5.2 Population Growth and Dynamics: Logistic Growth and Influencing Factors

Population Growth and Dynamics

Introduction to Population Growth Models

Population growth describes how the number of individuals in a population changes over time. Different models are used to understand and predict these changes, including geometric, exponential, and logistic growth. Logistic growth incorporates the concept of limiting resources and density dependence, making it a more realistic model for most natural populations.

• Geometric and Exponential Growth: Occur when resources are unlimited; population increases rapidly.

• Logistic Growth: Incorporates resource limitation and density dependence, resulting in an S-shaped (sigmoidal) growth curve.

• Open vs. Closed Populations: Open populations exchange individuals with other populations; closed populations do not.

Limits on Population Growth

Resource Limitation and Carrying Capacity

Population growth is limited by critical resources such as food, nesting sites, space, or water. As population density increases, these resources become scarce, leading to decreased reproduction and increased mortality.

• Carrying Capacity (K): The maximum population size that an environment can sustain indefinitely.

• Examples: Grazing animals limited by food, birds limited by nesting sites.

Population Regulation

Density-Independent and Density-Dependent Factors

Population size is regulated by both density-independent and density-dependent factors.

• Density-Independent Factors: Affect populations regardless of size (e.g., floods, hurricanes, extreme temperatures).

• Density-Dependent Factors: Effects increase or decrease with population size (e.g., competition, disease, predation).

Example: Hurricane Katrina reduced population sizes irrespective of their initial numbers (density-independent). Songbird reproductive success decreased as the number of breeding females increased, especially when food was limited (density-dependent).

Mathematical Models of Population Growth

Exponential Growth Equation

• Exponential growth assumes unlimited resources and is described by:

$dN/dt = rN$

• Where r is the intrinsic rate of increase and N is population size.

Logistic Growth Equation

• Logistic growth incorporates density dependence and carrying capacity:

$dN/dt = rN \left[1 - \frac{N}{K}\right]$

• K is the carrying capacity.

• When N = K, population growth rate is zero; the population is stable.

• Maximum growth rate occurs at N = K/2.

Derivation: The logistic equation is derived by incorporating density-dependent birth and death rates:

$\begin{align*} b &= b_0 - aN \\ d &= d_0 + cN \\ r &= (b_0 - d_0) - (a + c)N \end{align*}$

Where a and c are constants describing density dependence for births and deaths, respectively.

Carrying capacity is defined as:

$K = \frac{b_0 - d_0}{a + c}$

Example Calculations

• Cougar Population: If N = 100, r = -0.25, K = 100, then

$dN/dt = -0.25 \times 100 \times (1 - 100/100) = 0$

• Population is stable at carrying capacity.

• Beaver Population: If maximum growth rate occurs at N = 200, then K = 2N = 400.

r- and K-Selected Species

Life History Strategies

• r-Selected Species: Maximize intrinsic rate of increase (r), produce many offspring, low survival probability, exploit empty niches.

• K-Selected Species: Adapted to live near carrying capacity (K), produce few offspring, high survival probability, strong competitors in crowded environments.

Demography and Population Structure

Demography

Demography is the quantitative study of populations, focusing on factors such as age structure, sex ratio, and effective population size (number of individuals contributing offspring to the next generation).

• Age Structure: Populations with more individuals in reproductive age grow faster.

• Sex Ratio: The proportion of males to females; can be influenced by environmental factors (e.g., temperature-dependent sex determination in sea turtles).

Life Tables and Survivorship Curves

Life Tables

Life tables summarize demographic information such as survival and fecundity rates at different ages. They are used to estimate population growth rates and predict future changes.

Survivorship Curves

• Type I: Low mortality in early and middle life (e.g., humans).

• Type II: Constant mortality rate throughout life (e.g., some birds).

• Type III: High mortality in early life, lower in adulthood (e.g., many fish, plants).

Other Influencing Factors on Population Growth

Environmental Stochasticity

Random, unpredictable changes in the environment can cause fluctuations in population size and growth rate.

Delayed Density Dependence

Population responses to density can be delayed, leading to oscillations or cycles in population size. The length of the time lag affects whether the population stabilizes or continues to oscillate.

Allee Effect

The Allee effect describes a decrease in population growth rate as population density decreases. Small populations may struggle to survive and reproduce, increasing extinction risk.

• Example: Passenger pigeon extinction due to inability to survive at low densities.

Extinction Risk and Initial Population Size

Small populations are more likely to go extinct by chance (genetic drift, demographic stochasticity) than large populations.

Summary Table: Factors Influencing Population Growth

Factor	Type	Effect on Population	Example
Resource Limitation	Density-dependent	Slows/stops growth as resources become scarce	Food, nesting sites
Catastrophic Events	Density-independent	Reduces population regardless of size	Hurricanes, floods
Competition, Disease	Density-dependent	Increases with population size	Predation, epidemics
Sex Ratio	Demographic	Skewed ratios can limit reproduction	Sea turtles (temperature-dependent sex)
Allee Effect	Density-dependent (at low density)	Growth rate decreases at low population size	Passenger pigeon extinction
Environmental Stochasticity	Random	Unpredictable fluctuations	Climate variation

Key Equations

• Exponential Growth:

$dN/dt = rN$

• Logistic Growth:

$dN/dt = rN \left[1 - \frac{N}{K}\right]$

• Carrying Capacity:

$K = \frac{b_0 - d_0}{a + c}$

Summary

• Population growth is influenced by both density-dependent and density-independent factors.

• Logistic growth models are more realistic for natural populations due to resource limitation.

• Demographic factors such as age structure, sex ratio, and effective population size are crucial for understanding population dynamics.

• Environmental variability, delayed density dependence, and the Allee effect can cause fluctuations and increase extinction risk, especially in small populations.

Additional info: Some mathematical derivations and demographic concepts were expanded for clarity and completeness.