, Volume 180, Issue 3, pp 833–840 | Cite as

Enrichment scale determines herbivore control of primary producers

Ecosystem ecology - Original research


Anthropogenic nutrient enrichment stimulates primary production and threatens natural communities worldwide. Herbivores may counteract deleterious effects of enrichment by increasing their consumption of primary producers. However, field tests of herbivore control are often done by adding nutrients at small (e.g., sub-meter) scales, while enrichment in real systems often occurs at much larger scales (e.g., kilometers). Therefore, experimental results may be driven by processes that are not relevant at larger scales. Using a mathematical model, we show that herbivores can control primary producer biomass in experiments by concentrating their foraging in small enriched plots; however, at larger, realistic scales, the same mechanism may not lead to herbivore control of primary producers. Instead, other demographic mechanisms are required, but these are not examined in most field studies (and may not operate in many systems). This mismatch between experiments and natural processes suggests that many ecosystems may be less resilient to degradation via enrichment than previously believed.


Top-down versus bottom-up Eutrophication Consumer–resource dynamics Experimental bias Ideal free distribution Ecological resilience 


Ecologists have long sought to understand how communities are structured and how (or if) consumers [“top-down” effects (Hairston et al. 1960; Oksanen et al. 1981; Sih et al. 1985)] and resources [“bottom-up” effects (Ehrlich and Birch 1967; Osenberg and Mittelbach 1996)] control this process. This academic debate has pressing, real-world implications, because land and fertilizer use by humans increasingly alter the dynamics of nutrients that limit primary production in both terrestrial and aquatic systems (Elser et al. 2007; Nixon 2009; Vitousek et al. 1997). In aquatic systems, nutrient-stimulated primary production can lead to toxic algal blooms, hypoxia and fish kills (Breitburg et al. 2009). In marine systems, added nutrients can stimulate the growth of algae (Lapointe 1997; Lapointe et al. 2004, 1992), which can elicit a suite of negative ecological effects, including declines of corals (McCook et al. 2001; Rasher and Hay 2010; Smith et al. 2006), seagrasses (Burkholder et al. 2007), and kelp (Connell et al. 2008). Indeed, effects of algae on marine systems and associated losses in biodiversity and ecosystem services (e.g., fisheries, storm surge protection) are critical conservation concerns (Bellwood et al. 2004; Fabricius 2005; Hughes et al. 2010; Vitousek et al. 1997).

Field experiments that simultaneously manipulate consumer density (via exclusion designs) and nutrient availability have been used to evaluate the role of consumers in controlling primary producers in the face of eutrophication (Burkepile and Hay 2006; Gruner et al. 2008). Results from these studies suggest that herbivores partially or fully compensate for the deleterious effects of enrichment on algal biomass and thus may facilitate the persistence or recovery of valuable marine ecosystems (Burkepile and Hay 2006). However, in nature, enrichment often occurs at spatial scales that are many orders of magnitude larger than the scale at which field manipulations are conducted. For example, coastal eutrophication due to nutrient enrichment has been documented across ~30 km2 off the northeast coast of Italy (Penna et al. 2004), ~31 km2 of inner Kaneohe Bay, Hawai‘i, USA (Smith et al. 1981), >100 km2 of both the Chesapeake Bay and Pamlico Sound, USA (Paerl et al. 2006), and ≫100 km2 in the Great Barrier Reef, Australia (Bell et al. 2014). Yet, experiments are most often conducted at scales smaller than 1 m2 (Burkepile and Hay 2006; Gruner et al. 2008). This mismatch in scale is well known (Burkepile and Hay 2006), although the effect of this possible bias has not been explicitly addressed. Thus, we are left uncertain how the results of small-scale field experiments might inform management of real ecosystems. Large-scale field manipulations would help, but they are rare and often unfeasible for both ethical and logistical reasons. Fortunately, consumer–resource theory frees us from these constraints and allows us to explore how well consumers can reduce effects of enrichment on primary producers and how these effects depend on spatial scale.

