1
Running head: population growth and density dependence
VARIATION IN AND CORRELATION BETWEEN RATE OF INCREASE AND
CARRYING CAPACITY
Nora Underwood1
Department of Biological Science
Florida State University
Tallahassee, Florida 32306-1100 USA
1
E-mail: nunderwood@bio.fsu.edu; fax: (850) 644-9829
1
2
Abstract
Population growth rate and density dependence are fundamental components of population
dynamics. Theory suggests that variation in and correlations between these parameters among
individuals or among patches within a population can influence overall population size, but little
is known about the degree of variation or correlation in real organisms particularly . Replicate
populations of a specialist aphid (Chaetosiphon fragaefolii) were followed on 11 genotypes of its
host plant (Fragaria chiloensis) in the greenhouse. Population models were fit to these census
data to provide estimates of population growth rate and carrying capacity for aphid populations
on each plant genotype. Both growth rate and carrying capacity showed substantial variation
among plant genotypes and these two parameters were positively but not significantly correlated.
These results support the existence of spatial variation in population-dynamic parameters; data
on frequency distributions and correlations of these parameters in natural populations are needed
for evaluation of the importance of variation in growth rate and density dependence to population
dynamics in the field.
Key words: carrying capacity; population growth rate; Fragaria chiloensis; Chaetosiphon
fragaefolii; genetic variation; environmental variation; density dependence
2
3
Introduction
The long-term dynamics of populations are determined, at minimum, by two internal
factors: the rate of growth and the tendency for growth to be affected by crowding. In the
commonly used logistic model, these components of population dynamics are described by the
parameters r (intrinsic rate of increase) and K (carrying capacity), but similar terms exist in most
single-species models (e.g.,  and b in the Ricker and Beverton-Holt models (Turchin 2003).
Although we generally consider populations to be characterized by a single growth rate and
degree of density dependence, growth rate and density dependence can vary within individual
populations and models suggest that variation in and co-variance between growth rate and
density dependence should be play a role in determining long-term population dynamics. r-K
selection theory (MacArthur and Wilson 1967) provides expectations for how negative
correlations between r and K might arise between different populations, but little is known about
variation in and correlations between population dynamic parameters within populations.
For populations of a particular species, variation in growth rate and density dependence
within a population can come from two sources: genetic variation among individuals in traits
that influence per-capita growth rates at low and high density (see, e.g.,Bradshaw and Holzapfel
1989, Mueller 1997) and variation in the environment in which the population lives.
Environmental variation can be abiotic; for example, population growth rates of insects may be
higher where temperatures are higher (Agrawal et al. 2004). It can also arise from interactions
with other organisms. For example, for herbivorous insects, both growth rate (Agrawal et al.
2004, Helms et al. 2004) and carrying capacity (Underwood et al. 2000) can differ among hostplant genotypes or species.
3
4
When variation in growth rate and density dependence occurs within populations or
metapopulations (i.e., when individuals or patches with different parameters are part of the same
system), theory suggests that variance in and correlations between these factors can affect the
size of the population. For example, Bjornstad and Hansen (1994) show analytically that when
the relationship between population growth and density is nonlinear, variance among individuals
in traits that influence the rate of increase or response to density, such as clutch size, leads to
equilibrium population sizes different from those produced by the average trait values.
Simulation (Underwood 2004) and analytical (M. Donahue et al., manuscript in preparation)
models show that, when patches in space (e.g., individual host plants for an insect herbivore)
vary in population growth rate and carrying capacity, variance in these parameters influences
equilibrium population size; whether this effect is positive or negative depends on the correlation
between growth rate and carrying capacity across patches. If growth rate and carrying capacity
are positively correlated, such that patches that promote high growth also support large numbers
of organisms, equilibrium population sizes across all patches are larger than would be expected
without variance. When the two are negatively correlated, equilibrium population sizes are
smaller than would be expected without variance (Fig. 1).
Theory thus suggests that to fully understand the dynamics of a given population we need
information about the variation in and correlation between population dynamic parameters within
populations. Unfortunately, little is known about how these parameters are actually distributed
or correlated either among individuals or among environments within a population.
Performances in low- and high-resource environments are assumed to trade off across individual
genotypes, which would predict a negative correlation between growth at low densities and
carrying capacity across individuals within populations. Although some evidence indicates
4
5
trade-offs between traits that influence fecundity and traits that influence performance at high
density (Mueller et al. 1991), the relationship has been looked at thoroughly in few systems, and
the traits that have been considered may not actually determine carrying capacity (Mueller 1997).
