4/3: Notes on Population Dynamics and Growth Models (REC)(DOC)

Biological Hierarchy and Population Dynamics

• Overview of biological hierarchy: moving from individuals and organisms to populations and higher levels such as communities and ecosystems.

• Population Dynamics: Focus on population growth and factors influencing changes in population size.

Case Studies of Caribou Populations

• Comparison of two Caribou populations on St. Paul and St. George islands in the Pribilof Islands, Bering Sea.

  • St. Paul: 40 square miles, started with ~25 individuals in 1910. Population growth to over 2000 by 1930s and subsequent crash to extinction by 1960.

  • St. George: Similar habitat, also started with ~25 individuals but only reached a max of ~50-60 individuals and still exists today.

• Analyzing reasons for differing population dynamics:

  • Possible factors affecting growth rates on St. Paul: food availability, shelter, absence of predators, disease spread, and sudden predator introduction.

• Population Biologists: Study parameters of population changes and factors for growth or decline.

Exponential Growth

• Definition and characteristics of Exponential Growth:

  • All populations produce more offspring than environmental resources can support.

  • Darwin's natural selection highlighted this concept.

  • Example of elephant reproduction illustrating exponential growth: theoretically, population can increase drastically across generations leading to impracticable population sizes.

• Mathematical Example with Bacteria:

  • Starting with 1 bacterium, doubling every day:

  • Day 1: 1

  • Day 2: 2

  • Day 3: 4

  • Day 4: 8

  • Day 5: 16

  • Day 6: 32

  • After 7 days, population grows exponentially.

• Introduction of Geometric Growth as a specific form of exponential growth, though for this course the general term is sufficient.

Thought Experiment with Paper Folding

• Folding paper shows exponential growth potential:

  • After 3 folds: thickness of a fingernail.

  • After 7 folds: thickness of a notebook.

  • After 10 folds: 4-5 inches thick.

  • Amazing thickness calculations at 17 folds (2-story house height) and beyond (e.g., 30 folds reaching outer atmosphere).

• Benefits of exponential growth:

  • 50 folds: thickness to the sun (~95 million miles).

  • 100 folds: thickness equal to the radius of the universe (12 billion light years).

Mathematical Representation of Growth

• Population growth determined by:

  • Probability of birth (B)

  • Probability of death (D)

  • Per Capita Growth Rate (r): ( r = B - D )

• Changes in population size represented as:

  • ( rac{98N}{98t} = rN )

  • where N = population size, t = time, and ( 98 ) represents change.

Logistic Growth Model

• Logistic Growth connects early exponential growth to environmental limits.

• Characteristic S-shaped curve with:

  1. Initial exponential growth phase.

  2. Decelerating growth rate after the inflection point (switch from increasing to decreasing growth).

  3. Fluctuation around carrying capacity (K).

• Carrying Capacity (K): Maximum sustainable population size determined by environmental resources.

• Logistic growth equation:

  • ( rac{98N}{98t} = rN \left( \frac{K - N}{K} \right) )

• Growth stages include:

  • Density-independent growth (below inflection point).

  • Density-dependent growth (above inflection point).

Environmental Influences on Carrying Capacity

• Carrying capacity is dynamic, influenced by:

  • Food resources.

  • Disease presence.

  • Space, nesting materials, and potential mates.

• Population size responds to changing K:

  • Population decreases when N exceeds K and increases when N is less than K.

Population Inertia

• Definition: Persisting beyond carrying capacity due to delays in population adjustment.

• Example with St. Paul Caribou:

  • It experienced population inertia, exceeding carrying capacity before crashing.

Applications of Population Dynamics in Conservation

• Case of wolves in Wisconsin: shows the interaction between scientific understanding and politics in species management.

  • Endangered Species: early exponential growth stage.

  • Threatened Species: rapid growth nearing carrying capacity.

  • Management Goals: achieving sustainability while preventing overpopulation through regulated population control practices.

Summary

• Understanding population dynamics requires analyzing growth patterns, external influences on carrying capacity, and the sociopolitical context of wildlife conservation. Exponential and logistic growth models serve as frameworks to interpret and predict population changes.

Bergmann’s Law.

Endotherms living in warm climates are more likely to have problems shedding heat from their bodies to

maintain their optimum internal body temperature. Those living in colder climates are more likely to

have problems retaining heat and keeping their bodily temperatures elevated at their optimum.

Bergmann’s Law (sometimes referred to as Bergmann’s Rule) says that, within animals that have the

same overall body plan, those that live in warmer climates are likely to have smaller bodies than those

that live in colder climates. Having a smaller body, and corresponding greater surface area-to-volume

ratio, in warm climates helps the animal to lose heat to maintain that optimum temperature.

Conversely, having a large body, and smaller surface area-to-volume ratio, helps the animal to maintain

body heat in cold areas.

