Mathematical Model of Depletion of Forestry Resource: Effect of Synthetic Based Industries

Authors: Manisha Chaudhary, Joydip Dhar, Govind Prasad Sahu

Abstract:

A mathematical model is proposed considering the forest biomass density B(t), density of wood based industries W(t) and density of synthetic industries S(t). It is assumed that the forest biomass grows logistically in the absence of wood based industries, but depletion of forestry biomass is due to presence of wood based industries. The growth of wood based industries depends on B(t), while S(t) grows at a constant rate, independent of B(t). Further there is a competition between W(t) and S(t) according to market demand. The proposed model has four ecologically feasible steady states, namely, E1: forest biomass free and wood industries free equilibrium; E2: wood industries free equilibrium and two coexisting equilibria E∗1 , E∗2 . Behavior of the system near all feasible equilibria is analyzed using the stability theory of differential equations. In the proposed model, the natural depletion rate h1 is a crucial parameter and system exhibits Hopf-bifurcation about the non-trivial equilibrium with respect to h1. The analytical results are verified using numerical simulation.

Keywords: A mathematical model, Competition between wood based and synthetic industries, Hopf-bifurcation, Stability analysis.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1335720

References:

1] M. Agarwal, Tazeem Fatima, H.I.Freedman, Depletion of forestry resources biomass due to industrialization pressure: a ratio dependent mathematical model, Vol. 4 No. 4, pp. 381-396, 2010.
[2] J. Dhar, H. Singh, Modelling the depletion of forestry resource by wholly dependent industrialization in two adjoining habitat, Kobe J. Mathematics, Japan, 21(2-1), pp. 1-13, 2004.
[3] J. Dhar, H. Singh, Diffusion of population under the influence industrialization in a twin-city environment, Mathematical Modelling and Analysis, 9(3), pp.124-133, 2004.
[4] J. Dhar, Population model with diffusion and supplementary forestry resource in a two-patch habitat, applied mathematical modelling, Vol.32, pp. 1219-1235, 2008.
[5] J. Dhar, A.K. Sharma, The role of viral infection in phytoplankton dynamics with the inclusion of incubation class, Nonlinear Analysis: Hybrid Systems, Vol. 4 (1) , pp. 9-15, 2010.
[6] B. Dubey, S. Sharma, P. Sinha, J. B. Shukla, Modelling the depletion of forestry resources by population and population augmented industrialization, Vol. 33, Issue 7, pp. 3002-3014, 2009.
[7] D. Ghosh, A. K. Sarkar, Stability and oscillation in a resource based model of two interacting species with nutrient cycling, J. Ecological Modelling, 107, pp. 25-33, 1998.
[8] D. McHugh (Ed.) Forests and climate change. Special Report No. 6, World Wide Fund for Nature, WWF International, Gland Switzerland, pp. 16-18, 1991.
[9] O. Joshi, S. R. Mehmood, Factors affecting nonindustrial private forestry landowners willingness to supply woody biomass for bioenergy, biomass and bioenergy 35, pp. 186-192, 2011.
[10] S. Khare, O.P. Misra, J. Dhar, Role of toxin producing phytoplankton on a plankton ecosystem, Nonlinear Analysis: Hybrid Systems, Vol. 4 (3), pp. 496-502, 2010.
[11] W. Li, M. Ju, Le Liu, Y. Wang, T. Li, The Effects of Biomass Solid Waste Resources Technology in Economic Development, Energy Procedia 5 pp. 2455-2460, 2011.
[12] S.L. Naeem, J. Thompson, S.P. Lawler, G.H. Lawton and R.M. Wood fin, Decline biodiversity can alter the performance of ecosystem, Nature 368, pp. 734-737, 1994.
[13] Bob Persched, A. Evans, M. DeBonis, Forestry biomass Retantion and harvesting guidelines for the northeast, Forestry guild, 2010.
[14] G.P. Sahu, J. Dhar, Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate, Applied Mathematical Modelling, Vol. 36 (3), pp. 908-923, 2012.
[15] S. Sinha, O. P. Misra, J. Dhar, Modelling a predator-prey system with infected prey in polluted environment, Applied Mathematical Modelling 34 (7) , pp. 1861-1872, 2010.
[16] J. B. Shukla, O. P. Misra, M. Agarwal, A. Shukla, Effect of pollution and industrial development on degradation of biomass resource:A mathematical model with reference to Doon Valley, Mathl. Comput. Modell. Vol. 11, pp. 910-913, 1988.
[17] J. B. Shukla, H. I. Freedman, V. N. Pal, O.P. Misra,M. Agrawal,A. Shukla, Degradation and subsequent regeneration of a forestry resource: A mathematical model. Ecol. Modell. 44, pp. 219-229, 1989.
[18] J. B. Shukla, B. Dubey and H.I. Freedman, Effect of changing habitat on survival of species, Ecol. Model.87, pp. 205-216, 1996.
[19] J. B. Shukla and B. Dubey, Modelling the depletion and conservation of forestry resources: Effects of population and pollution, J. Math. Biol.36, pp. 7194, 1997.
[20] J. B. Shukla, A. K. Agrawal, P. Sinha, B. Dubey, Modelling effects of primary and secondary toxicants on renewable resources, Natural Resource Modelling, Vol. 16(1), pp. 1-22, 2003.
[21] J. B. Shukla, K. Lata, A. K. Misra, Modelling the depletion of a renewable resource by population and industrialisation: Effect of technology on its conservation, Natural Resource modelling, Vol. 24, 2011.