Population Ecology, part 1

Principles of Ecology Week 2

Reminder - self reflection due on Sunday

  • Prompt and submission available on Moodle.

  • For next week: See ā€œWeek 3 Activitiesā€ on class website or Moodle (read peer-reviewed paper and listen to a podcast episode) . . .

Reminder - complete weekly readings before class

  • Todayā€™s lecture covers material from Chapter 1 of Gotelliā€™s Primer of Ecology (link)

  • On Friday, we will begin to cover structured population growth, which is introduced in this video

Opportunity for logistical questions

Bird Walk at BREC this Saturday

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 ā€˜closedā€™ 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 \(r > 0\))

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

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 ā€œequilibriumā€ doesnā€™t mean ā€œnothing is changingā€: there will 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 (ā€˜doubling timeā€™):

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

The exponential growth model: summary

  • We can imagine that the dynamics of a population (i.e. how does population size change over time?) are strictly governed by two processes: births and deaths

  • We can express each of these in terms of the per-capita rates of birth and death: for a given unit of time, what is the per-capita rate of reproduction and of mortality?

  • The net population growth is governed by the difference in birth and death rate (\(r = b-d\))

  • This simplest model makes some unrealistic predictions, but some useful predictions

  • This simplest model also makes some assumptions

The exponential growth model: summary

Key assumptions of the exponential growth model

  • No immigration or emigration (Closed population)

  • Constant birth and death rate (\(b\) and \(d\) donā€™t vary with \(N\))

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

    • This implies that \(b\) and \(d\) donā€™t vary with age or stage

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

Semester project overview

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 \(d\) donā€™t vary with \(N\))

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

    • This implies that \(b\) and \(d\) donā€™t vary with age or stage

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

What if these assumptions donā€™t hold?

Key assumptions of the exponential growth model

  • No immigration or emigration (Closed population)

  • Constant birth and death rate (\(b\) and \(d\) donā€™t vary with \(N\))

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

    • This implies that \(b\) and \(d\) donā€™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 successfully hatch into juveniles, and \(50\%\) of juveniles survive and mature into adults. Adults die after reproduction.

  • How can we represent this in a diagram?

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 juveniles 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 distribution:

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

The value of a transition matrix

Matrix product of the transition matrix and the current distribution:

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

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), orā€¦.
  • Calculate the \(\mathrm{eigenvalue}\) of the transition matrix
  • The dominant eigenvalue reflects the long-term growth rate (\(\lambda\)).

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

Computing eigenvalues in R

The value of a transition matrix

  • The dominant Eigenvalue represents the expected long-term annual growth rate (\(\lambda\))
    • Over the long term, \(\frac{N_{t+1}}{N_t} = \lambda\)
  • Populations grow when \(\lambda \gt 1\)
  • Populations shrink when \(\lambda \lt 1\)
  • Population size is stable when \(\lambda = 1\)

Reminders for next week

  • Read Crouse et al. 1987 paper on using transition matrices to study turtle population dynamics
  • Listen to Sea Change podcast about sea turtle population dynamics in the 2020s
  • In-class activity on Friday next week.