Correlated Noises in Dynamical System with θ-Logistic Growth Process of Species

Bapi Saha¹*, Amiya Ranjan Bhowmick², Sabyasachi Bhattacharya², Joydev Chattopadhyay²

¹Government College of Engineering and Textile Technology, Berhampore, West Bengal

²Indian Statistical Institute, Kolkata, India

*Corresponding Author E-mail:

ABSTRACT:

Nonlinear dynamical system with correlated noise terms is an important and potential aspect in ecological problems. We consider a dynamical system where the growth process of the species follows θ-logistic model. Multiplicative and additive noises have been incorporated to the system. We study the steady state behaviour of the dynamical system in the presence of the two noises. The effect of the correlation coefficient between the two noises on steady state has been discussed. Apart from that the impact of θ and other related parameters on the steady state is studied using numerical methods aiming at the conservation strategy of the species.

KEYWORDS: Theta-logistic model, population dynamics, correlated noise, stationary distribution, steady state

1. INTRODUCTION:

Recently nonlinear dynamical systems with correlated noise terms have attracted attentions of ecologists. It is observed that in such systems the noise affects the dynamics through a system variable, i.e., the noise is both multiplicative and additive (Ai et al., 2002). In most of the literature the noises are generally considered to be uncorrelated, since they are assumed to be generated from independent sources. However, the noises in some stochastic process may have a common origin, and thus may be correlated (Fedchenia, 1988; Horsthemke and Lefever, 1984).

REFERENCES:

1. Ai, B., Wang, X., Liu, L., Nakano, M., Matsuura, H., 2002. Muscle’s motion in an overdamped regime. Commun. Theor. Phys. 37, 125.

2. Anderson, R., May, R., 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford, UK.

3. Banik, S. K., 2001. Correlated noise induced control of prey extinction. arXiv:physics/0110088.

4. Barker, D., Sibly, R. M., 2008. The effects of environmental perturbation and measurement error on estimates of the shape parameter in the theta-logistic model of population regulation. Ecological Modelling 219, 170–177.

5. Cameron, T., Benton, T., 2004. Stage-structured harvesting and its effects: an empirical investigation using soil mites. Journal of Animal Ecology. 73, 996–1006.

6. Cao, L., Wu, D. J., 1994. Phys. Lett. A 185, 59.

7. Caughley, G., Sinclair, A., 1994. Wildlife Ecology and Management. Blackwell Scientific, Boston, MA, USA.

8. Da-jin, W., Li, C., Sheng-zhi, K., 1994. Bistable kinetic model driven by correlated noises: Steady-state analysis. Phys. Rev. E 50, 2496-2502.

9. Diserud, O., Engen, S., 2000. A general and dynamic species abundance model, embracing the lognormal and the gamma models. Am. Nat. 155, 497–511.

10. Fedchenia, I. I., 1988. A two-dimensional Fokker-Planck equation degenerating on a straight line. J. Stat. Phys 52, 1005-1029.

11. Fulinski, A., Telejko, T., 1991. Phys. Lett. A 152, 11-14.

12. Gilpin, M. E., A. F. J., 1973. Global models of growth and competition. Proc. Nat. Acad. Sci. USA 70, 3590–3593.

13. Horsthemke, W., Lefever, R., 1984. Noise-induced Transitions. Springer-Verlag, Berlin.

14. Jia, Y., Cao, L., Wu, D. J., 1995. Phys. Lett. A 51, 3196.

15. Jia, Y., Li, J., 1996. yet to found. Phys. Rev. E 53, 5786.

16. Li, J., Huang, Z., 1996. Nonequilibrium phase transition in the case of correlated noises. Phys. Rev. E 3, 3315–3318.

17. Philippi, T., Carpenter, M., Case, T., Gilpin, M., 1987. Drosophila population dynamics: chaos and extinction. Ecology. 68, 154–159.

18. R., M. A. J., Hanggi, P., Wio, H. S., 1996. Giant suppression of the activation rate in the presence of correlated white noise sources. Physics Letters A 217, 248–252.

19. Reynolds, J. D., Freckleton, R. P., 2005. Population dynamics: Growing to extremes. Science 309, 567–568.

20. Saether, B. E., Engen, S., Grotan, V., Bregnballe, T., Both, C., Tryjanowski, P., Leivits, A., Wright, J., Moller, A. P., Visser, M. E., Winkel, W., 2008. Forms of density regulation and (quasi-) stationary distributions of population sizes in birds. Oikos 117, 1197–1208.

21. Sibly, R. M., Barker, D., Denham, M. C., Hone, J., Pagel, M., 2005. On the regulation of populations of mammals, birds, fish, and insects. Science. 309, 607–610.

22. Saether, B.-E., Engen, S., Matthysen, E., 2002. Demographic characteristics and population dynamical patterns of solitary birds. Scienc. 295, 2070–2073.

23. Zhu, S., 1993. Steady-state analysis of a single-mode laser with correlations between additive and multiplicative noise. Phys. Rev. A 47, 2405–2408.

Received on 15.01.2014 Accepted on 01.02.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 158-162