Population growth and logistic growth rates... ecology problem

Question

A survey of a chicago wood duck population shows that, on average, 52 individuals are born each year, and 37 die. the current population size is 160.

 

a) Assuming this population is growing exponentially, what will the population size be in 3 years?

 

b) what would the geometric growth of this population be?

 

c) if the same population of wood ducks had a carrying capacity of 265 individuals, what would the logistic growth rate be?

Answer 1

Hi! In this problem, you’re being asked to identify and apply different population growth models, which ecologists use to predict how populations change over time due to births, deaths, and environmental limits.

Given Information

Because we will be referencing the same scenario for all 3 parts, I would suggest writing out any relevant information that we were provided with in the problem’s introduction. This will help contextualize our values and streamline the substitution process in a way that makes it easier to understand our work and results as we work through each part.

1. Births per year = 52
2. Deaths per year = 37
3. Net increase per year = 52 births - 37 deaths = +15 individuals/year
4. Current population size N₀ = 160
5. Time t = 3 years
6. Carrying capacity (for part c) K = 265

Part a) Exponential Growth Model

Key Concept: Exponential growth assumes unlimited resources—there are no limits on food, space, or other needs. This is rare in real ecosystems, but useful for modeling populations in the early stages of growth, where environmental restraints have yet to materialize, as the numbers are still relatively small and have yet to exert major stresses on the larger ecosystem. 

In ecology, exponential growth can be modeled by the formula N(t) = N₀ * e^rt , where:

1. N(t) = population size at time t
2. N₀ = initial population size
3. r = per capita growth rate = (Births - Deaths) / N₀
4. t = time in years
5. e ≈ 2.718

Step 1: Calculate r, given the population had 52 births per year, 37 deaths per year, and an initial size of 160 individuals.

1. r = per capita growth rate = (Births - Deaths) / N₀ = (52-37)/160 = 15/160 = 0.09375 individuals per year
2. Note: Births - Death can also be expressed as the net increase per year, which we previously calculated as a part of the initial information section.

Step 2: Substitute values into exponential growth formula

N(t) = N₀ * e^rt

1. t = 3 years
2. N₀ = 160 individuals
3. r = 0.09375 individuals per year
4. e = 2.718 (constant)

N(3) = 160 * e^0.09375*3 = 160 * e^0.28125

1. Using a calculator, we can approximate e to be 1.3246
2. Therefore, N(3) = 160 * 1.3246 = 211.94

Because we cannot have .94 of a wood duck, we would round our answer up to 212 wood ducks. This means that the population after 3 years of exponential growth will be approximately 212 wood ducks.

Part b) Geometric Growth

Key Concept: Geometric growth is a discrete version of exponential growth. It assumes reproduction happens in pulses, such as once per year, rather than continuously. For example, many species have a particular “mating season,” such as birds and many mammals, in the Spring, when environmental conditions are most favorable for young life. At other parts in the year, population growth is stagnant, while the “boom” happens during a particular window of time.

In ecology, geometric growth can be modeled by the formula N(t) = N₀ + (1+r)^t, where:

1. r is the discrete growth rate (same as in part a, 0.09375)
2. t = time in years (same as in part a, 3 years)
3. Because we are primarily studying the effect of different growth formulas on population growth outcomes, all other variables, such as the amount of time we are studying, and the rate at which the population grows, will be held constant. This way, any changes in our outcome will be attributable to changes in modeling technique, as opposed to an additional factor.

We can now plug these values into the geometric growth formula as follows:

N(t) = N₀ + (1+r)^t

N(3) = 160 * (1+0.09375)³ = 160 * (1.09375)³ = 160 * 1.301 = 208.16

According to the rules of rounding, any decimal less than 5 gets rounded down to the lower whole number. Therefore, the geometric growth estimate after 3 years is approximately 208 wood ducks.

Part c) Logistic Growth Rate

Key Concept: Logistic growth considers environmental limits (like limited amounts of food and space), leading to a carrying capacity, K, the point above which population growth cannot be sustained in the long-term. As the population nears K, growth slows, as population reaches the limit at which the environment’s resources can reliably sustain their numbers. This leads to a plateau, in which exponential growth becomes restricted by environmental stressors and restraints on unlimited, unchecked growth.

The logistic growth rate (change in population size) is modeled as: ΔN = rN (1 - N/K), where:

1. ΔN = change in population size in one year
2. r = discrete growth rate
3. N = initial population size (same as N₀)
4. K = carrying capacity

In our problem, these values correspond with:

1. ΔN: needs to be solved for
2. r = 0.09375
3. N = 160
4. K = 265

With our variables defined, we can now evaluate the problem using the logistic growth formula and substitution:

ΔN = rN (1 - N/K)

ΔN = 0.09375 * 160 * (1 - 160/265) = 0.09375 * 160 * (1 - 0.6038) = 0.09375 * 160 * 0.3962 = 5.94

The logistic growth rate (change in number of wood ducks in one year) is approximately 5.94 individuals per year, meaning the population is growing more slowly due to environmental limits.

Answer summary:

A) 212 wood ducks using exponential growth

B) 208 wood ducks using geometric growth

C) +5.94 individuals per year using logistic growth (can be rounded to +6 per year)