Effects of light, nutrients, and food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models

Ecological Modelling 312 (2015) 125–135

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Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel

Effects of light, nutrients, and food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models Angela Peace ∗ National Institute for Mathematical and Biological Synthesis, University of Tennessee, USA

a r t i c l e

i n f o

Article history: Received 16 December 2014 Received in revised form 15 May 2015 Accepted 17 May 2015 Keywords: Ecological stoichiometry Food chain efficiency Carbon use efficiency Trophic transfer efficiency

a b s t r a c t Ecological trophic transfer efficiencies can provide meaningful measures of ecosystem function. Light levels, nutrient availability, and food chain length impact ecological interactions and can cause elemental imbalances between trophic levels which may lead to stoichiometric constraints on food chain efficiencies. Despite the important role that the chemical composition of primary producers and food quality plays in determining consumer productivity, most food chain models used to evaluate trophic transfer efficiencies neglect stoichiometric constraints. This study presents simple stoichiometric models of two and three trophic levels and investigates the effects of light and nutrient availability on ecological transfer efficiencies. The models predict that food chain efficiency is reduced when consumers are nutrient limited. Nutrient levels such that the primary producer and consumer have similar stoichiometric compositions provide conditions for high food chain efficiency. In fixed low nutrient environments, food chain efficiency is highest in light level conditions such that the primary producer and consumer have similar stoichiometric compositions. In fixed high nutrient environments, food chain efficiency is highest for intermediately low light levels such that the phosphorus:carbon ratio of the primary producer is higher than the phosphorus:carbon ratio of the consumer. Food chain efficiency is lower in tritrophic food chains than ditrophic food chains and consumer efficiency is lower in the presence of predation constraints. It is essential for future models to consider light and nutrient availability and the consequential stoichiometric constraints when predicting how energy and elements transfer across trophic levels and up food chains. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Considering the impacts of environmental factors on ecosystem function and their constraints on energy and nutrient transfer through food chains is important for gaining insight on many ecological processes. Predicting amounts of fish harvested as a function of primary production in aquatic systems requires knowledge of the efficiencies of transfer between trophic levels (Kemp et al., 2001). Ecological efficiencies have proved useful in the pursuit of understanding important influences on ecosystem function and the trophic transfer of nutrients and carbon up the food chain (Lindeman, 1942; Hairston Jr and Hairston Sr, 1993; Dickman et al., 2008; Tanaka and Mano, 2012). There are various measures of efficiency useful for understanding trophic transfer and ecological processes. Hairston Jr and Hairston Sr (1993) define the consumption efficiency as the percentage of net production of one trophic

∗ Tel.: +1 480 276 0367. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ecolmodel.2015.05.019 0304-3800/© 2015 Elsevier B.V. All rights reserved.

level that is consumed by the level above it. This efficiency is useful as it reflects the impact consumers have on the trophic levels they feed on. They define the assimilation efficiency as the percentage of consumed energy (carbon) that is assimilated into the trophic level. Classically, trophic transfer efficiency (also referred to as carbon use efficiency) is the rate of production of one trophic level divided by the rate of production of the trophic level immediately underneath it (Sterner and Elser, 2002). Food chain efficiency is the rate of production of the top trophic level divided by the rate of production of the lowest trophic level. Food chain efficiency can be an important indicator of ecosystem processes as it determines the productivity of the top trophic level given the amount of primary productivity (Tanaka and Mano, 2012). Nutrient and light availability can create drastic differences between the chemical compositions of primary producers and consumers which induce stoichiometric constraints on ecological efficiencies. The constraints on predators may be less important as their stoichiometric composition is more similar to that of their prey (Andersen, 1997; Sterner and Elser, 2002). Dickman et al. (2008) provide empirical evidence that light and nutrient

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A. Peace / Ecological Modelling 312 (2015) 125–135

