# Lotka Volterra multi-arcs problem

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Lotka Volterra multi-arcs problem
State dimension: 1
Differential states: 3
Discrete control functions: 1
Interior point equalities: 3

This site describes a Lotka Volterra variant with three singular arcs instead of only one in the standard variant.

## Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{llclr} \displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex] \mbox{s.t.} & \dot{x}_0 & = & x_0 - x_0 x_1 - \; c_0\; x_0 \; w, \\ & \dot{x}_1 & = & - x_1 + x_0 x_1 - \; c_1\; x_1 \; w, \\ & \dot{x}_2 & = & (x_0 - p_0(t))^2 + (x_1 - p_1(t))^2, \\[1.5ex] & x(0) &=& (0.5, 0.7, 0)^T, \\ & w(t) &\in& \{0, 1\}, \end{array}$

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.

$\begin{array}{rcl}[t_{0},t_{f}]&=&[0,18],\\(c_{0},c_{1},)&=&(0.4,0.2),\\p_{{0}}(t)&=&\left\{{\begin{array}[ll]1&{\text{for }}t\in [0,7.2]\end{array}}\right.\\p_{1}(t)&=&p_{0}(t).\end{array}$

## Reference Solutions

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

The optimal objective value of the relaxed problem with $n_t=12000, \, n_u=400$ is $x_2(t_f) =1.82875272$. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $x_2(t_f) =1.82878681$.