Concept of Ecological Efficiency
In a food chain, energy (or biomass) moves upward from producers to consumers. At each transfer, organisms use
a large portion of incoming energy for metabolism (respiration), movement, growth maintenance, and waste, so only a fraction
becomes new biomass available to the next trophic level. This fraction is called trophic transfer efficiency
(also called ecological efficiency in many introductory contexts).
Efficiency per step (η)
The calculator uses a step-by-step efficiency parameter η. You can enter η as a single constant percentage (for a simple model)
or provide a custom η for each transfer (to represent different diets or trophic pathways).
Conversions:
\[
\eta_{\text{fraction}} = \frac{\eta_{\%}}{100}
\]
Core transfer model
If Ek is the energy (or biomass) at trophic level k, then energy passed to the next level is modeled as:
\[
E_{k+1} = E_k \cdot \eta
\]
If efficiency is constant across all steps, the sequence is geometric:
\[
E_k = E_0 \cdot \eta^{k}
\]
If efficiencies differ by step (custom per transfer), use:
\[
E_{k} = E_0 \cdot \prod_{i=1}^{k}\eta_i
\]
Forward mode (start from producers)
In forward mode you provide producer energy/biomass E0 and the calculator computes each successive trophic level
up to the chosen chain length. This is useful for building an energy pyramid and visualizing how quickly energy declines with each transfer.
Loss at each transfer and cumulative loss
The loss from level k to k+1 is computed as the difference:
\[
\text{Loss}_k = E_k - E_{k+1}
\]
The cumulative loss up to level m (from producers to that level) can be expressed as:
\[
\text{Cumulative loss up to level } m = E_0 - E_m
\]
These values help interpret how much energy is “lost” (not passed upward) due to respiration and waste between trophic levels.
Reverse mode (start from a consumer and back-calculate producers)
In reverse mode you specify a target energy for the top consumer (the last trophic level in your chosen chain). The calculator then
estimates the producer energy required to support it, given the efficiencies.
For constant efficiency:
\[
E_0 = \frac{E_{n}}{\eta^{n}}
\]
For custom step efficiencies:
\[
E_0 = \frac{E_{n}}{\prod_{i=1}^{n}\eta_i}
\]
This is commonly used to estimate how much primary production is needed to support a predator population.
Optional: chain length given a minimum energy threshold
Sometimes we want to know how many trophic steps are possible before energy drops below a minimum viable threshold
Emin (for example, the minimum energy needed for a top predator to persist).
With constant efficiency, the inequality:
\[
E_0 \cdot \eta^{k} \ge E_{\min}
\]
can be rearranged to estimate the maximum number of transfers:
\[
k \le \frac{\ln\!\left(E_{\min}/E_0\right)}{\ln(\eta)}
\]
Because k must be a whole number of transfers, the result is typically rounded down to the nearest integer.
With custom efficiencies, there is no single closed-form expression; instead, the calculator checks levels iteratively
until the energy falls below Emin.
Interpreting the 10% rule
The often-cited “10% rule” is a simplified heuristic: many ecosystems show transfer efficiencies near 10% on average, but real values can
range widely depending on organism type, trophic pathway, food quality, temperature, and ecosystem productivity. Use the custom per-step η
option to represent more realistic chains (for example, higher efficiency in aquatic systems or lower efficiency in some terrestrial chains).
Units and what the calculator assumes
The calculator treats your input as a generic transferable quantity (energy in kJ/kcal/J or biomass such as g/kg). It assumes that the same
unit applies consistently at every level and that η represents the fraction transferred between levels. The model does not explicitly separate
assimilation efficiency, production efficiency, or consumption rates; η is a single effective efficiency per transfer.
How the visualizations connect to the math
The energy pyramid displays trophic levels as stacked trapezoids, where width is proportional to energy/biomass at that level.
The step-down bar chart plots the same values as bars to show absolute drops between levels. Hovering over shapes reveals exact numbers,
matching the computed table and the step-by-step calculations.
