Lecture 17

Question 1

population size symbol

Answer 1

N

Question 2

population density symbol

Answer 2

N/area

Question 3

why do we care about understanding population size, N?

Answer 3

• natural resources management (eg size of fish stocks in the ocean, abundance of outbreaking insect pests in forests)
• conservation: population decline of a species
• health: monitoring populations of viruses or bacteria in humans
• understanding and predicting human population growth
• basic science question of what limits population growth

Question 4

Population declines in Myotis lucifugous bats due to white nose syndrome (WNS)

Answer 4

• novel pathogen (fungus) emerged in bat population, causing a disease caused white nosed syndrome and really high mortality.
• steep decline in no of over-wintering bats

Question 5

HIV population dynamics in humans - draw graph of CD4 cells over time

Question 6

Malthus’ essay on population growth

Answer 6

in 1798, Malthus published an essay on the principle of population, arguing that the human population cannot grow faster than food production

Question 7

Paul Ehlrich

Answer 7

published The Population Bomb, arguing that explosive growth in the human population would have catastrophic social and environmental consequences

Question 8

how is the human population expected to change in the future?

Answer 8

demographers project that human population is soon going to peak, then fall dramatically (depopulation)

Question 9

goals of most population models

Answer 9

• predict the trajectory of population growth through time, i.e. N as a function of t
• how many individuals are in the population now? Nt
• how many individuals are in the population one step later? N(t+1)
• so the general model is N(t+1) = fN(f)
• challenge: choosing simple but realistic parameters for f

Question 10

when using differential equations, time steps are

Answer 10

infinitesimally small: use concept of limits and calculus; growth is smooth; best suited for species with continuous reproduction

Question 11

when using difference equations, time steps are

Answer 11

discrete units (days, years, etc); use iterated recursion equations; growth is stepwise and bumpy; best suited for episodic reproduction

Question 12

two types of time step approaches

Answer 12

continuous-time and discrete-time

Question 13

how do we pick between the two time-step approaches?

Answer 13

different organisms might be better fit by one or the other

Question 14

simple bookkeeping model: how can N change from Nt to Nt+1

Answer 14

D = number who die during one time step
B = number born during one time step
E - number who emigrate during one time step
I = number who immigrate during one time step

Nt+1 = Nt - D + B - E + 1

Question 15

what variables can we consider to be equivalent?

Answer 15

• birth and immigration (ie individuals added to the population)
• death and emigration (ie individuals that disappear from the population)

Question 16

geometric growth model

Answer 16

• assume no immigration or emigration
• treat birth and death during one time step as per capita rates that are fixed constants
• then, population changes by a constant factor each time step: N(t+1) = λNt
• λ is a multiplicative factor by which population changes over one time unit = ‘finite rate of increase’
• λ = Nt+1/Nt

Question 17

if λ>1,

Answer 17

birth exceed deaths and population grows

Question 18

if λ<1

Answer 18

deaths exceed births and population shrinks

Question 19

N1 =

Answer 19

λN0

Question 20

N2 =

Answer 20

λN1 = λλN0

Question 21

N3 =

Answer 21

λN2 = λλN1 = λλλN0

Question 22

so how can geometric growth be generalised

Answer 22

Nt = N0λ^t

Question 23

exponential growth

Answer 23

• instantaneous, fixed per-capita rates of birth and death (b and d)
• instantaneous, per-capita rate of population change = b-d=r (a constant)
• r = intrinsic rate of increase
• differential equation is dN/dt = rN
• this model is exponential growth

Question 24

draw a table comparing discrete-time and continuous time growth models

Question 25

find the relationship between r and λ

Answer 25

lnλ=r

Question 26

regardless of which model is adopted, the important consequence is the same

Answer 26

• in both models, the growth rate (λ or r) is a constant that simply reflects biology
• but a constant positive growth rate produces a population growth size that is not constant, but rather exploding in an exponential way

Question 27

all species…

Answer 27

• have the potential for positive population growth under good conditions (λ>1.0, births exceed deaths)
• have the potential for negative population growth under bad conditions (λ<1.0, deaths exceed births)
• but no species has ever sustained λ>1 or λ<1 for a long period

Question 28

why is exponential growth a bad model of reality over the long term?

Answer 28

• some factors use tend to keep populations from exploding or going extinct
• two kinds of factors may be acting: density dependent regulation (growth depends on N) or density-independent reduction

Question 29

how can we model the classically, density-dependent growth?

Answer 29

the logistic equation; an exponential growth with a new term added for brakes
dN/dt = rN(1-N/k)

Question 30

use bacteria as an example of two types of growth

Question 31

The logistic braking term models… (draw graph)

Answer 31

the simplest form of density dependence

Question 32

K

Answer 32

carrying capacity of the environment

Question 33

logistic trajectories are truly
S-shaped only when

Answer 33

starting from low numbers

Question 34

label an N vs t graph

Question 35

logistic model pros

Answer 35

• Mathematically tractable model of intraspecific competition for resources
• Simple (only one extra parameter, K, beyond exponential)
• Can be expanded to consider multispecies competition

Question 36

logistic podel cons

Answer 36

• Too simple: specifies one particular kind of
  density dependence
• Always a gradual approach to carrying
  capacity
• In reality, density-dependence is likely
  to be non-linear, may see overshoots of K