1. Building Models of Exponential Population Growth (Unbounded Growth) Change in population size = Births + Immigration − Deaths − Emigration
There is a population size that we are counting. The population is going to change in size due to births, immigration. death, and immigration.
*If immigration and emigration are ignored, a population’s growth rate (per capita increase) equals birth rate minus death rate
a. Exponential Growth
i. The parameter r is the instantaneous per capita rate of increase (r ins in the text) Don't forget about the age old question of What are the key features of the art "virgin and the child"?
1. r tells us at any given instant right at this second how fast the
population is changing
2. per capita - “per head” or “per individual”
a. Per the # of individuals that are already in the population.
Given the # of individuals here, what is the chance of the
population growing? It's basically the total growth rate
divided by the number of individuals.
N = # of individuals B = Birth AND = Death Don't forget about the age old question of How do we gather the information to reconstruct history?
r = RN
R = B - D
r = (b-m)n , this represents the population change over time
ii. The balance of births and determines r
1. b = the instantaneous chance that an individual will give birth. If
this value is high, this means there are individuals reproducing or
the individuals that are reproducing are giving birth to a lot of
2. m = the instantaneous chance that an individual will die
dN = Δt
ΔN = − m =
(b )N rN ; this means individuals are being dt born and dying at the same time We also discuss several other topics like What is the formula of a slope?
* In an open population, you would include immigration with births ( = gains per capita) and emigrants with deaths (= losses per capita)
We also discuss several other topics like If accumulation is more significant than ablation, what will happen at the front of the glacier?
Graph : dN/dT increases as population size increases. This means the growth accelerates which is why we have a J shaped curve!!
- If r, the instantaneous per capita rate of increase, is constant, the population grows exponentially We also discuss several other topics like What is a narrow band of electromagnetic radiation that can be conceptualized as a wave or a stream of electrically charged particles called photons?
- Larger r means the population is growing faster and Smaller r means the population is growing slower. THIS IS VERY IMPORTANT BECAUSE YOU NEED TO BE ABLE TO COMPARE CURVES
- r < 0 - more deaths than births so the population is declining
ΔN = 0
- r = 0 - no change ; ; flat line Δt
- r > 0 - more births than deaths so the population is increasing
- The rate of increase ( ) is a function of both r, the instantaneous rate of increase, and Δt N, the population size. r > 0
- In unrestrained growth, with constant r, increases as population size increases (i.e. Δt growth accelerates).
2. Logistic Population Growth (Growth with Limits)
a. Populations cannot continue to increase indefinitely
i. Exponential growth cannot be sustained for long in any population 1. In unmanaged populations there are a host of factors that will
cause individuals to stop reproducing or die. This causes the
population growth to be limited. If you want to learn more check out On a tissue level, what makes up the heart?
ii. A more realistic population model limits growth by incorporating carrying capacity (K), which is the maximum population size the
environment can support
1. So there is going to be a point at which the population is growing
and due to all of these different factors the population is going to
stop growing. This number at which the population stops growing
is the carrying capacity of the environment
b. The Logistic Growth Model
i. Starts with the exponential (r) model and adds an expression that reduces the instantaneous per capita rate of increase as N approaches K
r = ; determines the fraction of the carrying capacity of the population K ii. The instantaneous per capita rate of increase under logistic growth is (K−N)
always ≤ r : is ALWAYS < 1 K
c. The logistic model of population growth produces a sigmoid (S-shaped) curve
i. As the population size gets bigger, the growth rate is starting to slow down and eventually levels off at carrying capacity.
ii. The S shaped curve means the population growth is slowing down J shaped curve = Exponential Growth S shaped curve = Logistic Growth
d. Examples of limited growth
i. The growth of laboratory populations in constant environment lacking predators but with limited food
ii. Paramecium : The most common way these protist reproduce is by splitting itself in half and it does this about once a day. Let's say at Day 0 we have 1 paramecium. By Day 6, we have 64 paramecium which means its full. When is the paramecium population half full? Day 5! This is the power of exponential population growth. Each day the population is
increasing by an increasing amount.
e. Under the logistic growth model, the per capita instantaneous population growth rate is negatively density - dependent
1. As the population gets more crowded, meaning the density
increases, the population decreases. This is referred to as negative density dependent. This is a negative effect!
ii. Exponential growth
* = r
iii. Logistic growth
* = r
(K − N)
As N increases, the instantaneous growth rate decreases
f. Because instantaneous per capita growth is determined by the difference between the instantaneous per capita rates of births and deaths*, logistic growth implies density - dependence of birth rate, death rates or both.
