What is the equilibrium population density

Reference no: EM132365523

Module: Introduction to Modelling Population Dynamics

This lab is about modelling ecological population dynamics, which means the way that populations change over time. You are encouraged to collaborate with others in working through this lab, but do make sure that you fully understand what is going on. The assessment for this lab will be through entering answers into an online LMS ‘quiz'. As you go through this lab sheet, write down notes in the space provided for the questions marked with a *, either in pen or in the word document. These will help you check your understanding as you go, and form the basis of your answers for the quiz. You will not be able to access the quiz itself during the lab - it will open at the end of the lab.Pasting plots into this word doc can also be very useful, so you can look back at them to help with the quiz later. When you start writing your R script, make sure it is neat and structured, as you will have to submit that too. But you do not have to submit this document or any hard copy document.

Population Dynamics and Competition

Get the excel spreadsheet ‘intro to ecology modelling lab.xls' from LMS, save it on your home drive and/or your own thumb drive, and open it up. You should see two columns of numbers, headed ‘time' and ‘population density'.

While the ideas of modelling population dynamics will be very similar for many different species and situations, for this exercise we can assume that we are modelling the density of a species of mouse, that the time step is in months, and the density is the number of individuals per square kilometre. For other species we might use different spatial and temporal scales.

The number in cell B6 is the initial population density. Create a plot of time against density, so you can see how density is changing over time.
*What do you notice?

*What happens when you increase the reproduction rate?
* What happens if you reduce the reproduction rate? (Change it back to 0.1 when you're done.)
*Why is this not a realistic model of population growth?
*Why is this model of population growth more realistic?
*About how long does it take for the population to get close to its equilibrium population density?
*What happens if the initial population density is higher than the equilibrium population density?
*What happens if the initial population density is equal to the equilibrium population density?
*What happens now if you increase the ‘reproduction rate' parameter?
*What happens now if you decrease the ‘reproduction rate' parameter?
*What happens if you change the ‘death rate' to be 0.0002 times the population density instead of 0.0001 (ie change the ‘density dependence death rate parameter' to be equal to 0.0002)?
*How does the equilibrium population density depend on the ‘reproduction rate' parameter and the ‘density dependence death rate parameter'?? Can you write down an equation or give an explanation in wordsthat actually gives the equilibrium population density as a function of the ‘reproduction rate' parameter and the ‘density dependence death rate parameter'??
* If the population was twice as big on a given day, and the amount of food was the same, what would happen to the amount of food eaten, according to this equation (note that this question is just checking to see if you understand the equation above - you don't actually need to run or plot anything) ?
* If the population was twice as big, and the amount of food was also twice as big, what would happen to the amount of food eaten, according to this equation?
* What is happening on this island after the first mice arrive. What happens to the mice and what happens to the food?
* For mice to survive and reproduce, they need to find mates and companions. This means there is an ‘Allee Effect' - at small population densities, populations are no longer viable. We can represent this in a simple way by adding the following line of code at the end of the block:
if (pop[time]<3) pop[time]=0
* What is the minimum viable population density for this mouse?
* How long does it take for mice to die out completely? (You might need to extend the simulation.)
* If each mouse needs a little more food each day, does that make them die out earlier or later?
* If each mouse is a little better at finding food each day, does that make them die out earlier or later?
* If the growth rate is a little smaller, does that make them die out earlier or later?
* If the amount of food initially available is a little less than 100kg, does that make them die out earlier or later?
* What makes more difference to how long they last - halving the amount of food initially available, or doubling the initial population?
* Make sure you set all your parameter values back after changing them - have you done that?
Of course the grass on the island actually keeps on growing and producing seed. Each day the grass produces half a kilogram of edible seed. Make this change to the model.
* What is happening on this island after the first mice arrive now? What happens to the mice and what happens to the food?
* What happens to the population of the new mouse after colonisation?
* What happens to the population of the first mouse after colonisation by the new mouse?
* What happens to the amount of food available after colonisation by the new mouse?
* What happens to the population of the new mouse after colonisation?
* What happens to the population of the first mouse after colonisation by the new mouse?
* What happens to the amount of food available after colonisation by the new mouse?

