml.org/doc/222371>.

@article{Mandal2012,

abstract = {An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103–112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopf-bifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.},

author = {Mandal, P.S., Banerjee, M.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {Eco-epidemiology; stability; Hopf-bifurcation; chaos; stochasticity; extinction; eco-epidemiology},

language = {eng},

month = {6},

number = {3},

pages = {99-116},

publisher = {EDP Sciences},

title = {Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model},

url = {http://eudml.org/doc/222371},

volume = {7},

year = {2012},

}

TY - JOUR

AU - Mandal, P.S.

AU - Banerjee, M.

TI - Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model

JO - Mathematical Modelling of Natural Phenomena

DA - 2012/6//

PB - EDP Sciences

VL - 7

IS - 3

SP - 99

EP - 116

AB - An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103–112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopf-bifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.

LA - eng

KW - Eco-epidemiology; stability; Hopf-bifurcation; chaos; stochasticity; extinction; eco-epidemiology

UR - http://eudml.org/doc/222371

ER -

## References

top- E. Allen. Modeling with Itô Stochastic Differential Equations. Springer, The Netherlands, 2007. Zbl1130.60064
- L. J. S. Allen. An Introduction to Stochastic Processes with Applications to Biology. Pearson Eduction Inc., New Jercy, 2003. Zbl1205.60001
- L. J. S. Allen, M. A. Jones, C. F. Martin. A discrete-time model with vaccination for a measles epidemic. Math. Biosci., 105 (1991), 111–131. Zbl0728.92023
- O. Arino, A. El. Abdllaoui, J. Mikram, J. Chattopadhyay. Infection on prey population may act as a biological control in ratio-dependent predator-prey model. Nonlinearity, 17 (2004), 1101-1116. Zbl1062.34050
- E. J. Allen, L. J. S. Allen, A. Arciniega, P. Greenwood. Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl., 26 (2008) 274-297. Zbl1137.60026
- F. G. Ball. Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci., 156 (1999) 41–67. Zbl0979.92033
- E. Beltrami, T. O. Carroll. Modelling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol., 32 (1994) 857-863. Zbl0825.92122
- F. Brauer, C. Castillo-Chàvez. Mathematical Models in Population Biolgy and Epidemiology Springer-Verlag, New York, 2001. Zbl0967.92015
- T. Britton. Stochastic epidemic models : A survey. Math. Biosci., 225 (2010) 24–35. Zbl1188.92031
- T. Britton, D. Lindenstrand. Epidemic modelling : Aspects where stochasticity matters. Math. Biosci., 222 (2009) 109-116. Zbl1178.92046
- J. Chattopadhyay, N. Bairagi. Pelicans at risk in Salton Sea - an eco-epidemiological model. Ecol. Model., 136 (2001) 103–112.
- M. S. Chan, V. S. Isham. A stochastic model of schistosomiasis immuno-epidemiology. Math. Biosci., 151 (1998) 179–198. Zbl0933.92029
- H. I. Freedman. A model of predator-prey dynamics as modified by the action of parasite. Math. Biosci., 99 (1990) 143–155. Zbl0698.92024
- T. C. Gard. Introduction to Stochastic Differential Equations. Marcel Decker, New York, 1987. Zbl0628.60064
- C. W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, New York, 1983. Zbl0515.60002
- D. T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phy., 22 (1976) 403–434.
- D. T. Gillespie. The chemical Langevin equation. J. Chem. Phy., 113 (2000) 297–306.
- N. S. Goel, N. Richter-Dyn. Stochastic Models in Biology. Academic Press, New York, 1974.
- D. Greenhalgh, M. Griffiths. Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model. J. Math. Biol., 59 (2009) 1–36. Zbl1204.92059
- K. P. Hadeler, H. I. Freedman. Predator-prey population with parasitic infection. J. Math. Biol., 27 (1989) 609–631. Zbl0716.92021
- M. Haque, D. Greenhalgh. A predator-prey model with disease in prey species only. M2AS, 30 (2006) 911–929. Zbl1115.92049
- D. J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 43 (2001) 525–546. Zbl0979.65007
- W. O. Kermack, A. G. McKendrick. A Contribution to the Mathematical Theory of Epidemics. Proc. Roy. Soc. Lond. A.115 (1927) 700–721. Zbl53.0517.01
- P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1999. Zbl0752.60043
- M. Kot. Elements of Mathematical Biology. Cambridge University Press, Cambridge, 2001.
- Y. A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, Berlin, 1997.
- A. J. Lotka. Elements of physical biology. Williams & Wilkins Co., Baltimore, 1925. Zbl51.0416.06
- J. Marsden, M. McCracken. The Hopf Bifurcation and its Applications. Springer, New York, 1976. Zbl0346.58007
- H. Malchow, S. V. Petrovskii, E. Venturino. Spatiotemporal Patterns in Ecology and Epidemiology : Theory, Models and Simulations. Chapman & Hall, London, 2008. Zbl1298.92004
- J. D. Murray. Mathematical Biology. Springer, New York, 1993. Zbl0779.92001
- R. J. Serfling. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York, 1980. Zbl0538.62002
- D. Stiefs, E. Venturino, U. Feudel. Evidence of chaos in eco-epidemic models. Math. Biosci. Eng., 6 (2009) 857–873. Zbl1178.92057
- R. K. Upadhyay, N. Bairagi, K. Kundu, J. Chattopadhyay. Chaos in eco-epidemiological problem of the Salton Sea and its possible control. Appl. Math. Comput., 196 (2008) 392–401. Zbl1131.92070
- E. Venturino. The influence of diseases on Lotka-Volterra systems. Rocky Mountain Journal of Mathematics.24 (1994) 381–402. Zbl0799.92017
- E. Venturino. Epidemics in predator-prey models : disease in the prey, In ‘Mathematical Population Dynamics, Analysis of Heterogeneity’. 1, O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds), Wnertz Publisher Ltd, Canada, 381–393, 1995.
- V. Volterra. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. 2. Mem. R. Accad. Naz. dei Lincei. Ser. VI, 1926. Zbl52.0450.06

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.