A dynamical programming approach to populations management

In biology complex population models can usually attain several equilibrium states. Sometimes they are linked to each other via transcritical bifurcations. The ultimate system behavior can thus be represented by a graph in which nodes stand for equilibria and the arcs for the transcritical bifurcations that link them. The transitions from one equilibrium to a neighboring one is obtained by suitable changes in the model parameters, through the crossing of critical thresholds. Thus properly acting on these populationrelated parameters, if at all possible, allows to attain from the configuration at which the system is found, a desirable state, for instance a pest-free situation in an agrocosystem. However, in practical situations to each transition a cost is associated, as acting on the model parameters involves some kind of human effort. The problem of optimizing the total cost of the operations is here approached as a shortest path problem, through a dynamic programming procedure. The main problem is divided into smaller subproblems, of which the solution at each stage is incorporated into the next stage. This procedure is based on the optimality principle, stating that the best policy - the minimum cost path from the initial node to the final node - is also the best policy starting from an intermediate node. We illustrate this application through some examples, related to systems already investigated in the literature. The Matlab code takes as input the existing links between nodes and constructs the graph incidence matrix, identifying then the sequence of the stages originating from the initial configuration. It is adapted to account also for the case in which a path reaches the final node, representing the desired configuration, before the last stage: the program is devised to generate a sequence of dummy nodes linking up at cost zero to the final node, to move it in the last stage of the sequence.