LV Competitive - Revision history

RobertLampel: /* Parameters */

2026-01-29T10:05:04Z

Parameters

← Older revision		Revision as of 10:05, 29 January 2026
Line 31:		Line 31:
	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	[t_0, t_f] &=& [0, 20],\\		[t_0, t_f] &=& [0, 40],\\
	(c_{1}, c_{2}) &=& (0.1, 0.4),\\		(c_{1}, c_{2}) &=& (0.1, 0.4),\\
	x_0 &=& (0.5, 1.5) \text{ or } (1.5, 0.5),\\		x_0 &=& (0.5, 1.5) \text{ or } (1.5, 0.5),\\

RobertLampel: /* Mathematical formulation */

2026-01-29T09:58:37Z

Mathematical formulation

← Older revision		Revision as of 09:58, 29 January 2026
Line 17:		Line 17:
	\mbox{s.t.}		\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}_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, \\		& x(0) &=& x_0, \\
	& u(t) &\in& [0,1], \\		& u(t) &\in& [0,1], \\

RobertLampel: /* Mathematical formulation */

2026-01-29T09:49:19Z

Mathematical formulation

← Older revision		Revision as of 09:49, 29 January 2026
Line 14:		Line 14:
	<math>		<math>
	\begin{array}{llclr}		\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.}		\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}_0(t) & = & x_0(t) \left(1 - \frac{x_0(t) + \alpha x_1(t)}{K} \right) - c_1 x_0(t) u(t), \\

RobertLampel: /* Reference Solutions */

2025-11-28T10:45:45Z

Reference Solutions

← Older revision		Revision as of 10:45, 28 November 2025
Line 41:		Line 41:
	== Reference Solutions ==		== 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_Comp_init_1.png\| Local optimum for a direct approach and start values <math>x_0 = (0.5, 1.5)</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_Comp_init_2.png\| Local optimum for a direct approach and start values <math>x_0 = (1.5, 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>.

RobertLampel: /* Reference Solutions */

2025-09-04T06:30:51Z

Reference Solutions

← Older revision		Revision as of 06:30, 4 September 2025
Line 42:		Line 42:

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:LV_Comp_init_1.png\| Local optimum a direct approach ~~for~~ start values <math>x_0 = (0.5, 1.5)</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_Comp_init_2.png\| Local optimum a direct approach ~~for~~ start values <math>x_0 = (1.5, 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>		</gallery>

RobertLampel at 08:00, 25 August 2025

2025-08-25T08:00:29Z

← Older revision		Revision as of 08:00, 25 August 2025
Line 1:		Line 1:
	{{Dimensions		{{Dimensions
	\|nd = 1		\|nd = 1
	\|nx = 3		\|nx = 2
	\|nw = 1		\|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 ==		== Mathematical formulation ==

RobertLampel: /* Reference Solutions */

2025-08-25T07:56:20Z

Reference Solutions

← Older revision		Revision as of 07:56, 25 August 2025
Line 42:		Line 42:

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" 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 a direct approach for 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 a direct approach for start values <math>x_0 = (1.5, 0.5)</math>.
	</gallery>		</gallery>

RobertLampel: /* Parameters */

2025-08-25T07:55:33Z

Parameters

← Older revision		Revision as of 07:55, 25 August 2025
Line 33:		Line 33:
	[t_0, t_f] &=& [0, 20],\\		[t_0, t_f] &=& [0, 20],\\
	(c_{1}, c_{2}) &=& (0.1, 0.4),\\		(c_{1}, c_{2}) &=& (0.1, 0.4),\\
	x_0 &=& (1.5, 0.5~~, 1~~) \text{ or } (1.5~~, 1~~, 0.5),\\		x_0 &=& (0.5, 1.5) \text{ or } (1.5, 0.5),\\
	\alpha &=& 1.2.		\alpha &=& 1.2 \\
			K &=& 1.8.
	\end{array}		\end{array}
	</math>		</math>

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 3 |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 ODE model. == Mathematical formulation == The optimal control problem is given by
$\begin{array}{llclr} \displaystyle \min_{u} & \int_0^{t_f} && (x_0(t) - 1)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)..."$

2025-08-25T07:34:29Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |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 ODE model. == Mathematical formulation == The optimal control problem is given by <p> <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)..."

New page

{{Dimensions
|nd = 1
|nx = 3
|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]].

== Mathematical formulation ==

The optimal control problem is given by

<p>
<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]
\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]
& x(0) &=& x_0, \\
& u(t) &\in& [0,1], \\
& \alpha &>& 1.
\end{array}
</math>
</p>

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 20],\\
(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}
</math>

== Reference Solutions ==

<gallery caption="Reference solution plots" widths="180px" heights="140px" 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_Shared_init_2.png| Local optimum a direct approach for start values <math>x_0 = (1.5, 1, 0.5)</math>.
</gallery>


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]