Research projects

Project CEMAPRE internal

TitleSDEFish (II)
ParticipantsNuno Brites (Principal Investigator), Manuel Guerra, João Janela
SummaryStochastic 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
of random fluctuations by using deterministic growth models and adjusting them to data by
(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.
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.
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
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,
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
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.