Traditional models of closed plant–herbivore systems posit that enrichment increases herbivore population growth, and, consequently, herbivore density. Increased density of herbivores then drives the abundance of primary producers back to the pre-enrichment state (Mittelbach et al. 1988; Oksanen et al. 1981; Rosenzweig 1971). Support for this mechanism has been provided by field experiments in closed systems with rapid consumer turnover [e.g., zooplankton in lakes (Carpenter and Kitchell 1988; Leibold 1989)]. However, in many other cases, experiments are conducted in open systems, with small plots accessible to herbivores that move over much larger spatial scales (Fig. 1). In these cases, herbivores can migrate in from surrounding areas, decoupling local resource dynamics from herbivore dynamics that operate at the regional scale. Furthermore, a true numerical demographic response by herbivores to enrichment would be unlikely or unimportant in many experiments, because: (1) the generation time of the focal herbivore often exceeds the duration of the experiment (Online Resource 1, Fig. A1); or (2) the offspring produced by local herbivores are dispersed over very large scales, as in many coastal marine systems (Hixon et al. 2002). Thus, the conditions of many field experiments preclude the application of consumer–resource theory developed for closed systems. In contrast, patch- or habitat-selection models may be more appropriate in these contexts.
Fig. 1

Herbivore movement range compared to experimental scale of enrichment in previous top-down versus bottom-up field experiments (nutrient addition fully crossed with herbivore exclusion) across systems and taxa (see Online Resource 1 for literature survey methods). The line indicates the 1:1 relationship, and overlapping points were jittered to show all points. The observed ratio of experimental scale to herbivore movement range was minuscule, with a median of 5.2 × 10−6

Herbivores that are free to move between enriched and unenriched habitats may exhibit an “ideal free” distribution (Fretwell and Lucas 1970), in which herbivores have equal fitnesses in different habitats but their densities across the landscape are heterogeneous, reflecting underlying variation in resource production (Nicotri 1980; Oksanen et al. 1995; Power 1984; Sutherland 1983). In such scenarios, patch selection by herbivores (i.e., between enriched and unenriched patches) could drive responses of primary producers in top-down versus bottom-up field experiments, and the magnitude of these responses could depend on the spatial scale of the studies.

We used a mathematical model to evaluate whether the spatial scale of nutrient addition (from small experimental scales to larger, more natural scales) affects the ability of herbivores to control primary producers in the face of enrichment. We solved our model analytically and found that across parameters space, increasing the scale of enrichment weakens herbivore control of primary producers. Our findings reveal a bias in previous short-term field experiments and suggest that many systems may be more vulnerable to ecologically harmful effects of nutrient enrichment than previously believed.

Model formulation and analysis

We used a mathematical model that simulates typical top-down versus bottom-up field experiments that consist of four treatments: caged plots that exclude herbivores versus open plots that permit herbivore access, crossed with ambient versus elevated nutrients. We designated Pi,j as the density of the primary producer in the ith nutrient treatment (unenriched = U or enriched = E) and the jth herbivore treatment [indicating the presence (+), or absence (−) of herbivores]. To examine how increasing the size of experimental plots influences enrichment effects, we made the following assumptions:
  1. 1.

    The sessile primary producer grew logistically, with its intrinsic growth rate (r) and carrying capacity (K) greater in enriched versus unenriched habitat (i.e., rE > rU and KE > KU).

  2. 2.

    Experimental plots were independent with respect to herbivore or nutrient treatments, i.e., plots were sufficiently separated to prevent the sharing of individual herbivores or nutrients.

  3. 3.

    Herbivores exhibited an ideal free distribution (Fretwell and Lucas 1970) over uncaged habitat within an area ST, within which a single experimental plot was located. In effect, ST represents the movement range of the herbivore. Thus, when a plot was uncaged and enriched, herbivore densities could differ between the enriched plot area, SE, and the area of the surrounding, unenriched habitat, SE,out (Nicotri 1980; Oksanen et al. 1995; Power 1984; Sutherland 1983).

  4. 4.

    Herbivore density is sufficiently high that when herbivores redistribute themselves following enrichment, some herbivores remain in the unenriched areas surrounding the enrichment plot. As a result, and given assumptions 2 and 3, the equilibrium densities of primary producers in an enriched plot and the unenriched surrounding habitat within an area ST will be equivalent in the presence of herbivores: i.e., \(P_{{{\text{E}}, + }}^{*} = P_{{{\text{E}}, + ,{\text{out}}}}^{*}\), where “out” denotes unenriched habitat outside the enriched plot. We later relax this assumption.

  5. 5.

