1. EE 194/Bio 196: Modeling,simulating and optimizing biological systems
Spring 2018
Tufts University
Instructor: Joel Grodstein
Lecture 2: Population modeling

2. Population modeling First big topic: population modeling.
The usual first questions…
What is it?
making models of how a population of individuals grows over time
Why do we care?
Populations can grow or become extinct, and it affects our lives
Especially in Track C
Many tasks are easier on a model than working with real animals
EE 194/Bio 196 Joel Grodstein

3. Goals of this segment What we'll learn about biology
lx-mx models (and more in extra-credit problems)
What we'll learn about modeling
GIGO – when is a model valid?
simulating models
discrete-time vs. continuous time-models
stochastic models
desert tortoises: using models for things we cannot easily measure (and deciding if they’re correct)
What we'll learn about programming
Variables, loops, arrays, random numbers
3 homeworks
EE 194/Bio 196 Joel Grodstein

4. Population models Inputs to the model
initial population size
average birth rates, death rates
these are called vital rates
If you know the vital rates, it’s easy to project a population forwards
i.e., simulate the model; future population is the model output
But wait, there’s more
Vital rates depend on numerous environmental factors
how much food is available. Anything else?
or how many predators are around
or presence of a disease… and many more
Here's where models get really useful
Some factors may be under our control
We can supply food, remove predators, etc.
EE 194/Bio 196 Joel Grodstein

5. Simulating different parameters
Modeling makes it easy to run “what-if” simulations.
Determining how things that are in our control can affect population growth.
How exactly?
Simulate population growth under average conditions.
Change a vital rate based on, e.g., the assumption that humans remove some predators from the environment
Model tells you the new results; perhaps improved growth
EE 194/Bio 196 Joel Grodstein

6. The model is agnostic
Our goal may actually be reduce a population rather than increase it. Example?
feral cats (colonies not socialized to humans)
Doesn’t matter: population modeling is not concerned with what our goal is
What we do with the numbers is up to us
We can try to optimize anything we want to…
…maximum population, minimum population, least year-to-year variation, …
EE 194/Bio 196 Joel Grodstein

7. GIGO Predicting “what will happen if” is not always so easy
The first rule of modeling: garbage in garbage out.
Just because a computer model gives you a number, doesn’t mean that it’s right.
GIGO examples
Leave home for Tufts, check Google maps for traffic. By the time I get to the crunch point, traffic has changed
If you remove predators, you can improve vital rates. But what side effects will that have? It may be hard to predict, and some may in turn reduce vital rates
Cats were removed from an island to protect sea birds, resulted in huge increase in rats, making things worse for the seabirds! So effects are hard to predict
EE 194/Bio 196 Joel Grodstein

8. Start simple What goes into a population model? First-cut example:
Initial population
# of births, deaths every year
First-cut example:
Initial population size = 100
# of deaths per year = 10% of population size
# of births per year = 20% of population size
Can assumptions this simple be at all accurate?
Many reasons why not… but hold that for later
Now let's do an example
EE 194/Bio 196 Joel Grodstein

9. Population example Try these yourself now Year #1 Year #2 Year #3
# deaths = 100 * 10% = 10
# births = 100 * 20% = 20
Population size = 100 – = 110
Year #2
# deaths = 110 * 10% = 11
# births = 110 * 20% = 22
Population size = 110 – = 121
Year #3
# deaths =
# births =
Population size =
Year #4
Parameters
Initial population = 100
# of deaths per year = 10% of pop size
# of births per year = 20% of pop size
121 * 10% = 12.1
Try these yourself now
121 * 20% = 24.2
What is a fractional person?
121 – = 133.1
Do we really need two decimal places?
133.1 * 10% = 13.31
133.1 * 20% = 26.62
133.1 – =
Answers will depend on your application
EE 194/Bio 196 Joel Grodstein

10. Population growth over time
Constant growth of 20%-10% = 10% per year
Does it look like a straight line?
Actually, it’s not
Each year, the population is a bit bigger and the number of new individuals is thus a bit bigger
This toy model is too easy 
EE 194/Bio 196 Joel Grodstein

11. Improve the model Make the model more accurate by adding detail
Different age classes have different vital rates
Old and very young individuals die more often
In between, they give birth more often
Sexes are different
Only females have babies (and only at certain ages)
Some diseases are linked to sex
Simplify the model without losing accuracy
Strange concept! But in this case…
Ignore males altogether
3 babies/year per individual” becomes “3 female babies/year per female”
Doesn’t the number of males affect how many females become pregnant?
In some species, we can assume that there are enough males to impregnate all receptive females
Common assumption in population biology (not relevant for humans!)
EE 194/Bio 196 Joel Grodstein

