Lecture 17 - Population Ecology

What is a population?

collection of individuals living in an area

What are the units of population size and population density?

population size: N (number of individuals)

population density: N/area

Why do we care about population size (N)?

- natural resource management
- size of fish stocks to see how much is available to us
- abundance of outbreaking insect pests in forests

- conservation
- population extinctions and declines of species

- health
- populations of viruses and bacteria in humans and animals

- predicting population growth
- what limits population growth

What is the Pink salmon graph depicting and what type of reasoning of understanding N, does it reflect?

- natural resource management
- graph should us when there was a high abundance of salmon
- represents abundance of salmon in that square area in BC

What is the decline in bats due to white-nose syndrome depicted?

What type of reasoning of understanding N, does it reflect?

- for conservation
- y-axis shows the number of bats in a cave
- x-axis is the number of years
- you can see their population size growing quickly over a period of time and but all off the sudden they decline
- this is where we see a huge decline in 2010 because of this disease in bats

What is the graph of HIV population dynamics depicting?

What type of reasoning of understanding N, does it reflect?

- health for humans
- y-axis in blue: quality of human immune system
- y-axis in red: population size of HIV viral particles

- the number of viruses can predict how sick a person might be
- can help us predict what might be the population size in an infected person with not treatment

What is the graph of HIV population dynamics depicting?

What type of reasoning of understanding N, does it reflect?

- health in humans
- number of confirmed cases since 2020
- to make predictions about the public health risk posed by covid-19

What did Malthus say about human population growth?

- principle of population
- human growth cannot grow faster than food production
- populations can't grow faster than the resources that support them

What did Paul Ehrlich say about human populations growth?

- explosive growth would have negative and catastrophic social and environmental consequences

What does this graph on population growth show us?

- that there was rapid population growth since the Industrial Revolution

What does this graph of human population growth prediction show us?

- that we will encounter a carrying capacity of approx.10B people

What do all of these time series graphs have in common?

they are plotting the same thing
- y-axis is population size
- x-axis is across a period of time

What is the goal of projection models?

- predict the population growth of N over time
- determine how many individuals there are Nt
- how many individuals are in the population one step later

What is the general model for population growth? What is a challenge?

Nt+1 = f(Nt)
- population one step later = population now times f

- challenge is deciding what f represents

What are time steps? What is the difference between differential equations and difference equations?

Differential Equations: continuous-time approach
- small time steps
- limits
- growth is smooth
- suited for species with CONTINUOUS REPRODUCTION

Different Equations: discrete time approach
- time steps are discrete units: day, month, year
- iterated recursion equations: constant in one time step until you change to another
- growth is stepwise/bumpy
- suited for episodic reproduction

What is Nt? What is No?

population size at time t

initial population size at t=0
- starting population size

What does variables D, B, E, and I represent? How does this affect N change?

D - # of people who die during one time step
B - # of people born during one time step
E - # of people who emigrate during one time step
I- # of people who immigrate during one time step

Nt+1 = Nt - D + B - E + I

What are the two steps of a simplified population growth model?

- simplify and convert changes to per capita rates
- think of birth and death as constant
- population changes by constant factor each time step

What is the constant factor that changes the population in each time step? What is it? What is the growth model equation now? What is this model called?

- λ (lamda)
- factor of population change over one time unit
- "finite rate of increase"

λ=Nt+1/Nt
Nt+1 = λNt

- geometric growth

What happens when λ is less than or greater than 1?

λ > 1 means more births than deaths; population increases
λ < 1 means more deaths than births; population decreases

What is the generalized geometric growth model for many different years (time steps)? What do these notations represent?

Nt = No λ^t

- Nt is the population at the time step
- No is the initial population
- λ is constant of birth and death rates
- t is the number of time steps

What is the difference between continuous time models? What is this new common variable? What does it represent?

instantaneous fixed capita
- the instantaneous rate of change instead of at each time step

- b-d = r (constant)
r - is the intrinsic rate of increase

What is the differential equation? What is this growth called?

dN/dt = rN
model is exponential growth

What is the relationship between discrete time and continuous time equations?

discrete time: different equation
- Nt+1 = λNt
- Nt = Noλ^t
- geometric growth (step function)

continues time: differential equation
- dN/dt = rN
- Nt = Noe^rt
- exponential growth (smooth function)

Noλ^t = Noe^rt
λ = e^r
ln λ = r

What does this graph show of graphing geometric growth?

- the greater the λ>1 value, the faster the growth is
- the λ

What does this graph show of graphing exponential growth?

- r=ln(>1), the faster the growth is
- r=ln(

What are consequences of both these functions?

- they are both constants that simply reflect biology
- constant + growth rate produces population that grows in exponential way

What is a result of species growing to infinity and going extinct?

- no species have sustained λ >1 for long time (infinite growth)
- no species have sustained λ

What are the staggering implications of these growth models?

- bad for long term growth
- doesn't account for factors that keep populations from going extinct or growing to infinity

What are two factors that keep populations from growing infinity and going extinct?

density- dependant regulation (growth depends on N)
- depends on population size and resources available

density-independent reduction
- wipe out populations independant of how many there are

What does the graph of growing bacteria in nutrient-rich broth look like?

- exponential growth but they don't grow like that for long

What does the bacteria growth actually look like?

- S-shaped curve
- growth rate slows down because they run out of food
- logistic growth
- density-dependant

How is density-dependant growth modelled by logistic equation?

dN/dt = rN(1-N/K)

- N is far from K, its growing as RN
- N is big as K, population growth stops and goes to 0

- K is carrying capacity when populations get too high

What does this image/graph depict?

no braking at low density

complete breaking when N approaches K

What is the logistic growth model? What does it graphs look like? What are the variables in this function?

Nt = KNoe^rt / K + No(e^rt -1)

- S-shaped curve
- N is population at time t
- K is carrying capacity
- r is intrinsic growth rate
- t are the number of timesteps

What does K the carrying capacity mean?

- maximum number of individuals that there can be fore resources
- inflection point is halfway to the carrying capacity
- this is when populations are growing the fastest
- after the inflection point, population growth slows down because they are running out of food

What does the graph of logistic functions show us?

- they are only S-shaped when starting from low umbers
- when r and K are constant but the initial population changes, it changes the graph

- N=K, straight line
- N>K starts higher and goes to carrying capacity
- N- N approaching K comes higher to carrying capacity

What does this graph tell us about the slopes of a logistic function?

- dN/dt is moderate
- dN/dt is highest
- dN/dt is moderate
- dN/dt is zero

What are pros and cons of logistic models?

pros:
- tractable model of INTRAspecific competition for resources
- simple

Cons:
- too simple; one kind of density-dependance
- gradual approach to K
- not linear!

How can we add complexity into our models?

- different forms of density-dependance factors
- time lags before population sizes decrease, cause overriding K
- incorporation of species interactions

What are Allee effects?

- negative effects of low density from social benefits: mating finding, group living, and defence
- populations fluctuate btw K and lower limit (set by all effects)
- dropping below lower limit goes to extinction
- important in convection