    Total herbivore abundance (NT) was fixed within an area ST (i.e., there was no reproduction, mortality, immigration or emigration). In other words, the timescale of the model relative to herbivore generation time matched the short experimental duration that is characteristic of past studies (Online Resource 1, Fig. A1). Thus, for an unenriched open plot, herbivore density is HU = HT = NT/ST (and homogeneous throughout ST). In contrast, enriched, open plots create a locally heterogeneous landscape and thus the total herbivore abundance, NT, must be partitioned between the enriched plot and the unenriched habitat that surrounds the plot: i.e., NT = SEHE + SE,outHE,out = SEHE + (ST − SE)HE,out, where HE and HE,out are the herbivore densities in the enriched plot and the surrounding unenriched habitat, respectively (and both are >0 due to assumption 4).

  6. 6.

    Herbivores had a type I functional response, i.e., feeding rate increased linearly with the density of food (Holling 1966). The per capita consumption rate (α) was equal for herbivores feeding in enriched and unenriched habitat. (In Online Resource 2, we show that a type II functional response yields the same qualitative results, albeit more complex to solve.)

Given these assumptions, in the absence of herbivores (i.e., in caged plots), the equilibrium density of primary producers is set by their carrying capacity:
$$P_{{{\text{U}}, - }}^{*} = K_{\text{U}} ,{\text{ and}}$$
$$P_{{{\text{E}}, - }}^{*} = K_{\text{E}} .$$
In the presence of herbivores, the dynamics of primary producers is set by the balance between logistic growth and herbivore consumption. For unenriched plots accessible to herbivores,
$$\frac{{{\text{d}}P_{{{\text{U}}, + }} }}{{{\text{d}}t}} = r_{\text{U}} P_{{{\text{U}}, + }} \left( {1 - \frac{{P_{{{\text{U}}, + }} }}{{K_{\text{U}} }}} \right) - \alpha H_{\text{U}} P_{{{\text{U}}, + }} ,$$
and at equilibrium,
$$P_{{{\text{U}}, + }}^{*} = K_{\text{U}} \left( {1 - \frac{{\alpha H_{\text{U}} }}{{r_{\text{U}} }}} \right) .$$
For enriched plots accessible to herbivores, we have to consider the dynamics of the enriched plot, as well as the adjacent unenriched habitat, because herbivores can distribute themselves between the two habitats. Thus, we have:
$$\frac{{{\text{d}}P_{{{\text{E}}, + }} }}{{{\text{d}}t}} = r_{\text{E}} P_{{{\text{E}}, + }} \left( {1 - \frac{{P_{{{\text{E}}, + }} }}{{K_{\text{E}} }}} \right) - \alpha H_{\text{E}} P_{{{\text{E}}, + }} \quad {\text{and}}$$
$$\frac{{{\text{d}}P_{{{\text{E}}, + ,{\text{out}}}} }}{{{\text{d}}t}} = r_{\text{U}} P_{{{\text{E}}, + ,{\text{out}}}} \left( {1 - \frac{{P_{{{\text{E}}, + ,{\text{out}}}} }}{{K_{\text{U}} }}} \right) - \alpha H_{{{\text{E}},{\text{out}}}} P_{{{\text{E}}, + ,{\text{out}}}} .$$
Next, we set Eqs. 5 and 6 equal to 0 and solve for \(P_{{{\text{E}}, + }}^{*} \;{\text{and}}\;P_{{{\text{E}}, + ,{\text{out}}}}^{*}\), but based on assumptions of ideal free distribution (i.e., assumption 4: \(P_{{{\text{E}}, + }}^{*} = P_{{{\text{E}}, + ,{\text{out}}}}^{*}\)), we can set these solutions equal to one another, and after a few more steps (involving substitutions for HE,out and HE), we get the final solution:
$$P_{{{\text{E}}, + }}^{*} = \frac{{K_{\text{E}} K_{\text{U}} \left[ {S_{\text{E}} r_{\text{E}} + \left( {S_{\text{T}} - S_{\text{E}} } \right)r_{U} - \alpha N_{\text{T}} } \right]}}{{S_{\text{E}} r_{\text{E}} K_{\text{U}} + \left( {S_{\text{T}} - S_{\text{E}} } \right)r_{\text{U}} K_{\text{E}} }} .$$
From this analytical solution, we see that the equilibrium density of primary producers (PE,+*) in uncaged, enriched plots always increases with the spatial scale of enrichment (SE):
$$\frac{{\partial P_{{{\text{E}}, + }}^{*} }}{{\partial S_{\text{E}} }} = \frac{{S_{\text{T}} K_{\text{E}} K_{\text{U}} r_{\text{E}} r_{\text{U}} \left( {K_{\text{E}} - K_{\text{U}} } \right)}}{{\left[ {S_{\text{E}} r_{\text{E}} K_{\text{U}} + \left( {S_{\text{T}} - S_{\text{E}} } \right)r_{\text{U}} K_{\text{E}} } \right]^{2} }} > 0 .$$

