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. 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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