Lotka Volterra fishing problem

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 dimensions:

$\begin{array}{rcl} n_{x} & = & 3 \\ n_{z} & = & 0 \\ n_{u} & = & 0 \\ n_{w} & = & 1 \\ n_{p} & = & 0 \\ n_{ρ} & = & 0 \\ n_{c} & = & 0 \\ n_{r^{i}} & = & 0 \\ n_{r^{e}} & = & 3 \end{array}$

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

Mathematical formulation

For $t \in [t_{0}, t_{f}]$ the mixed-integer optimal control problem is given by

$\begin{array}{llcl} \min_{x, w} & x_{2} (t_{f}) \\ s.t. & {\dot{x}}_{0} (t) & = & x_{0} (t) - x_{0} (t) x_{1} (t) - c_{0} x_{0} (t) w (t), \\ {\dot{x}}_{1} (t) & = & - x_{1} (t) + x_{0} (t) x_{1} (t) - c_{1} x_{1} (t) w (t), \\ {\dot{x}}_{2} (t) & = & (x_{0} (t) - 1)^{2} + (x_{1} (t) - 1)^{2}, \\ x (0) & = & x_{0}, \\ w (t) & \in & {0, 1} . \end{array}$

Initial values and parameters

$\begin{array}{rcl} t_{0} & = & 0 \\ t_{f} & = & 12 \\ c_{0} & = & 0.4 \\ c_{1} & = & 0.2 \\ x_{0} & = & (0.5, 0.7, 0)^{T} \end{array}$

Reference Solutions

The two differential states and corresponding adjoint variables in the indirect approach

Source Code

  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);

Miscellaneous

Testing Graphviz

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

Testing Syntax Highlighting

// Hello World in Microsoft C# ("C-Sharp").

using System;

class HelloWorld
{
    public static int Main(String[] args)
    {
        Console.WriteLine("Hello, World!");
        return 0;
    }
}

External references