untitled

(Fig. 8.13), so the population declines rapidly. Repetition of this boom–bust pattern

of overshooting the carrying capacity and subsequent decline to levels below the

carrying capacity results in the erratic fluctuations of deterministic chaos seen in

Fig. 8.17. For lower rates of increase (2.0 <rmax<2.7) the pattern of fluctuation

would be regular cycles, rather than deterministic chaos, but the underlying cause

is still overcompensation.

The underlying cause of instability due to overcompensatory density dependence

can be appreciated better by plotting the population dynamics over time on a graph

with Nton the horizontal axis and Nt+ 1 on the vertical axis (Fig. 8.18). The diagonal

identifies potential points of equilibria, at which Nt+ 1 =Nt. We will also plot the recruit-

ment curve. Dynamics are plotted by starting at a particular value of N 0 , projecting

upwards to the recruitment curve, that identifies the next year’s population density.

Then we project horizontally to the broken equilibrium line, before repeating the

process. At modest values of rmax, the recruitment curve is low and has a shallow

angle of incidence as it intersects the equilibrium line. The result is that the

population trajectory becomes pinched between the recruitment curve and the equi-

librium line as it converges on K. This leads to stability.

126 Chapter 8

300

200

100

0

0 5 10 15 20

t

Population density

Initial N = 2.0

Initial N = 2.1

Fig. 8.17Simulated
dynamics over time of
two different populations
growing according to the
Ricker logistic equation,
with rmax=3.3 and K=

100. The first population
     was initiated at a density
     of 2.0 individuals per unit
     area, whereas the second
     population was initiated
     at a slightly higher density
     of 2.1 individuals per unit
     area. The rapid divergence
     in population dynamics due
     to slight changes in starting
     conditions is typical of
     deterministic chaos.

300

200

100

0

0 50 100 150 200 250 300

N

t+1

Nt

Fig. 8.18Plot of
predicted recruitment
(Nt+ 1 ) relative to Nt
(the heavy curve),
equilibrium line at
which Nt+ 1 =Nt(thin
broken line), and
trajectory of population
dynamics over time for
a simulated population
following the Ricker
logistic model, with rmax
=1.3 and K=100 (thin
solid line).