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Last updated

Sep 13, 2024
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- R Practice: Using Pattern Matching and Metadata to Connect with Local Ecology (Part I: Locally Endangered Fishes)
- Front Matter

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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
$F_x$	$F_x = S_x m_{x + 1}$	age-specific fecundity
$T_x$	$T_x = x L_x$	$T_x$ = years left to live
$E_x$	$E_x = T_x n_x$	$E_x$ = life expectancy
$L_x$	$L_x = (n_x + n_x + 1) / 2$	$L_x$ = 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
alpha beta gamma	average number of species gamma / alpha total # of species	diversity within a specific habitat/ecosystem comparison of diversity within habitats measure of diversity at landscape level (in several habitats within a region)

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