Surface Area to Volume Ratios

The surface area and volume of animals increase at different rates with an increase in animal size. For

example, as an approximation, when considering cubes of increasing edge size, the surface area of the

cube increases at a rate of the edge increase squared, but the volume increases as a function of the

edge increase cubed. So, as animals grow, their inside (volume) increases at a faster rate than their

outside (surface area) does. Although we can say that a larger animal has a larger surface area than a

smaller animal does, as well as a larger volume, often it is the ratio of surface area to volume that

matters in determining how well an animal functions in a particular environment. Because of the

difference in how surface area and volume increase with increased body size, larger bodies are

described as having a smaller surface area-to-volume ratio than smaller bodies have.

Lecture week 11 (4/3)

Population Ecology – Exponential and Logistic Models

I. Intro to population Ecology

a. What is it?

i. Population ecology is the study of how and why population size changes over

time and the effects that population change has on the population. Population

genetics answered how populations change genetically over time, and how

new populations may get started. The two fields are often linked in

discussions, but populations may get larger or smaller without genetic change,

and genetic change may occur without population size changes.

b. Why study it?

i. Understanding how populations change, and what is causing those changes,

provides us with important information. For example, knowing how

populations of food plants grow, and are limited by density, helps us to better

plan our farms, and knowing how insect populations grow and decline helps

us to maintain local ecosystems that keep our water and air clean.

c. How is it done?

i. A typical problem for conservation biologists is to figure out how many

individuals there are in a population and why the population is growing or

declining.

ii. This central question is answered by monitoring the changes of specific

elements of a population (e.g., numbers, ages, sex ratio, etc.) over time,

keeping track of the factors that affect these elements, and making predictions

about and studying outcomes of population changes.

iii. Example: Case Study of Caribou on Pribilof Islands.

II. Exponential Growth

a. Elephants Galore!

b. Imagine a population of organisms living in an unlimited environment, with unlimited

resources, and unlimited space. Imagine that the organism divides into 2 every 24

hours. You could then chart its population growth:

Day Population Size

1 1

2 2

3 4

4 8

5 16

6 32

This is exponential growth

When these data are graphed, you get a curve that starts out with a relatively slow

increase that quickly turns to a rapid increase in numbers with very little change in

time.

c. Paper-folding Demo

d. We can describe these patterns mathematically.

i. Individuals have the probability of dividing/giving birth (b)

ii. Individuals have the probability of dying (d)

iii. Instantaneous rate of growth per individual (or per capita) is b-d, which is

typically known as r. Rate of growth (r) = b - d (typically considered as a

maximum r or rmax).

iv. If the population size = N, then the rate of population growth = rN

e. Formally written as the Exponential Growth Equation:

(ΔN/Δt) = rN

Note: not only should you know this equation, but, more importantly, you should

know what it means and how to describe it in terms other than the variables listed.

For example, the above exponential growth equation can be translated as: the change

in population size per change in time (rate of growth) equals the rate of population

growth for the individual multiplied by the number of individuals in the population.

III. Logistic Growth

a. Consider the sheep of South Australia.

b. Logistic or Sigmoid growth involves three stages:

i. Initial exponential growth.

ii. Decelerating growth rates.

iii. Fluctuations around some “average” population size, often called K or

carrying capacity of the environment.

c. Again, we can describe this pattern mathematically.

d. The changing rate of growth of a population showing sigmoid growth can be modeled

by the Logistic equation:

(ΔN/Δt) = rN((K-N)/K)

e. Where:

i. N = current pop. size

ii. K = the highest value that N can take or the “carrying capacity” of the

environment

iii. Watch out for r. Where r was r-max in the exponential equation, r here is

relative (rrel). How an individual can reproduce relative to the influence

population size has on the individual.

iv. How can the equation be translated into ideas and descriptions from variables

(as we did in IIe above)?

f. Overall, the logistic model comes closer to predicting real populations than the

exponential model does. The exponential model doesn’t take into account the limit

that resources place on populations.

g. Notes about Carrying Capacity (K).

i. It can be defined as the point at which the population size is in equilibrium

with resources

ii. Or the number of individuals of a species that the environment can support.

iii. Or the number of individuals that can survive in the environment.

iv. K is not constant! As environmental conditions change, so does K. That is, as

time goes by, and the environment changes (food base changes, other

resources change, etc.) the K for a given population in a given environment

shifts.

v. Note: the differences between i, ii, and iii above are based on point of view.

One is the population and environment together (i), the environment (ii), and

the population (iii).

h. Using the logistic model

i. If N > K:

ii. If N < K:

iii. If N = K:

i. Example Case Study of Gray Wolf populations in Wisconsin

IV. Problems with the models.

a. What are some of the problems with these models?