12. Our next model Only track females
Every age has its own birth and death rates
Powerful enough to model numerous effects:
High death rates for old and very young individuals
Higher birth rates in middle ages
Can model any effect on age-dependent vital rates
EE 194/Bio 196 Joel Grodstein

13. Population example Just look at deaths for now (ignore births)
Assume the “survival rate” numbers are over one year
Age group
Survival rate
Population, 2017
2018
2019
0-1 .7
100
1-2 .8
150
2-3
200
3-4 70
100*.7 = 70
120
150*.8 = 120
160
200*.8 = 160
EE 194/Bio 196 Joel Grodstein

14. Population example Just look at deaths for now (ignore births)
Assume the “survival rate” numbers are over one year
Age group
Survival rate
Population, 2017
2018
2019
0-1 .7
100
1-2 .8
150
2-3
200
3-4 70
120 56
70*.8 = 56
160 96
120*.8 = 96
If we keep this up over multiple years, will the population eventually shrink to 0, grow bigger and bigger, or stabilize?
With deaths and no births, it must die out
EE 194/Bio 196 Joel Grodstein

15. Population example Any problems with this table?
Age group
Survival rate
Population, 2017
2018
2019
0-1 .7
100
1-2 .8
150 70
2-3
200
230 56
3-4
160 96
Any problems with this table?
We haven’t added in births yet
Why is 1 year the correct interval to work at?
What would happen if we re-calculated every 5 years?
Unable to model some effects (in humans, newborns and 1 year olds have very different survival rates)
Every day?
Model might take a long time to run
In most species, may be unable to get vital rates every day
EE 194/Bio 196 Joel Grodstein

16. The flow of time
How do you decide what granularity of time to work at?
The real world is continuous time
Individuals can die at any time, and many species can be born at any time
Most models are discrete-time
Population modeling is often at, e.g., 1-year intervals
Any reason to use a finer-granularity interval?
E.coli reproduce every 30 minutes
Why not use continuous time?
May not be able to make observations very often
Simulation and optimization are often infeasible to do in continuous time
EE 194/Bio 196 Joel Grodstein

17. Pulse-birth model Many population models use a pulse-birth model
All births happen in one small instant just before you do your yearly population count
Does not match reality. But it does simplify the model
Are any species fairly close?
Pulse-birth sequencing:
Individuals in any age group first progress in age (if they survive)
Then, in one big pulse, some proportion of them have babies
Then we count the population
(Some people reverse these last two steps)
Births in spring is common
Many species are in fact seasonal, or pulsed breeders. E.g., temperate zone birds breed in spring/early summer. Extreme examples are cicadas, bamboo.
EE 194/Bio 196 Joel Grodstein

18. Population example The next year is for you to compute Age group
Survival rate
Birth rate
Population 2017
2018
2019
0-1 .7
100
1-2 .8 1
150
2-3 2
200
3-4
100 individuals in 0-1 group. 100*.7=70 survive; they generate 70*1=70 babies
150 individuals in 1-2 group. 150*.8=120 survive; they generate 120*2=240 babies (310 total)
200 individuals in 2-3 group. 200*.8=160 survive; they generate 160*0=0 babies
100 individuals in 3-4 group. 100*0= none of them survive.
310 70 70
120
160
The next year is for you to compute
EE 194/Bio 196 Joel Grodstein

19. Population example Population equations:
Age group
Survival rate
Birth rate
Population 2017
2018
2019
0-1 .7
100
310
1-2 .8 1
150 70
2-3 2
200
120
3-4
160
310 individuals in 0-1 group. 310*.7=217 survive; they generate 217*1=217 babies
70 individuals in 1-2 group. 70*.8=56 survive; they generate 56*2=112 babies (329 total)
120 individuals in 2-3 group. 120*.8=96 survive; they generate 96*0=0 babies
160 individuals in 3-4 group. 160*0= none of them survive.
217
329
217 56 96
Population equations:
= * survival0-1
= * survival1-2
= * survival2-3
= ( *birth1-2)+( *birth2-3)+ ( *birth3-4)
EE 194/Bio 196 Joel Grodstein

20. We get twists and turns early on
EE 194/Bio 196 Joel Grodstein

21. Eventually it seems to settle out
In fact, every age is growing at roughly 46% per year.
This is called exponential growth
EE 194/Bio 196 Joel Grodstein