Expectations about correlations between population growth rate and density dependence across
biotic and abiotic environments are less clear. We might predict positive correlations because,
for example, environments with faster resource renewal should promote rapid growth at low
density and support higher densities of organisms, but other conceivable scenarios might produce
negative correlations. To my knowledge, no data exist that bear on these correlations among
environments within populations or metapopulations.
Data addressing variance in and correlations between population dynamic parameters are
likely to be logistically difficult to gather in many systems, because they will require measuring
growth and density dependence in different patches in isolation; migration among patches would
alter the observed population-dynamic parameters within patches. Because plant-insect herbivore
systems can often be relatively small scale with rapid population dynamics, these systems are a
good tool for looking at these correlations. As a step toward examining the relationship between
growth rate and carrying capacity across environments, I measured the intrinsic rate of increase
(r) and carrying capacity (K) of a specialist aphid across genotypes of its host plant. Like many
herbivorous insects, these aphids move among plants in the field (Underwood et al., in review),
so individual plants represent environmental patches within the insect population. Using
greenhouse experiments and model fitting, I asked (1) whether aphid r and/or K vary across the
environments represented by different host-plant genotypes and (2) what the magnitude of
correlation between these parameters is across host-genotype environments.
5
6
Methods
Wild strawberry (Fragaria chiloensis) and the strawberry aphid (Chaetosiphon
fragaefolii, a specialist on Fragaria) are native to the Pacific coast of the Americas. Strawberry
clones for this experiment were either collected in the field (Sonoma Co., California, clones
“Wrights 2,” “Doran 1,” “Doran 3”) or obtained from the National Clonal Germplasm
Repository (Corvallis, Oregon, clones “34,” “37,” “39,” “59,” “55,” “340,” “361,” “1312”).
Some clones from the Germplasm Repository were chosen for either high or low resistance to
arthropods, but most (including those from the field) were chosen without any knowledge of
their effects on aphids. Fragaria chiloensis is a perennial herb that is easily cloned from asexual
runners. All clones used in this experiment resulted from at least two rounds of asexual
propagation in a common greenhouse environment. Chaetosiphon fragaefolii for this experiment
were collected from several locations in the field (Sonoma Co., California) and reared for at least
6 months in the greenhouse, a minimum of 10 overlapping aphid “generations.” The aphid
colony I used was initiated and occasionally refreshed with a mixture of genotypes from different
field populations and was reared on a mixture of strawberry genotypes not used in the study, to
prevent the development of a feeding bias within the aphid population.
I rooted six individuals of each of the strawberry clones listed above in separate 3.8- 
21-cm tubes (Cone-tainers, Stuewe & Sons, Corvallis, Oregon, USA) filled with UC Davis
research mix in late fall 1999. Plants were watered and fertilized as necessary with Miracle-Gro
(The Scotts Company, Marysville, Ohio, USA). The tubes were placed in 6 racks (each rack
constituted a spatial block) on benches at the University of California, Davis, greenhouses, with
one individual of each clone per rack. Each individual plant was enclosed in a mesh bag (Fibe-
6
7
Air Sleeve, Kleen Test Products, Milwaukee, Wisconsin, USA) that prevented aphid movement
among plants. I began the experiment by placing either one (for two of the six blocks) or two
(for the other four blocks) adult aphids on each plant. Aphid populations were counted biweekly
for 10 to 14 censuses; 2 blocks were started later than the other blocks, and in 8 cases individual
plants died before the end of the experiment, resulting in shorter time-series. If an aphid
population went extinct while the plant was alive, the population was restarted with either one or
two new adult aphids, as appropriate for that block.
N
I fit the logistic model ( dN
dt  rN (1  K ) ), where N is population size, r is intrinsic rate of
increase, and K is carrying capacity, to the time-series of aphid population sizes for each
strawberry clone separately. I fit two alternative maximum-likelihood models, one with
observation noise and one with process noise (de Valpine and Hastings 2002). Poisson error
resulted in the best model fits (compared to normal or lognormal error). This resulted in
estimates of population growth rate (r) and carrying capacity (K) for aphids on each of the
strawberry clones. Estimates using process and observation noise were significantly correlated
for both r (Pearson correlation coefficient r = 0.77, P = 0.02) and K (r = 0.85, P = 0.003), so
results are shown only for observation error. I calculated contours around the best-fit estimates
of r and K that indicate significant differences at the P = 0.05 level for likelihood ratio tests.
