FW364 Ecological Problem Solving

1. FW364 Ecological Problem Solving Class 5: Population Growth Sept. 16, 2013

2. Outline for Today Goal for Today: Introduction to population growth – Discrete Growth Objective for Today: Derive a simple model of discrete population growth between consecutive time periods (Nt and Nt+1) Objective for Next Class: Derive an equation to forecast population growth (still discrete growth) Objective for Class after Next: Derive continuous population growth equation Text (optional reading): Next three classes correspond to Chapter 1

3. Muskox Case Study 1700-1800s: Excessive hunting eliminated muskox from most of their natural North American range 1850-1860: Last individuals killed in Alaska 1930: Territory of Alaska Legislature authorized re-introduction from Greenland to Nunivak Island Ovibosmoschatus 1936: 31 muskox introduced to Nunivak Island 1965: Population size of 514 muskox! 1965-1968: 48 animals removed for re-introductions elsewhere in Alaska Great opportunity for population analysis because: Closed population Regular census from 1936-1968 Nunivak Island

4. Populations • What is a population? • Individuals of the same species in a defined area • A group of interbreeding organisms • Why is it important to define “the population” for modeling and management? • Definition of the population in part determines whether immigration & emigration need to be accounted for •  The larger the area over which the population is defined, the less important immigration and emigration become

5. Populations • What is a population? • Individuals of the same species in a defined area • A group of interbreeding organisms • Why is it important to define “the population” for modeling and management? • Definition of the population in part determines whether immigration & emigration need to be accounted for •  The larger the area over which the population is defined, the less important immigration and emigration become • Today’s Focus: • Modeling (i.e., building an equation) to describe • how populations grow (or shrink)

6. Basic Population Growth • General form: • Density at time in future = function of density at present time • Nt+1= f(Nt) • Nt= density of the population at time t (e.g., year 3 or day 12) • Nt+1= density of the population at time t+1(e.g., year 3+1 or day 12+1) • f(Nt) means a function “dependent on Nt” • Note: It is common to refer to Ntand Nt+1 as “densities” • because we are working with abundance in a pre-determined area • [remember the definition of a population] • However, the spatial aspect is implicit in the model • … We’ll treat Nt and Nt+1 as abundances (i.e., numbers of individuals)

7. Basic Population Growth • General form: • Density at time in future = function of density at present time • Nt+1= f(Nt) • Nt= density of the population at time t (e.g., year 3 or day 12) • Nt+1= density of the population at time t+1(e.g., year 3+1 or day 12+1) • f(Nt) means a function “dependent on Nt” • What are general factors that might affect abundance at t+1 • if you know abundance at time t? • Inputs to populationOutputs from population • Births (B) Deaths (natural, harvesting, other) (D) • Immigration (I) Emigration (E) • f(Nt) • Nt+1 = Nt + B – D + I – E

8. Basic Population Growth • Nt+1 = Nt + B – D + I – E • Remember from last week: • When trying to understand an ecological system, should start simple • For now, will assume we are working with closed population: • No immigration or emigration; only births and deaths • Nt+1 = Nt + B – D • Let’s start by building an equation for discrete growth • Discrete models – recap: • Useful for predicting quantities over fixedintervals • Time is modeled in discrete steps; Intervening time is not modeled • Good for populations that reproduce seasonally, like moose, salmon… • … and muskox

9. Basic Population Growth • 1965-1968: Removals • Population closed to immigration / emigration • So we can use • 1936: First introduction • Nt+1 = Nt + B – D • to model muskox population size at discrete intervals • e.g., one year to the next • Fig 1.3 in text • Plot curves upward: • Suggestive of “multiplicative growth”, but not diagnostic (we’ll check later) • Multiplicative (geometric) growth: population size increases (or decreases) by a constant fraction per year (rather than adding, e.g. 50 individuals, per year)

10. Birth Rate • Nt+1 = Nt + B – D • Let’s look closer at B, the birth term • What two most general factors determine B, the number of births? •  The number of individuals (or just number of females), N • Whether all individuals or just females are considered varies (we’ll do all individuals) • (males are often ignored in models of population growth... sorry guys) •  The number of offspring per individual in one time step, b’ • Per capita birth rate (i.e., per individual) • Prime distinguishes discrete rate from instantaneous (continuous) rate (used later) • Birth rate is a population parameter: an average rate for the population • (all individuals will not produce the same number of offspring in any time step) • b’ = (total number of births/time step) / total number of individuals • For a yearly time step: • b’ = (total number of births/year) / total number of individuals