This result (Eq. 8) was obtained assuming plots were independent (i.e., herbivores could not travel between plots: see assumption 2 above). If we relax this assumption (e.g., assume that all plots occur within the herbivore’s foraging range), the qualitative result still holds (i.e., \(\frac{{\partial P_{{{\text{E}}, + }}^{*} }}{{\partial S_{\text{E}} }} > 0\)).

To quantify the scale dependence of the effect on primary producer biomass, we used results from our model (Eqs. 57), analyzed over a range of spatial scales and parameter values, to calculate the effectiveness of herbivores in controlling the enrichment effect on primary producers:
$${\text{Relative effectiveness of herbivores}} = 1 \, {-}\left[ {{{\left( {P_{{{\text{E}}, + }}^{*} {-}P_{{{\text{U}}, + }}^{*} } \right)} \mathord{\left/ {\vphantom {{\left( {P_{{{\text{E}}, + }}^{*} {-}P_{{{\text{U}}, + }}^{*} } \right)} {\left( {P_{{{\text{E}}, - }}^{*} {-}P_{{{\text{U}}, - }}^{*} } \right)}}} \right. \kern-0pt} {\left( {P_{{{\text{E}}, - }}^{*} {-}P_{{{\text{U}}, - }}^{*} } \right)}}} \right].$$

This metric describes how well herbivores control the response of primary producers to added nutrients (numerator in second term) relative to the response of primary producers to nutrients in the absence of herbivores (denominator in second term). We then varied experimental plot size (SE) to determine its effect on the equilibrium density of primary producers (Eqs. 1, 2, 4, 7) and the relative effectiveness of herbivores (Eq. 9).

When the scale of nutrient addition is small relative to the movement range of the herbivore (a ubiquitous characteristic of field experiments; Fig. 1), the density of primary producers in the enriched plots is low (Fig. 2a, b), and herbivore control of primary producers (Eq. 9) is very high (Fig. 2d, e). This is because herbivores move into the enriched plots from surrounding areas in response to localized increases in primary production. Thus, although production increases in the enriched plots, the increased density of the herbivores (via immigration) completely prevents an increase in the density of primary producers (when plots are very small). In contrast, primary producers increase in density (or biomass) in response to enrichment in the absence of herbivores.
Fig. 2

The equilibrium density of primary producers in enriched plots with herbivores (PE,+; from Eq. 7; ac) and the relative effectiveness of herbivores in preventing increased density of primary producers in response to enrichment (Eq. 9; df), as a function of the scale of enrichment relative to the movement range of the herbivore. acy-intercepts of curves reflect the densities of primary producers in unenriched plots with herbivores (PU,+ from Eq. 4). Curves were generated by changing the effect of enrichment on carrying capacity (K; a, d) or intrinsic growth rate (r; b, e) or altering the herbivore population feeding rate (c, f). Otherwise, default parameter values were: KU = 10, KE = 40, rU = 1, rE = 2, α × NT = 50. Vertical lines indicate the median scale of enrichment (scaled to herbivores movement range) from field experiments (i.e., 5.2 × 10−6; Online Resource 1). For df, a response of 1 indicates that herbivores completely prevent an increase in primary producers following enrichment. The code used to generate this figure is available in Online Resource 3

However, as the scale of enrichment increases, herbivore density cannot respond to the same degree because of the limited foraging area over which migration occurs. As a result, the density of primary producers increases with experimental scale (Fig. 2a–c) and herbivore control decreases (Fig. 2d–f). At very large scales (i.e., as the scale of the experiment approaches the scale of the herbivore’s foraging range), the density of primary producers increases to its maximum (Fig. 2a–c), and herbivore control is reduced to its minimum (Fig. 2d–f). This large scale better matches real-world enrichment scenarios (Fig. 1; Bell et al. 2014; Paerl et al. 2006; Penna et al. 2004; Smith et al. 1981).