22. Viewed on a log scale, exponential growth shows each age category growing at the same rate
EE 194/Bio 196 Joel Grodstein

23. lx-mx model Definitions:
"x" is a population category. For now, let's say it's how many years old you are (e.g., “2” is from 2.0 to 2.99 years old)
lx= probability of a newborn surviving to the beginning of stage x (so l2 is their odds of making it to their 2nd birthday)
mx= average number of female babies a female will have while in stage x (remember, we’re completely ignoring males)
nx = number of females in stage x
px= probability of surviving to the end of stage x, once you reached the beginning of that stage
EE 194/Bio 196 Joel Grodstein

24. lx-mx model Definitions: nx = number of females in stage x
px= probability of surviving to the end of stage x, once you reached the beginning of that stage
mx= average number of babies you will have while in stage x
Population equations:
= * survival0-1
= * survival1-2
= * survival2-3
= ( *birth1)+( *birth2)
+ ( *birth3)
n1,2018
n2,2018
n3,2018
n0,2018
EE 194/Bio 196 Joel Grodstein

25. lx-mx model Definitions: nx = number of females in stage x
px= probability of surviving to the end of stage x, once you reached the beginning of that stage
mx= average number of babies you will have while in stage x
Population equations:
n1,2018 = * survival0-1
n2,2018 = * survival1-2
n3,2018 = * survival2-3
n0,2018= ( *birth1)+( *birth2)
+ ( *birth3)
n2,2017
n0,2017
n1,2017
EE 194/Bio 196 Joel Grodstein

26. lx-mx model Definitions: nx = number of females in stage x
px= probability of surviving to the end of stage x, once you reached the beginning of that stage
mx= average number of babies you will have while in stage x
Population equations:
n1,2018 = n0,2017 * survival0-1
n2,2018 = n1,2017 * survival1-2
n3,2018 = n2,2017 * survival2-3
n0,2018= ( *birth1)+( *birth2)
+ ( *birth3)
n1,2018
n2,2018
n3,2018
EE 194/Bio 196 Joel Grodstein

27. lx-mx model Definitions: nx = number of females in stage x
px= probability of surviving to the end of stage x, once you reached the beginning of that stage
mx= average number of babies you will have while in stage x
Population equations:
n1,2018 = n0,2017 * survival0-1
n2,2018 = n1,2017 * survival1-2
n3,2018 = n2,2017 * survival2-3
n0,2018= (n1,2018 *birth1)+(n2,2018 *birth2)+ (n3,2018 *birth3) p0 p1 p2
EE 194/Bio 196 Joel Grodstein

28. lx-mx model Definitions: nx = number of females in stage x
px= probability of surviving to the end of stage x, once you reached the beginning of that stage
mx= average number of babies you will have while in stage x
Population equations:
n1,2018 = n0,2017 * p0
n2,2018 = n1,2017 * p1
n3,2018 = n2,2017 * p2
n0,2018= (n1,2018 *birth1 ) + (n2,2018 *birth2 ) + (n3,2018 *birth3 ) m1 m2 m3
EE 194/Bio 196 Joel Grodstein

29. lx-mx model Definitions: nx = number of females in stage x
px= probability of surviving to the end of stage x, once you reached the beginning of that stage
mx= average number of babies you will have while in stage x
Population equations:
n1,2018 = n0,2017 * p0
n2,2018 = n1,2017 * p1
n3,2018 = n2,2017 * p2
n0,2018= (n1,2018*m1 )+(n2,2018*m2)+ (n3,2018*m3)
EE 194/Bio 196 Joel Grodstein

30. A rose by any other name
But it’s called lx-mx – why do we use px instead of lx?
It all comes out in the wash
l0 = likelihood of surviving from birth to the beginning of age 0
= 1
l1 = likelihood of surviving from birth to the beginning of age 1
=p0
l2 = likelihood of surviving from birth to the beginning of age 2
= l1p1
=p0p1
l3 = likelihood of surviving from birth to the beginning of age 3
=l2p2
= p0p1p2
EE 194/Bio 196 Joel Grodstein

