Life Tables and Population Growth Indices

Cohort: A group of individuals born at the same time period. A cohort life table follows these individuals from birth through death.
Subscript x ( $x$ ): Refers to the age or stage class (e.g., eggs, instars, adults).
Observed Numbers ( $A_x$ ): Actual number of individuals counted. This value is case-specific and difficult to compare across studies.
Standardized Survivorship ( $L_x$ ): Calculated as $L_x = \frac{A_x}{A_0}$ . This represents the proportion surviving relative to the start ( $t = 0$ ), enabling inter-study comparisons.
Standardized Mortality ( $D_x$ ): Calculated as $D_x = L_x - L_{x+1}$ . These values can be summed to determine total mortality up to a specific stage.
Age-Specific Mortality ( $Q_x$ ): Calculated as $Q_x = \frac{D_x}{L_x}$ . It indicates the intensity of mortality at each stage. It cannot be summed.
Age-Specific Survivorship ( $P_x$ ): Calculated as $P_x = 1 - Q_x$ or $P_x = \frac{A_{x+1}}{A_x}$ . It indicates the probability of surviving to the next stage.
Killing Power ( $K$ ): Calculated as $K = \log_{10}(A_x) - \log_{10}(A_{x+1})$ . It combines the summable attribute of $D_x$ with the intensity reflection of $Q_x$ .

Fecundity Indices: - $F_x$ : Total offspring produced per age/stage class. - $M_x$ : Mean offspring per individual of age $x$ , calculated as $M_x = \frac{F_x}{A_x}$ . - $L_x M_x$ : Offspring produced per original individual in the cohort.
Basic Reproductive Rate ( $R_0$ ): Calculated as $R_0 = \sum (L_x M_x)$ or $R_0 = \frac{\sum F_x}{A_0}$ . It is the mean offspring produced per individual over its lifetime.
Generation Time ( $T$ ): Calculated as $T = \frac{\sum (x L_x M_x)}{R_0}$ . Units are consistent with the study intervals (e.g., six-month units for the cane rat).
Intrinsic Rate of Natural Increase ( $r$ ): Calculated as $r = \frac{\ln(R_0)}{T}$ . This is used for populations with overlapping generations.
Fundamental Reproductive Rate ( $R$ ): Calculated as $R = e^r$ . It represents the multiplier for population growth over one time interval ( $N_{t+1} = N_t \times R$ ).
Growth Projection: For population size at time $t$ , the formula is $N_t = N_0 \times R^t$ .

Life Expectancy ( $e_x$ ): The average time an individual of age $x$ is expected to live. - Calculation Steps: 1. Calculate $L_{upper}$ (mean survivors): $L_{upper} = \frac{A_x + A_{x+1}}{2}$ . 2. Calculate $T_{upper}$ (cumulative survivors): Sum of $L_{upper}$ from age $x$ to the end. 3. $e_x = \frac{T_{upper}}{A_x}$ .
Finite Rate Conversions: To adjust finite rates between scales (e.g., annual to monthly), use the formula: $\text{Adjusted Rate} = (\text{Observed Rate})^{\frac{T_{standardized}}{T_{observed}}}$ .
Instantaneous Mortality ( $Z$ ): Calculated as $Z = \ln(P_x)$ . These rates are always negative.

Static Life Table: A "snapshot" of a population at one time. It assumes the current age distribution reflects historical processes. Used for long-lived species like elephants or red deer on the Isle of Rum.
Smoothing Data: Static tables may show impossible increases in numbers between age classes due to environmental fluctuations. Techniques include: - General Smoothing: Applying running averages (e.g., 3-point or 5-point) to the entire dataset using MS Excel. - Restricted Smoothing: Averaging only specific problematic data points.
Survivorship Curves: - Type I: High survival early in life; mortality concentrated at the end (e.g., Humans, primates). - Type II: Constant mortality rate throughout life (e.g., Birds, some reptiles). - Type III: High early mortality; survivors have high late-life survival (e.g., Cod spawning 2,000,000 eggs).
Reproductive Strategies: - Iteroparity: Repeated reproduction over time (e.g., Cod, deer). - Semelparity: Single reproductive event at the end of life (e.g., Salmon, grasshoppers).