Population Dynamics: Age Distribution and Rates of Population Change

Population Dynamics

Introduction

Population dynamics is the study of how and why the number of individuals in a population changes over time. It involves understanding the factors that influence population size, structure, and growth, including birth rates, death rates, immigration, and emigration.

Age Distribution

Definition and Importance

• Age distribution refers to the proportion of individuals of different ages within a population.

• It reflects the population's history of survival, reproduction, and future growth potential.

• Age distribution is highly informative for predicting population trends and assessing population health.

Examples of Age Distribution

• White Oak Population: Age distribution is biased towards young trees, indicating sufficient reproduction for replacement and a stable population.

• Rio Grande Cottonwood (Populus deltoides wislizeni): Populations are declining due to the absence of seasonal floods, which are necessary for seed germination. The lack of young trees suggests poor recruitment and potential population decline.

• Galapagos Finches (Grant and Grant Study): Age structure can shift dramatically in response to environmental variation, such as droughts, which affect the survival and recruitment of different age classes.

Table: Age Distribution Comparison

Population	Age Structure	Implication
White Oak	Many young, few old	Stable, sufficient reproduction
Rio Grande Cottonwood	Few young, many old	Declining, poor recruitment
Galapagos Finches	Variable, responsive to environment	Population structure shifts with conditions

Rates of Population Change

Key Variables Affecting Population Size

• Birth rate (natality): Number of young born per female in a given period.

• Death rate (mortality): Number of individuals dying per unit time.

• Immigration rate: Number of individuals entering the population.

• Emigration rate: Number of individuals leaving the population.

Fecundity Schedule

• A fecundity schedule is a table showing birth rates for females of different ages.

• It is used to calculate reproductive rates and predict population growth.

Life Table and Fecundity Schedule Metrics

• Net reproductive rate ($R_0$): Average number of daughters produced per female during her lifetime.

• Geometric rate of increase ($\lambda$): Ratio of population size at two points in time. $\lambda = \frac{N_{t+1}}{N_t}$

• Generation time ($T$): Average time between the birth of individuals and the birth of their offspring.

• Per capita rate of increase ($r$): Difference between birth rate and death rate. For overlapping generations, $r = \frac{\ln R_0}{T}$

Table: Life Table Example (P. drummondii)

Age (days)	Number surviving to day x ($n_x$)	Proportion surviving ($l_x$)	Average seeds produced ($m_x$)	$l_x m_x$
0-29	996	1.000	0.000	0.000
30-34	184	0.185	0.000	0.000
35-39	154	0.155	2.094	0.324
40-44	145	0.146	2.594	0.379
45-49	140	0.141	3.571	0.504
50-54	138	0.139	4.007	0.557
55-59	132	0.133	4.007	0.533
60-64	72	0.072	4.007	0.289
65-69	0	0.000	0.000	0.000
Total $R_0 = \sum (l_x m_x)$				2.4177

Additional info: Table values are based on the provided slides and standard life table structure.

Estimating Population Growth Rates

Non-Overlapping Generations (e.g., Annual Plants)

• Net reproductive rate ($R_0$): $R_0 = \sum (l_x m_x)$

• Geometric rate of increase ($\lambda$): $\lambda = \frac{N_{t+1}}{N_t}$

• Population size at time t: $N_t = N_0 \lambda^t$

• Example: For P. drummondii, $R_0 = 2.4177$ means each individual produces on average 2.4177 offspring.

Overlapping Generations (e.g., Animals)

• Average generation time ($T$): $T = \frac{\sum (x l_x m_x)}{R_0}$, where x = age in years.

• Per capita rate of increase ($r$): $r = \frac{\ln R_0}{T}$

• Interpretation:

  • If $r = 0$, the population is stable.

  • If $r < 0$, the population is declining.

  • If $r > 0$, the population is increasing.

Example Calculation (Mud Turtle)

• $R_0 = 0.601$

• $\sum (x l_x m_x) = 6.4$

• $T = 10.6$

• $r = \frac{\ln 0.601}{10.6} = -0.05$

• Since $r < 0$, the population is declining.

Summary Table: Population Growth Metrics

Population Type	Key Metrics	Formulas
Annual plants (non-overlapping)	$R_0$, $T$, $\lambda$	$R_0 = \sum (l_x m_x)$ $\lambda = \frac{N_{t+1}}{N_t}$ $T = \frac{\sum (x l_x m_x)}{R_0}$
Animals (overlapping)	$R_0$, $T$, $r$	$R_0 = \sum (l_x m_x)$ $T = \frac{\sum (x l_x m_x)}{R_0}$ $r = \frac{\ln R_0}{T}$

Key Terms

• Net reproductive rate ($R_0$): The average number of offspring produced per individual during its lifetime.

• Geometric rate of increase ($\lambda$): The ratio of population size at two points in time for populations with non-overlapping generations.

• Generation time ($T$): The average period between the birth of individuals and the birth of their offspring.

• Per capita rate of increase ($r$): The intrinsic rate of population increase, accounting for overlapping generations.

• Fecundity schedule: A table showing the number of offspring produced by females of different ages.

• Life table: A demographic table showing survival and reproductive rates for each age class.

Conclusion

Understanding age distribution and rates of population change is essential for predicting population trends and managing species conservation. Life tables and fecundity schedules provide the quantitative foundation for these analyses, allowing biologists to estimate growth rates and assess population health under varying environmental conditions.