QUESTION 1

Part 1: Population Dynamics and Competition
Get the excel spreadsheet 'intro to ecology modelling lab.xls' from LMS, save it on your home drive and/or your own thumb drive, and open it up. You should see two columns of numbers, headed 'time' and 'population density.

The number in cell 86 is the initial population density. Create a plot of time against density, so you can see how density is changing over time.

'What do you notice?

Population density is

increasing linearly
decreasing linearly
staying constant
increasing exponentially
decreasing exponential

QUESTION 2
Look at the formula in cell B7 i.e. "-B6+136*$C$1". This means that the new density is the old one plus a certain proportion of the old one. This proportion (the value in cell CI) is the reproduction rate; a value of 0.1 means that, on average, each mouse in the population produces 0.1 babies per month. Try changing this rate and see what happens in the graph.
*What happens when you increase the reproduction rate?

Population density changes than before
faster
slower
at the same rate

QUESTION 3

*why is this not a realistic model of population growth?
the population is increasing too slowly
real populations would be expected to increase linearly
the population keeps increasing without bound
population fluctuates
the population becomes negative

QUESTION 4
One thing we haven't accounted for is the fact that animals die as well as reproduce. Add a death rate parameter equal to 0.05 to the
spreadsheet. Adapt the equationin B7 to account for this death rate as well as the reproduction (birth) rate, and then copy it down the column.

*What happens?

Population density changes ___ than before
faster
slower
at the same rate

QUESTION 5
Then the population density ____ begin to stabilise.
does
does not

QUESTION 6
At the moment we are assuming that death rate is constant, but in reality it is likely to increase as the population gets large and resources
become limiting. Instead of a constant death rate, add an extra column for a death rate that changes with population size. Assume that
death rate will be equal to 0.0001 times the population density. Let's call the parameter 0.0001 the 'density dependence death rate parameter'. Adapt the formulas in other columns as required.

*What happens?

Population density starts increasing
linerly
exponentially

QUESTION 7

Then the population density begin to stabilise.
does
does not

QUESTION 8
*Why is this model of population growth more realistic?
Because

the values don't go negative
populations can't keep growing forever
real populations can't grow exponentially

QUESTION 9
Note that the population tends towards a certain density that we can call the equilibrium population density.

What is the equilibrium population density?

QUESTION 10
*About how long (how many months) does it take for the population to get close to its equilibrium population density?

QUESTION 11
*What happens if the initial population density is higher than the equilibrium population density?

The population

increases continuously from its initial value
increases from its initial value and then stabilises
decreases from its initial value and then stabilises decreases continuously from its initial value

QUESTION 12

If the initial population is less than the equilibrium population then changing from 0.0001 to 0.0002 makes the population initially
increase faster
increase slower
increase at the same rate

QUESTION 13

Then
Stabilise at a lower equilibrium population
Stabilise at a higher equilibrium population
stabilise at the same equilibrium population
not stabilise at an equilibrium population

QUESTION 14
*How does the equilibrium population density depend on the 'reproduction rate parameter' (RRP) and the 'density dependence death rate parameter' (DDDRP)?? Can you write down an equation or give an explanation in words that actually gives the equilibrium population density as a function of the 'reproduction rate' parameter and the 'density dependence death rate parameter'??

The equilibrium population density is equal to

QUESTION 15
Now we want to do something similar, but taking resource levels and consumption into account more explicitly, so we can start to think about competition. (For the mouse - the main limiting resource is food, edible seeds produced by plants in its environment).

We'll create a new model, representing a situation where a small population of the mouse species colonises a new environment (an island perhaps). It is probably easiest to start by copying and pasting the code you wrote already (your model) and add a comment for yourself so you know this is the mouse competition model when you look back later. We are changing the time step for this new model - the time step is now days.

