Population Ecology, part 1

Principles of Ecology Week 2

Reminder - self reflection due on Sunday

  • Prompt and submission available on Moodle.

Reminder - complete weekly readings before class

  • Calendar available on Moodle

  • Also on Moodle: Please complete “Who’s in class” survey and fill out coworking hours poll (these will start next week, I will announce time on Friday).

Opportunity for logistical questions

Population ecology

What defines a population?

  • Individuals of the same species living together

  • Individuals interact with one-another
    e.g. mating, facilitating, competing

Photo by Jeffrey Hamilton on Unsplash

Photo by Dawn W on Unsplash

Photo by National Cancer Institute on Unsplash

How does a population grow or shrink?

  • Birth (+)
  • Death (-)
  • Immigration (individuals coming in, +)
  • Emigration (individuals leaving, -)

How does a population grow or shrink?

  • Birth (+)
  • Death (-)
  • Immigration (individuals coming in, +)
  • Emigration (individuals leaving, -)

For now, we will consider a ‘closed’ population

How does a population grow or shrink?

In a closed population, population size (\(N\)) only changes due to births and deaths

\[N_{t+1} = N_{t} + B - D\]

\[N_{t+1} - N_{t} = B-D\]

\[\Delta N = B-D\]

\[\frac{dN}{dt} = B-D\]

How does a population grow or shrink?

\[\frac{dN}{dt} = B-D\]

We need to know: What determines \(B\) and \(D\)?

Depends on the birth rate and mortality (death) rate:

  • Total births = per-capita birth rate * number of individuals

\[B = bN\]

  • Total deaths = per-capita mortality rate * number of individuals

\[D = dN\]

How does a population grow or shrink?

\[\frac{dN}{dt} = B-D\]

\[B = bN \text{ and } D = dN\]

\[\frac{dN}{dt} = bN - dN\]

\[\frac{dN}{dt} = (b- d) N\]

\[\boxed{\frac{dN}{dt} = rN}\]

Populations grow when there are more births than deaths (\((b-d) > 0\); aka \(r > 0\))

sim_df_pos <- run_exponential_model(time = 50, params = c(r = 0.1), init = c(N1 = 100))
plot_continuous_population_growth(sim_df_pos) + 
  scale_x_continuous(expand = c(0,0)) +
  labs(title = "r = 0.1")

Populations shrink when there are more deaths than births (\((b-d) < 0\); aka \(r < 0\))

sim_df_neg = run_exponential_model(time = 50, params = c(r = -0.1), init = c(N1 = 100))
plot_continuous_population_growth(sim_df_neg) +
  scale_x_continuous(expand = c(0,0)) +
  labs(title = "r = -0.1")

The magnitude of \(r\) determines the rate of growth

How can such a simple population model help?

Source: Wikimedia

Recall that the magnitude of \(r\) determines the rate of growth

What happens when \(r = 0\)?

Equilibrium in ecology

\(r = 0\), when the rates of birth and death are equal to one another (\(b-d = 0\))

When this is true, the total number of births cancels out the total number of deaths, meaning the population size does not change.

\[\frac{dN}{dt} = rN = 0\]

This is the equilibrium state of the system.

Note that “equilibrium” doesn’t mean “nothing is changing”: there might be new births, and new deaths.

But they cancel each other, and the population size remains constant.

Equilibrium in ecology

\[\frac{dN}{dt} = rN = 0\]

This is the equilibrium state of the system.

The exponential growth model has two equilibrium conditions. One is that \(r = 0\). What is the other?

We will be exploring the equilibrium conditions for many of the models we introduce through the semester, because it has been a central concept in ecology.

How math helps us derive insights from the model

\[\frac{dN}{dt} = rN\]

  • If we integrate this model, we can predict population size at any time in the future:

\[N_t = N_0*e^{rt}\]

  • We can also predict how long it takes a population to double in size (‘doubling time’):

\[t_{\text{doubling}} = \frac{ln(2)}{r}\]

Semester project overview

Activity 1 due next Sunday Sept 15

Population Ecology, pt 2:
Structured populations

Exponential growth model

\[\frac{dN}{dt} = rN\]

Key assumptions of the exponential growth model

  • No immigration or emigration (Closed population)

  • Constant birth and death rate (\(b\) and \(d\) don’t vary with \(N\))

  • No variation within population (all individuals have similar \(b\) and \(d\))

    • This implies that \(b\) and \(d\) don’t vary with age or stage

  • Continuous population growth without time lags
    (e.g. no seasonality)

What if these assumptions don’t hold?

Key assumptions of the exponential growth model

  • No immigration or emigration (Closed population)

  • Constant birth and death rate (\(b\) and \(d\) don’t vary with \(N\))

  • No variation within population (all individuals have similar \(b\) and \(d\))

    • This implies that \(b\) and \(d\) don’t vary with age or stage

  • Continuous population growth without time lags
    (e.g. no seasonality)

Organisms with stage structure

Photo by Mike Erskine on Unsplash

Modeling structured populations

Consider a bird species where each (female) adult lays \(10\text{ eggs}\). Only \(10\%\) of eggs sucessfully hatch into juveniles, and \(50\%\) of juveniles survive and mature into adults. Adults die after reproduction.

  • How can we represent this in a diagram?

