Testing the r' method of estimating per capita growth rate in
Aedes albopictus
Matthew Chmielewski, Camilo Khatchikian and Todd Livdahl
Department of Biology, Clark University, Worcester MA
Results
We used laboratory populations of the Asian Tiger Mosquito
Aedes albopictus to compare Livdahl and Sugihara’s (1984) method
of projecting population growth rate (r') with the derived population
growth rate (r) generated using life table methods. Additionally, we
tested r’ for possible density effects by raising populations of three
different larval densities. r’ was found to be a significant predictor of
trends in r, although accuracy was low. Additionally, r’ was not found
to be susceptible to density effects, supporting the robustness of this
method regardless of the level of larval competition.
The per capita rate of change was found to be an accurate
predictor of trends in instantaneous growth rate when r was
regressed on r′ (F2,28 = 9.14, p<0.01, Figure 1). Despite an
accurate correlation, r’ tended to overpredict r. Analysis of
covariance among density groups failed to show a significant
interaction between r′ and density (ANCOVA, F2,24=0.72, p=0.50),
indicating that the predictive capacity of r′ was robust to differences
in larval density conditions. Further support for the robustness of r′
as a predictor is indicated by no significant departure from the
overall model for any of the density groups after the interaction term
had been removed (ANCOVA, F2,26=0.27, p=0.76).
Regressions of r and r′ on density (n) were both significant (r:
F2,28=7.02, p=0.01; r′: F2,28=166.45, p<0.01, Figure 2). Once
regressed, the predicted values of the maximum per capita rate of
increase (r: 0.056; r’: 0.11) and carrying capacity (r: 21.35; r’: 34.74)
were found to be quite variable, indicating that r’ is prone to
producing inflated values in these populations statistics.
The net reproductive rate (Ro) generated using the r’ method
tended towards overpredicting r-derived Ro (Figure 3). Alternately, r’
tended to underpredict the cohort generation time () when
compared with r (Figure 4).
Livdahl and Sugihara’s1 method of projecting population growth
rate (r’),
0.1
0.15
[1]
x
Figure 1: r (derived population growth rate) regressed on r′
(projected population growth rate (y = 0.8831x - 0.0389, R2 =
0.246, F2,28 = 9.14, p<0.01). The diagonal line through the
origin represents the ideal correlation (1:1) between r and r′.
Those values below this line represent an overestimate of r by
r′, and those values above the line represent an underestimate.
x
x
x
F1 generation Aedes albopictus eggs (adult collection in
Bermuda) were hatched and the larvae subsequently sorted into
three different densities (low=5, medium=10, and high=15 individuals
30 mL-1), with ten replicates each. Adults were kept in 20 x 20 x 20
cm mesh cages and given the opportunity to blood feed every two d.
On feeding days, egg-trap liners were removed so that eggs could be
counted. Additionally, dead females were removed and wing lengths
were measured.
Age-specific survival fractions (lx) and fecundity values (mx)
were constructed in order to iteratively find the derived population
growth rate (r) using the Lotka-Euler equation2:
[2]
x
A projected population growth rate (eqn. [1]) was derived for
each replicate, and was regressed against the derived population
growth rate. An analysis of covariance was conducted to observe
possible density effects that might change the predictive capability of r’
in various density regimes. These two growth rate measures were
then regressed on larval density, allowing for a measure of potential
carrying capacity (K) and maximum growth rate (rmax) of the study
species.
The net reproductive value (Ro) and cohort generation time ()
for both methods of generating population growth rate were compared
with one another to determine where variation in the prediction of r by
r’ might be generated.
-1
Per capita rate of change (r) d
Methods
40
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
0
5
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15
-0.04
-0.06
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120
10
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0
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Per capita rate of change (r') d-1
allows for a less time consuming and less resource intensive method,
when compared to traditional life table (r) calculations, of
understanding population growth. Due to the accessibility of this
method, a number of studies have gone on to employ r’ as a measure
of population success. Despite the ease of use, it has been unclear
whether or not r’ might be sensitive to density effects.
