# Lotka Volterra multi-arcs problem

Lotka Volterra multi-arcs problem | |
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State dimension: | 1 |

Differential states: | 3 |

Discrete control functions: | 1 |

Interior point equalities: | 3 |

This site describes a Lotka Volterra variant with three singular arcs instead of only one in the standard variant.

## Mathematical formulation

The mixed-integer optimal control problem is given by

Here the differential states describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation . This problem variant allows to choose between three different fishing options.

## Parameters

These fixed values are used within the model.

## Reference Solutions

If the problem is relaxed, i.e., we demand that be in the continuous interval instead of the binary choice , the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with is . The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is .