A Mathematical Model for Optimal Management and Utilization of a Renewable Resource by Population

Abstract

A dynamical model is proposed and analyzed to study the effect of the population on the resource biomass by taking into account the crowding effect. Biological and bionomical equilibria of the system are discussed. The global stability behavior of the positive equilibrium is studied via the output feedback control. An appropriate Hamiltonian function is formed for the discussion of optimal harvesting of resource which is utilized by the population using Pontryagin's Maximum Principal. A numerical simulation is performed on the model to analyze the theoretical results.

Observer design for a mathematical model for the management of a renewable resource population

“…Then the dynamics of the biomass resource is described by (see Dubey and Patra, 2013): Then the dynamics of the biomass resource is described by (see Dubey and Patra, 2013):…”

“…Keeping the above aspect in view, the dynamics of the system can be governed by the system of the following differential equations (see Dubey and Patra, 2013):…”

An application of optimal control to the effective utilization of a renewable resource

“…The most common growth function favoured by most researchers is the Pearl-Verhulst logistic function; see, for example, Clark (2010); Clark and Munro (1975);Craven (1995); Dubey and Patra (2013); Dubey et al (2003); Tar and Chakraborty (2009). In a pioneering work, Clark and Munro (1975) proposed a nonlinear optimal control model that permitted the rate of fishing effort to be nonlinear, thereby introducing nonlinear costs into the model.…”

“…Additionally, in determining the net revenue for the harvested fish, the model not only considered the unit price and unit cost of the fish but also the diminishing returns when there is a large amount of fish to sell. Dubey and Patra (2013) proposed a model that involves the interaction of the human population and the resource population. They considered the human population to be partially dependent on the resource, which is then harvested for the benefit of society.…”

Complex Dynamics of Wetland Ecosystem with Nonlinear Harvesting: Application to Chilika Lake in Odisha, India

“…Optimal harvesting policy was introduced by Clark [1976]. Many authors [Kar & Chaudhuri, 2004;Gibbons, 1996;Dubey & Patra, 2013;Kar et al, 2009] have discussed the optimal harvesting policy in their model. Dubey et al [2003] discussed the optimal harvesting policy in the model of fishery resource with reserve area.…”