Optimal Management of Forestry Biomass Affected with Toxicant

International Journal of Research and Scientific Innovation (IJRSI) | Volume VIII, Issue VIII, August 2021 | ISSN 2321–2705

Optimal Management of Forestry Biomass Affected with Toxicant

Prayga Mishra^1*, Vimlesh Singh², Rajeev Kumar Singh³ & Akhilesh Kumar^4*
1Department of Mathematics, Pt. Deen Dayal Upadhyay Government Girl’s P.G. College, Rajaji Puram, Lucknow-226017, U. P. India.
²Department of Mathematics and Statistical Sciences, Shri Ramswaroop Memorial University
Dewa Road-Lucknow, Barabanki-225003, U.P., India.
³Department of Mathematics, Pratap Bahadur Post Graduate College, Pratapgarh City-2300002, India.
⁴Department of Physics, Pt. DeenDayal Upadhyay Government Girl’s P.G. College, Rajaji Puram, Lucknow-226017, U. P. India.
*Corresponding Author

Abstract: The optimum harvest strategy for forestry biomass influenced by a toxicant is suggested and studied by using a non-linear mathematical model. The nature and uniqueness of equilibrium, conditions for existence of their local and global equilibrium points, are all established. Both equilibrium levels of biomass and total sustainable yield decrease as toxicant concentrations rise, as per the analysis. The optimum harvest approach is frequently discussed using Pontryagin’s Maximum principle. Numerical analysis is implemented to validate the mathematical findings.

Keywords: Forestry biomass, stability, optimal harvesting, toxicant; Pontryagin’s Maximum principle

AMS subject classification: 92D25.

I. INTRODUCTION

The toxicant and contaminants released from various industries, vehicles and other man-made projects leave deleterious effect on biological and forestry biomass. Some studies on the impact of toxicants on biological populations have been conducted in recent years. In particular, (Hallam et al., 1984) in a series of their papers studied qualitative approach of toxicants on populations. They assumed growth rate density of single-spaces population as decreasing function of concentration of toxicant but the corresponding carrying capacity being unaffected by the presence of toxicant in environment. Taking this inconsideration, (Freedman and Shukla, 1991) studied the effect of toxicant on a single species and on a predator-prey system by takinginto account the introduction of toxicant from an external source. Further (Shukla et al., 2009) studied the effects of toxicants on resource dependent population where they considered toxicant emitted from the external sources and by its precursor. The effect of environmental toxicant on resource biomass has been studied by (Gakkhar and Sahani, 2009) and (Naresh et al., 2014), and it was found that growth rate of plants was affected because of uptake of pollutants which lead to a decrease in resource biomass. (Shukla and Dubey, 1997) in their paper discussed the combined effect of population and pollution on depletion of forestry biomass. (Lata. K et al., 2016) studied the effect of industrialization on forestry resources in which they assess the effect of wood and

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