Key Concepts of Population Growth Models

Why This Matters

Population growth models are fundamental to understanding how evolution actually works in real populations. When the AP exam tests you on Hardy-Weinberg equilibrium, natural selection, or genetic drift, you need to understand why population size matters—and that's exactly what these models explain. The concepts here connect directly to population genetics (Topic 7.4), Hardy-Weinberg assumptions (Topic 7.5), and the conditions that allow evolution to occur.

You're being tested on your ability to distinguish between idealized mathematical models and real-world biological constraints. The exam loves asking about the relationship between population size and evolutionary mechanisms like genetic drift, or why carrying capacity matters for allele frequency changes. Don't just memorize the shapes of growth curves—know what each model assumes, when those assumptions break down, and how population dynamics connect to genetic variation, natural selection, and ecosystem stability.

---

Mathematical Models of Population Growth

These two foundational models describe how populations change over time. The key difference lies in their assumptions about environmental limits—exponential growth assumes unlimited resources, while logistic growth incorporates environmental resistance.

Exponential Growth Model

• Assumes unlimited resources and ideal conditions—populations grow at a constant per capita rate without any environmental constraints
• Equation: $\frac{dN}{dt} = rN$ where $N$ is population size and $r$ is the intrinsic rate of increase
• Rarely sustained in nature but occurs temporarily when populations colonize new habitats or recover from bottlenecks—directly relevant to founder effect scenarios

Logistic Growth Model

• Incorporates carrying capacity (K)—growth rate slows as population approaches the maximum sustainable size
• Equation: $\frac{dN}{dt} = rN\frac{(K-N)}{K}$ where the term $(K-N)/K$ represents environmental resistance
• More realistic for natural populations—predicts stabilization around carrying capacity, which connects to long-term allele frequency dynamics

---

Growth Curve Visualizations

The shape of a population's growth curve tells you which model applies and what stage of growth the population is experiencing. These visual patterns appear frequently on multiple-choice questions.

J-Shaped Growth Curve

• Represents exponential growth graphically—steep upward curve that accelerates continuously without leveling off
• Indicates minimal environmental resistance—resources abundant, predators absent, disease limited
• Warning sign for ecological instability—populations showing J-curves often crash dramatically when resources deplete

S-Shaped Growth Curve (Sigmoid)

• Represents logistic growth graphically—initial exponential phase transitions to a plateau near carrying capacity
• Three distinct phases: lag phase (slow start), log phase (rapid growth), and stationary phase (stabilization)
• Reflects density-dependent regulation—the curve bends because competition, predation, or disease intensifies as density increases

---

Factors Limiting Population Growth

Understanding what limits growth is essential for predicting population dynamics and connecting to evolutionary concepts. These factors determine whether populations remain large enough to avoid genetic drift or small enough for random events to dominate.

Density-Dependent Factors

• Impact scales with population size—competition for food, disease transmission, and predation all intensify as individuals crowd together
• Create negative feedback loops—high density → increased mortality or decreased reproduction → population decline → reduced density
• Examples include intraspecific competition, parasitism, and waste accumulation—these regulate populations toward carrying capacity

Density-Independent Factors

• Affect populations regardless of size—natural disasters, temperature extremes, and habitat destruction kill the same proportion whether population is large or small
• Can cause population bottlenecks—sudden reductions that dramatically alter allele frequencies through genetic drift
• Examples include hurricanes, droughts, volcanic eruptions, and human habitat destruction—critical for understanding founder effects and bottleneck scenarios on the AP exam

Carrying Capacity (K)

• Maximum sustainable population size—determined by limiting resources like food, water, space, and nesting sites
• Dynamic, not fixed—environmental changes, seasonal variation, and human activity can shift K up or down
• Central to Hardy-Weinberg assumptions—populations at or near K are more likely to be "large" enough to minimize genetic drift

---

Key Parameters in Population Models

These quantitative measures allow biologists to predict and compare population growth across species. Understanding what these values mean helps you interpret data tables and graphs on the exam.

Intrinsic Rate of Increase (r)

• Maximum per capita growth rate under ideal conditions—calculated as birth rate minus death rate when resources are unlimited
• Species-specific trait shaped by life history—organisms with high fecundity and short generation times have higher $r$ values
• Appears in both growth equations—$r$ drives the exponential term in $\frac{dN}{dt} = rN$ and $\frac{dN}{dt} = rN\frac{(K-N)}{K}$

---

Life History Strategies

These contrasting reproductive strategies reflect evolutionary trade-offs shaped by natural selection in different environments. This connects population ecology directly to evolution and adaptation.

r-Selected Species

• Maximize reproductive rate—produce many offspring with minimal parental investment, favoring quantity over quality
• Adapted to unstable or unpredictable environments—rapid reproduction allows quick colonization and recovery after disturbances
• Examples: insects, bacteria, annual plants, small rodents—populations fluctuate dramatically and rarely reach carrying capacity

K-Selected Species

• Maximize competitive ability near carrying capacity—produce fewer offspring with extensive parental care, favoring survival over numbers
• Adapted to stable, competitive environments—success depends on outcompeting others for limited resources
• Examples: elephants, whales, primates, large trees—populations remain relatively stable near environmental limits

---

Quick Reference Table

---

Self-Check Questions

1. Which two concepts both involve the variable $r$, and how does each model use it differently?

2. A volcanic eruption kills 90% of a rabbit population. Is this a density-dependent or density-independent factor, and how might this event affect allele frequencies in the surviving population?

3. Compare and contrast r-selected and K-selected species in terms of their typical growth curves and environmental conditions.

4. If a population's growth curve shows an inflection point where the slope begins decreasing, what model does this represent, and what biological factors are likely causing the change?

5. An FRQ asks you to explain why small populations are more susceptible to evolutionary change than large populations. How would you connect carrying capacity, population bottlenecks, and genetic drift in your response?