Bionomic equilibrium of two-species system. I

The concept of bionomic equilibrium is used to obtain limits of cost per unit effort and maximum value of effort for the joint harvesting of three types of two-species system: (1) a logistic growth model of two ecologically independent species, (2) a logistic growth model of two species that have competitive interactions, and (3) a Lotka-Volterra model of one prey and one predator. In each case, the existence of feasible bionomic equilibrium points and that of partially feasible bionomic equilibrium points are considered separately. First it is shown that cost per unit effort has both upper and lower threshold values if feasible bionomic equilibrium points occur, whereas it has only the upper threshold values if partially feasible bionomic equilibrium points occur. Further, the threshold values depend solely on the catchability coefficients, the selling prices of unit biomasses, and the parameters of the given model, namely the biotic potentials, carrying capacities, etc. Next, it is proven that for the first type of feasible point, if the cost per unit effort is kept under proper threshold values, then corresponding to each such value, the maximum value of the efforts exerted can be calculated in terms of the given parameters only. Moreover, the set of all such maximum values of the efforts for different choices of cost per unit effort is always bounded and the supremum of these maximum efforts is independent of the choice of cost per unit effort.