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

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

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

/* steady state with u == 0 */
double ref0 = 1, ref1 = 1;
 
/* Biomass of prey */
rhs[0] =   xd[0] - xd[0]*xd[1] - p[0]*u[0]*xd[0];
/* Biomass of predator */
rhs[1] = - xd[1] + xd[0]*xd[1] - p[1]*u[0]*xd[1];
/* Deviation from reference trajectory */
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

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