Bio-economics of a renewable resource in a seasonally varying environment

Optimal Harvest Problem for Fish Population—Structural Stabilization

The influence of environmental conditions and fishery on a typical pelagic or semi-pelagic fish population is studied. A mathematical model of population dynamics with a size structure is constructed. The problem of the optimal harvest of a population in unstable environment conditions is investigated and an optimality system to the problem research is constructed. The solutions properties in various cases have also been investigated. Environmental conditions influence the fish population through recruitment. Modelling of recruitment rate is made by using a stochastic imitation of environmental conditions. In the case of stationary environment, a population model admits nontrivial equilibrium state. The parameters of fish population are obtained from this equilibrium condition. The variability of environment leads to large oscillations of generation size. The fluctuations of the fish population density follow the dynamics of recruitment rate fluctuations but have smaller gradients than recruitment. The dynamics of the optimal fishing effort is characterized by high variability. The population and the average size of individuals decrease under the influence of fishery. In general, the results of computer calculations indicate the stabilization of the population dynamics under influence of size structure. Optimal harvesting also contributes to stabilization.

Optimal harvesting for a single-species population governed by Gompertz law: Influence of environmental fluctuation and limited harvesting capacity

This work presents an optimal harvesting problem associated with a single-species population governed by Gompertz law in a seasonally fluctuating environment. The influence of environmental fluctuation is accommodated by choosing the coefficients in the differential equation to be periodic functions with the same period and restriction on the harvesting effort is accommodated by considering binding constraints on the control variable. Hence, a linear optimal control problem has been considered where the state dynamics is governed by Gompertz equation and the control variable is subject to the binding constraints. With the help of maximum principle and the concept of blocked intervals, an optimal periodic solution has been obtained which is followed by the construction of optimal solution using the theory of most rapid approach. Important results of the study are demonstrated through numerical simulations.

Renewable resource management in a seasonally fluctuating environment with restricted harvesting effort

This paper presents bio-economics of a renewable resources in a seasonally changing environment in which the resource exploitation is subjected to restrictions on harvesting effort. The dynamics of the resource is assumed to be governed by the logistic equation. Seasonality is incorporated into the system by choosing the coefficients in the growth equation to be periodic functions with the same period. A linear optimal control problem involving binding constraints on the control variable has been considered. As a result the concept of blocked interval plays a key role in the construction of optimal solution. In view of the periodicity associated with the considered problem, we first construct an optimal periodic solution. The optimal solution is established using the most rapid approach path to the said optimal periodic solution. The global asymptotic stability property of the optimal periodic solution enables construction of a suboptimal solution. The optimal solution is found to be periodic after some finite time and suboptimal solution approaches the optimal periodic solution asymptotically. Key results are illustrated through numerical simulation.

Impact of seasonal forcing on reactive ecological systems

Our focus is on the short-term dynamics of reactive ecological systems which are stable in the long term. In these systems, perturbations can exhibit significant transient amplifications before asymptotically decaying. This peculiar behavior has attracted increasing attention. However, reactive systems have so far been investigated assuming that external environmental characteristics remain constant, although environmental conditions (e.g., temperature, moisture, water availability, etc.) can undergo substantial changes due to seasonal cycles. In order to fill this gap, we propose applying the adjoint non-modal analysis to study the impact of seasonal variations of environmental conditions on reactive systems. This tool allows the transient dynamics of a perturbation affecting non-autonomous ecological systems to be described. To show the potential of this approach, a seasonally forced prey-predator model with a Holling II type functional response is studied as an exemplifying case. We demonstrate that seasonalities can greatly affect the transient dynamics of the system.