Lotka Volterra fishing problem

2008-07-06T18:30:48Z

88.64.190.171: /* Miscellaneous */

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This problem was set up as a simple benchmark problem. Despite of its simple structure, the optimal solution contains a singular arcs, making the Lotka Volterra fishing problem an ideal candidate for benchmarking of algorithms.

In this problem the Lotka Volterra equations for a predator-prey system have been augmented by an additional linear term, relating to fishing by man.

== Model dimensions and properties ==

The model has the following [[model dimensions|dimensions]]:

<math>
\begin{array}{rcl}
n_x &=& 3\\
n_z &=& 0\\
n_u &=& 0\\
n_w &=& 1\\
n_p &=& 0\\
n_{\rho} &=& 0\\
n_c &=& 0\\
n_{r^\mathrm{i}} &=& 0\\
n_{r^\mathrm{e}} &=& 3
\end{array}
</math>

It is thus an [[ordinary differential equation|ODE]] model with a single integer control function. The interior point equality conditions fix the initial values of the differential states.

== Mathematical formulation ==

For <math>t \in [t_0, t_f]</math> the mixed-integer optimal control problem is given by

<math>
\begin{array}{llcl}
\displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex]
\mbox{s.t.} & \dot{x}_0(t) & = & x_0(t) - x_0(t) x_1(t) - \; c_0 x_0(t) \; w(t), \\
& \dot{x}_1(t) & = & - x_1(t) + x_0(t) x_1(t) - \; c_1 x_1(t) \; w(t), \\
& \dot{x}_2(t) & = & (x_0(t) - 1)^2 + (x_1(t) - 1)^2, \\[1.5ex]
& x(0) &=& x_0, \\
& w(t) &\in& \{0, 1\}.
\end{array}
</math>

== Initial values and parameters ==

<math>
\begin{array}{rcl}
t_0 &=& 0\\
t_f &=& 12\\
c_0 &=& 0.4\\
c_1 &=& 0.2\\
x_0 &=& (0.5, 0.7, 0)^T
\end{array}
</math>

== Reference Solutions ==
<div style="float:right;text-align:center;padding-left:10px">
[[Image:lotkaindirektStates.png|thumb|340px|States]]
<br />
''The two differential states and corresponding adjoint variables in the indirect approach''</div>

== Source Code ==

<source lang="cpp">
double ref0 = 1, ref1 = 1; /* steady state with u == 0 */

rhs[0] = xd[0] - xd[0]*xd[1] - p[0]*u[0]*xd[0];
rhs[1] = - xd[1] + xd[0]*xd[1] - p[1]*u[0]*xd[1];
rhs[2] = (xd[0]-ref0)*(xd[0]-ref0) + (xd[1]-ref1)*(xd[1]-ref1);
</source>

== Miscellaneous ==
Testing Graphviz

<graphviz border='frame' format='svg'>
digraph G {Hello->World!}
</graphviz>

Testing bibwiki
<bibref>Sager2006</bibref>

== External references ==

[[Category:ODE Model]]

mintOC - User contributions [en]

Lotka Volterra fishing problem