A handling tool to estimate upper bounds of environmental fluctuations

Abstract

The construction of the bound for the intensity of the environmental fluctuations, based on the Lyapunov function (LF), is one of the important problem in dynamical system. But the method of identifying proper LF is really very difficult and some time might be impossible, as it involves complete blind search procedure which is arbitrary and ad hoc. In addition, the random fluctuations or in statistical terminology the error terms involved in the above experiments are completely unknown to us. In usual practice, researchers assume the simple independent structure of the error terms. But for the sake of completeness of the model as well as for the realistic environmental set-up (where the error terms or the equations are dependent), the study of more complicated and generalized structure of the random environmental fluctuations is demanding. In this paper, we propose a probabilistic bound for the intensity of the environmental fluctuations. The proposed method has several key advantages – (a) it is simple and can be used as a handling tool or a working formula for the experimental biologists, (b) can be used regardless the situation where the identification of proper LF is either difficult or not possible, (c) it can be applied for a system of any dimensions, and moreover (d) applicable under more general set-up of error terms.

Keywords:

• migratory bird
• eco-epidemiological model
• stochastic differential equations
• Lyapunov function
• upper bound
• conjecture

AMS Subject Classifications:

• 34F05
• 60H10
• 92B05
• 92D25

Acknowledgments

The authors are grateful to the learned reviewer for his/her useful comments and suggestions in the previous version of the paper.

Notes

Samrat Chatterjee’s current address is Drug Discovery Research Centre, Translational Health Science & Technology Institute, Gurgaon-122016, India.