Genotypes for which the confidence contours do not overlap can thus be considered significantly
different at the P = 0.05 level. I calculated a Pearson correlation coefficient as a description of
the strength of association between the most likely growth rate and carrying capacity parameters
across plant genotypes. I also tested whether this correlation is significantly different from zero,
but note that results of this test should be interpreted with caution because it is unlikely that all
the assumptions of the test (e.g. homogeneity of variances) are met by these data.
7
8
Results and Discussion
Aphid populations on different strawberry clones varied substantially in estimates of
population growth rate and carrying capacity (Fig. 2). The confidence contours around estimates
for the parameters for individual clones did not overlap, indicating that the differences in
parameters among strawberry clones were statistically significant. The confidence with which
these parameters could be estimated varied widely (cf. the estimates of r for clones “39” and
“361”). Because the study was carried out in the greenhouse, after several generations of
greenhouse propagation, these differences are probably due to genetic differences among the
plants, although lingering “maternal” effects of the environments from which the clones were
collected are possible. Although evidence is abundant that plant traits can influence herbivore
performance and behavior (Fritz and Simms 1992), few studies have shown that differences
between host species or genotypes influence longer-term population dynamics (both growth rate
and density dependence). Many studies examining plant effects on herbivore populations have
demonstrated differences in population growth rates across either plant species or genotypes (e.g.
Underwood et al. 2000, Agrawal et al. 2004, Helms et al. 2004). Fewer have attempted to
measure differences in carrying capacity or density dependence (but see Underwood et al. 2000,
Agrawal et al. 2004).
The plant genotypes used in the present study were collected from many different natural
populations across several states, so the results cannot tell us the distribution of populationdynamic parameters in any one population in the field. However, two of the genotypes in the
experiment (“Doran 1” and “Doran 3”) were collected from the same population (Doran Park,
Sonoma Co., California, USA) and differ significantly in both r and K, so some variation in these
8
9
parameters within populations in the field does exist. Although F. chiloensis appear to reproduce
primarily asexually by means of runners, their populations contain substantial genetic variation
(Alpert et al. 1993). Aphid growth rate and carrying capacity also varied in a field experiment
on a subset of the same strawberry genotypes used in the present study (Underwood, manuscript
in preparation), suggesting that the kind of differences in parameters observed in the greenhouse
can exist even with natural variation in weather, predators, and competitors.
Results of this study suggest that variation in population dynamic parameters among wild
strawberries has the potential to influence aphid population size in the field; whether that
influence is positive or negative will depend on the correlations between population growth rate
and carrying capacity across plants (Fig. 2). There was no significant correlation between aphid
growth rate and carrying capacity across wild strawberry genotypes in this study (r = 0.41, P =
0.20). Because the variation in r and K measured in the present study is a property of the aphids’
environment (the plant) rather than of traits of the aphids themselves, one could argue that either
negative or positive correlations between r and K across plants could have been expected. For
example, plants with high throughput of nitrogen might support both fast aphid reproduction at
low densities and larger aphid densities. Alternatively, trade-offs between inducible and
constitutive plant resistance might produce a negative correlation between r and K. Plants with
high constitutive resistance should produce aphid populations with low growth rates, and
carrying capacities should be set by other factors in the environment (constitutive resistance is by
definition unresponsive to herbivore density and thus cannot contribute to density dependence).
Plants with inducible resistance, on the other hand, might allow high growth rates for herbivores
at low density (r), when resistance has not yet been induced, but could also contribute to low
9
10
ering herbivore carrying capacity because inducible resistance can be density dependent
(Underwood 2000).
The trend in this study was toward a positive correlation between r and K across
strawberry genotypes, and one might imagine that with a larger sample of genotypes a significant
correlation might emerge. However, it is clear from this study that the correlation between r and
K across these plant genotypes is not very tight. This suggests that any particular population of
wild strawberry plants could have anything between a positive and a negative association
between r and K. That is, one could construct a group of plant clones with a negative
correlation, and thus a smaller-than-expected equilibrium aphid population size, from a larger
pool of plant clones that had no correlation or an overall positive correlation. Unless correlations
between r and K across environments turn out to be stronger than indicated by this study,
prediction of the influence of such correlations within any particular population may not be
possible without characterizing the distribution of population dynamic parameters for that
particular population.
It is important to remember that in addition to environmentally driven variation in growth
rate and density dependence, there can also be variation among individuals in traits contributing
to these forces. The actual dynamics of any given population should be a function of the mean
and variance of parameters both among environments and among individuals. A full
understanding of population dynamics may thus require describing the distributions of
population-dynamic parameters both among individuals due to their own traits, and among
environmental patches within populations. The results reported here suggest that variation in
these parameters across environments can exist. However, these parameters will have to be
10
11
measured in real populations (a natural history of population dynamic parameters) to determine
the true importance of variance in and correlations between parameters for population dynamics.