availability constrain trophic transfer efficiencies. Their study found that increased nutrient availability and/or decreased light availability improves food chain efficiency. This empirical evidence is supported by another study where zooplankton (consumer) production increased as light level decreased in a phosphorus limited lake (Urabe et al., 2002). Nutrient/light balance plays an important role in ecological interactions and influences food chain efficiencies. Trophic transfer efficiencies depend on food quality and organismal composition. Despite this fact, most food chain models that evaluate trophic transfer efficiencies focus on how a single constituent, usually carbon or energy, is transferred up the food chain. Tanaka and Mano (2012) built a minimal model of pelagic ecosystems with primary producers, consumers, and predators which predicted that the conversion efficiency of the consumer was one of the most important and general factors for determining food chain efficiencies. A major component of consumer efficiency is the stoichiometric composition of the primary producer, which is highly dependent on nutrient and light availability. Incorporating the effects of multiple constituents and nutritional food quality into food chain models may more accurately capture population dynamics and lead to improved predictions and understandings of trophic transfer efficiencies and ecosystem processes. The theory of ecological stoichiometry, which considers the balance of energy and multiple chemical elements in ecological interactions, provides new constraints and mechanisms that can be formulated into mathematical models (Sterner and Elser, 2002; Andersen et al., 2004; Moe et al., 2005; Hessen et al., 2013). Modeling under the framework of ecological stoichiometric allows the investigation of food quantity, as well as, food quality on food web population dynamics. Many models that incorporate stoichiometric constraints have produced rich dynamics, making qualitatively different predictions on population dynamics, community structure, and the effects of environmental perturbations compared to nonstoichiometric models (Andersen, 1997; Loladze et al., 2000; Muller et al., 2001; Grover, 2004; Andersen et al., 2004; Hall et al., 2007; Elser et al., 2012; Peace et al., 2013; Hessen et al., 2013). A stoichiometric food chain model can incorporate the consequences of elemental imbalances between trophic levels when determining ecological efficiencies. This study considers simple stoichiometric models of two and three trophic levels to investigate the impacts of environmental factors on ecosystem function. The models use ecological trophic transfer efficiencies as important gauges of ecosystem function in order to determine the effects of nutrient enrichment, light availability, and food chain length. The models are used to test two hypotheses: (1) food chain efficiency is highest under light and nutrient conditions such that the stoichiometric composition of the primary producer is near that of the consumer; and (2) transfer efficiency from primary producer to consumer is lower in chains of three trophic levels than in chains of only two trophic levels due to predation constraints.

2. Methodology 2.1. Mathematical models Loladze et al. (2000) formulated a producer–consumer Lotka–Volterra type model (LKE model) of the first two trophic levels of an aquatic food chain (algae-Daphnia) incorporating the fact that both producers and consumers are chemically heterogeneous organisms composed of two essential elements, carbon (C) and phosphorus (P). The model allows the phosphorus to carbon ratio (P:C) of the producer to vary above a minimum value. This variable P:C ratio of the producer, denoted as Q, brings food

quality into the model. Below is the LKE model from Loladze et al. (2000):



dx x = bx 1 − dt min{K, (P − y)/q} dy = eˆ min dt



1,

Q 





− f (x)y

f (x)y − dy

(1a) (1b)

where Q =

P − y . x

x(t) and y(t) are the biomass of the producer and consumer respectively, measured in terms of C. b is the maximum growth rate of producer, K is the light dependent producer carrying capacity in terms of C, P is the total phosphorus in the system,  is the consumer’s constant P:C, Q is the producer’s variable P:C ratio, q is the producer’s minimal P:C, eˆ is the maximum production efficiency, and d is the consumer loss rate. The consumer’s ingestion rate, f(x) is taken to be a monotonic increasing and differentiable function, f (x) ≥ 0, f(0) = 0. f(x) is saturating with limx→∞ f (x) = fˆ . The model makes the following three assumptions. A1: The total mass of phosphorus in the entire system is fixed, i.e., the system is closed for phosphorus with a total of P (mg P/L). A2: P:C ratio in the producer varies, but it never falls below a minimum q (mg P/mg C); the consumer maintains a constant P:C,  (mg P/mg C). A3: All phosphorus in the system is divided into two pools: phosphorus in the consumer and phosphorus in the producer. Here, a minimum function is used to describe the producer carrying capacity, min {K, (P − y)/q}. The first input, K, is the carrying capacity determined by light availability. The second input, (P − y)/q is the carrying capacity determined by phosphorus availability. Another minimum function is used to describe the consumer growth rate, min {1, Q/}. The first input, 1, is used when consumer growth is limited by carbon. The second input, Q/ is used when consumer growth is limited by phosphorus. 2.1.1. Ditrophic model In order to investigate the effects of light and nutrients on systems of two trophic levels this study uses the LKE model (1) with a slight modification. A portion of the ingested carbon is used for the consumer’s metabolic costs, such as respiration. Let ey be the consumer’s maximal production efficiency in terms of carbon. Then Q/ey is the P:C ratio of the post-ingested producer representing the amount of P and C available for consumer growth. Let  y be the constant P:C ratio of the consumer and g(x, y) the consumer growth rate. When (Q/ey ) >  y , the growth of the consumer is limited by carbon and satisfies g(x, y) = f(x)ey . However when (Q/ey ) <  y , the growth of the consumer is limited by phosphorus and satisfies g(x, y) y = f(x)Q. Modifying the consumer’s growth yields the following ditrophic model: dx = bx dt



dy = min dt

1−



x min{K, (P − y y)/q}

Q ey , y



− f (x)y

(2a)



f (x)y − dy y

(2b)

where Q =

P − y y , x

ey is the consumer maximal production efficiency in terms of carbon, dy is the consumer loss rate, and  y is the consumer’s constant P:C ratio. Its important to note that this ditrophic model is simply