Add a new variable called 'food' - this is the biomass of edible seeds available (kg per square kilometre). Set the initial value for this variable to be 100.

This mouse needs at least 10g of food per day to just stay healthy. Within the loop, we can represent this by adding the following line of R code at the start of the ( ) block:
foodneeded = pop(time-1) • 0.01
The amount of food eaten on any day will depend on how much food the mouse finds - more mice will find more food, and if there is more food then more will be found. We can represent this with the following line of code:
foodeaten = food(time-1) • pop(time-1) • 0.001

*If the population was twice as big on a given day, and the amount of food was the same, what would happen to the amount of food eaten, according to this equation (note that this question is just checking to see if you understand the equation above - you don't actually need to run or plot anything) ?

The amount of food would be
a quarter as much
the same
twice as much
half as much
four times as much

QUESTION 16
* What is happening on this island after the first mice arrive? What happens to the mice and what happens to the food?

The mouse population at first, at a(n) ______ rate, while the amount of food available at first, at a(n) rate.

increases
decreases
increasing
decreasing

QUESTION 17
Then the rate of change of the mouse population and the rate of change in the amount of food available

speeds up
slows down

QUESTION 18
The amount of mice reaches a after around days, and then starts to .

maximum
minimum
20
40
60
80
100
increases
decreases

QUESTION 19

365 days after the start, the amount of food is than it was at the start, and than it was after 50 days, while the amount of mice is than it was at the start, and than it was after 50 days.

QUESTION 20

* For mice to survive and reproduce, they need to find mates and companions. This means there is an Allee Effect' - at small population
densities, populations are no longer viable. We can represent this in a simple way by adding the following line of code at the end of the block:

if (pop(time)<3) pop(time)=0
What is the minimum viable population density for this mouse.

QUESTION 21
How many days does it take for mice to die out completely? (You might need to extend the simulation.)

QUESTION 22
* If each mouse needs a little more food each day, does that make them die out earlier or later?
If each mouse is a little better at finding food each day, does that make them die out earlier or later?
* If the growth rate is a little smaller, does that make them die out earlier or later?
* If the amount of food initially available is a lithe less than 100kg, does that make them die out earlier or later?

QUESTION 23
What makes more difference to how long they last - halving the amount of food initially available, or doubling the initial population?
Changing the initial population makes a difference in absolute terms.

larger
smaller

QUESTION 24

" Make sure you set all your parameter values back after changing them - have you done that?

Of course the grass on the island actually keeps on growing and producing seed. Each day the grass produces half a kilogram of edible seed. Make this change to the model.

What is happening on this island after the first mice arrive now? What happens to the mice and what happens to the food?

QUESTION 25
Now we want to model what happens if a second species of mouse colonises the island a year after the first one arrives. To do this we need a new population variable (call it `pop2'). The initial population of the second species is zero, but at the end of day 365, some mice arrive. Let's say 60 arrive, and the area of grassland on our island is 10 square kilometres, so on day 365 the population density of the new species jumps to 6. Include this in your model using an 'if' command.
At this point, run the model, and add another line to the output plot showing the density of the second species over time. You see it jump to 6 on day 365, then remain constant.
This second species eats the same seeds as the first one. It needs the same amount of seeds per day to just remain healthy. It is exactly as good at finding food, has the same growth rate, and this growth rate depends on the food it finds and eats in the same way.
Adapt the model as required - you will need to copy and paste and modify most of the lines of code - be careful that you change all relevant variable names in each line.

When you've done that, run the model and check it is working correctly.
What happens to the population of the new mouse after colonisation?

it eventually dies out
it persists

QUESTION 26
*What happens to the amount of food available after colonisation by the new mouse?
it decreases.
it increases
it stays the same
it first decreases, then returns to the same level
it first increases, then returns to the same level

QUESTION 27

Now change the model so that the new mouse is bigger than the old one and thus needs more feed each day ie 15g instead of 10g. When you've done that, run the model and check it is working correctly.

