Lotka Volterra absolute fishing problem

Lotka Volterra absolute fishing problem
State dimension:	1
Differential states:	3
Discrete control functions:	5
Interior point equalities:	3

This site describes a Lotka Volterra variant with five binary controls that all represent fishing of an absolute biomass.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{ll}\underset{x,w}{\min }& x_2(t_f)\\ \text{s.t.}& {\dot{x}}_0& =& x_0-x_0{x}_1-\sum _{i=1}^5{c}_{0,i}{w}_i,\\ & {\dot{x}}_1& =& -x_1+x_0{x}_1-\sum _{i=1}^5{c}_{1,i}{w}_i,\\ & {\dot{x}}_2& =& (x_0-1{)}^2+(x_1-1{)}^2,\\ & x(0)& =& (0.5,0.7,0{)}^T,\\ & \sum _{i=1}^5{w}_i(t)& =& 1,\\ & w_i(t)& \in & \{0,1\},i=1\dots 5.\end{array}$

Here the differential states $(x_0,x_1)$ describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation $\min x_2(t_f)$. This problem variant allows to choose between three different fishing options.

Parameters

These fixed values are used within the model.

$\begin{array}{rcl}[t_0,t_f]& =& [0,12],\\ (c_{0,1},c_{1,1})& =& (0.2,0.1),\\ (c_{0,2},c_{1,2})& =& (0.4,0.2),\\ (c_{0,3},c_{1,3})& =& (0.01,0.1),\\ (c_{0,4},c_{1,4})& =& (0,0),\\ (c_{0,5},c_{1,5})& =& (-0.1,-0.2).\end{array}$

Reference Solutions

If the problem is relaxed, i.e., we demand that $w(t)$ is in the continuous interval $[0,1]$ rather than being binary, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with $n_t=12000,n_u=150$ is $x_2(t_f)=0.345768563$. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is $x_2(t_f)=0.348617982$.

• Reference solution plots
• 
  Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=12000,n_u=400$.
• 
  Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=12000,n_u=400$. The relaxed controls were approximated by Combinatorial Integral Approximation.