A. Peace / Ecological Modelling 312 (2015) 125–135

the LKE model (1) with a slight modification of the assimilation efficiency in the expression for consumer’s growth. 2.1.2. Tritrophic model In order to investigate the effects of light and nutrients on systems of three trophic levels, the ditrophic models (System (2)) is expanded to incorporate a predator (z) with a fixed P:C ratio ( z ). The second and third assumptions are modified: A2: P:C ratio in the producer varies, but it never falls below a minimum q (mg P/mg C); the consumer maintains a constant P:C,  y (mg P/mg C); the predator maintains a constant P:C,  z (mg P/mg C). A3: All phosphorus in the system is divided into three pools: phosphorus in the producer, phosphorus in the consumer, and phosphorus in the predator. The tritrophic model takes the following form: dx = bx dt



dy = min dt dz = min dt

x 1− min{K, (P − y y − z z)/q}



ey ,



Q y

y ez , z

− f (x)y

(3a)



f (x)y − g(y)z − dy y

(3b)

Q¯ =

g(y)z − dz z

P − y y¯ x¯

and

Q¯ =

P − y y¯ − z z¯ x¯

(4)

for the ditrophic (System (2)) and tritrophic (System (3)) models respectively. 2.3.1. Trophic transfer efficiencies Let CE be the consumer efficiency, defined as the consumer’s rate of production in terms of carbon divided by the producer’s rate of production in terms of carbon: min{ey , Q¯ /y }f (¯x)y¯ . b¯x

(5)

Let PE be the predator efficiency, defined as the predator’s rate of production in terms of carbon divided by the consumer’s rate of production in terms of carbon: PE =



¯ z min{ez , y /z }g(y)¯ . ¯ min{ey , Q /y }f (¯x)y¯

(6)

(3c)

where Q =

These trophic efficiency definitions assume equilibrium con¯ and (¯x, y, ¯ z¯ ) be stable equilibrium values for ditions. Let (¯x, y) the ditrophic (System (2)) and tritrophic (System (3)) models respectively. Scenarios when there are no stable equilibria and ¯ and (¯x, y, ¯ z¯ ) are taken as the systems exhibit limit cycling, (¯x, y) averages of the values from the cycles. Using these conditions, the producer P:C ratio at equilibrium conditions becomes

CE =



127

P − y y − z z , x

ez is the predator maximal production efficiency in terms of carbon, dz is the predator loss rate, and  z is the predator’s constant P:C ratio. The predator’s ingestion rate, g(y) is assumed to have a similar form as the consumer’s ingestion rate, f(x). In the following analysis, both f(x) and g(y) are taken to be holling type II functions. Its important to note that this tritrophic model is simply an expansion of the two species LKE model (1) to include three species in a linear food chain. While the above ditrophic model (2b) and tritrophic model (3) only slightly modify the model presented by Loladze et al. (2000), the following investigation of ecological efficiencies in stoichiometric models is novel to this work. 2.2. Parameterization The models are parameterized for simple aquatic food chains. The primary producer x is parameterized for algae, consumer y is the zooplankton Daphnia, and predator z is assumed to be planktivorous fish, such as larval gizzard shad which feeds on zooplankton (Dickman et al., 2008). Parameter values and sources are presented in Table 1. 2.3. Ecological efficiencies Trophic transfer efficiency is defined as the rate of production of one trophic level divided by that rate of production of the trophic level immediately underneath it (Sterner and Elser, 2002). This is used to define trophic transfer efficiencies between the producer and consumer, as well as, between the consumer and the predator. Food chain efficiency (FCE) is the proportion of carbon fixed by the primary producer that is transferred to the top trophic level (Hairston Jr and Hairston Sr, 1993; Dickman et al., 2008). Since this carbon is transferred through the food chain to the top trophic level it depends on the trophic transfer efficiencies at each level.