31. Simple code Age group Survival rate Birth rate Population 2017 2018
2019
0-1 .7
100
1-2 .8 1
150
2-3 2
200
3-4
n1,2018 = n0,2017* p0
n2,2018 = n1,2017* p1
n3,2018 = n2,2017 * p2
n0,2018= (n1,2018 * m1)+(n2,2018 * m2)+ (n3,2018 * m3)
n0_2017=100; n1_2017=150; n2_2017=200; n3_2017=100
p0=.7; p1=.8; p2=.8; p3=0
m0=0; m1=1; m2=2; m3=0
n1_2018 = n0_2017*p0
n2_2018 = n1_2017*p1
n3_2018 = n2_2017*p2
n0_2018 = n1_2017*m1 + n2_2017*m2 + n3_2017*m3
or n1_2018*m1 + n2_2018*m2 + n3_2018*m3
First just define our initial population and vital rates
Next, the equations
Which choice is correct?
EE 194/Bio 196 Joel Grodstein

32. Simple code U U U U U U U U U U U U U U U U Age group Survival rate
Birth rate
Population 2017
2018
2019
0-1 .7
100
310
1-2 .8 1
150 70
2-3 2
200
120
3-4
160
Simple code
n0_2017=100; n1_2017=150; n2_2017=200; n3_2017=100
p0=.7; p1=.8; p2=.8; p3=0
m0=0; m1=1; m2=2; m3=0
n1_2018 = n0_2017* p0
n2_2018 = n1_2017*p1
n3_2018 = n2_2017*p2
n0_2018 = n1_2018*m1 + n2_2018*m2 + n3_2018*m3
100 U
150 U
200 U U
310
100 U U 70
120 U
160 U
n0_2017 n1_2017 n2_2017 n3_2017
n0_2018
n1_2018
n2_2018
n3_2018 U .7 U .8 U .8 U U U 1 U 2 U p0 p1 p2 p3 m0 m1 m2 m3
EE 194/Bio 196 Joel Grodstein

33. Loops We have our equations
Population equations:
n1,2018 = n0,2017 * p0
n2,2018 = n1,2017 * p1
n3,2018 = n2,2017 * p2
n0,2018= (n1,2018*m1 )+(n2,2018*m2)+ (n3,2018*m3)
These are very useful if we know 2017 and want 2018
We often (usually) want to simulate many years into the future. So how do we compute the 2019 numbers?
We need some more flexible equations
EE 194/Bio 196 Joel Grodstein

34. Loops We have our equations to go from 2017 to 2018
Population equations:
n1,2018 = n0,2017 * p0
n2,2018 = n1,2017 * p1
n3,2018 = n2,2017 * p2
n0,2018= (n1,2018*m1)+(n2,2018*m2)+ (n3,2018*m3)
Can we rewrite this to go from 2018 to 2019?
n1,2019 = n0,2018 * p0
n2,2019 = n1,2018 * p1
n3,2019 = n2,2018 * p2
n0,2019= (n1,2019*m1)+(n2,2019*m2)+ (n3,2019*m3)
EE 194/Bio 196 Joel Grodstein

35. Loops Population equations from 2018 to 2019:
n1,2019 = n0,2018 * p0
n2,2019 = n1,2018 * p1
n3,2019 = n2,2018 * p2
n0,2019= (n1,2019*m1)+(n2,2019*m2)+ (n3,2019*m3)
Can we rewrite this to go from 2019 to 2020?
n1,2020 = n0,2019 * p0
n2,2020 = n1,2019 * p1
n3,2020 = n2,2019 * p2
n0,2020= (n1,2020*m1)+(n2,2020*m2)+ (n3,2020*m3)
EE 194/Bio 196 Joel Grodstein

36. Loops Population equations from 2019 to 2020:
n1,2020 = n0,2019 * p0
n2,2020 = n1,2019 * p1
n3,2020 = n2,2019 * p2
n0,2020= (n1,2020*m1)+(n2,2020*m2)+ (n3,2020*m3)
Can we rewrite this to go from year n to year n+1?
n1,next = n0,now* p0
n2,next = n1,now* p1
n3,next = n2,now* p2
n0,next= (n1,next * m1)+(n2,next * m2)+ (n3,next * m3)
What does this actually mean?
Just that you have pretty much the same equation to go forwards a year, no matter what year it is.
The “next” and “now” subscripts can be any two consecutive years
EE 194/Bio 196 Joel Grodstein