11
12
Acknowledgements
I thank B. D. Inouye and M. Donahue for helpful conversations, A. A. Agrawal for
impetus to think about these issues, R. Karban for greenhouse space and discussion, and N.
Hummel for assistance with censuses. E. Crone, S. Halpern, and reviewers provided helpful
comments. This work was supported by NRI Competitive Grants Program/USDA grant 9835302-6984, NSF DEB grant 0089570, and an FSU First Year Assistant Professor Award to N.
Underwood.
12
13
References
Agrawal, A. A., N. Underwood, and J. Stinchcombe. 2004. Intraspecific variation in the strength
of density dependence in aphid populations. Ecological Entomology 29:521-526.
Alpert, P., R. Lumaret, and F. Di Giusto. 1993. Population structure inferred from allozyme
analysis in the clonal herb Fragaria chiloensis (Rosaceae). American Journal of Botany
80:1002-1006.
Bjornstad, O. N., and T. F. Hansen. 1994. Individual variation and population dynamics. Oikos
69:167-171.
Bradshaw, W. E., and C. M. Holzapfel. 1989. Life-historical consequences of density-dependent
selection in the pitcher-plant mosquito, Wyeomyia smithii. The American Naturalist
133:869-887.
de Valpine, P., and A. Hastings. 2002. Fitting population models incorporating process noise and
observation error. Ecological Monographs 72:57-76.
Fritz, R. S., and E. L. Simms. 1992. Plant resistance to herbivores and pathogens: ecology,
evolution, and genetics. University of Chicago Press, Chicago Illinois USA.
Helms, S. E., S. J. Connelly, and M. D. Hunter. 2004. Effects of variation among plant species
on the interaction between a herbivore and its parasitoid. Ecological Entomology 29:4451.
Inouye, B. D. 2005. The importance of the variance around the mean effect size of ecological
processes. Ecology 86:262-265.
MacArthur, R. H., and E. O. Wilson. 1967. The theory of island biogeography. Princeton
University Press, Princeton New Jersey USA.
Mueller, L. D. 1997. Theoretical and empirical examination of density-dependent selection.
Annual Review of Ecology and Systematics 28:269-288.
Mueller, L. D., P. Guo, and F. J. Ayala. 1991. Density-dependent natural selection and trade-offs
in life history traits. Science 253:433-435.
Tuljapurkar, S., C. C. Horvitz, and J. B. Pascarella. 2003. The many growth rates and elasticities
of populations in random environments. The American Naturalist 162:489-502.
Turchin, P. 2003. Complex population dynamics: a theoretical/empirical synthesis. Princeton
University Press, Princeton, USA.
Underwood, N. 2000. Density dependence in induced plant resistance to herbivore damage:
threshold, strength and genetic variation. Oikos 89:295-300.
Underwood, N. 2004. Variance and skew of the distribution of plant quality influence herbivore
population dynamics. Ecology 85:686-693.
Underwood, N., W. Morris, K. Gross, and J. R. I. Lockwood. 2000. Induced resistance to
Mexican bean beetles in soybeans: variation among genotypes and lack of correlation
with constitutive resistance. Oecologia 122:83-89.
Figure Legends
Fig. 1 (redrawn from Underwood 2004). The effect of variation in carrying capacity (K) among
patches on the equilibrium population size across patches in a simulation model. The
13
14
observed equilibrium population size (N*obs) in simulations is expressed relative to the
expected population size (N*exp) in the absence of an effect of variation (i.e., average
carrying capacity across patches). Population dynamics are modeled as logistic equations
in each patch, linked by global migration. Each point indicates equilibrium population
size in a single run of the model.  indicates the correlation between r and K.
Fig. 2. Maximum-likelihood estimates of strawberry-aphid population growth rates and carrying
capacities on 11 different genotypes of wild strawberry. Each point indicates a different
genotype. Contours indicate parameter values significantly different from the maximumlikelihood estimates at the P = 0.05 level, on the basis of likelihood-ratio tests.
Figures
(N*obs – N*exp ) / N*exp
Fig. 1
0.1
-0.1
=
-0.3
0.04
0.02
0
-0.02
-0.04
-0.5
-0.7
0
0.5
1
Variation in plant quality (CVK)
14
15
Fig. 2
Carrying capacity (K)
1000
361
59
800
55
340
600
Wrights 2
400
34
37
200
0
Doran1 Doran 3 1312 39
0.0
0.05
0.10
0.15
0.20
0.25
0.30
Rate of increase (r)
15