2.3.2. Food chain efficiency Let FCE be the food chain efficiency, defined as the top predator’s rate of production in terms of carbon divided by the primary producer’s rate of production in terms of carbon. Since the ditrophic model (System (2)) assumes only two trophic levels, the FCE is equivalent to the CE (Eq. (5)). For the tritrophic model (System (3)) the FCE takes the following form: FCE =

¯ z min{ez , y /z }g(y)¯ . b¯x

(7)

3. Model analysis 3.1. Analysis of ditrophic model The ditrophic model (System (2)) exhibits similar dynamics to the LKE model (System (1)), since there is simply a small modification between the two. Loladze et al. (2000) provide good details on the analysis of the LKE model. They prove boundedness and invariance of the system as well as investigate the complex dynamics that lead to multiple positive equilibria and show that bistability and deterministic extinction of the consumer are possible. Li et al. (2011) presented further analysis of the LKE model including a global analysis for the model and a robust bifurcation analysis of the light dependent carrying capacity, K. The effects of the nutrient and light enrichment on the dynamics of the ditrophic model (System (2)) can be seen in the phase portrait (Fig. 1). The number and nature of the equilibria are captured in the intersections of the two nullclines, where the size of the producer and consumer populations remain constant. Similar to the producer nullcline, the consumer nullcline is humped-shaped due to the stoichiometric constraints incorporated into the model. P enrichment raises both nullclines, increasing their respective maxima (Fig. 1a). Light (K) enrichment only affects the producer nullcline (Fig. 1b). Under high K the intersection of the nullclines are in ranges of high producer density. We refer to Loladze et al. (2000) and Li et al. (2011) for a detailed bifurcation analysis under light enrichment. Varying nutrients (P) and/or light (K) change the producer P:C quota (Q) which results in bifurcations to the system. Increasing P increases Q, whereas, increasing K decreases Q. For high values

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A. Peace / Ecological Modelling 312 (2015) 125–135

Table 1 Model parameters.  Values are biologically realistic values obtained from Andersen (1997) and Urabe et al. (1996) and used by Loladze et al. (2000) and Peace et al. (2013). * Values are realistic for most planktivorous fish according to Leidy and Jenkins (1977). ♦ Parameterization of  z is in the range of juvenile gizzard shad P:C ratios presented in Dickman et al. (2008). • Values are in the ranges of parameter values compiled by MacKenzie et al. (1990) for larval fish ingestion rates. ◦ Parameterization of dz includes both natural mortality and respiration rates from Leidy and Jenkins (1977). Parameter b K dy ey ez P y z q

Maximal growth rate of producer Producer carrying capacity Consumer loss rate Maximal consumer production efficiency Predator maximal production efficiency Total phosphorus Consumer constant P:C Predator constant P:C Producer minimal P:C

f(x)

Consumer ingestion rate

cy ay

Maximal ingestion rate of the consumer Half saturation of the consumer ingestion response

Value

Source

1.2/day 0–3 mg C/L 0.25/day 0.8 (unitless) 0.8 (unitless) 0.01–0.08 mg P/L 0.03 mg P/mg C 0.04 mg P/mg C 0.0038

mg P/mg C

   

0.81/day

0.25 mg  C/L

 

0.81/d 0.25 mg C/L 0.25/day

• • ◦

cy x ay +x

cz y az +y

g(y)

Predator ingestion rate

cz az dz

Maximal ingestion rate of the predator Half saturation of the predator ingestion response Predator loss rate

*

  ♦ 

/day

/day

(a) (b)

Fig. 1. Phaseplane of ditrophic model (System (2)) for (a) varying nutrient levels (P = 0.01, 0.02, 0.03 mg P/L) with constant light level (K = 1.5 mg C/L) and (b) varying light levels (K = 0.25, 0.5, 1, 1.5 mg C/L) with constant nutrient level (P = 0.03). Dashed lines are producer nullclines, solid lines are consumer nullclines. Other parameter values are listed in Table 1.

of Q the system exhibits a positive stable coexistence equilibrium. As Q (food quality) decreases there is a Hopf bifurcation and limit cycles emerge. As Q continues to decrease the limit cycles collapse near a saddle-node bifurcation and there is a second coexistence equilibrium. Decreasing Q further will decrease the consumer population density until food quality becomes low enough to lead to deterministic extinction of the consumer. Consumer efficiency for varying nutrient and light enrichment are presented in Fig. 2. Under low P availability CE is zero since there is not enough P to support consumer growth. As P increases the CE begins to increase slowly. As P continues to increase there is drastic increase is the CE then it quickly levels off above 0.4 (Fig. 2a). Under low light availability (low K) the CE is zero since there is not enough carbon to support consumer growth. As K increases the CE increases rapidly to its maximum value. As K continues to increase the CE begins to decrease. The CE decreases rapidly near K = 0.75 where there is a saddle-node bifurcation and continues to decrease to very low values for high levels of light (Fig. 2b). 3.2. Analysis of tritrophic model Similar to the phase plane analysis of the ditrophic model (Fig. 1), the effects of nutrient and light enrichment on the dynamics of the tritrophic model (System (3)) can be seen in the three dimensional phase portraits (Figs. 3 and 4). The nullsurfaces are also humped shaped. P enrichment increases the maximum peaks

