SOLUTION: Population Dynamics Worksheet - Studypool - Free Printable

I'll work through this problem step by step to help you understand the differences between dN/dt and r in population growth models.

Let me think through this carefully:

Step 1: Understanding what each symbol represents

- dN/dt: This is the instantaneous rate of change of population size with respect to time. It tells us how many individuals are being added (or removed) from the population at any given moment.

- r: This is the per capita growth rate, also called the intrinsic rate of increase. It represents the average contribution of each individual to population growth.

Step 2: Looking at the mathematical relationship

In exponential growth, the equation is:
- dN/dt = rN

This shows that:
- dN/dt depends on BOTH the growth rate (r) AND the current population size (N)
- r is just the per capita rate - it doesn't depend on how many individuals there are

Step 3: Understanding the practical difference

- dN/dt tells you the TOTAL number of new individuals added per unit time
- Example: If a population has 100 bacteria and dN/dt = 50/day, then 50 new bacteria are added each day

- r tells you the GROWTH RATE PER INDIVIDUAL
- Example: If r = 0.5/day, each bacterium contributes 0.5 new bacteria per day on average
- This means if you have 100 bacteria, dN/dt = 0.5 × 100 = 50/day
- But if you have 200 bacteria, dN/dt = 0.5 × 200 = 100/day

Step 4: Key distinction

The main difference is:
- dN/dt changes as the population size changes (it's an absolute rate)
- r stays constant regardless of population size (it's a per capita rate)

Think of it like this:
- r is like your hourly wage (fixed amount per hour worked)
- dN/dt is like your total earnings (depends on both your wage AND how many hours you work)

Final Answer:

dN/dt represents the actual number of individuals added to or removed from the population per unit time (the total population growth rate), while r represents the per capita growth rate (the average contribution of each individual to population growth). The key difference is that dN/dt depends on both the growth rate AND the current population size (dN/dt = rN), so it changes as the population grows larger. In contrast, r is a constant rate per individual that doesn't change with population size. For example, if r = 0.1/day, each individual contributes 0.1 offspring per day, but the total number of new individuals (dN/dt) will be higher when the population is larger.