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									                                                                  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

Abstract                                                                                                                                                                                                               Results
                                                                                                                                                                                                                               The per capita rate of change was found to be an accurate
       We used laboratory populations of the Asian Tiger Mosquito

                                                                                                                                                             0.15                                                      predictor of trends in instantaneous growth rate when r was                                              Given the ability of r’ to correctly predict trends in r, this study
Aedes albopictus to compare Livdahl and Sugihara’s (1984) method

                                                                                                                       Per capita rate of change (r) d
                                                                                                                                                                                                                       regressed on r′ (F2,28 = 9.14, p<0.01, Figure 1). Despite an                                     further supports the use of r’ as a method of predicting population
of projecting population growth rate (r') with the derived population
                                                                                                                                                                                                                       accurate correlation, r’ tended to overpredict r. Analysis of                                    growth rates. The lack of accuracy in the values of r’ relative to r
growth rate (r) generated using life table methods. Additionally, we
tested r’ for possible density effects by raising populations of three                                                                                        0.1
                                                                                                                                                                                                                       covariance among density groups failed to show a significant                                     can be further diagnosed by studying the r’ equation itself. In order
                                                                                                                                                                                                                       interaction between r′ and density (ANCOVA, F2,24=0.72, p=0.50),                                 to determine why the r′ values were not accurate, it is important to
different larval densities. r’ was found to be a significant predictor of
                                                                                                                                                                                                                       indicating that the predictive capacity of r′ was robust to differences                          look at the equation for r′ in terms of numerator (Ro) and
trends in r, although accuracy was low. Additionally, r’ was not found
to be susceptible to density effects, supporting the robustness of this                                                                                                                                                in larval density conditions. Further support for the robustness of r′                           denominator () driving the overall value of r′ .
                                                                                                                                                                                                                       as a predictor is indicated by no significant departure from the                                         The conjunction of an overestimate of Ro (Figure 3)and
method regardless of the level of larval competition.
                                                                                                                                                                                                                       overall model for any of the density groups after the interaction term                           underestimate of t (Figure 4) is the driving factor in the inflation of r′
                                                                                                                                                                                                                       had been removed (ANCOVA, F2,26=0.27, p=0.76).                                                   relative to r. The overprediction of Ro most likely has to do with the
                                                                                                                                                                0                                                              Regressions of r and r′ on density (n) were both significant (r:                         assumption that a given female will have a reproductive output that
Introduction                                                                                                                                                         0         0.05   0.1        0.15                  F2,28=7.02, p=0.01; r′: F2,28=166.45, p<0.01, Figure 2). Once                                    can be predicted by her size. The amount of variation in egg laying
        Livdahl and Sugihara’s1 method of projecting population growth                                                                                                                                                 regressed, the predicted values of the maximum per capita rate of                                based on size may be large enough in this population to cause the
rate (r’),                                                                                                                                                   -0.05
                                                                                                                                                                                                                       increase (r: 0.056; r’: 0.11) and carrying capacity (r: 21.35; r’: 34.74)                        overprediction.
                                                                                                                                                                                                                       were found to be quite variable, indicating that r’ is prone to                                          The underestimation of t may stem from the same issue, but
                          1                                                                                                                                          Estimated per capita rate of                      producing inflated values in these populations statistics.                                       it may also come from the assumed value of D, the time lag
                               A          x   f (w x )                                                                                                                    change (r') d-1                                     The net reproductive rate (Ro) generated using the r’ method                             between emergence and the beginning of oviposition. The
                                                                                                                                                                                                                       tended towards overpredicting r-derived Ro (Figure 3). Alternately, r’
                r'                x
                                                                                                                                                                                                                                                                                                                        assumed value of D may be too low, although it is more likely that D
                           xA f (w )  x           x
                                                                                                                     Figure 1: r (derived population growth rate) regressed on r′
                                                                                                                                                                                                                       tended to underpredict the cohort generation time () when
                                                                                                                                                                                                                       compared with r (Figure 4).
                                                                                                                                                                                                                                                                                                                        is not constant with every population. Because of the consistency of
                                                                                                                                                                                                                                                                                                                        prediction across densities, it is tempting to suggest a simple
                       D      x
                                                                                                                     (projected population growth rate (y = 0.8831x - 0.0389, R2 =                                                                                                                                      corrective calibration of r’ to adjust for the overestimation. We do
                           A f (w )   x           x                                                                 0.246, F2,28 = 9.14, p<0.01). The diagonal line through the                                                                                                                                        not advocate this because of the likely difference between survival
                               x                                                                                     origin represents the ideal correlation (1:1) between r and r′.                                                                                                                                    and reproductive rates under field and laboratory conditions. As
                                                                                                                     Those values below this line represent an overestimate of r by                                                                                                                                     survival in laboratory conditions seems likely to exceed field
allows for a less time consuming and less resource intensive method,                                                 r′, and those values above the line represent an underestimate.                                                                                                                                    survival, and blood meal access in field conditions is likely to be
when compared to traditional life table (r) calculations, of                                                                                                                                                                                                                                                            much more difficult, the lengthy period of reproduction observed in
understanding population growth. Due to the accessibility of this                                                                                                                                                                                                                                                       this study may not be approached by field populations. As such, a
method, a number of studies have gone on to employ r’ as a measure                                                                                                                                                                                                                                                      major source of error in r’ estimates could be reduced, and the

