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AN INTRODUCTION TO DYNAMICAL SYSTEMS APPLIED TO EXPLOITED
POPULATIONS
THE FIGURES IN THE ARTICLE
Logistic Curve
dn/dt
400
dn/dt=rn(1-n/K)
MSY
carrying
capacity
0
0
nmsy
K
Figure 1. corresponds to (2.2). The natural growth of the population, dn/dt is
shown as a function of n. The growth of the population is positive whenever
it is less than K (dn/dt>0 if 0<n<K). K is the environmental carrying capacity
and r the intrinsic growth rate. The parabola reaches its maximum value,
MSY when n=nmsy. MSY = rK/4 and nmsy = K/2.
300
n
logistic curve with constant harvest
500
H>rK/4
H,
dn/dt
H=rK/4
H<rK/4
0
0
A
K/2
C
K
300
Figure 2. corresponds to (3.1.1). The net growth of the population, dn/dt is shown as a
function of n. K is the environmental carrying capacity and r the intrinsic growth rate.
The parabola reaches its maximum when n = K/2. The parabola is the natural growth
of the population while the horizontal lines are the constant yields, H at three different
predating levels. Where the line lies above the parabola; the net growth rate is
negative. Where the line lies below the parabola, the net growth rate is positive. The
points of intersection correspond to possible equilibria and the arrows indicate the
stability of the equilibria.
logistic curve with Type I
Figure 3. corresponds to (3.2.2). The net growth of the population, dn/dt is shown as a
function of n. K is the environmental carrying capacity, r is the intrinsic growth rate, e
is the effort and q is the capturability. The parabola is the natural growth of the
population while lines are loss due to predation using constant effort, h=qe(t)n. When
e(t) is increased, h1 moves to the left to h2. Where the line lies above the parabola; the
net growth rate is negative. Where the line lies below the parabola, the net growth rate
is positive. The points of intersection correspond to possible equilibria and the arrows
indicate the stability of the equilibria.
n
Logistic curve with Type II
600
h5
H(n),
dn/dt
h4
h3
h2
h1
K
0
0
260
n
Figure 4. corresponds to (3.3.2) and (3.3.3). The net growth of the population, dn/dt
is shown as a function of n. K is the environmental carrying capacity and H(n) is
Holling Type II functional response. The points of intersection correspond to the
possible equilibria. Where the parabola is above the curve, the net growth rate is
positive (the flow along the x-axis is to the right). Where the curve is above the
parabola, the net growth rate is negative (the flow along the x-axis is to the left).
h(1),h(2),h(3),h(4),h(5) are predation levels with a =0.12 and η =0.048, 0.136, 0.224,
0.312 and 0.48 respectively.
Logistic curve with Type III
350
H(n),
dn/dt
H5
H4
H3
H2
H1
0
0
K
260
n
n
Figure 5. corresponds to (3.4.2) and (3.4.3). The net growth of the population, dn/dt is shown as
a function of n. K is the environmental carrying capacity and H(n) is Holling Type III functional
response. The points of intersection correspond to the possible equilibria. Where the parabola is
above the curve, the net growth rate is positive (the flow along the x-axis is to the right). Where
the curve is above the parabola, the net growth rate is negative (the flow along the x-axis is to
the left). H(1),H(2),H(3),H(4),H(5) are predation levels with a =1.2 and η =0.12, 0.24 , 0.36,
0.48 and 0.64 respectively.
logistic curve with Type III
500
H(n),
dn/dt
h5
h4
h3
h2
h1
0
0
260
K
Figure 6. corresponds to (3.4.2) and (3.4.3). The net growth of the population, dn/dt is shown as
a function of n. K is the environmental carrying capacity and H(n) is Holling Type III functional
response. The points of intersection correspond to the possible equilibria. Where the parabola is
above the curve, the net growth rate is positive (the flow along the x-axis is to the right). Where
the curve is above the parabola, the net growth rate is negative (the flow along the x-axis is to
the left). h(1),h(2),h(3),h(4),h(5) are predation levels with a =0.12 and η =0.048, 0.136, 0.224,
0.312 and 0.48 respectively.
n