*In an open population, you would include immigrants with births (= gains per capita) and emigrants with deaths (= losses per capita)
i. Be able to tell if the population is density dependent or density
independent through the given curve!!
Interactions Between Species
1. What is a Community?
a. Community : an assemblage of species living in close enough proximity for potential interaction
i. Communities have emergent properties such as species, diversity, tropic structure and stability over time
2. Interactions Between Species
a. Direction of Effect ( + or -)
b. Mechanism of Interaction
3. Species Interactions
a. Competition (- / -)
i. Mechanisms of Competition
1. Exploitation - individuals deplete resources by consuming or using
2. Interference - aggressive encounters among individuals
3. Competition between ecologically similar species can lead to
4. Differentiation in resource can allow similar species to coexist in a
community. This is called resource partitioning
ii. Character Displacement
b. Consumption (+ / -)
i. Mechanism of Consumption
1. Predation - a predator species kills and consumes the prey species
2. Herbivory - herbivores eat parts of a plant or algae
3. Parasitism - parasites live in or on their host for an extended period
a. Symbiosis : species live in direct and intimate contact with
ii. Predator - Prey Dynamics
1. Hares increase rapidly
2. Lynx increase due to abundant hares
3. Hares decline due to predation, starvation
4. After lag, lynx decline because few hares
5. Hare population stabilizes, food plants recover
6. Cycle starts anew!
iii. Consumers and consumed : A evolutionary arms race
1. Prey display various defense adaptations
2. Behavioral defenses include hiding, fleeing, forming herds or
schools, self-defense and alarm calls
3. Animals also have morphological and physiological defence
4. Mechanical and chemical defenses protect species such as
porcupines and skunks
5. Structural / Physical Defense
a. Structural / physical defense : the way an organism looks
can also be an important part of the predator/prey
i. Acosabotic Coloration (also called warning
coloration). This can warm the predator that their
prey is poisonous.
ii. Cryptic Coloration (camouflage). This allows an
animal to blend in with their surroundings and hide
6. Chemical Defense
c. Facilitation / Cooperation (+ / +)
1. Obligate vs. Facultative
2. Symbiotic vs. Non-symbiotic
1. An interaction in which one species has positive effects on another
species without direct and intimate contact
1. Community Assembly
a. The assembly of communities results from multiple processes
i. Evolutionary History
iii. Abiotic Factors
iv. Biotic Factors
2. Measuring Diversity
a. Species Diversity
i. The species diversity of a community is the variety of species that make up the community (How many species there are in a given community).
There are two patterns of species diversity :
1. Species Richness : the TOTAL number of species in a community
2. Relative Abundance : the proportion of each species represents of
all individuals in the community (evenness)
ii. Example : Which forest has a higher species diversity?
1. Two communities can have the same species richness (number of
species) but different relative abundances (evenness)
a. Relative abundances are more equal in community 1 than
iii. Ecologists quantity species diversity using indices that combine species richness and relative abundance (evenness)
1. The Simpson’s diversity index (D) where P is the relative i
abundance of species i, and S is the number of species
3. Trophic Structure
a. A food web is branching food chain with complex trophic interactions
1. Patterns of Diversity
a. Communities with higher diversity are :
i. More productive and more stable in their productivity
ii. Better able to withstand and recover from environmental stresses
iii. More resistant to invasive species
i. The species - area curve quantifies that idea that, all other factors being equal, a larger geographic area has more species
1. Larger areas tend to have higher colonization rates and lower
ii. The species - area relationship can be described mathematically
i. A disturbance can be any given event that changes a community by removing organisms from it or altering resource availability
1. Disturbances vary in both frequency and intensity
2. Species diversity may be maximum at intermediate levels of