of the producer nullsurfaces with respect to consumer dynamics (Fig. 3a), as well as, increases the maximum peaks of the consumer nullsurfaces with respect to predator dynamics (Fig. 3b). Under P enrichment, nullsurfaces will intersect in ranges with higher consumer and predator biomass. Therefore, as P increases, the biomass of consumers and predators at interior equilibria increases. Light enrichment only affects the producer nullsurface (Fig. 4). Under high K the intersection of the nullsurfaces are in ranges of high population densities. Similar to the dynamics of the ditrophic model, varying nutrients (P) and/or light (K) change the producer P:C quota (Q) which results in bifurcations to the system. The bifurcation diagrams shown in Fig. 5 exhibit period doubling and chaotic dynamics as light levels vary. These complex dynamics can make it difficult to empirically measure, predict and understand FCEs for systems with multiple trophic levels. Ecological transfers efficiencies under varying nutrients and light availability for the tritrophic model are presented in Fig. 6. While CE increases with P enrichment for both models (Figs. 6a and 2a), CE is higher for the ditrophic model. The qualitative dynamics of CE under light K enrichment are similar for both models (Figs. 6d and 2b), however CE is higher for the ditrophic model. These results support the second hypothesis as CE is lower when consumers are constrained by predators. Under low nutrient availability PE and FCE are zero since there is not enough P to support predator growth. As P increases both PE and FCE begin to increase then decrease before leveling off at

A. Peace / Ecological Modelling 312 (2015) 125–135

129

Fig. 2. Food chain efficiency (Eq. (5)) of the ditrophic model (System (2)) for (a) varying nutrient levels with fixed light K = 1.5 mg C/L and (b) varying light levels with fixed nutrient P = 0.02 mg P/L. Here CE assumes equilibrium conditions. Black points are when stable equilibrium values were found. Gray points are when the system exhibits limit cycling and averages over the cycles are used to define the CE. Since there are only two trophic levels here, the CE is equivalent to the FCE. Other parameter values are listed in Table 1.

Fig. 3. Producer (a) and consumer (b) nullsurfaces of tritrophic model (System (3)) for low nutrient levels (P = 0.02 mg P/L) dark gray surface, intermediate nutrient levels (P = 0.0275 mg P/L) gray surface, and high nutrient levels (P = 0.04 mg P/L) and K = 1.5 mg C/L light gray surface. Other parameter values are listed in Table 1.

intermediate values (Fig. 6b and c). Under low light availability PE and FCE are zero since there is not enough carbon available to support predator growth. As K increases both PE and FCE increase, as K continues to increase the dynamics get complicated, however, generally PE and FCE begin to decrease as K increases to high values (Fig. 6e and f). 3.3. Model predictions compared to data

Fig. 4. Phasespace of tritrophic model (System (3)) for varying light levels (K = 0.5 dark gray surface, K = 1 mg C/L gray surface, K = 1.5 mg C/L light gray surface) with constant nutrient level (P = 0.04).Other parameter values are listed in Table 1.

Dickman et al. (2008) present empirical data from aquatic mesocosm studies that explicitly quantifies how light and nutrients interactively influence trophic transfer efficiencies. Throughout their experiments, they manipulated light levels, nutrient supply, and the presence of planktivouros fish (predator) in 5000 L mesocosms containing local phytoplankton (producer) and zooplankton

Fig. 5. Bifurcation diagrams of the tritrophic model (System (3)) with bifurcation parameter K (light level) for (a) producer, (b) consumer and (c) predator. Parameter values are listed in Table 1 and P = 0.04 mg P/L.

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 6. Consumer efficiency (Eq. (5)), Predatory efficiency (Eq. (6)), and Food chain efficiency (Eq. (7)) of the tritrophic model (System (3)) for varying nutrient levels with fixed light K = 1.5 mg C/L (a–c) and for varying light levels with fixed nutrient P = 0.02 mg P/L (d–f). Trophic transfer efficiencies assume equilibrium conditions. Black points are when stable equilibrium values were found. Gray points are when the system exhibits limit cycling and averages over the cycles are used to define the trophic transfer efficiencies. Other parameter values are listed in Table 1.

