Formula Sheet: Ecological Values and Equations

Formula	Meaning	Variables
r = b - d	Intrinsic rate of natural increase	b = birth rate; d = death rate
dN/dt = rN	Exponential growth rate	r = intrinsic rate; N = population size
N_t = $N_{0}$ e^(rt)	Exponential growth (integrated)	$N_{0}$ = initial size; t = time; e ≈ 2.718
dN/dt = rN(K-N)/K	Logistic growth rate	K = carrying capacity
Max growth rate at N = K/2	Inflection point of sigmoid	dN/dt maximized when N = K/2
N = (n_{1} × n_{2})/m	Mark-recapture population estimate	n_{1}, n_{2} = sample sizes; m = recaptured marked

Value	Meaning	Source
~10%	Energy transferred between trophic levels	Lindeman's efficiency (10% law)
~90%	Energy lost as heat at each trophic level	Respiration and metabolic processes
T_(n) = $T_{1}$ × (0.1)^(n-1)	Energy at trophic level n	n = trophic level number
$T_{4}$ energy from $T_{1}$	$T_{1}$ × (0.1)^{3} (three transfers)	Three steps from $T_{1}$ to $T_{4}$

Formula	Meaning
NPP = GPP - Respiration	Net Primary Productivity
GPP = NPP + Respiration	Gross Primary Productivity (rearranged)

Value	Ecological meaning
71%	Ocean stores 71% of Earth's $CO_{2}$
K/2	Population size for maximum logistic growth rate
1,000 characters	Approximate chunk size in document processing (irrelevant to ecology — do not confuse)
10%	Lindeman's efficiency (energy transfer)