The name is absent



which means a* = 0 or:

<i(t) =


--------x(t) (eɪnɪ + С2П2) -
n1 + n2


b+ρ-


2x(t)
(x
2(t) + 1)2


=0


And so the steady-state Pareto-optimal solution is:

(nɪ + n2) b + ρ -  25)

*
a*


(x2(t)+1)

(19)


2x(t) (eɪnɪ + n2c2)

3.2 Graph and Analysis of the Steady-state Dynamic Pareto-
optimal Solution

This solution can be plotted in the (x,a)-plane together with the phase plot for
the steady-states of the lake when dx/dt = 0, given by equation (3). The inter-
section of the two curves gives society’s optimal phosphorous loading solution.
Using the hysteretic lake value b = 0.6, ρ = 0.03, n
ɪ = 2, n2 = 2, c1 = 0.2 and
c
2 = 2, the graphs intersect at (x*, a*) = (0.3472, 0.1007), as shown in Figure 1
below, thus giving us the Pareto-optimal steady-state equilibrium. Note that
this result is below the point at which the lake flips from an oligotrophic to a
eutrophic state, i.e. for the selected constants, society prefers an oligotrophic
lake.

Figure 1: Pareto-optimal Loading with Two Welfare Functions


10




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