(consumer) communities for a duration of eight weeks. They used laval gizzard shad (Dorosoma cepedianum) as the predator, which are carnivorous during early life stages and mostly feed on zooplankton. In total the experiment had eight treatments: two levels

(b)

(a)

(c)

of light, two levels of nutrients, and two or three trophic levels, each replicated three times. They measured production of each trophic level and calculated trophic transfer efficiencies. Fig. 7 shows production data from Dickman et al. (2008) for each trophic level under

(d)

(e)

Fig. 7. Production of (a) producer x and (b) consumer y in a ditrophic system. Production of (c) producer x, (d) consumer y, and (e) predator z in a tritrophic system. Black dots are empirical data of phytoplankton (producer), zooplankton (consumer), and larval gizzard shad (predator) production from mesocosm experiments by Dickman et al. (2008). Gray bars show model predictions under high light K = 1.5 mg C/L, low light K = 0.5 mg C/L, high nutrient P = 0.06 mg P/L, and low nutrient P = 0.02 mg P/L conditions. Model (2b) was used for (a and b) and model (3c) was used for (c–e). Model predictions are at equilibrium conditions.

A. Peace / Ecological Modelling 312 (2015) 125–135

131

(a)

(c)

(b)

(d)

Fig. 8. Trophic transfer efficiencies. (a) Food chain efficiency Eq. (7) of tritrophic system. Consumer efficiency Eq. (5) of (b) tritrophic system and (c) ditrophic system. (d) Predator efficiency Eq. (6) of tritrophic system. Black dots are calculated from empirical data of phytoplankton (producer), zooplankton (consumer), and larval gizzard shad (predator) production from mesocosm experiments by Dickman et al. (2008). Gray bars show model predictions under high light K = 1.5 mg C/L, low light K = 0.5 mg C/L, high nutrient P = 0.06 mg P/L, and low nutrient P = 0.02 mg P/L conditions. Model (2b) was used for (a, b and d). Model was (3c) used for (c). Model predictions are at equilibrium conditions.

the varying light, nutrient, and food chain length conditions along with the predictions from the ditrophic and trophic models (2) and (3). While the model predictions and data are within similar biological ranges (see vertical axes scales in Fig. 7), quantitatively, the models predict higher production than measured in the mesocosm experiments. It is important to note that these predictions depend on the values of K and P used for high/low light and nutrient conditions. Some of the discrepancies between the model predictions and data may be due to some of the simplifying assumptions of the models. The models assume light and nutrient conditions are constant and everything is well-mixed, neglecting depth shading. The model only considers the effects of phosphorus whereas Dickman et al. (2008) manipulated both phosphorus and nitrogen across nutrient treatments.

The largest discrepancies between the model and data are seen for producer (x) production under high light and low nutrient conditions (Fig. 7a and d), where the model predicts highest producer production. These high light and low nutrient conditions correspond to a low producer P:C ratio (Q) and thus nutrient deficient low quality food (Sterner and Elser, 2002; Loladze et al., 2000; Urabe et al., 2002; Hessen et al., 2002). This low quality food results in low consumer (y) growth and low predation pressure on the producer (x), allowing for high producer production. This is not observed in the data, perhaps due to community diversity. The models assume the producer and consumer trophic levels are simply composed of only one species each, whereas the empirical data is from trophic communities consisting of multiple species. Dickman et al. (2008) found the consumer zooplankton community diverse, including cladocerans, copepods, nauplii, and rotifers.

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Fig. 9. Food chain efficiencies (FCE) for the ditrophic and tritrophic models for varying nutrient (P) and under low (K = 0.5 mg C/L) and high (K = 1.5 mg C/L) light levels. Dashed lines represent the P values when producer P:C ratio at equilibrium conditions is equal to the constant consumer P:C ratio. The shaded regions are when consumer growth is limited by P. FCEs assume equilibrium conditions. Black points are when stable equilibrium values were found. Gray points are when the system exhibits limit cycling and averages over the cycles are used to define the trophic transfer efficiencies. Parameter values are listed in Table 1.

The models were parameterized for a single consumer, Daphnia, with a fixed stoichiometric ratio, . The diversity of the consumer trophic levels in the experiment may result in less sensitivity to low quality food conditions. There are similar differences between the model assumptions and the empirical data for the producer trophic level. Dickman et al. (2008) calculated trophic transfer efficiencies from the collected production data. Fig. 8 presents both the trophic transfer efficiencies calculated from the data and those predicted by the models for each trophic level under the varying light, nutrient, and food chain length conditions. The model predictions of the efficiencies are in the same range as those calculated by Dickman et al. (2008) from their empirical data (see vertical axes scales in Fig. 8). The data and model produce comparable values for FCE (Fig. 8a). The second hypothesis is supported by both the model and the data as CE is lower when consumers are constrained by predators (see Fig. 8b for tritrophic CE and Fig. 8c for ditrophic CE). The biggest discrepancies between the data and model are for the PE (Fig. 8d) and the ditrophic CE under low light conditions (Fig. 8c). In both these cases, the data show efficiencies greater than one. Dickman et al. (2008) explain that they calculated PE > 1 possibly because toward the end of the experiment the larval gizzard shad may have been consuming some algae, since mature gizzard shad are typically omnivores. They calculated CE > 1 in low light conditions, possibly because the zooplankton were

