A Level Biology - How to Solve: Population Growth (r- and K-Selection, Logistic Growth, Predator-Prey Graphs)

How to Solve: Population Growth (r- and K-Selection, Logistic Growth, Predator-Prey Graphs)

Complete Guide For GCSE/A-Level Biology (AQA, Edexcel, OCR, WJEC)

Introduction

"Mastering population growth models can earn you 10–15% of your Biology exam marks—and help you predict real-world crises like pandemics, endangered species recovery, or even zombie outbreaks. Today, you’ll learn the exact steps to solve any question on r/K-selection, logistic growth, or predator-prey cycles."

WHAT YOU NEED TO KNOW FIRST

1. Exponential growth: Unlimited resources → population doubles at a constant rate.
2. Carrying capacity (K): Maximum population size an environment can sustain.
3. Density-dependent vs. density-independent factors: How population size affects survival (e.g., disease vs. natural disasters).

KEY TERMS & FORMULAS

1. Logistic Growth Formula

Formula: [ \frac{dN}{dt} = rN \left( \frac{K - N}{K} \right) ]

Variables: - ( \frac{dN}{dt} ) = population growth rate (change in population over time) - ( r ) = intrinsic growth rate (max growth rate without limits) - ( N ) = current population size - ( K ) = carrying capacity (max sustainable population)

MEMORISE THIS – You’ll need to interpret graphs and calculate growth rates.

2. r-Selection vs. K-Selection Traits

MEMORISE THIS TABLE – Examiners love comparing traits.

3. Predator-Prey Graphs

• Key Features:
• Prey population peaks first, followed by predator population.
• Lag time: Predators increase after prey increase (due to reproduction delay).
• Oscillations: Populations cycle due to feedback loops.

MEMORISE THIS PATTERN – You’ll be asked to sketch or interpret these graphs.

STEP-BY-STEP METHOD

Step 1: Identify the Model

• Exponential growth? → Unlimited resources, J-shaped curve.
• Logistic growth? → S-shaped curve, carrying capacity (K).
• Predator-prey? → Oscillating cycles, lag time.

Step 2: Extract Key Values

• For logistic growth:
• Find ( r ) (growth rate), ( N ) (current population), ( K ) (carrying capacity).
• For predator-prey graphs:
• Note peak populations and lag time.

Step 3: Apply the Correct Formula

• Logistic growth: Use ( \frac{dN}{dt} = rN \left( \frac{K - N}{K} \right) ).
• Exponential growth: Use ( N_t = N_0 e^{rt} ) (given on exam sheet).
• Predator-prey: Describe trends (e.g., "Predators increase after prey due to reproduction delay").

Step 4: Interpret the Graph

• Logistic growth:
• Exponential phase: ( N ) is small → ( \frac{K - N}{K} \approx 1 ) → fast growth.
• Plateau phase: ( N \approx K ) → growth slows to zero.
• Predator-prey:
• Prey increases first → predators follow.
• Prey decreases → predators decrease later.

Step 5: Answer the Question

• Calculation? Plug numbers into the formula.
• Explanation? Use key terms (e.g., "density-dependent factors limit growth").
• Graph sketch? Label axes, peaks, and lag time.

WORKED EXAMPLES

Example 1 – Basic (Logistic Growth Calculation)

Question: A population of rabbits has: - ( r = 0.2 ) per year - ( K = 500 ) rabbits - Current population ( N = 100 )

Calculate the growth rate ( \frac{dN}{dt} ).

Solution:
1. Identify model: Logistic growth (given ( K )).
2. Extract values: ( r = 0.2 ), ( N = 100 ), ( K = 500 ).
3. Apply formula: [ \frac{dN}{dt} = 0.2 \times 100 \times \left( \frac{500 - 100}{500} \right) ] [ = 20 \times \left( \frac{400}{500} \right) ] [ = 20 \times 0.8 = 16 ]
4. Answer: The population grows by 16 rabbits per year.

What we did and why: - Used the logistic growth formula because the question gave a carrying capacity (( K )). - Multiplied ( r ), ( N ), and the fraction ( \frac{K - N}{K} ) to find the growth rate.

Example 2 – Medium (r/K-Selection Comparison)

Question: A species of fish produces 1,000 eggs per year but only 5 survive to adulthood. Is this an r-selected or K-selected species? Explain.

Solution:
1. Identify traits: - Many offspring (1,000 eggs) → r-selected trait. - Low survival (only 5 adults) → r-selected trait.
2. Compare to table: - r-selected species have many offspring, low survival. - K-selected species have few offspring, high survival.
3. Answer: This is an r-selected species because it produces many offspring with low survival rates.

What we did and why: - Matched the given data to the r/K-selection table. - Justified the answer using specific traits (offspring number and survival).

Example 3 – Exam-Style (Predator-Prey Graph Interpretation)

Question: The graph below shows lynx and hare populations over 20 years. a) Which line represents the predator? Explain. b) Why does the predator population peak after the prey?

Solution:
1. Identify predator/prey: - The higher peak is the prey (hares). - The lower peak is the predator (lynx). - Answer (a): The red line (lower peak) is the predator because predators are less abundant than prey.
2. Explain lag time: - Predators reproduce more slowly than prey. - When prey increase, predators take time to find food, reproduce, and increase. - Answer (b): The predator population peaks after the prey because it takes time for predators to reproduce and respond to increased food availability.

What we did and why: - Used graph trends (peaks and timing) to identify predator/prey. - Explained the biological reason (reproduction delay) for the lag time.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

"Here’s what you need to remember for your exam:
1. Logistic growth = S-shaped curve, carrying capacity (( K )). Use the formula ( \frac{dN}{dt} = rN \left( \frac{K - N}{K} \right) ).
2. r-selected species = many babies, low survival (e.g., insects). K-selected = few babies, high survival (e.g., elephants).
3. Predator-prey graphs = prey peaks first, predators follow. Always mention lag time.
4. Common mistakes: Mixing up ( r ) and ( K ), forgetting units, and ignoring lag time.
5. Exam traps: Examiners love hiding ( K ) in graphs or asking for trends without labels.

Now go practice—you’ve got this!"