LV Shared Resource

LV Shared Resource
State dimension:	1
Differential states:	3
Discrete control functions:	1

This Lotka Volterra problem with explicit inclusion of a shared resource is a variant of the Lotka Volterra fishing problem. Its dynamics are given via a three-dimensional ODE model.

Mathematical formulation

The optimal control problem is given by

$\begin{array}{llcl}\underset{u}{\min }& \underset{0}{\overset{t_f}{\int }}& & (x_0(t)-1.5{)}^2+(x_1(t)-1{)}^2+(x_2(t)-1{)}^{2}dt\\ \text{s.t.}& {\dot{x}}_0(t)& =& x_0(t)-x_0(t)x_1(t)-x_0(t)x_2(t),\\ & {\dot{x}}_1(t)& =& -x_1(t)+x_0(t)x_1(t)-c_1{x}_1(t)u(t),\\ & {\dot{x}}_2(t)& =& -x_2(t)+\alpha x_0(t)x_2(t)-c_2{x}_2(t)u(t),\\ & x(0)& =& x_0,\\ & u(t)& \in & [0,1],\\ & \alpha & >& 1.\end{array}$

Parameters

These fixed values are used within the model.

$\begin{array}{rcl}[t_0,t_f]& =& [0,40],\\ (c_1,c_2)& =& (0.1,0.4),\\ x_0& =& (1.5,0.5,1)\text{ or }(1.5,1,0.5),\\ \alpha & =& 1.2.\end{array}$

Reference Solutions

• Reference solution plots
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  Local optimum a direct approach for start values $x_0=(1.5,0.5,1)$.
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  Local optimum a direct approach for start values $x_0=(1.5,1,0.5)$.