eating foods other then phytoplankton, such as other zooplankton, periphyton, and bacteria. 4. Discussion It is understood that heterotrophic consumers tend to be richer in P and have smaller stoichiometric variability than autotrophs (Sterner and Elser, 2002; Sterner and Hessen, 1994). Stoichiometric composition of primary producers can vary widely compared to consumers. Indeed, it is well known that algal P:C ratios (Q) vary with P and light availability whereas the P:C ratios of Daphnia ( y ) are generally higher and less variable (Sterner and Elser, 2002). Increasing P causes Q to increases, whereas, increasing light (K) causes Q to decrease (Loladze et al., 2000; Urabe et al., 2002; Hessen et al., 2002). 4.1. Effects of nutrient enrichment When nutrient levels are low, consumers and predators are limited by low quality food. Low P levels reduce Q¯ so that consumers’ growth is limited by P, which affects FCEs. Fig. 9 shows FCE for both models under P enrichment for low and high light environments. FCE is lowest in the shaded region where consumer growth is limited by P since Q¯ < y . In this shaded region, Increasing

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133

Fig. 10. Food chain efficiencies (FCE) for the ditrophic and tritrophic models for varying light (K) levels under low (P = 0.02 mg P/L) and high (P = 0.04 mg P/L) nutrient levels. Dashed lines represent the K values when producer P:C ratio at equilibrium conditions is equal to the constant consumer P:C ratio. The shaded regions are when consumer growth is limited by P. FCEs assume equilibrium conditions. Black points are when stable equilibrium values were found. Gray points are when the system exhibits limit cycling and averages over the cycles are used to define the trophic transfer efficiencies. Parameter values are listed in Table 1.

P increases Q toward  y and FCE increases. Under low light, FCE quickly reaches a plateau after Q¯ = y in both models. Under high light in the tritrophic model, FCE reaches a maximum values near Q¯ = y then declines and levels off. These results provide partial support for hypothesis 1; FCE is highest for nutrient conditions such that the stoichiometric composition of the primary producer is near that of the consumer.

4.2. Effects of light enrichment Fig. 10 shows FCE for both models under K enrichment for low and high nutrient environments. Under very low light availability (low K) consumer and predator biomass and thus FCE are limited by carbon availability. Increasing K quickly raises FCE to its maximum values, which occurs in both models near the Hopf bifurcations where limit cycles emerge. In low nutrient environments, FCE is highest for light conditions such that the stoichiometric composition of the primary producer is near that of the consumer, supporting hypothesis 1. In high nutrient environments, while there are some complex dynamics during the limit cycles, generally increasing K reduces FCE. Here increasing K decreases Q¯ toward  y . Once K is large enough that Q¯ = y the consumer becomes limited by P and FCE reduces rapidly (shaded region). In this shaded region, the high light conditions cause growth to be limited by P since Q¯ < y . The behavior of the FCE in this region partially

supports hypothesis 1, since FCE decreases as Q¯ reduces away from y . The behavior of the FCE for low levels of K in high nutrient environments does not support the hypothesis. Here, the largest values of FCE are achieved under intermediately low light conditions near the Hopf bifurcations. This optimum occurs when Q¯ > y . These results may be partially explained by the stoichiometric constraints in the models. These models incorporate the consequences of Plimitation on consumer and predator growth, however, they do not incorporate any consequences for excess levels of P. Recent evidence strongly indicates there are reductions in consumer growth due to nutrient-rich food (Boersma and Elser, 2006; Elser et al., 2006, 2012). Stoichiometric models that include consequences on growth due to excess nutrients have been developed (Peace et al., 2013, 2014). Such models will likely predict reduced FCEs for high values of Q¯ .

