Dynamic Reaction Model

Population dynamics has attracted interest from the commercial harvesting industry and from many scientific communities including biology, ecology and economics. Population ecologists study how births and deaths affect the dynamics of populations and communities, while ecosystem ecologists study how species control the flux of energy and materials through food webs and ecosystems. Although all these processes occur simultaneously in nature, the mathematical frameworks bridging the two disciplines have developed independently.

The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance (Berryman,

In an open-access fishery of a fully dynamic model, the level of fishing effort expands or contracts according as the net economic revenue (i.e., perceived rent) to the fisherman is positive or negative. A model reflecting this dynamic interaction between the perceived rent and the effort in a fishery is called a dynamic reaction model. The imposition of tax acts as a deterrent to the fishermen and helps to control harvesting of prey fish and in turn, it helps the predator to grow. As it has been noted, IWC (International Whaling Commission) has imposed a complete ban on killing whales. Whales predate on kills which are being heavily harvested nowadays. To ensure normal growth of whales, krill harvesting must be controlled. The main aim of this paper is to find the proper taxation policy which would give the best possible benefit through harvesting to the society while preventing extinction of the predator. This is different from the usual optimal harvesting problem whose objective is purely monetary. The existence of the possible steady states and local stability at interior point is discussed. I also derived condition for global stability of the interior equilibrium. Optimal tax policy is obtained using the maximum principle.

It is well known that a linear term leads to unstable equilibrium followed by a limit cycle and therefore the incorporation of a linear term is not appropriate for many marine species (Steela and Henderson [20]). Again if the abundance of predators is high, density-dependent effects start to play an important role towards the stability of the proposed system. Of several possible alternatives, the form is considered here as the most economical in parameters. Thus, the growth of the predator species in the second equation is limited due to the presence of the term and can be non-positive for large enough value of , even if the size of the prey is very high. Here as the focal prey population increases, the predator uses less alternative prey and when the mass of alternative prey consumed tends to zero, and conversely, as the focal prey decreases, the predators increase their feeding on alternative prey. Thus even in case of extinction of the focal prey the predator population maintains its growth rate, varying linearly with its density Here are positive