## Project CEMAPRE internal

Title | SDEFish (II) |

Participants | Nuno Brites (Principal Investigator), Manuel Guerra, João Janela |

Summary | Stochastic differential equations (SDE) adequately model the behaviour of dynamical phenomena permanently influenced by environmental stochasticity by incorporating the effect of such random fluctuations in the dynamics. On the contrary, the current paradigm in many areas ignores the effect of random fluctuations by using deterministic growth models and adjusting them to data by regression (treating deviations as independent measurement errors), thus leading to inaccurate predictions. Stochastic differential equations have been applied to the growth dynamics of harvested populations living in a randomly varying environment, with the purpose of obtaining optimal fishing policies. In a random environment, the typical approach to obtain optimal harvesting policies uses stochastic optimal control to maximize the expected accumulated discounted profit over a time horizon. However, contrary to the deterministic case, the population cannot be kept at an equilibrium size and will rather keep experiencing random fluctuations. Therefore, the optimal fishing effort must be adjusted at every instant, so that the size of the population is below (and close to) some threshold value. So, the optimal fishing effort will have very frequent transitions between maximum/high efforts and low/null efforts. These transitions are not compatible with the logistics of fisheries. Besides, the period of low/no harvesting poses socioeconomic undesirable implications (intermittent unemployment is just one of them). In addition, these optimal policies require the knowledge of the population size at every instant to define the appropriate level of effort. The estimation of the population size is a difficult, costly, time consuming and inaccurate task. For all these reasons, such policies should be considered unacceptable and inapplicable. Considering a constant fishing effort will result in a stochastic sustainable behaviour. The probability distribution of the population size at time t converges, as t goes to infinity, to an equilibrium probability distribution having a probability density function. Such policies are extremely easy to implement and lead to a stochastic steady state. The constant effort that maximizes the expected profit per unit time at the steady state, considering general population SDE growth models (with and without the presence of Allee effects) and also specific models like the logistic and the Gompertz models are quite easy to obtain. One might think that a constant effort policy would result in a substantial profit reduction compared with the optimal variable effort policy, but that profit reduction is quite small. This new easily implementable policy, rather than switching between large and small or null fishing efforts, keeps a constant effort and is therefore compatible with the logistics of fisheries and does not pose socioeconomic problems. Furthermore, this alternative policy does not require knowledge of the population size. To obtain a policy based on variable effort we will consider a partial differential equation which is non-linear and highly complex. Therefore, the possibility of an analytical solution is ruled out and numerical methods should be used to solve the partial differential equation and to obtain the optimal solution. This partial differential equation is parabolic and, therefore, to approximate the spatial and temporal derivatives we will use a Crank-Nicolson difference scheme. The numerical scheme will be deeply studied in terms of stability and convergence. |