# Lotka Experimental Design

Lotka Experimental Design | |
---|---|

State dimension: | 1 |

Differential states: | 11 |

Discrete control functions: | 2 |

Path constraints: | 4 |

Interior point equalities: | 11 |

The **Lotka Experimental Design problem** looks for an optimal fishing strategy to be performed on a fixed time horizon to control the biomasses of both predator as prey fish as described in Lotka Volterra fishing problem. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. In addition to the fishing decision the question when measurements of the two state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale ODE model. The ODE from Lotka Volterra fishing problem is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows bang bang behavior.

## Contents

## Mathematical formulation

We are interested in estimating the parameters and of the Lotka-Volterra type predator-prey fish initial value problem

where is a fishing control that may or may not be fixed. The other parameters, the initial values and are fixed. We are interested in how to fish and when to measure, with an upper bound on the measuring time. We can measure the states directly, and . We use two different sampling functions, and in the same experimental setting. This can be seen either as a two-dimensional measurement function , or as a special case of a multiple experiment, in which , and are identical. The experimental design problem then reads

with , , , , and , .

Note that the state has been left out for reasons of symmetry.

## Parameters

We use , , and , . The upper bound on the measurement time intervals is chosen as .

## Reference Solutions

Here is one local solution to the above control problem.

## Source Code

Model descriptions are available in

## Variants

There are several alternative formulations and variants of the above problem, in particular

- a prescribed time grid for the control function [Sager2006]
**Address:***Heidelberg***Author:***S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder***Booktitle:***Recent Advances in Optimization***Editor:***A. Seeger***Note:***ISBN 978-3-5402-8257-0***Pages:***269--289***Publisher:***Springer***Series:***Lectures Notes in Economics and Mathematical Systems***Title:***Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem***Volume:***563***Year:***2009*

, see also Lotka Experimental Design (AMPL), - no fishing, i.e., ,
- different fishing control functions for the two species,
- different parameters and start values.

## Miscellaneous and Further Reading

The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper [Sager2006]**Address: ** *Heidelberg***Author: ** *S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder***Booktitle: ** *Recent Advances in Optimization***Editor: ** *A. Seeger***Note: ** *ISBN 978-3-5402-8257-0***Pages: ** *269--289***Publisher: ** *Springer***Series: ** *Lectures Notes in Economics and Mathematical Systems***Title: ** *Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem***Volume: ** *563***Year: ** *2009*

and revisited in his PhD thesis [Sager2005]**Address: ** *Tönning, Lübeck, Marburg***Author: ** *S. Sager***Editor: ** *ISBN 3-89959-416-9***Publisher: ** *Der andere Verlag***Title: ** *Numerical methods for mixed--integer optimal control problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2005.pdf***Year: ** *2005*

. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, [Sager2011d]**Author: ** *S. Sager***How published: ** *University of Heidelberg***Month: ** *August***Note: ** *Habilitation***Title: ** *On the Integration of Optimization Approaches for Mixed-Integer Nonlinear Optimal Control***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2011d.pdf***Year: ** *2011*

.

## References

[Sager2005] | S. Sager (2005): Numerical methods for mixed--integer optimal control problems. (%edition%). Der andere Verlag, Tönning, Lübeck, Marburg, %pages% | |

[Sager2006] | S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder (2009): Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem. Springer, Recent Advances in Optimization | |

[Sager2011d] | S. Sager: On the Integration of Optimization Approaches for Mixed-Integer Nonlinear Optimal Control, 2011 |