Formulas Summary
- Page ID
- 134119
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
|
Name |
Formula |
Interpretation |
|
Fundamental equation of population growth |
\(N_{t+1} = N_t + B_t - D_t + I_t- E_t\) |
\(N_t\) = population size at time t B = births; D = deaths; I = immigrants; E = emigrants |
|
Net reproductive rate - average total number of female offspring per female over the course of her lifetime |
\(R_0 = I_x m_x\) |
increasing pop: \(R_0\) > 1 stable population: \(R_0\) = 1 decreasing pop: \(R_0\) < 1 |
|
Generation time |
\(T = x I_x m_x R_0\) |
T = average age of reproduction |
|
Intrinsic/ per capita growth rate |
\(r = T ln (R_0)\) |
increasing pop: r > 0 stable population: r = 0 decreasing pop: r < 0 |
|
Rule of 70 |
\(t = 70 (100r)\) |
t = time (in years) for population size to double |
|
Fx |
\(F_x = S_x m_{x + 1}\) |
age-specific fecundity |
|
Tx |
\(T_x = x L_x\) |
\(T_x\) = years left to live |
|
Ex |
\(E_x = T_x n_x\) |
\(E_x\) = life expectancy |
|
Lx |
\(L_x = (n_x + n_x + 1) / 2\) |
\(L_x\) = number surviving |
|
Geometric population growth - growth rate of population with pulsed (seasonal) reproduction patterns |
\(\lambda = N_{t + 1} / N_t\) |
increasing pop: > 1 stable population: = 1 decreasing pop: < 1 |
|
Geometric growth model future population estimation |
\(N_t = N_0 \lambda^t\) |
\(N_t\) = population size at time t |
|
Exponential population growth - growth by a population with continuous reproduction (rate of population size change over time) |
\(dN/dt = rN\) |
increasing population: r > 0 stable population: r = 0 decreasing population: r < 0 |
|
Exponential growth model future population estimation |
\(N_t = N_0 e^{rt} \) |
N = population size at time t \((e \approx 2.71828 )\) |
|
Logistic population growth - exponential growth limited by carrying capacity |
\(\frac{dN}{dt} = rN \frac{(K-N_{t})}{K}\) |
increasing pop: N < K stable population: N = K decreasing pop: N > K |
|
Bioenergetics model |
\(S = IE - (FE + UE) - M\) |
S = energy storage (growth and reproduction) |
|
Lincoln-Peterson Index (Mark-Recapture Model) |
\(N = \frac{(M * S)}{R}\) |
N = population size estimate M = # of animals marked and released S = # of animals recaptured R = size of sample on 2nd visit |
|
Leslie Matrix formula |
\(N_{t + 1} = L * N^t\) |
\(N_{t+1}\) = population size at time t L = Leslie Matrix \(N_t\) = age-specific population at initial start time |
|
Simpson’s Index of Diversity (measure of probability): the less diversity, the greater the probability that two randomly selected individuals will be the same |
\(D = \sum_{i=1}^{S}\left(\frac{n_{i}}{N}\right)^{2}\) |
\(n_i\) = number of individuals of species i N = total number of individuals of all species 1 - D: if D is closer to zero, then less diversity; if D is closer to 1, then more diversity |
|
Shannon-Wiener Diversity Index (measure of certainty): more common species, more uncertain which one will be selected |
\(H = -\sum_{i=1}^{S} p_{i} * \ln p_{i}\) |
\(n_i / N = p_i\) = proportion of individuals of species i H = 0 in the absence of diversity |
|
Species richness |
\(H_{\max} = ln(S)\) |
\(H_{max}\) = maximum number of different species H can reach S = number of different species |
|
Species evenness |
\(J = \frac{H}{H_{\max }}\) |
if J is closer to 0, then less evenness; if J is closer to 1, then more evenness |
|
|
|