J. Nonl. Evol. Equ. Appl. 2014 (6), pp. 77-100, published on January 12, 2015:

Trophic cascades scenario of a detritus-based stage-structured model in marine ecology

K. Das

National Institute of Food Technology Entrepreneurship and Management; Department of Mathematics, Kundli-131 028, Haryana, India.

A. Sabarmathi, M. N. Srinivas

School of Advanced Sciences; VIT University, Vellore-632 014, Tamil Nadu, India

N. H. Gazi

Department of Mathematics, Aliah University; DN-47, Sector-V, Salt Lake City, Kolkata-700 091, India

B. R. Kumar

School of Advanced Sciences; VIT University, Vellore-632 014, Tamil Nadu, India

Received on January 7, 2013, revised version on July 17, 2013
Accepted on July 23, 2013

Communicated by Ti-Jun Xiao

Abstract.  In the recent years, we have seen that human activity has been changing in the local and the global environment at an unprecedented rate. Some of the significant changes include widespread nutrition enrichment and biotic impoverishment are beneficial besides the occurrence of global warming and sea level rise that prove to be hazardous in the long run. When the impact of these multiple stresses on the biosphere is considered, we face a challenging and an urgent problem of environmental science. This is a vital step towards sustainable management in ecology. Hence it is particularly important to consider the coastal salt marsh estuarine system because it is one amongst the most biologically productive areas in the world and is used by human beings for a variety of purposes. The areas under discussion are now experiencing changes in nutrient loading, species composition and sea level raise which are 5-10 fold higher in comparison to that which was measured over the last century. This is undoubtedly affecting the productivity and sustainability of coastal regions. In this paper, the mathematical model of a detritus-based ecosystem with stage-structure and selective harvesting which is mainly found in Sunderban Mangrove area in India is considered. At the positive stationary state, the local and global stability due to discrete time delay and stochastic perturbation is analyzed. The exploitation of the prey is controlled by a regulatory agency by imposing a tax per unit biomass of the detritus and the optimal harvesting policy is achieved by using Pontryagin's maximum principle. It is very interesting that the time delay has a great role in the real ecology by inducing a stable equilibrium into an unstable one. Finally numerical simulations are carried out to compare the analytical results.
Keywords:Detritus; detritivorus; stage structure; stability; time delay; stochastic perturbation; harvesting.
2010 Mathematics Subject Classification:   70K, 92D25, 92D25, 92B99, 34L30, 11S82.

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