What happens to the population of the new mouse after colonisation?

it eventually dies out
it persists

QUESTION 28
* What happens to the amount of food available after colonisation by the new mouse?
it decreases
it increases
it stays the same
it first decreases, then returns to the same level
it first increases, then returns to the same level

QUESTION 29
Set the two species to be equal sizes again. Now change the model so that the new mouse is better at finding food than the old one ie increase the relevant parameter by 50%.
When you've done that, run the model and check it is working correctly.
* What happens to the population of the new mouse after colonisation?

it eventually dies out
it persists

QUESTION 30
What happens to the amount of food available after colonisation by the mouse?

QUESTION 31
Keep the parameter value you just changed at the same value ie second mouse 50% better at finding food than the first. Can you now change the amount of food needed by the first species to a value where the two species can co-exist in the long-term?

* Does such a value exist? If so, what is the value? (Enter 0 if no value exists, if it exists, enter to 4 decimal places)
(Don't waste too much time on this one; come back to it later if you get stuck.)

QUESTION 32
Now set the values for the mice back to the way they were at the start le exactly the same.

Seed production by plants is likely to vary over the year, according to the seasons (rainfall, temperature etc). Model this by setting the amount of food produced in a day to be:
1+sin(time/365*pi*2)
When you've done that, run the model over six years and check it is working correctly.
What day of the simulation within the first year is the day of maximum food production?

QUESTION 33
* After things stabilise after colonisation by the new mouse, what happens to the amount of food available over each year?
The amount of food available peaks the mouse population peaks.

before
after

QUESTION 34
From then until the next annual peak it changes its direction (increasing to decreasing or vice versa)____ times, not including the maximum annual peaks.

QUESTION 35

* After things stabilise after colonisation by the new mouse, what happens to the first mouse over each year?
It increases quickly, peaks, then decreases a little, then increases again, peaks again, then decreases... the second peak is than

higher
lower

QUESTION 36
In the long run, over a few hundred years, what happens to the population of the new mouse? Is this a different result than when the food
doesn't vary? Why do you think this is the case?

The population

dies out
persists at a much lower level than the old mouse
persists at a much higher level than the old mouse
persists at a similar level as the old mouse

QUESTION 37

The result ___ diffrent

is
is not

QUESTION 38
Part 2: Population Dynamics and Chaos

Ok, now let's make things really simple again. We are now considering the population density of a kind of fly. Our time step will be a month. The fly is living in an environment where food is supplied at a constant rate. This means that if the population density of the fly is low, then its population will increase. At higher densities, the growth rate slows, and if it gets too high, then it will run out of food, and crash. The higher it is, the lower it will crash.
We can represent this simply with the following equation: nt+1= g nt (1- nt/1000)
which looks like this when r3.
The parameter g represents the maximum growth rate at low densities, and also how hard the population crashes. The height of the top of the peak (750 in this case) is the maximum sustainable population density.
How does the maximum sustainable population density depend on the value of g?
The maximum sustainable population density is equal to

QUESTION 39
Write some R code that simulates how the density of this population will change over 100 months, using a loop. Set the initial population density to 7 and the growth rate to 2. Run the simulation and plot the population density changing over time. You should find that the population increases exponentially at first, then levels out towards an equilibrium population density of 500.
Change the growth rate to 0.2 and run the simulation. What happens now?

The population
increases smoothly to an equilibrium population density
decreases to an equilibrium density
decreases towards zero
stays constant at its initial density
increases, reaches a maximum, then decreases to an equilibrium population density increases, reaches a maximum, then fluctuates without tending towards an equilibrium
increases, reaches a maximum, then fluctuates, with fluctuations of decreasing size, tending towards an equilibrium population density very slowly
increases, reaches a maximum, then fluctuates. with fluctuations of decreasing size, tending towards an equilibrium population density relatively quickly

QUESTION 40