Lotka Volterra fishing problem

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

The Lotka Volterra fishing problem looks for an optimal fishing strategy to be performed on a fixed time horizon to bring the biomasses of both predator as prey fish to a prescribed steady state. The problem was set up as a small-scale benchmark problem. The well known Lotka Volterra equations for a predator-prey system have been augmented by an additional linear term, relating to fishing by man. The control can be regarded both in a relaxed, as in a discrete manner, corresponding to a part of the fleet, or the full fishing fleet.

It is thus an ODE model with a single integer control function. The interior point equality conditions fix the initial values of the differential states.

The optimal solution contains a singular arc, making the Lotka Volterra fishing problem an ideal candidate for benchmarking of algorithms.

Mathematical formulation

For $t \in [t_0, t_f]$ almost everywhere the mixed-integer optimal control problem is given by

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

Initial values and parameters

These fixed values are used within the model.

$\begin{array}{rcl} [t_0, t_f] &=& [0, 12],\\ (c_0, c_1) &=& (0.4, 0.2),\\ x_0 &=& (0.5, 0.7, 0)^T. \end{array}$

Reference Solutions

If the problem is relaxed, i.e., we demand $w(t) \in [0, 1]$ instead of $w(t) \in \{0,1\}$, the optimal solution can be determined by means of Pontryagins maximum principle. The optimal solution contains a singular arc, as can be seen in the plot of the optimal control. The two differential states and corresponding adjoint variables in the indirect approach are also displayed.

• 

  Optimal relaxed control determined by an indirect approach and by a direct approach on different control discretization grids.

• 

  Differential States and corresponding adjoint variables in the indirect approach.

Source Code

States

C

The differential equations in C code:

/* steady state with u == 0 */
double ref0 = 1, ref1 = 1;
 
/* Biomass of prey */
rhs[0] =   xd[0] - xd[0]*xd[1] - p[0]*u[0]*xd[0];
/* Biomass of predator */
rhs[1] = - xd[1] + xd[0]*xd[1] - p[1]*u[0]*xd[1];
/* Deviation from reference trajectory */
rhs[2] = (xd[0]-ref0)*(xd[0]-ref0) + (xd[1]-ref1)*(xd[1]-ref1);

AMPL

The model in AMPL code with a collocation method. We need a model file lotka_ampl.mod,

# ----------------------------------------------------------------
# Lotka Volterra fishing problem with collocation (explicit Euler)
# (c) Sebastian Sager
# ----------------------------------------------------------------
 
param T    > 0;
param nt   > 0;
param c1   > 0;
param c2   > 0;
param ref1 > 0;
param ref2 > 0;
param dt := T / (nt-1);
set I:= 0..nt;
 
var x {I, 1..2} >= 0;
var w {I} >= 0, <= 1 binary;
 
minimize Deviation:
	  0.5 * dt * ( (x[0,1] - ref1)*(x[0,1] - ref1) + (x[0,2] - ref2)*(x[0,2] - ref2) )
	+ 0.5 * dt * ( (x[nt,1] - ref1)*(x[nt,1] - ref1) + (x[nt,2] - ref2)*(x[nt,2] - ref2) )
	+ dt * sum {i in I diff {0,nt} } ( (x[i,1] - ref1)*(x[i,1] - ref1) 
	                                 + (x[i,2] - ref2)*(x[i,2] - ref2) ) ;
 
subj to ODE_DISC_1 {i in I diff {0}}:
   x[i,1] = x[i-1,1] + dt * ( x[i-1,1] - x[i-1,1]*x[i-1,2] - x[i-1,1]*c1*w[i-1] );
 
subj to ODE_DISC_2 {i in I diff {0}}:
   x[i,2] = x[i-1,2] + dt * ( - x[i-1,2] + x[i-1,1]*x[i-1,2] - x[i-1,2]*c2*w[i-1] );

a data file lotka_ampl.dat,

# ------------------------------------
# Data: Lotka Volterra fishing problem
# ------------------------------------
 
# Algorithmic parameters
param nt := 100;
 
# Problem parameters
param T := 12.0;
param c1 := 0.4;
param c2 := 0.2;
param ref1 := 1.0;
param ref2 := 1.0;
 
# Initial values differential states
let x[0,1] := 0.5;
let x[0,2] := 0.7;
fix x[0,1];
fix x[0,2];
 
# Initial values control
let {i in I} w[i] := 0.0;
fix w[nt];
 
param mysum;

and a running script lotka_ampl.run,

# ----------------------------------------------------
# Solve Lotka Volterra fishing problem via collocation
# ----------------------------------------------------
 
model ampl_lotka.mod;
data ampl_lotka.dat;
 
option solver bonmin;
 
solve;

Miscellaneous and Further Reading

The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bibref>Sager2006</bibref> and revisited in his PhD thesis <bibref>Sager2005</bibref>. These are also the references to look for more details.

References

<bibreferences/>