Although herbivore control of primary producers (Eq. 8) is qualitatively consistent across parameter space, the strength of that control depends upon parameters that govern the dynamics of the system (Fig. 2). In particular, herbivore control of primary producers is greater when enrichment causes a smaller increase in r or a greater increase in K of the primary producer, or there is a greater density (HT) or individual feeding rate (α) of the herbivore. The effect of increasing KE is perhaps counterintuitive, but it arises because increased KE causes the numerator in Eq. 5 to increase less rapidly than the denominator; i.e., enrichment increases primary producer density more in the absence than in the presence of herbivores.

We derived the above results based on assumption 4. However, when herbivory is very weak relative to the effect of enrichment on primary producers, the immigration response of herbivores may be insufficient to keep the density of primary producers equal inside and outside of the enriched plot. This happens when all of the herbivores aggregate inside the enriched plot and none remain in the surrounding habitat. Thus, we can substitute \(H_{\text{E}} = \frac{{N_{\text{T}} }}{{S_{\text{E}} }}\) into Eq. 5, resolve for the equilibrium and obtain:
$$P_{{{\text{E}}, + }}^{*} = K_{\text{E}} \left( {1 - \frac{{\alpha N_{\text{T}} }}{{r_{\text{E}} S_{\text{E}} }}} \right).$$

Under this condition, \(\frac{{\partial P_{{{\text{E}}, + }}^{*} }}{{\partial S_{\text{E}} }} > 0\): i.e., the previous qualitative result (Eq. 8) still holds. Indeed, this relaxation of assumption 4 only reinforces our main finding: enrichment effects increase (and herbivore effects decline) with increasing plot size.


Here, we provide the first explicit demonstration that the mismatch between the scales of experimental enrichment studies and the scale of herbivore movement (Fig. 1) can create the (potentially false) perception that herbivores can prevent increased biomass of primary producers: i.e., given the small size of experimental plots, herbivores can aggregate in response to increased food production. However, at larger enrichment scales, which are more indicative of real-world enrichment scenarios (but not field experiments; Fig. 1), herbivores are less able to control primary producers because the potential immigration response is reduced. This mechanism is similar to that proposed by Englund (1997), in which prey migration into/out of cages could mask effects of predators when experiments were conducted in small plots. As cage size increased, the importance of movement decreased, and the within-plot manipulation (predator presence/absence) became relatively more important in driving observed effects. Our system differed somewhat from Englund’s (1997), however, because the consumer (herbivore) was mobile, but the resource (prey) was sessile. Our quantitative results also echo the conceptual arguments of Van de Koppel et al. (2005) in their general discussion of scale mismatch in consumer–resource interactions. Collectively, these studies indicate that when key players or processes operate at scales that exceed the grain of observation (e.g., plot size; Fig. 1), results can change significantly (Fig. 2), simply because observational scales fail to match the natural contexts that they are meant to represent (see Levin 1992).

This mismatch may have important practical implications. For example, top-down versus bottom-up field experiments in marine systems suggest that mobile herbivores can mitigate, or even prevent, increases in algal biomass following enrichment (Burkepile and Hay 2006). As a result, coral reef managers may conclude that herbivores alone (if they are not over-exploited) can protect marine systems and their associated services from harmful effects of nutrient enrichment (Bellwood et al. 2004; Burkholder et al. 2007; Hughes et al. 2010). However, our results indicate that observed herbivore control of algal biomass in marine systems could be an artifact of the small spatial scale of field experiments relative to the large movement range of dominant herbivores (Figs. 1, 2).

These experimental biases can be reduced by improving the match between the experimental enrichment and movement patterns of herbivores. For example, systems in which herbivores move over smaller scales (e.g., small-bodied invertebrates) could be studied with less bias. Similarly, increasing the scale of experiments also could reduce the bias [e.g., as in whole lake or watershed experiments (Carpenter and Kitchell 1988; Schindler et al. 2008)], but this remains impractical in many systems, or can be difficult or impossible to replicate. Furthermore, alternative experimental approaches also could reduce bias: e.g., inclusion (rather than exclusion) cages (e.g., Ghedini et al. 2015; Silliman and Bertness 2002) can impose realistic consumer densities and eliminate the influx of consumers from the surrounding landscape into small enriched plots. This approach, however, remains impractical for many systems (e.g., those with large-bodied herbivores) and other potential problems may arise by confining herbivores (e.g., Quinn and Keough 1993).

