The name is absent

3.3 Non-cooperative Equilibria

In the absence of management, however, each community maximizes its own
utility according to its welfare function, and therefore the agricultural commu-
nities each

s.t. x(t) = a(t) — bx(t) H—x (^)— = 0,
x² (t) + 1

n1           n2

^a(t) = £ ai(^t) + £ aj^(t)
i=1         j=1

and the green communities each

x² (t)

X(t) = a(t) — bx(t) + 2 ^l = 0,
x² (t) + 1

n1           n2

^a(t) = £ ai(^t) ⁺ £ aj^(t)

i=1         j=1

Setting up a current value Hamiltonian and solving first order conditions for
these two problems yields:

Solving for constant phosphorous loading, that is, dai/dt = 0 and daj /dt = 0,
one obtains the steady-state open-loop Nash equilibrium for total loading a:

3.4 Graph and Analysis of the Nash Equilibrium Solution

Again, this solution is 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 the Nash equilibrium phosphorous loading solu-
tions for the state of the lake. Using the hysteretic lake value b = 0.6, ρ = 0.03
and n1 = 2, n2 = 2, c1 = 0.2 and c2 = 2, the curves intersect at (0.4485, 0.1016),
(0.7402, 0.0902) and (3.1832, 0.9976), as show in Figure 2. Point (0.7402, 0.0902)
is an unstable skiba point, i.e., a small variation in loading will cause the equi-
librium to shift to either the lower equilibrium at (0.4485, 0.1016) or the higher

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