11. Birth Rate • Nt+1 = Nt + B – D • Let’s look closer at B, the birth term • What two most general factors determine B, the number of births? •  The number of individuals or just number of females, N •  The number of offspring per individual in one time step, b’ • We can now create an equation for B: • B = b’ Nt • where: • Bis the number of births • Nt is population size at time, t • b’ is per capita birth rate • In other words, the number of births is the product of the number of individuals and the average number of offspring each individual has

12. Birth Rate • B = b’ Nt • Concept check: What are some of the assumptions we have made for B? • (a)-(e) are possible b’ values • b • many • e • c • Which of these options could fit our equation for B? • B • a • d • Which of these options is/are impossible to ever have? • 0 • 0 • many • Nt

13. Birth Rate • B = b’ Nt • Concept check: What are some of the assumptions we have made for B? • (a)-(e) are possible b’values • b • many • e • c • Which of these options could fit our equation for B? • B • Either (b) or (c): linear function • a • d • Which of these options is/are impossible to ever have? • 0 • (a): cannot have births at Nt = 0 • 0 • many • Nt

14. Birth Rate • B = b’ Nt • Concept check: What are some of the assumptions we have made for B? • (a)-(e) are possible b’values • b • many • e • c • What is the difference between (b) and (c)? • B • a • d • Which option(s) make the most sense ecologically? • 0 • 0 • many • Nt

15. Birth Rate • B = b’ Nt • Concept check: What are some of the assumptions we have made for B? • (a)-(e) are possible b’values • b • many • e • c • What is the difference between (b) and (c)? • B • b’ in (b) is larger than (c) • a • d • Which option(s) make the most sense ecologically? • (e): # births decreases as a function of density (carrying capacity) • (d): # births increases as a function of density (finding mates)… • (d) realistic a low pop size, not high • 0 • 0 • many • Nt

16. Death Rate • Nt+1 = Nt + B – D • Let’s look closer at D, the death term • What two most general factors determine D, the number of deaths? •  The number of individuals, N • The average death rate, d’ • Per capita death rate; Probability that one individual will die during the time step • Proportion of all individuals from population dying • d’ = (total number of deaths/time step) / total number of individuals • Are there bounds on d’? • Can d’ be negative? • Can d’ be > 1?

17. Death Rate • Nt+1 = Nt + B – D • Let’s look closer at D, the death term • What two most general factors determine D, the number of deaths? •  The number of individuals, N • The average death rate, d’ • Per capita death rate; Probability that one individual will die during the time step • Proportion of all individuals from population dying • d’ = (total number of deaths/time step) / total number of individuals • Are there bounds on d’? • Can d’ be negative?  No (that would be inverse death) • Can d’ be > 1? •  No, d’must be ≤ 1 (cannot have >100% of population dying)

18. Death Rate • Nt+1 = Nt + B – D • Let’s look closer at D, the death term • What two most general factors determine D, the number of deaths? •  The number of individuals, N • The average death rate, d’ • Per capita death rate; Probability that one individual will die during the time step • Proportion of all individuals from population dying • d’ = (total number of deaths/time step) / total number of individuals • Are there bounds on d’? • Can d’ be negative?  No (that would be inverse death) • Can d’ be > 1? • Are there bounds on b’? • Can b’be negative? • Can b’ be > 1? •  No, d’must be ≤ 1 (cannot have >100% of population dying)

19. Death Rate • Nt+1 = Nt + B – D • Let’s look closer at D, the death term • What two most general factors determine D, the number of deaths? •  The number of individuals, N • The average death rate, d’ • Per capita death rate; Probability that one individual will die during the time step • Proportion of all individuals from population dying • d’ = (total number of deaths/time step) / total number of individuals • Are there bounds on d’? • Can d’ be negative?  No (that would be inverse death) • Can d’ be > 1? • Are there bounds on b’? • Can b’be negative?  No (cannot have negative births) • Can b’be > 1? •  No, d’must be ≤ 1 (cannot have >100% of population dying) •  Absolutely! Individuals can have > 1 offspring

20. Death Rate • Nt+1 = Nt + B – D • Let’s look closer at D, the death term • What two most general factors determine D, the number of deaths? •  The number of individuals, N • The average death rate, d’ • Per capitadeath rate; Probability that one individual will die during the time step • Proportion of all individuals from population dying • d’ = (total number of deaths/time step) / total number of individuals • Like for B, we can create an equation for D: • D = d’ Nt • where: • Dis total number of deaths • Nt is population size at time, t • d’ is per capita death rate

