4.5 Marine Food Webs

Step 1: Energy budget of a consumer

Consider a consumer at trophic level $n+1$ that ingests food at rate $I_n$ (energy per unit time from trophic level $n$). The ingested energy is partitioned into assimilated energy and egestion (feces):

$$I_n = A + F_{\text{eg}}$$

where $A$ is assimilated energy and $F_{\text{eg}}$ is egested energy. The assimilation efficiency is $\text{AE} = A / I_n \approx 0.6\text{-}0.9$.

Step 2: Partition assimilated energy

The assimilated energy is further divided into respiration $R$ (metabolic costs), excretion $U$ (dissolved losses), and production $P_{n+1}$ (growth + reproduction):

$$A = R + U + P_{n+1}$$

Step 3: Define trophic transfer efficiency

The trophic transfer efficiency (TTE) is the ratio of production at level $n+1$ to production at level $n$. It can be decomposed into three component efficiencies:

$$\text{TTE} = \frac{P_{n+1}}{P_n} = \underbrace{\frac{I_n}{P_n}}_{\text{exploitation}} \times \underbrace{\frac{A}{I_n}}_{\text{assimilation}} \times \underbrace{\frac{P_{n+1}}{A}}_{\text{net growth}}$$

Step 4: Estimate typical values

Each component typically contributes a fraction less than 1. For a marine zooplankton grazer: exploitation efficiency $\approx 0.4$, assimilation efficiency $\approx 0.7$, net growth efficiency $\approx 0.35$:

$$\text{TTE} \approx 0.4 \times 0.7 \times 0.35 \approx 0.10$$

This yields the classic "10% rule" -- only about 10% of energy at one trophic level is converted to production at the next level.

Step 5: Production available at trophic level $k$

After $k$ trophic transfers from a base of primary production $P_0$, the production available is:

$$P_k = P_0 \cdot \text{TTE}^k = P_0 \cdot (0.10)^k$$

For example, from 50 Gt C/yr of marine primary production: tuna at TL 4 receive $50 \times 10^{-3} = 0.05$ Gt C/yr. This explains why shorter food chains (e.g., sardines at TL 2-3) support far greater fishery yields than long chains (e.g., tuna at TL 4-5).

Step 1: Lindeman's progressive efficiency

Lindeman (1942) defined the efficiency at each trophic level as the ratio of energy intake at level $n$ to energy intake at level $n-1$:

$$\lambda_n = \frac{\Lambda_n}{\Lambda_{n-1}}$$

where $\Lambda_n$ is the total energy assimilated at trophic level $n$.

Step 2: Thermodynamic constraint (Second Law)

The Second Law of Thermodynamics requires that entropy increase with each energy transformation. Not all assimilated energy can be converted to biomass -- a substantial fraction must be dissipated as heat through respiration:

$$\Lambda_n = R_n + P_n + U_n, \quad \text{with } R_n > 0 \quad \Longrightarrow \quad \lambda_n < 1$$

Step 3: Relate Lindeman efficiency to food web length

The total energy reaching trophic level $N$ from the primary production base $\Lambda_1$ is:

$$\Lambda_N = \Lambda_1 \prod_{n=2}^{N} \lambda_n$$

Step 4: Maximum food chain length

A trophic level can only sustain a viable population if $\Lambda_N$ exceeds some minimum threshold $\Lambda_{\min}$. This sets a maximum chain length:

$$N_{\max} = 1 + \frac{\ln(\Lambda_1 / \Lambda_{\min})}{\ln(1/\bar{\lambda})}$$

where $\bar{\lambda}$ is the geometric mean Lindeman efficiency. With $\bar{\lambda} \approx 0.10$, each additional trophic level requires 10x more primary production, limiting most marine food chains to 4-5 levels.

Step 5: Lindeman's observation of increasing efficiency

Lindeman noted that $\lambda_n$ tends to increase with trophic level (e.g., $\lambda_2 \approx 0.05$, $\lambda_3 \approx 0.15$, $\lambda_4 \approx 0.20$). This arises because higher predators are more mobile and better at locating and consuming prey, leading to higher exploitation efficiency. The overall pattern is bounded by:

$$\lambda_n \leq \text{AE}_n \times \text{NGE}_n \leq 0.9 \times 0.5 = 0.45$$

where AE is assimilation efficiency (bounded by digestive capacity) and NGE is net growth efficiency (bounded by metabolic costs). In practice, marine Lindeman efficiencies range from 5% to 25%.

Step 6: Implications for fisheries yield

Ryther (1969) used Lindeman efficiency to estimate potential fisheries production. Starting from net primary production and applying ecosystem-specific chain lengths and efficiencies:

$$Y = \text{NPP} \times e \times \prod_{n=2}^{N} \lambda_n$$

where $e$ is the export ratio and $Y$ is fisheries yield. Upwelling systems (short chains, $N = 3$, high $\lambda$) produce ~50% of world fish catch from <1% of ocean area, while open-ocean gyres (long chains, $N = 5$, low $\lambda$) are comparatively unproductive.