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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 [[:Category:ODE model|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 [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>p_2</math> and <math>p_4</math> of the Lotka-Volterra type predator-prey fish initial value problem
<p>
<math>
\begin{array}{rcl}
\dot{x_1}(t) &=& p_1 \; x_1(t) - p_2 x_1(t) x_2(t) - p_5 u(t) x_1(t), \; t \in [0,t_f], \quad x_1(0) = 0.5, \\
\dot{x_2}(t) &=& - p_3 \; x_2(t) + p_4 x_1(t) x_2(t) - p_6 u(t) x_2(t), \; t \in [0,t_f], \quad x_2(0) = 0.7,
\end{array} 
</math>
</p>
where <math>u(\cdot)</math> is a fishing control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 12</math> are fixed. We are interested in how to fish and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w^1(\cdot)</math> and <math>w^2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical. The experimental design problem then reads

<p>
<math>
 \begin{array}{lll}
 \displaystyle \min_{x,G,F,z^1,z^2,u,w^1,w^2} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
 \text{subject to} \\
\quad \dot{x_1}(t) & = &  p_1 \; x_1(t) - p_2 x_1(t) x_2(t) - p_5 u(t) x_1(t),\\
\quad \dot{x_2}(t) & = &  - p_3 \; x_2(t) + p_4 x_1(t) x_2(t) - p_6 u(t) x_2(t),\\
\quad \dot{G_{11}}(t) & = & f_{x11}(\cdot) \; G_{11}(t) + f_{x12}(\cdot) \; G_{21}(t) + f_{p12}(\cdot), \\
\quad \dot{G_{12}}(t) & = & f_{x11}(\cdot) \; G_{12}(t) + f_{x12}(\cdot) \; G_{22}(t), \\
\quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\
\quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\
\quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{21}(t)^2, \\
\quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{21}(t) G_{22}(t), \\
\quad \dot{F_{22}}(t) & = & w^1(t) G_{12}(t)^2 + w^2(t) G_{22}(t)^2, \\
\quad \dot{z^1}(t) & = & w^1(t), \\
\quad \dot{z^2}(t) & = & w^2(t), \\[1.5ex]
\quad x(0) &=& (0.5, 0.7), \\
\quad G(0) &=& F(0) = 0, \\
\quad z^1(0) &=& z^2(0) = 0, \\ [1.5ex]
\quad u(t) & \in & \mathcal{U}, \; w^1(t) \in \mathcal{W}, \; w^2(t) \in \mathcal{W}, \\
\quad 0    & \le & M - z(t_f)
  \end{array}
</math>
</p>

with 
<math>f_{x11}(\cdot) = \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t)</math>, 
<math>f_{x12}(\cdot) = - p_2 x_1(t)</math>, 
<math>f_{x21}(\cdot) = p_4 x_2(t)</math>, 
<math>f_{x22}(\cdot) = -p_3 + p_4 x_1(t) - p_6 u(t)</math>, and 
<math>f_{p12}(\cdot) = \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t)</math>, 
<math>f_{p24}(\cdot) = \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t)</math>.  

Note that the state <math>F_{21}(\cdot) = F_{12}(\cdot)</math> has been left out for reasons of symmetry.

== Parameters ==

We use <math>t_f=12</math>, <math>p_1 = p_2 = p_3 = p_4 = 1</math>, and <math>p_5 = 0.4</math>, <math>p_6 = 0.2</math>. The upper bound on the measurement time intervals is chosen as <math>M=4</math>.

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 Image:vplotkaUStates.png| States and fishing control.
 Image:vplotkaUSensitivities.png| Sensitivities G().
 Image:vplotkaUMeas1.png| Sampling function for first state.
 Image:vplotkaUMeas2.png| Sampling function for second state.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:AMPL | AMPL]] at [[Lotka Experimental Design (AMPL)]]
* [[:Category: VPLAN | VPLAN code]] at [[Lotka Experimental Design (VPLAN)]]

== Variants ==

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

* a prescribed time grid for the control function <bib id="Sager2006" />, see also [[Lotka Experimental Design (AMPL)]],
* no fishing, i.e., <math>u \equiv 0</math>,
* 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 <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />

<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]