Kunwer Singh Jatav and Poonam Sinha/ Elixir Appl. Math. 86 (2015) 35377-35382 35377 Introduction Depletion of forest resource by industrialization and rapid growth of population, particularly in the third world country is of grave concern. A typical example in this regard is the degradation of the forestry resources in the Doon Valley located in the foothills of Himalayas, Uttaranchal, India. Here the degradation of forest has been caused mainly by limestone quarries, paper, other wood-based industries and associated population growth [14]. [21] have proposed a Mathematical model for forest degradation caused by resource independent industrialization population by considering the spatial distribution of both the forest biomass as well as the density of industrialization by studying the behavior of uniform steady state solution. It may be pointed out here that in real ecological situations when forest is degraded by industrialization distributed spatially, patchiness is caused in the forest habitat. It is worth nothing that little efforts have been made to study such systems using mathematical models [6, 16, 21, and 23]. How ever in [21], has not considered the effect of patchiness caused by industrialization. Further [2-4,7] studied a single species diffusion model by assuming that the habitat consists of two adjoining patches and studied the behavior of steady state distribution and local asymptotically stable conditions. Biotic populations are usually distributed non-uniformly in their habitat, and the distribution is often patchy, due to patchiness of the habitat which arises from a variety of mechanisms and processes under various conditions including deforestation in the case of a forest habitat. It would, thus, seem natural to study the population dynamics of a single species by including diffusion effects, in a patchy habitat. Many investigators [1,5,6,8-13] have shown that in a homogeneous habitat, the diffusion , increases the stability of the system, but this may not always be true, if the habitat is patchy [7,15,17,18]. A model of a single species population living in two patch habitats with migration between them across a barrier was proposed by Freedman and Waltman [5]. The model was extended in [17,19] to include the case where animal species leaving one habitat does not necessarily reach the other habitat, the existence of a positive equilibrium as a function of barrier strengths was examined. Also Freedman [7] studied a single species diffusion model by assuming that the habitat consists of two patches and has shown that there exists a positive, monotonic, continuous non uniform steady state solution that is linearly asymptotically stable under both reservoir and no-flux boundary conditions. Mathematical Model We consider a forest habitat 2 0 L s   linearly distributed, where forest resources are depleted by different levels of industrialization in the above adjoining regions 1 0 L s   and 2 1 L s L   where 0 0  L and 1 L is the interface of two regions. Let ) , ( t s R i and ) , ( t s I i , (i=1,2) be respectively the densities of resource biomass and industrialization (population) pressure at location s and time t in the above mentioned i th regions [ see figure 1]. It is assumed that ) , ( t s R i grows logistically in both the region with the same intrinsic growth rate and carrying capacity, i.e. in absence of industrialization leading to a uniform spatial distribution. Since the levels of industrialization are assumed to be different in these two regions, the growth rate i  and carrying capacity Tele: E-mail addresses: sing1709@gmail.com © 2015 Elixir All rights reserved Mathematical study of the effect of industrialization on the resource biomass under going harvesting and diffusion in heterogeneous habitat Kunwer Singh Jatav 1,* and Poonam Sinha 2 1 Department of Applied Sciences, ABV-Indian Institute of Technology and Management, Gwalior. 2 Department of Mathematics, Govt. Model Science College, Gwalior. ABSTRACT In this chapter, a Mathematical model is proposed to the study of depletion of a uniformly distributed forest biomass caused by different levels of industrialization and population in two adjoining regions of the habitat. Industrialization dependent, constant, instantaneous, and periodic emissions of pollutant into the environment are taken into consideration. Criteria for local stability, instability, and global stability of non-negative equilibrium are obtained in the absence of diffusion and in presence of diffusion. A model of a single species population living in two patch habitats with migration between them across a barrier was proposed by Freedman and Waltman [5]. The model was extended in [17,19] to include the case where animal species leaving one habitat does not necessarily reach the other habitat, the existence of a positive equilibrium as a function of barrier strengths was examined. Also Freedman [7] studied a single species diffusion model by assuming that the habitat consists of two patches and has shown that there exists a positive, monotonic, continuous non uniform steady state solution that is linearly asymptotically stable under both reservoir and no-flux boundary conditions. © 2015 Elixir All rights reserved. ARTICLE INFO Article history: Received: 13 May 2012; Received in revised form: 25 September 2015; Accepted: 30 September 2015; Keywords Biomass, Diffusion, Haditat. Elixir Appl. Math. 86 (2015) 35377-35382 Applied Mathematics Available online at www.elixirpublishers.com (Elixir International Journal)