$arr
  row col Angle Value       rad    ArrowX    ArrowY     TextX     TextY
1   2   1     0   0.1 0.1666667 0.3325479 0.2333352 0.3333333 0.2133333
2   3   2     0   0.5 0.1666667 0.6658813 0.2333352 0.6666667 0.2133333
3   1   3     0    10 0.3333333 0.5005236 0.7333329 0.5000000 0.7533333

$comp
             x   y
[1,] 0.1666667 0.4
[2,] 0.5000000 0.4
[3,] 0.8333333 0.4

$radii
       x         y
[1,] 0.1 0.1189873
[2,] 0.1 0.1189873
[3,] 0.1 0.1189873

$rect
          xleft      ybot    xright      ytop
[1,] 0.06666667 0.2810127 0.2666667 0.5189873
[2,] 0.40000000 0.2810127 0.6000000 0.5189873
[3,] 0.73333333 0.2810127 0.9333333 0.5189873

Photo by Mike Erskine on Unsplash

Consider a short-lived plant that lives for 6 years. Each year, only \(1\%\) of seeds survive and germinate into seedlings. (The other seeds are eaten by squirrels and jays). \(50\%\) of seedlings survive to a juvenile stage, and \(95\%\) of juvenlies transition into reproductive sub-adults. Each sub-adult makes \(10\text{ seeds}\) per year, and \(95\%\) of sub-adults transition into fully-mature adults. Each mature adult makes \(450\text{ seeds}\) per year, and \(10\%\) of adults survive to a post-reproductive adult stage, which are incapable of making new seeds.

  • Draw the transition diagram.

Transition matrices

These transition diagrams can be represented mathematically as transition matrices.

For a population with \(n\) stages, we can capture the demographic transitions in an \(n\mathrm{-by-}n\) matrix.

  • Each element captures the contribution from the current column, to the current row.
    • The top row always reflects new births
    • The first (leftmost) column reflects the youngest individuals
    • The last (rightmost) column reflects the oldest individuals

Worked example

Consider a bird species where each (female) adult lays \(10\text{ eggs}\). Only \(10\%\) of eggs successfully hatch into juveniles, and \(50\%\) of juveniles survive and mature into adults. Adults die after reproduction.

\[ \begin{bmatrix} ~ & ~ & ~ & ~ & ~& ~ & ~ \\ ~ & ~ & ~ & ~ & ~& ~ & ~ \\ ~ & ~ & ~ & ~ & ~& ~ & ~ \\ ~ & ~ & ~ & ~ & ~& ~ & ~ \\ ~ & ~ & ~ & ~ & ~& ~ & ~ \\ ~ & ~ & ~ & ~ & ~& ~ & ~ \end{bmatrix} \]

Worked example

Consider a bird species where each (female) adult lays \(10\text{ eggs}\). Only \(10\%\) of eggs successfully hatch into juveniles, and \(50\%\) of juveniles survive and mature into adults. Adults die after reproduction.

\[ \begin{bmatrix} 0 & 0 & 10 \\ 0.1 & 0 & 0 \\ 0 & 0.5 & 0 \end{bmatrix} \]

Your turn

  • Consider a species of fish whose individuals live for 3 years (age classes 0, 1, 2, and 3). Newborn fish have a \(30\%\) survival rate to year 1; year 1 fish have a \(80\%\) survival rate to year 2; year 2 fish have a \(50\%\) survival rate to year 3; and all fish die in their third year.

  • Newborn fish are sexually immature and cannot give birth. Year 1 fish can give birth to \(1\) newborn per year; Year 2 fish can give birth to \(8\) newborns per year; and Year 3 fish can only give birth to \(0.5\) newborns per year.

  • Draw a transition diagram and write the transition matrix for this population.

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

The value of a transition matrix

  • The product of the current population distribution and the transition matrix tells you the future population distribution

The value of a transition matrix

  • From our worked example, recall the transition matrix

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \]

  • At time \(t=0\), the population has \(52\) individuals in stage 0, \(10\) in stage 1, \(15\) in stage 2, and \(30\) in stage 3.

  • What is the expected distribution of individuals at \(t=1\)?

The value of a transition matrix

Matrix product of the transition matrix and the current distrubition:

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 52 \\ 10 \\ 16 \\ 30 \end{bmatrix} \]

The value of a transition matrix

Matrix product of the transition matrix and the current distrubition:

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 52 \\ 10 \\ 16 \\ 30 \end{bmatrix} = \begin{bmatrix} 153 \\ 15 \\ 8 \\ 8 \end{bmatrix} \]

The value of a transition matrix

For timestep \(2\):

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 153 \\ 15 \\ 8 \\ 8 \end{bmatrix} \]

The value of a transition matrix

For timestep \(2\):

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 153 \\ 15 \\ 8 \\ 8 \end{bmatrix} = \begin{bmatrix} 83 \\ 45.9 \\ 12 \\ 4 \end{bmatrix} \]

The value of a transition matrix

For timestep \(3\):

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \times \begin{bmatrix} 83 \\ 45.9 \\ 12 \\ 4 \end{bmatrix} = \dots \]

The value of a transition matrix

  • We can keep iterating over and over (and over), or….
  • Calculate the \(\mathrm{eigenvalue}\) of the transition matrix
  • The dominant eigenvalue reflects the long-term growth rate.

\[ \begin{bmatrix} 0 & 1 & 8 & 0.5 \\ 0.3 & 0 & 0 & 0 \\ 0 & 0.8 & 0 & 0 \\ 0 & 0 & 0.5 & 0 \end{bmatrix} \xrightarrow[]{\text{Eigenvalue}} 1.33 \]

The value of a transition matrix

  • The dominant Eigenvalue represents the expected long-term annual growth rate (\(\lambda\))
  • Populations grow when \(\lambda \gt 1\)
  • Populations shrink when \(\lambda \lt 1\)
  • Population size is stable when \(\lambda = 1\)