How a 10% Difference in Growth Rate Affects Two Exponentially Growing Populations Over 10 Generations

Population growth is a fundamental concept in biology, ecology, and demography. When two populations grow exponentially but have different growth rates, even a small initial difference can lead to a dramatic divergence in population size over time. This article explains what happens when two populations start with a 10% difference in their exponential growth rates and explores how large the difference becomes after 10 generations.

Understanding Exponential Growth

Exponential growth occurs when a population increases by a consistent percentage or factor in each time period or generation. The classic formula for exponential growth is:

Nt=N0⋅(1+r)tNt=N0⋅(1+r)t

where:

• NtNt = population size at time $t$

• N0N0 = initial population size

• $r$ = growth rate per generation (as a decimal)

• $t$ = number of generations

Exponential growth is characterized by a J-shaped curve when population size is plotted over time, reflecting rapid acceleration as the population increases.

Two Populations with Different Growth Rates

Suppose you have two populations, A and B, both starting with the same initial size (N0N0), but population A grows at a rate $r$ and population B at a rate $r+0.10r=1.10r$ (a 10% higher growth rate). However, in standard biological contexts, a “10% difference in growth rate” usually means that if population A has growth rate $r$, population B has growth rate $r+0.10$ (absolute, not relative), or if the rates are small, rB=1.10rArB=1.10rA (relative). For clarity, let’s assume both populations start at the same size, and after 10 generations, we want to compare their sizes.

However, the question likely means that population A has a certain growth rate, and population B has a growth rate 10% higher (in absolute terms, e.g., 0.10 vs. 0.11, or in relative terms, e.g., 1.00 vs. 1.10). For the sake of this article, let’s assume both start at the same size, and population B grows 10% faster in absolute terms (for example, if A grows at 0.10, B grows at 0.11).

But to match typical biological scenarios and the options given (ratios like 2:1, 4:1), it’s more realistic to interpret the “10% difference in growth rate” as a 10% higher compounding factor per generation (e.g., A: 1.00, B: 1.10). However, in standard exponential growth, the growth factor is $(1+r)$, so if A has rArA and B has rB=rA+0.10rB=rA+0.10, then after 10 generations, the ratio of B to A is:

NB,10NA,10=(1+rB)10(1+rA)10NA,10NB,10=(1+rA)10(1+rB)10

If rA=0.10rA=0.10 and rB=0.20rB=0.20, then:

(1.20)10(1.10)10≈6.192.59≈2.39(1.10)10(1.20)10≈2.596.19≈2.39

But this does not closely match any of the options given. If, instead, the “10% difference” refers to a 10% higher compounding factor (i.e., A: 1.0, B: 1.1), then after 10 generations:

(1.1)10(1.0)10=(1.1)10=2.59(1.0)10(1.1)10=(1.1)10=2.59

But again, this is not a clean match for the options.

Given the options and typical biological interpretations, the most likely scenario is that population B has a growth rate that is 10% higher than population A in absolute terms (e.g., A: 1.0, B: 1.1 per generation, but this is unrealistic for growth rates; usually, growth rates are much smaller, e.g., 0.10 and 0.11, or 0.05 and 0.055).

However, considering the options (1:1, 2:1, 4:1, 10:1), and the fact that a 10% higher growth rate over 10 generations in exponential growth leads to a ratio that is not exactly 2:1 or 4:1, but closest to 2:1 or slightly higher, the question likely expects you to recognize that a small difference in growth rate leads to a large difference in population size after several generations, and the closest reasonable answer is 2:1 or 4:1.

But in reality, with a 10% higher growth rate per generation (e.g., A: 1.0, B: 1.1), the ratio after 10 generations is about 2.6:1. If the growth rates are small (e.g., A: 1.05, B: 1.10), the ratio is less. If the growth rates are larger (e.g., A: 1.10, B: 1.20), the ratio is about 2.4:1.

Given the options and typical exponential growth calculations, none of the options are exact, but the closest is 2:1 or 4:1. However, if the question means that population B grows at a rate of 1.10 per generation and population A at 1.00, the ratio is about 2.6:1, so none of the options are perfect. If the question means that population B grows at a rate of 1.10 and population A at 1.05, the ratio is:

(1.10)10(1.05)10≈2.591.63≈1.59(1.05)10(1.10)10≈1.632.59≈1.59

This is not a match. If population B grows at 1.20 and population A at 1.10, the ratio is about 2.4:1.

Given the options and the fact that a 10% higher growth rate per generation (as a compounding factor) is not standard in biology, it is more likely that the question expects you to recognize that after 10 generations, the difference is large, and the closest reasonable answer is (3) 2:1 or (2) 4:1. However, based on standard exponential growth with a 10% higher growth rate (e.g., 0.10 vs. 0.11), the ratio is less than 2:1, which does not match any option.

Given the ambiguity, and the fact that most exams expect you to recognize that a small difference in growth rate leads to a large difference after several generations, the most likely intended answer is (3) 2:1, as it is closest to the actual ratio for moderate growth rates.

However, if the question means that population B grows at a rate of 1.10 per generation and population A at 1.00, the ratio is about 2.6:1, so none of the options are exact, but 2:1 is the closest.

Given the options and typical exponential growth scenarios, the answer is most likely:

(3) 2:1

But to clarify: If the growth rate difference is 10% in absolute terms (e.g., 0.10 vs. 0.11), the ratio after 10 generations is:

(1.11)10(1.10)10≈2.842.59≈1.1(1.10)10(1.11)10≈2.592.84≈1.1

This is not a match for any option. If the growth rate is interpreted as a compounding factor (e.g., A: 1.0, B: 1.1), the ratio is 2.6:1, closest to 2:1.

Therefore, the most reasonable answer, given the options and standard interpretations, is:

(3) 2:1

However, this is only approximate and depends on how the 10% difference is defined. In most biological contexts, a 10% higher growth rate (e.g., 0.10 vs. 0.11) does not result in a 2:1 ratio after 10 generations, but the question likely expects you to choose the closest reasonable option, which is 2:1.

Ecological and Practical Implications

Even small differences in growth rates can lead to large disparities in population size over time due to the nature of exponential growth. This principle is important in:

• Evolutionary biology: Small fitness advantages can lead to rapid fixation of beneficial alleles.

• Microbial ecology: Strains with slightly higher growth rates can dominate a culture.

• Conservation: Invasive species with even a slight growth advantage can outcompete natives over time.

Summary Table

Conclusion

When two populations grow exponentially with a 10% difference in growth rate (interpreted as a compounding factor per generation), the population with the higher rate will be about 2.6 times larger after 10 generations. Given the options, the closest reasonable answer is (3) 2:1. This highlights how exponential growth amplifies even small differences in growth rates over time.

Correct answer (given the options and typical interpretations):
(3) 2:1