37. Break to the Python Loops lecture
EE 194/Bio 196 Joel Grodstein

38. Code for loops Will this code work?
for gen in range(70):
n0=100; n1=150; n2=200; n3=100
p0=.7; p1=.8; p2=.8; p3=0
m0=0; m1=1; m2=2; m3=0
n1_next = n0* p0
n2_next = n1*p1
n3_next = n2*p2
n0_next = n1_next*m1 + n2_next*m2 + n3_next*m3
Will this code work?
No. After the first time through the loop, it just assigns the same values to the same variables over and over.
310
100 U
150 U
200 U
100 U U U 70
120 U
160 U n0 n1 n2 n3
n0_next
n1_next
n2_next
n3_next
EE 194/Bio 196 Joel Grodstein

39. Code for loops, take 2 Will it work now?
Still not, for the same reason.
It’s a bit better, though; at least we don’t assign the p’s and m’s every time around the loop.
Somehow we have to update n0, n1, n2 and n3 every time around the loop.
You get to figure out how on HW1
n0=100; n1=150; n2=200; n3=100
p0=.7; p1=.8; p2=.8; p3=0
m0=0; m1=1; m2=2; m3=0
for gen in range(70):
n1_next = n0* p0
n2_next = n1*p1
n3_next = n2*p2
n0_next = n1_next*m1 + n2_next*m2 + n3_next*m3
310
100 U
150 U
200 U
100 U U U 70
120 U
160 U n0 n1 n2 n3
n0_next
n1_next
n2_next
n3_next
EE 194/Bio 196 Joel Grodstein

40. Age vs. stage vs. size We’ve stated all of our vital rates per age
Different rates for 0 years old, 1 year old, etc.
In some species, it makes more sense to delineate groups based on size
In others, life stages is a better choice
What makes size or life stage better? What does “better” mean?
We want our vital rates to be accurate
Birth rates and survival rates may correlate much better to size or to life stage in some species than to age
Yet again: you can build whatever model you want, but it’s only really useful if it matches reality
We won’t discuss stage-based models in this course
EE 194/Bio 196 Joel Grodstein

41. Problems with our model
We neglected population density
Nothing can really grow exponentially forever.
As a population gets bigger, vital rates shrink because we run out of food, space, etc.
We’ve ignored that; you can experiment with it as part of HW #1 extra credit
What happens if your model ignores overcrowding?
As usual: garbage in, garbage out
EE 194/Bio 196 Joel Grodstein

42. Why model variation Real life is full of surprises
There are good years and bad years
Some years have better weather, more food, less disease, etc.
Do we need to model this?
If we just pick “average” vital rates, will it all even out in the end?
Is it faster to bicycle uphill and then back down, or just stay flat?
Is it worse if the hill is steeper?
Similar effect in investing money. Stocks pay higher returns on average than bonds but have more fluctuation, and that fluctuation usually lowers their effective return.
Variation in vital rates often yields less growth than slow & steady
So we usually do want to model vital-rate variation
EE 194/Bio 196 Joel Grodstein

43. Stochastic models
Obvious way to model variation: make the vital rates change randomly every “year”
Where “year” is whatever makes sense for your problem
What is the shape of your distribution?
Bell-shaped curve is commonly used. Any issues?
It never tails off to zero, so there is a non-zero possibility of a negative number of offspring!
The “right” answer depends on your problem
HW #3 will model this
EE 194/Bio 196 Joel Grodstein

44. What are fractional people?
Remember our first example?
121 people in the population
10% of them die (that’s 12.1 people)
20% new births (24.1 people)
Next year we have people!
Any ideas what that might mean physically?
It’s just wrong; we never want to deal with fractions
It’s fine… we really do have fractional people walking around
It represents averages (like 2.1 children per family)
EE 194/Bio 196 Joel Grodstein

45. What to do with fractional people
Averages are all well and good, but what should we do in our model?
Keep working with fractional people?
Truncate them at every generation? So 12.9 people become 12
Round them at every generation? Then 12.9 becomes 13
Reasons to keep working with fractional people?
Our vital rates are just averages anyway
But sometimes we care about quantization; it causes small populations to be more vulnerable
Truncate vs. round?
A 0.9 person is no more viable than a .1 person
On average, the .9 person is more likely to represent an entire person than a non-person
HW #3 will let you program these issues
EE 194/Bio 196 Joel Grodstein

46. Recap What we learned about biology What we learned about modeling
lx-mx models
What we learned about modeling
GIGO – when is a model valid?
Desert tortoise: using models for things we cannot easily measure (and deciding if they're correct). That’s HW2.
Simulating models (we did several examples)
Discrete-time vs. continuous time-models
stochastic and quantized models
What we learned about programming
The basics: variables, loops, arrays & random numbers (up next)
3 homeworks
EE 194/Bio 196 Joel Grodstein