Lotka Volterra fishing problem

Lotka Volterra fishing problem
State dimension:	1
Differential states:	3
Algebraic states:	0
Continuous control functions:	0
Discrete control functions:	1
Continuous control values:	0
Discrete control values:	0
Path constraints:	0
Interior point inequalities:	0
Interior point equalities:	3

The Lotka Volterra fishing problem looks for an optimal fishing strategy to be performed on a fixed time horizon to bring the biomasses of both predator as prey fish to a prescribed steady state. The problem was set up as a small-scale benchmark problem. The well known Lotka Volterra equations for a predator-prey system have been augmented by an additional linear term, relating to fishing by man. The control can be regarded both in a relaxed, as in a discrete manner, corresponding to a part of the fleet, or the full fishing fleet.

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.

The optimal solution contains a singular arc, making the Lotka Volterra fishing problem an ideal candidate for benchmarking of algorithms.

Mathematical formulation

For $t\in [t_0,t_f]$ almost everywhere the mixed-integer optimal control problem is given by

$\begin{array}{ll}\underset{x,w}{\min }& x_2(t_f)\\ \text{s.t.}& {\dot{x}}_0(t)& =& x_0(t)-x_0(t)x_1(t)-c_0{x}_0(t)w(t),\\ & {\dot{x}}_1(t)& =& -x_1(t)+x_0(t)x_1(t)-c_1{x}_1(t)w(t),\\ & {\dot{x}}_2(t)& =& (x_0(t)-1{)}^2+(x_1(t)-1{)}^2,\\ & x(0)& =& x_0,\\ & w(t)& \in & \{0,1\}.\end{array}$

Initial values and parameters

These fixed values are used within the model.

$\begin{array}{rcl}[t_0,t_f]& =& [0,12],\\ (c_0,c_1)& =& (0.4,0.2),\\ x_0& =& (0.5,0.7,0{)}^T.\end{array}$

Reference Solutions

States

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

The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bibref>Sager2006</bibref> and revisited in his PhD thesis <bibref>Sager2005</bibref>. These are also the references to look for more details.

References

<bibreferences/>