We aimed to examine the applicability of r’ in predicting r in the
Asian Tiger Mosquito Aedes albopictus. In particular we were
interested in how well population growth rates generated using r’
methods reflect changes in population growth derived via life table
methods in the presence of multiple larval densities.
1   l x m x e  rx
0.05
-0.05
f (w x )
 xA f (w )
D
 A f (w )
x
0
Estimated per capita rate of
change (r') d-1
x
x
0
methods
x
0.05
Value of Ro derived using life table (r)
r' 
A
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10
15
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40
Value of Ro using r' methods
0.1
0.08
Figure 3: The net reproductive value (R0) derived from the life table
study regressed on the net reproductive value (R0) estimated from the
r′ calculations (y = 0.2016x + 8.184, R2 = 0.0103, F2,28=0.34, p=0.57).
Any values below the 45 degree x=y line represent an overestimation
of R0 by the r′ calculation. Any values above the line represent
underestimates.
0.06
0.04
0.02
References
0
0
5
10
15
Larval density (individuals 30mL-1)
Figure 2: r and r’ regressed by larval density (r: y = -0.0026x + 0.0555,
R2 = 0.2005, F2,28=7.02, p=0.01; r’: y= -0.0031x+0.1077, R2 = 0.856,
F2,28=166.45, p<0.01). The y intercept predicts rmax (r 0.056; r’ 0.11),
the maximum per capita rate of increase one could expect these
populations to exhibit. The x intercept predicts K (r 21.35; r’: 34.74), the
carrying capacity for this species under similar environmental
conditions.
Given the ability of r’ to correctly predict trends in r, this study
further supports the use of r’ as a method of predicting population
growth rates. The lack of accuracy in the values of r’ relative to r
can be further diagnosed by studying the r’ equation itself. In order
to determine why the r′ values were not accurate, it is important to
look at the equation for r′ in terms of numerator (Ro) and
denominator () driving the overall value of r′ .
The conjunction of an overestimate of Ro (Figure 3)and
underestimate of t (Figure 4) is the driving factor in the inflation of r′
relative to r. The overprediction of Ro most likely has to do with the
assumption that a given female will have a reproductive output that
can be predicted by her size. The amount of variation in egg laying
based on size may be large enough in this population to cause the
overprediction.
The underestimation of t may stem from the same issue, but
it may also come from the assumed value of D, the time lag
between emergence and the beginning of oviposition. The
assumed value of D may be too low, although it is more likely that D
is not constant with every population. Because of the consistency of
prediction across densities, it is tempting to suggest a simple
corrective calibration of r’ to adjust for the overestimation. We do
not advocate this because of the likely difference between survival
and reproductive rates under field and laboratory conditions. As
survival in laboratory conditions seems likely to exceed field
survival, and blood meal access in field conditions is likely to be
much more difficult, the lengthy period of reproduction observed in
this study may not be approached by field populations. As such, a
major source of error in r’ estimates could be reduced, and the
predictive ability of r’ could actually be improved under field
conditions.
1. Livdahl, T.P. and Sugihara, G. (1984) Non-linear interactions of
populations and the importance of estimating per capita rates of
change. Journal of Animal Ecology 53: 573-580.
2. Lotka, A. J. 1907. Studies on the mode of growth of material
aggregates. American Journal of Science 24:199-216.
Acknowledgments
We thank the Department of Biology, Clark University, the National
Institutes of Health (R15 AI062712-01) and the Keck Foundation for
supporting this project.
t
Per capita rate of change (r) d
Introduction
1
ln
N0
0.15
Discussion
Value of τ derived from life table (r)
methods
-1
Abstract
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t
Value of τ estimated using r' methods
Figure 4: The net cohort generation time (t) derived from the life table
study regressed on the cohort generation time (t) estimated from the r′
calculations (y = 0.6912x + 50.739, R2 = 0.1056, F2,28=2.64, p=0.12).
Any values below the 45 degree x=y line represent an overestimation
of t by the r′ calculation. Values above the line represent
underestimates