4.3. Effects of food chain length Food chain length determines predation constraints on consumers, which has implications for ecological transfer efficiencies. There are several qualitative similarities between the behaviors of FCEs in the ditrophic and tritrophic models under nutrient and light enrichment (Figs. 9 and 10), however the FCEs for the tritrophic model are an order of magnitude lower than the FCEs in the

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ditrophic model (note the difference in scale on the vertical axes in Figs. 9 and 10). FCE depends on the ecological efficiencies are every trophic coupling (Hairston Jr and Hairston Sr, 1993; Dickman et al., 2008). The number of trophic couplings affects transfer efficiencies between each trophic level and this influences FCE. The models predict that CE is higher in the absence of predators, supporting hypothesis 2 (Figs. 2 and 6a, b). Transfer efficiency from primary producer to consumer, CE, decreases as predation constraints on consumers increase. In a two level food chain, consumers are not constrained by predators and their rate of production relative to the rate of production of primary producers is high. In a three level food chain, predation pressure mediates consumer density and this mediates CE. In a four level food chain, consumers are likely less constrained by predators than the three level food chain due to predation constraints on the third level. Ecological transfer efficiencies depend on food chain length and whether the chain is made of an odd or even number of levels (Hairston Jr and Hairston Sr, 1993). Overfishing and harvesting practices and environmental conditions may lead to changes in food chain length by creating environments where species are vulnerable to extinction or invasion, and thus lead to possible implications on transfer efficiencies. 4.4. Model limitations The above models allow the stoichiometric composition of primary producers, Q, to vary depending on nutrient and light availability. However, they assume that the stoichiometric composition of higher trophic levels,  y , and  z are constant (Assumption 2). This strict homeostasis assumption is based on the fact that, although consumers and predators have variable stoichiometries, the range of variability is much smaller compared to the range of variability for stoichiometries of primary producers. It has been suggested that changes in consumer P:C ratio influences PE and thus FCE (Malzahn et al., 2007; Dickman et al., 2008). Wang et al. (2012) relax this strict homeostasis assumption and investigate how variable consumer stoichiometries affect population dynamics. Developing stoichiometric food chain models following the approach of Wang et al. (2012) may lead to further insight on how ecological transfer efficiencies depend on nutrients, light, and food chain length. The simple aquatic food chains models used here do not incorporate detritus and pelagic–benthic dynamics, which may play an important role for aquatic ecosystems (Vadeboncoeur et al., 2002; Vander Zanden et al., 2006). Stoichiometric food chain models that incorporate this pathway may lead to further insight. The ditrophic and tritrophic models assume that all phosphorus in the systems in located inside the organisms and do not allow for free nutrients to be in the environment (Assumption 3). This assumption is based on the fact that algae take up nutrient quickly and P only spends a short time in the environment. This assumption may lead to consequences when nutrient pools in the environment are important to the dynamics, as in terrestrial food chains. In such cases, stoichiometric models that explicitly track free nutrients in the environment may be more appropriate (Kuang et al., 2004; Wang et al., 2008; Peace et al., 2014). The models here made use of minimum function as thresholds to incorporate limitations by carbon or phosphorus. These minimum functions follow from Justin Leibig’s law of the minimum, which states that an organism’s growth will be limited by whichever resource is in lowest supply relative to the organism’s needs (Sterner and Elser, 2002). This minimum law applies to an individual organism and our models are at the scale of a population, therefore, our uses of minimum functions present approximations of the population dynamics. An alternative approach is to use synthesizing units (Muller et al., 2001; Kooijman et al., 2004). The behavior of a synthesizing unit is similar to Leibig’s law of the

minimum except for a narrow region near the threshold where several substrates can limit growth simultaneously (Kooijman et al., 2004).

5. Conclusion This study provides evidence that nutrient, light, and food chain length affect ecological transfer efficiencies. Regardless of food chain length, both nutrient and light conditions such that the stoichiometric composition of primary producers is phosphorus limited compared to the stoichiometric composition of consumers (Q¯ < y ) reduce FCEs. The models predict that FCE is highest under nutrient conditions such that the stoichiometric composition of primary producers is near that of consumers (Q¯ ≈ y ). In low nutrient environments the models predict that FCE is highest under light conditions such that these trophic levels are similar in stoichiometric composition, however, in high nutrient environments the models predict that FCE is highest under intermediately low light conditions, where Q¯ > y . The models predict lower FCE in three level food chains then two level food chains. It is becoming crucial to be able to accurately predict how ecological transfer efficiencies will response to climate perturbations. Light levels, nutrient availability, and food chain length need to be considered when making predictions as they play an important role in determining stoichiometric constraints that influence the transfer of energy and elements across trophic levels and up food chains.

Acknowledgements This work was conducted while a Postdoctoral Fellow at the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation through NSF Award #DBI-1300426, with additional support from The University of Tennessee, Knoxville. The author thanks two anonymous reviewers for their helpful comments that improved this manuscript and MJ Vanni for providing data from Dickman et al. (2008).

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