Spatial scale is just one dimension of the potential problem. Similar challenges, as we have articulated, also exist with respect to the timescale of experiments, which often are much shorter than the timescale of population dynamics (Online Resource 1, Fig. A1). This temporal mismatch probably acts in the opposite direction than the spatial scale mismatch. For example, we would expect herbivore control to increase as their demographic rates change and drive changes in density; i.e., short-term experiments (which preclude demographic responses) likely underestimate potential control of primary producers by herbivores. Thus, the short timescale and small spatial scale of experiments could compensate for one another. Because field exclusion studies typically do not allow for population dynamics of herbivores (e.g., Online Resource 1, Fig. A1), we did not incorporate these population responses into our model. The extent to which the inference provided by our short timescale model will provide realistic insights about natural systems will depend on the potential for responses in the density of herbivores to enrichment. In natural systems, these considerations include:
  1. 1.

    Dispersal of herbivore offspring: if offspring are dispersed widely [as they are in many marine systems (Hixon et al. 2002)] then the local benefits of enrichment will be less likely to translate into local increases in herbivore density.

  2. 2.

    Trophic complexity: the numerical response of herbivores to increased primary production depends on the structure of the upper trophic levels: e.g., in a three-level food chain, increased production results in no change in herbivore density, but instead, an increase in their predator (Oksanen et al. 1981).

  3. 3.

    Interference among herbivores: higher herbivore interference (e.g., due to competition or territoriality) can cause herbivore populations to grow less at higher herbivore densities (Gresens 1995), restricting the numerical response to enrichment over timescales that extend well beyond herbivore generation times. However, interference may similarly limit the immigration and recruitment response to enriched patches over experimental timescales.

  4. 4.

    Feedbacks on consumer behavior and recruitment: herbivory can induce plant defenses (Agrawal 1998) and select for unpalatable or defended plant species (Augustine and McNaughton 1998). Shifts toward less edible plants will reduce herbivory rates as herbivore density increases, and thus limit further increases in herbivore density. Furthermore, nutrients may exacerbate these effects. For example, recent work in coral reefs show that primary producers (i.e., benthic algae) can reach a size at which they become unpalatable to most herbivores (Bellwood et al. 2012; Nyström et al. 2012) and can even reduce herbivore recruitment by producing negative settlement cues and degrading settlement habitat (Dixson et al. 2014; Paddack et al. 2009).


Our study highlights the need for future work to examine numerical responses, both via migration and via population dynamics, in a context that matches data to the scale of interest. Indeed, our results suggest that in many systems herbivores may be less capable of controlling primary producer biomass in natural settings [i.e., when enrichment occurs over square kilometers to hundreds of square kilometers (Bell et al. 2014; Paerl et al. 2006; Penna et al. 2004; Smith et al. 1981)] than would be expected based upon small-scale experiments [with enrichment manipulated at the ≤squared meter scale (Burkepile and Hay 2006; Gruner et al. 2008)]. Our model may further provide a mechanism to explain observed, long-term phase shifts from coral to algae in enriched coral reefs with intact herbivore communities (Hatcher and Larkum 1983; Ledlie et al. 2007; Walker and Ormond 1982), when such responses are not expected based upon experimental studies (e.g., Burkepile and Hay 2006). Consequently, anthropogenic nutrient enrichment, which continues to increase globally (Nixon 2009; Vitousek et al. 1997), could pose a greater threat to natural ecosystems, particularly coastal marine systems, than we previously believed.



Support was provided by a National Science Foundation (NSF) Graduate Research Fellowship (DGE-0802270), a Florida Sea Grant Fellowship, and NSF Grant OCE-1130359. We thank B. R. Silliman, T. Frazer, R. Fletcher, and anonymous reviewers for constructive comments on previous versions of this manuscript, and N. Hackney for assistance with our literature review.

Author contribution statement

M. A. G. and C. W. O. conceptualized the study; M. A. G., C. W. O. and J. J. developed the model; M. A. G. drafted the manuscript, and all authors revised the text.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

442_2015_3505_MOESM1_ESM.pdf (427 kb)
Supplementary material 1 (PDF 427 kb)


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Michael A. Gil
    • 1
  • Jing Jiao
    • 1
  • Craig W. Osenberg
    • 1
    • 2
  1. 1.Department of BiologyUniversity of FloridaGainesvilleUSA
  2. 2.Odum School of EcologyUniversity of GeorgiaAthensUSA

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