LV Shared Resource: Difference between revisions

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

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Latest revision as of 13:43, 28 November 2025

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

$\begin{array}{llclr} \min_{u} & \int_{0}^{t_{f}} & (x_{0} (t) - 1.5)^{2} + (x_{1} (t) - 1)^{2} + (x_{2} (t) - 1)^{2} d t \\ s.t. & {\dot{x}}_{0} (t) & = & x_{0} (t) - x_{0} (t) x_{1} (t) - x_{0} (t) x_{2} (t), \\ {\dot{x}}_{1} (t) & = & - x_{1} (t) + x_{0} (t) x_{1} (t) - c_{1} x_{1} (t) u (t), \\ {\dot{x}}_{2} (t) & = & - x_{2} (t) + α x_{0} (t) x_{2} (t) - c_{2} x_{2} (t) u (t), \\ x (0) & = & x_{0}, \\ u (t) & \in & [0, 1], \\ α & > & 1 . \end{array}$

Parameters

These fixed values are used within the model.

$\begin{array}{rcl} [t_{0}, t_{f}] & = & [0, 40], \\ (c_{1}, c_{2}) & = & (0.1, 0.4), \\ x_{0} & = & (1.5, 0.5, 1) or (1.5, 1, 0.5), \\ α & = & 1.2 . \end{array}$

Reference Solutions

Reference solution plots
Local optimum a direct approach for start values $x_{0} = (1.5, 0.5, 1)$ .
Local optimum a direct approach for start values $x_{0} = (1.5, 1, 0.5)$ .

@@ Line 2: / Line 2: @@
 |nd        = 1
 |nx        = 3
-|nw        = 5
+|nw        = 1
-|nre       = 3
 }}
-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 [[Category:OED_model]].
+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 [[:Category:ODE model|ODE model]].
 == Mathematical formulation ==
-The mixed-integer optimal control problem is given by
+The optimal control problem is given by
 <p>
 <math>
 \begin{array}{llclr}
-  \displaystyle \min_{x, w} & x_2(t_f)   \\[1.5ex]
+  \displaystyle \min_{u} & \int_0^{t_f} && (x_0(t) - 1.5)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)^2 \ dt \\[1.5ex]
   \mbox{s.t.}
-  & \dot{x}_0 & = &  x_0 - x_0 x_1 - \; \sum\limits_{i=1}^{5} c_{0,i}\;  w_i, \\
+  & \dot{x}_0(t) & = &  x_0(t) - x_0(t) x_1(t) - x_0(t) x_2(t), \\
-  & \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{5} c_{1,i}\;  w_i,  \\
+  & \dot{x}_1(t) & = & - x_1(t) + x_0(t) x_1(t) - c_1 x_1(t) u(t),  \\
-  & \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2,  \\[1.5ex]
+  & \dot{x}_2(t) & = & -x_2(t) + \alpha x_0(t) x_2(t) - c_2 x_2(t) u(t),  \\[1.5ex]
-  & x(0) &=& (0.5, 0.7, 0)^T, \\
+  & x(0) &=& x_0, \\
-  & \sum\limits_{i=1}^{5}w_i(t) &=& 1, \\
+  & u(t) &\in& [0,1], \\
-  & w_i(t) &\in&  \{0, 1\}, \quad i=1\ldots 5.
+  & \alpha &>&  1.
 \end{array}
 </math>
 </p>
-Here the differential states <math>(x_0, x_1)</math> 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 <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between five different fishing options.
 == Parameters ==
@@ Line 35: / Line 32: @@
 <math>
 \begin{array}{rcl}
-[t_0, t_f] &=& [0, 12],\\
+[t_0, t_f] &=& [0, 40],\\
-(c_{0,1}, c_{1,1}) &=& (0.2, 0.1),\\
+(c_{1}, c_{2}) &=& (0.1, 0.4),\\
-(c_{0,2}, c_{1,2}) &=& (0.4, 0.2),\\
+x_0 &=& (1.5, 0.5, 1) \text{ or } (1.5, 1, 0.5),\\
-(c_{0,3}, c_{1,3}) &=& (0.01, 0.1),\\
+\alpha &=& 1.2.
-(c_{0,4}, c_{1,4}) &=& (0, 0),\\
-(c_{0,5}, c_{1,5}) &=& (-0.1, -0.2).
 \end{array}
 </math>
@@ Line 46: / Line 41: @@
 == Reference Solutions ==
-If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, the optimal solution can be determined by means of direct optimal control.
+<gallery caption="Reference solution plots" widths="400px" heights="240px" perrow="2">
+  Image:LV_Shared_init_1.png| Local optimum a direct approach for start values <math>x_0 = (1.5, 0.5, 1)</math>.
-The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=150  </math> is <math>x_2(t_f) =0.345768563</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>x_2(t_f) =0.348617982</math>.
+  Image:LV_Shared_init_2.png| Local optimum a direct approach for start values <math>x_0 = (1.5, 1, 0.5)</math>.
-<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
-  Image:Lotka_abs_fish_relaxed_12000_80.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=150</math>.
-  Image:Lotka_abs_fish_CIA_states_12000_80.pdf| Differential states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
- Image:Lotka_abs_fish_CIA_controls_12000_80.pdf| Binary control determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation. The fishing control shows a lot of chattering.
 </gallery>
@@ Line 67: / Line 51: @@
 [[Category:ODE model]]
 [[Category:Tracking objective]]
-[[Category:Chattering]]
 [[Category:Sensitivity-seeking arcs]]
 [[Category:Population dynamics]]