LV Competitive: Difference between revisions

LV Competitive
State dimension:	1
Differential states:	2
Discrete control functions:	1

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Latest revision as of 10:05, 29 January 2026

This Competitive Lotka Volterra problem is a variant of the Lotka Volterra fishing problem. Its dynamics are given via a two-dimensional ODE model.

Mathematical formulation

The optimal control problem is given by

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

Parameters

These fixed values are used within the model.

$\begin{array}{rcl} [t_{0}, t_{f}] & = & [0, 40], \\ (c_{1}, c_{2}) & = & (0.1, 0.4), \\ x_{0} & = & (0.5, 1.5) or (1.5, 0.5), \\ α & = & 1.2 \\ K & = & 1.8 . \end{array}$

Reference Solutions

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

@@ Line 1: / Line 1: @@
 {{Dimensions
 |nd        = 1
-|nx        = 3
+|nx        = 2
 |nw        = 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 [[:Category:ODE model|ODE model]].
+This '''Competitive Lotka Volterra problem''' is a variant of the [[:Lotka Volterra fishing problem]]. Its dynamics are given via a two-dimensional [[:Category:ODE model|ODE model]].
 == Mathematical formulation ==
@@ Line 14: / Line 14: @@
 <math>
 \begin{array}{llclr}
-  \displaystyle \min_{u} & \int_0^{t_f} && (x_0(t) - 1)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)^2 \ dt \\[1.5ex]
+  \displaystyle \min_{u} & \int_0^{t_f} && (x_0(t) - 1)^2 + (x_1(t) - 1)^2 \ dt \\[1.5ex]
   \mbox{s.t.}
   & \dot{x}_0(t) & = &  x_0(t) \left(1 - \frac{x_0(t) + \alpha x_1(t)}{K} \right) - c_1 x_0(t) u(t), \\
-  & \dot{x}_1(t) & = &  x_1(t) + \left(1 - \frac{x_0(t) + x_1(t)}{K} \right) - c_2 x_1(t) u(t),  \\[1.5ex]
+  & \dot{x}_1(t) & = &  x_1(t) \left(1 - \frac{x_0(t) + x_1(t)}{K} \right) - c_2 x_1(t) u(t),  \\[1.5ex]
   & x(0) &=& x_0, \\
   & u(t) &\in& [0,1], \\
@@ Line 31: / Line 31: @@
 <math>
 \begin{array}{rcl}
-[t_0, t_f] &=& [0, 20],\\
+[t_0, t_f] &=& [0, 40],\\
 (c_{1}, c_{2}) &=& (0.1, 0.4),\\
 x_0 &=& (0.5, 1.5) \text{ or } (1.5, 0.5),\\
@@ Line 41: / Line 41: @@
 == Reference Solutions ==
-<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
+<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>.
+  Image:LV_Comp_init_1.png| Local optimum for a direct approach and start values <math>x_0 = (0.5, 1.5)</math>.
-  Image:LV_Shared_init_2.png| Local optimum a direct approach for start values <math>x_0 = (1.5, 1, 0.5)</math>.
+  Image:LV_Comp_init_2.png| Local optimum for a direct approach and start values <math>x_0 = (1.5, 0.5)</math>.
 </gallery>