21. Putting Everything Together • Let’s plug in our new equations! • Nt+1 = Nt + B – D • Nt+1 = Nt + b’Nt – d’Nt • Rearrange to get: • B = b’ Nt • D = d’ Nt • Nt+1 = Nt (1 + b’ – d’) • (note: birth and death rates are “additive”) • Now, we can define a new parameter, r’ • r’ = b’ – d’ (r’ is net population change) • and plug r’into equation: • Nt+1 = Nt (1 + r’) • We can define another new parameter, λ(lambda) λ = 1 + r’ • and plug λinto equation: • Nt+1 = Ntλ • Note: Text refers to λ as R, then later calls it λ. We’ll use λalways.

22. Putting Everything Together • Nt+1 = Ntλ • We have created a useful model of population growth! • Let’s take a moment to think about what we have just done… • …We derived an important equation using “first principles” • …We started by thinking about simple additions and losses to populations • …and have arrived at a powerful (though still simple) equation

23. Population Growth Break! • Let’s think about birth and death rates in an example population…humans! • http://www.prb.org/pdf12/2012-population-data-sheet_eng.pdf

24. Pick a Country Card • Each card has a country with its birth and death rate. • Using the birth and death rate, calculate net population change and be ready to comment and share: • r’ = b’ – d’

25. More on λ • Nt+1 = Ntλ • Simple model of multiplicative population growth (discrete type) • Multiplicativemeans the population increases in proportion to its size • Equation allows us to predict this year from last year, or next year from this year • Let’s talk about λ(lambda) • λis the finite growth rate of population • λis the factor by which the population grows (or shrinks) each year • Like r’, λ is a net rate: the net of inputs (b’) and outputs (d’) • λ is also the ratio of population sizes for consecutive time steps: λ = Nt+1 / Nt

26. More on λ • Nt+1 = Ntλ λ = Nt+1 / Nt • What is happening to a population with a growth rate, λ, of: • λ= 1 ? • λ> 1 ? • 0 < λ< 1 ? • λ< 0 ?

27. More on λ • Nt+1 = Ntλ λ = Nt+1 / Nt • What is happening to a population with a growth rate, λ, of: • λ= 1 ? Population is stable (not changing, Nt+1= Nt) • λ> 1 ? Population is increasing (growing) • 0 < λ< 1 ? Population is decreasing (shrinking) • λ< 0 ? Not possible; cannot have negative population size

28. More on λ • Nt+1 = Ntλ λ = Nt+1 / Nt • By what percentage is a population changing size with a growth rate, λ, of: • λ= 1.01 ? • λ= 1.23 ? • λ= 0.95 ?

29. More on λ • Nt+1 = Ntλ λ = Nt+1 / Nt • By what percentage is a population changing size with a growth rate, λ, of: • λ= 1.01 ? Population increasing by 1% per time step • λ= 1.23 ? Population increasing by 23% per time step • λ= 0.95 ? Population decreasing by 5% per time step

30. More on λ • Nt+1 = Ntλ λ = Nt+1 / Nt • By what percentage is a population changing size with a growth rate, λ, of: • λ= 1.01 ? Population increasing by 1% per time step • λ= 1.23 ? Population increasing by 23% per time step • λ= 0.95 ? Population decreasing by 5% per time step Recall that: λ= 1 + r’ so r’is the proportion by which the population changes (grows or shrinks) each time step • From above: • whenλ= 1.01 , r’ = 0.01 or 1% per time step • whenλ= 1.23 , r’ = 0.23 or 23% per time step • whenλ= 0.95 , r’ = -0.05 or -5% per time step

31. Correspondence with Book • Note: The book uses different symbols! • The book uses this equation for “long-lived” populations: • Nt+1 = Nt(s + f) • where: sis survival rate (proportion surviving) • fis the birth rate (fecundity rate) • f is equivalent to our b’ (we use b’because birth begins with “b”) • s is equivalent to 1 – d’(proportion surviving is 1 - proportion dying) • So our equation: • Nt+1 = Nt(1 + b’ – d’) • is equivalent to the book's equation: • Nt+1 = Nt(s + f), when substituting b’ = f and s = 1 – d’

32. Looking Ahead Next Class: Derive an equation to forecast population growth (still discrete growth) so we can predict population growth between non-consecutive time steps (e.g., 10 years in the future, not just last year to this year, or this year to next year) Tomorrow: Lab 3 Bring exercise, pencil, paper, and calculator