MATH3104 Lecture 2 The Population Bomb

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```					           MATH3104                                               Ecological modelling
Lecture 2: The Population Bomb                         • Explore how to model populations with dynamic models
• Density independent models
• Density dependent models
• Look at simple models for meta-populations
• Look at simple models for environmental stochasticity
• Look at harvesting from density dependent models – why
is this interesting?
• What are key population processes for biology – birth,
death, immigration and emigration
• Use Matlab throughout

Anthony J. Richardson
anthony.richardson@csiro.au

Outline                                       The population bomb
•   What is the population bomb?                       • First one E. coli then 2, 4, 8, 16,…
•   Populations in discrete time                       • Divide every 20 mins, so after 32 h entire
•   Populations in continuous time                       earth covered 1 m deep!
•   Age structured models – the Leslie Matrix          • Like compound interest – E. coli interest
with discrete data                                   rate is 100% every 20 mins!
• Ecological interest rates rarely as high –
most struggle to exist with rates ~0
• Ecological interest rates differ temporally
and spatially

Why is it a population bomb?                                          Discrete time
• cf nuclear bomb – particles split and collide with   • Time step anything convenient: day, month, year
other particles and continue splitting – chain         and census population at beginning of step
reaction like exponential growth
• Rate of reaction not determined by number of
B = Number of births each animal has at beginning of time step
particles present
that live to beginning of next time step
• Growth rate (or ecological interest rate – i.e.
birth and death rates) not dependent upon             D = Probability an animal dies during time step
number of organisms - Density independent                               Nt +1 = N t + BN t − DN t
• Geometric (discrete) and exponential
(continuous) growth same process                                      N t +1 = N t (1 + B − D)
Nt +1 = RN t
R = 1 + ( B − D)

1
Discrete time…                                                               Continuous time
N t +1 = RN t
• Counterpart of geometric growth in discrete time
N at t = 0 is N 0
is exponential growth in continuous time
N1 = RN 0
b = average birth rate per unit time per organism
N 2 = RN1                                  d = average death rate per unit time per organism
= R ( RN 0 )
dN
= R N0
2                                                                 = bN − dN
dt
Nt = R N 0
t

birth and death usually combined into “intrinsic rate of increase”, r = b - d
M 2.1      Let’s plot discrete-time geometric growth equation
dN
= rN
M 2.2      Plot represents an “explosion”, but more useful to                                                     dt
plot on a semi-log scale
dN
Ex 2.1     Show that slope of semi-log plot of population size            Ex 2.2           Solve         = rN         where N (0) = N 0
dt
vs time is log(R)

Continuous time…                                                              Continuous time…
Compare continuous time with discrete
time formulation                                               M. 2.4          Let’s also plot this on a semi-log scale

N (t ) = N 0e rt            Continuous
Ex. 2.3          Show that slope of semi-log plot is r
Nt = N0 Rt                 Discrete
R = er                                 • Geometric and exponential growth same
r = log(R )                                 thing, except how time kept
• Distinction biologically important when
M. 2.3     Let’s plot continuous-time exponential growth with               density dependence considered
r = log( R ) = log(2) ≈ 0.69
– Geometric and exponential growth linear as
population size (N) never have N2 term

Leslie Matrix for Discrete Data                                          Leslie Matrix with 3 Age Classes
• Describes population age groups                                      Fx = fertility of an organism in age class x, where x = 1,2 or 3
(demography)                                                                = number offspring born per parent of age class x surviving to census
• Is population dominated by teeny boppers
Px = probability individual starting age class x surviving entire step
or old, wise individuals
• Easier and more useful in discrete time                              n x ,t = number of individuals of age class x at time step t
• Simplest to have age-class width same as
time-step (1-yr time step = 1-yr age class)                                                   n1,t +1 = F1n1,t + F2 n2,t + F3 n3,t
• All individuals from just born to 364 d old at                                                n2,t +1 = P n1,t
1
census are age 1                                                                              n3,t +1 = P2 n2,t

2
Leslie Matrix…
Written more compactly as:
⎛ n1,t +1 ⎞ ⎛ F1     F2   F3 ⎞ ⎛ n1,t ⎞
⎜         ⎟ ⎜                ⎟ ⎜         ⎟        Here Leslie Matrix
⎜ n2,t +1 ⎟ = ⎜ P1   0    0 ⎟ × ⎜ n2 , t ⎟        constant over t
⎜n        ⎟ ⎜                ⎟ ⎜n ⎟
⎝ 3,t +1 ⎠ ⎝ 0       P2   0 ⎠ ⎝ 3, t ⎠

nt +1 = m × nt               m is Leslie matrix

All age 3 individuals die at end of each time step

3

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