Population Growth Model Examples with Step-by-Step Solutions

📚 Quick Study Guide

• 🌍 Exponential Growth: Population increases at a constant rate. Formula: $N(t) = N_0e^{rt}$, where $N(t)$ is the population at time $t$, $N_0$ is the initial population, $r$ is the growth rate, and $e$ is the base of the natural logarithm.
• 🚧 Logistic Growth: Population growth slows as it reaches carrying capacity. Formula: $\frac{dN}{dt} = rN(1 - \frac{N}{K})$, where $K$ is the carrying capacity.
• ➕ Per Capita Growth Rate (r): The contribution of each individual to population growth. Calculated as $r = b - d$, where $b$ is the birth rate and $d$ is the death rate.
• ⏱️ Discrete-Time Models: Population changes occur in discrete intervals (e.g., annually). Formula: $N_{t+1} = N_t + rN_t$ or $N_{t+1} = \lambda N_t$ where $\lambda$ is the finite rate of increase.
• ⚖️ Carrying Capacity (K): The maximum population size that an environment can sustain.

Practice Quiz

1. A population of bacteria doubles every hour. If the initial population is 100, what is the population after 3 hours, assuming exponential growth?

   1. 200
   2. 400
   3. 800
   4. 1600

2. Which of the following factors is NOT typically included in a basic exponential growth model?

   1. Initial population size
   2. Growth rate
   3. Carrying capacity
   4. Time

3. In the logistic growth model, what happens to the population growth rate as the population size approaches carrying capacity?

   1. It increases exponentially
   2. It remains constant
   3. It decreases
   4. It fluctuates randomly

4. If a population has a birth rate of 0.2 and a death rate of 0.1, what is the per capita growth rate (r)?

   1. 0.1
   2. 0.2
   3. 0.3
   4. 0.02

5. A population follows a discrete-time growth model with $N_{t+1} = 1.2N_t$. If the initial population is 50, what will the population be after 2 time steps?

   1. 60
   2. 72
   3. 86.4
   4. 100

6. What does the carrying capacity (K) represent in population growth models?

   1. The minimum population size
   2. The maximum population size that the environment can sustain
   3. The rate of population growth
   4. The time it takes for a population to double

7. Which model is most suitable for describing the growth of a population in a resource-limited environment?

   1. Exponential growth model
   2. Linear growth model
   3. Logistic growth model
   4. Geometric growth model