                                                                                                                                                                                                             Value of Ro derived using life table (r)
                                                                                                                     0.12                                                                                                                                         35
                                                                                Per capita rate of change (r) d -1

of population success. Despite the ease of use, it has been unclear                                                                                                                                                                                                                                                     predictive ability of r’ could actually be improved under field
whether or not r’ might be sensitive to density effects.                                                              0.1                                                                                                                                                                                               conditions.
       We aimed to examine the applicability of r’ in predicting r in the                                                                                                                                                                                         30
Asian Tiger Mosquito Aedes albopictus. In particular we were
interested in how well population growth rates generated using r’                                                    0.06                                                                                                                                         25

methods reflect changes in population growth derived via life table                                                  0.04
methods in the presence of multiple larval densities.
                                                                                                                                                         0                 5                10          15
                                                                                                                     -0.02                                                                                                                                        10

                                                                                                                                                                                                                                                                                                                        Value of τ derived from life table (r)
Methods                                                                                                                                                                                                                                                           5                                                                                              100
       F1 generation Aedes albopictus eggs (adult collection in
Bermuda) were hatched and the larvae subsequently sorted into                                                                                                                                                                                                     0
                                                                                                                                                                                                                                                                       0   5   10    15     20     25    30   35   40                                            80
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                                                 0.12                                                                                                                                                      Value of Ro using r' methods

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                                                    0.1
                                                                             Per capita rate of change (r') d-1

counted. Additionally, dead females were removed and wing lengths
were measured.                                                                                                       0.08
                                                                                                                                                                                                                       Figure 3: The net reproductive value (R0) derived from the life table                                                                     40
       Age-specific survival fractions (lx) and fecundity values (mx)

                                                                                                                     0.06                                                                                              study regressed on the net reproductive value (R0) estimated from the
were constructed in order to iteratively find the derived population
                                                                                                                                                                                                                       r′ calculations (y = 0.2016x + 8.184, R2 = 0.0103, F2,28=0.34, p=0.57).
growth rate (r) using the Lotka-Euler equation2:
                                                                                                                     0.04                                                                                              Any values below the 45 degree x=y line represent an overestimation                                                                       20

    1   l x mx e      rx
                                                                                                                                                                                                                       of R0 by the r′ calculation. Any values above the line represent
                                                                                                                     0.02                                                                                              underestimates.
           x                                                                                                                                                                                                                                                                                                                                                      0
        A projected population growth rate (eqn. [1]) was derived for                                                                                                                                              References                                                                                                                                          0   20       40      60       80      100   120

each replicate, and was regressed against the derived population                                                                                                                                                   1. Livdahl, T.P. and Sugihara, G. (1984) Non-linear interactions of                                                                                     Value of τ estimated using r' methods
                                                                                                                                                         0                  5               10          15
growth rate. An analysis of covariance was conducted to observe                                                                                                                                                    populations and the importance of estimating per capita rates of
possible density effects that might change the predictive capability of r’                                                                                       Larval density (individuals 30mL-1)               change. Journal of Animal Ecology 53: 573-580.
in various density regimes. These two growth rate measures were                                                                                                                                                    2. Lotka, A. J. 1907. Studies on the mode of growth of material                                      Figure 4: The net cohort generation time (t) derived from the life table
then regressed on larval density, allowing for a measure of potential               Figure 2: r and r’ regressed by larval density (r: y = -0.0026x + 0.0555,                                                      aggregates. American Journal of Science 24:199-216.                                                  study regressed on the cohort generation time (t) estimated from the r′
carrying capacity (K) and maximum growth rate (rmax) of the study                   R2 = 0.2005, F2,28=7.02, p=0.01; r’: y= -0.0031x+0.1077, R2 = 0.856,                                                                                                                                                                calculations (y = 0.6912x + 50.739, R2 = 0.1056, F2,28=2.64, p=0.12).
species.                                                                            F2,28=166.45, p<0.01). The y intercept predicts rmax (r 0.056; r’ 0.11),                                                                                                                                                            Any values below the 45 degree x=y line represent an overestimation
        The net reproductive value (Ro) and cohort generation time ()
                                                                                                                                                                                                                   Acknowledgments                                                                                      of t by the r′ calculation. Values above the line represent
                                                                                    the maximum per capita rate of increase one could expect these
for both methods of generating population growth rate were compared                                                                                                                                                We thank the Department of Biology, Clark University, the National
                                                                                    populations to exhibit. The x intercept predicts K (r 21.35; r’: 34.74), the                                                                                                                                                        underestimates
with one another to determine where variation in the prediction of r by                                                                                                                                            Institutes of Health (R15 AI062712-01) and the Keck Foundation for
                                                                                    carrying capacity for this species under similar environmental
r’ might be generated.                                                                                                                                                                                             supporting this project.

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