MATHEMATICAL MODELING OF RENEWABLE RESOURCES

Abstract

The main motive of the present thesis is to study the dynamical behavior of some bio-economical and ecological models in which to preserve the species from extinction. This thesis comprise of eight chapters. The Chapter 1 includes the literature survey, basic de nitions and technical tools which will be used throughout the thesis. The chapter wise summary of the thesis is given below: In Chapter 2, a nonlinear harvesting of a modi ed Leslie{Gower type predatorprey dynamical system is studied where harvesting e ort is taken as a dynamic variable. The conditions of existence and local asymptotic stability of various equilibrium states have been obtained. The dynamical behavior of the interior state of the system shows that it is locally and globally asymptotically stable under certain condition. It is established that the coexistence of prey and predator population depend upon the proper harvesting strategies. Using analytical and numerical results, it is examined that for a xed value of price per unit mass and other parameters of the system, as the value of cost per unit mass is increasing, the level of harvesting start increases. After some times, a level of cost is obtained where harvesting e ort will tend to zero. Accordingly, for the coexistence of prey predator population along with e ort, optimal level of cost is obtained. In Chapter 3, model of chapter 2 is extended incorporating taxation as a control instrument. The existence of interior steady state of this system is strongly depends on range of taxation. The bionomic equilibrium of the system provides the range of harvesting rate which may be useful for a harvesting agency to get the pro table i ii yields. The su cient condition for global stability of unique interior equilibrium point provides a domain for global solutions. The conditions of persistence for the system are derived. It is also investigated that coexistence of prey and predator population depends upon the proper harvesting strategies such as the risk of extinction of species can be avoided. The objective of this work includes both ecological and economic aspects. The economic objective is to maximize the net economic revenue and ecologically, want to keep the prey and predator population from extinction. In Chapter 4, a two dimensional predator{prey dynamical system where predator is provided an additional food resource, is studied incorporating combined harvesting. In this chapter, prey and predator both are a ected by some external toxicant substances which are harmful for both species. The steady states of the system and their stability analysis have been carried out for all possible feasible equilibrium points. The system undergoes some local bifurcations i.e., trans-critical, Hopf, saddle node bifurcations for a threshold level of parametric values. Also, some global bifurcations i.e., Bogdanov{Taken bifurcation (BT) and Generalized Hopf bifurcation (GH) are detected w.r.t. di erent parametric values. The su cient condition for the bionomic equilibrium and optimal harvesting policy for the model is obtained. Chapter 5 analyzed a mathematical model of a dynamical Stock-E ort system with nonlinear harvesting of species where taxation is used as a control instrument. This system is considered in two di erent shing zones. The migration rates of shing vessels between patches is assumed to be stock dependence. In this model, we have assumed that there are two time scales, a fast one for movement sh and boats between patches, and slow one corresponding to sh population growth and shery dynamics. The aggregated method is used to simplify the mathematical analysis of the complete model. Qualitative analysis of this system reveals that nonlinear harvesting term plays an important role to determine the dynamics and bifurcation of system. Existence of bifurcations indicates that the high taxes will cause closed of shery. However, the Maximum Sustainable yield (MSY) and Optimal Taxation Policy is discussed for the aggregated model. iii In Chapter 6, a spatial prey predator mathematical model has been proposed and analyzed. This system is based upon the two time scales: fast one and slow one. Therefore, to simplify the mathematical analysis, an aggregation method is used. The e ect of toxicity is considered in the system. The unique interior equilibrium point exists under certain condition and it is globally asymptotically stable. Bendixon{Dulac Criteria con rms that there does not exist any periodic solutions in the interior. Numerically it is also shown that the trajectories of aggregated model remain close to the trajectories of the complete model. Chapter 7 investigates a predator{prey dynamic reaction model in a heterogeneous water body where only prey is subjected to harvesting. The consequences of prey refuge, availability of alternative food resource for predators and e ects of harvesting e ort on the dynamics of prey predator system are explored. The two time scales are considered in the dynamics of the model. The reduced aggregated model is analyzed analytically as well as numerically using dynamic of harvesting e ort. Numerical simulation shows that introducing the dynamics of harvesting e ort can destabilize prey predator system. This is con rmed by the bifurcation diagrams and dynamics of Lyapuonov exponent w.r.t. bifurcation parameter. The Chapter 8 is about the achievements and future scope and possible extension of the present work.