Models of Population Growth
• Our models of population growth have assumed that
growth rates are constant:
– Continuous: r
– Discrete: R
• Imply
I l th
thatt populations
l ti
grow without
ith t b
bound.
d
– Unrealistic assumption.
– Populations typically limited by available space and
resources.
Limited population growth
• Thomas Malthus (1798):
– Populations can increase exponentially.
– Food supplies generally increase arithmetically.
arithmetically
– Population growth will eventually be limited.
• Charles Darwin ((1859)) used Malthus’ concept
p in
formulation of theory of natural selection.
– Most organisms produce many more offspring than can
survive
i tto breed.
b d
• Estimated that a single pair of elephants would produce
nearly 19 million individuals in 750 years.
– Competition for limited resources favors individuals better
‘adapted’ to their environment.
– Better-adapted individuals produce
produce, on average
average, more
offspring then typical for the population.
Limited population growth
• Many
M
ways in
i which
hi h population
l ti growth
th might
i ht b
be lilimited:
it d
– Shortage of food
– Lack of space
– Accumulation of toxic substances in the environment
• Population increases may increase competition for
common resources:
– Directly: overt interactions between individuals.
– Indirectly: via passive depletion of the environment
(interference competition).
• Such factors might lead to:
– Increased mortality rates
– Reduced reproduction rates
• Result: population growth rate decreases with increasing
population size.
Limited population growth
• Also
Al applies
li tto molecular
l
l and
d cellular
ll l llevel:
l
– Chemical reactions:
• Increase in concentration of a product may be limited
by amount of reactant present.
– Nutrient uptake by cell:
• May be limited by the concentration in the
surrounding medium.
• P
Processes att allll llevels
l llead
d tto lilimited
it d growth
th iin
which the growth (or production) rate declines
ssteadily
ead y with popu
population
a o ssize
e (o
(or a
amount
ou p
produced).
oduced)
• Let’s look at some mathematical models for limited
population growth that apply to a wide range of
biological processes and mechanisms.
Closed-system model for limited growth
in continuous time
• Basic pattern of limited population growth:
– Growth rate declines steadily towards zero
zero.
– The larger the population, the more slowly it grows.
• Monomolecular model: used extensively to describe:
– Increase in concentration of a product formed in a
1st-order chemical reaction.
– Nutrient uptake by a cell in a constant medium
medium.
– In both cases, medium is not refreshed.
• Consider chemical reaction represented
p
by
y R  P.
– R = reactant, P = product.
– Assumption: concentration of reactant is diminished linearly
by the reaction,
reaction and is not replaced (closed system)
system).
– [R] decreases, while [P] increases proportionately.
Monomolecular model for limited growth
R  P.
• R
Reaction:
ti
• Assumption: rate of reaction at any instant is
proportional to the concentration of reactant remaining
at that time.
– Let R(t) be concentration of reactant at time t.
– Instantaneous rate of decline of R:
dR
 kR
dt
– As in the model of exponential growth, k is reaction rate
constant measured in units of concentration per unit time
time.
• This is the familiar differential equation for exponential
decline,, with solution:
R  t   Ae  kt
where A (=all) denotes amount of
reactant available at time t = 0: R(0).
Monomolecular model for limited growth
• By definition: R  t   C  t   A
– R(t) = concentration of reactant at time t.
– C(t) = concentration of product at time t.
• Then: R  t   Ae  kt
A  C  t   Ae  kt
C  t   A   Ae kt
C  t   A  Ae  kt
C  t   A 1  e kt 
• This is the monomolecular growth model.
– Growth rate is proportional to resources yet to be
achieved.
– C(t) is analogous to a population size.
Monomolecular model for limited growth
• Monomolecular growth model for closed systems:
R t   C t   A
dR
 kt
 kR  k  A  C 
R
t

Ae


dt
dC
C  t   A 1  e kt 
  kR  k  A  C 
dt
• Extends to cases in which g
growth rate declines linearly
y
with population size.
– R(t) is analogous to resources currently available.
– C(t) is analogous to a population size.
– 2-parameter model: A, k
• Also
Al kknown as: negative
ti exponential
ti l growth
th model,
d l
= von Bertalanffy growth model.
Monomolecular model for limited growth
Conce
entration of product
[P
Populatio
on size]
• Monomolecular growth curves for various values of
the rate constant k:
Monomolecular model for limited growth
• Ex: Conversion of N-chloroacetanilide into
p-chloroacetanilide:
– Observed data points and fitted model:
Open-system model for limited growth
in continuous time
• Most basic observed pattern of limited population
growth in nature is a sigmoidal (‘S-shaped’)
( S shaped ) curve:
Open system: resources replenished, but limited.
Logistic model for limited population growth
in continuous time
• Many different models for sigmoidal growth curves.
• Particular case: logistic model:
– Widely used in theoretical work.
– Empirically found to describe population growth in the
field and in laboratory conditions.
• To develop: as before,
– Set up a differential equation for the instantaneous
growth rate.
– Solve
S l (i
(integrate)
t
t ) the
th equation
ti to
t find
fi d the
th pattern
tt
off
population size over time.
Logistic model for limited population growth
• Recall, for exponential growth in continuous time:
– Instantaneous growth rate for a population of size N(t)
at time t:
dN
 rN
(1-parameter model)
dt
– r is the exponential rate constant, = growth rate.
– Assumed to be constant over time
time.
• Characteristic feature of limited population growth:
– Growth rate is not constant over time.
– Decreases as the population size increases.
– Population size ‘feeds back’ to the growth rate.
Logistic model for limited population growth
• There are an infinite number of nonlinear models
that might be assumed
assumed.
– Varying numbers of parameters.
– More complicated
p
mechanistic models involve
postulates about underlying processes and
mechanisms.
• Si
Simplest
l t case: assume that
th t the
th growth
th rate
t constant
t t
(r) declines linearly with increasing population size.
– Linearity is a parsimonious assumption:
• ‘Assumption of ignorance’, as in linear regression.
Logistic model for limited population growth
• Rate of change of population size then given by differential
equation:
dN
 N    N 
 r0 N 1     r0 1    N
dt
 K    K 
– 2-parameter
p
model: r0, K.
– Parameter K is called the carrying capacity of the
environment, = equilibrium level.
– Parameter r0 called the initial growth rate
rate, = intrinsic growth
rate, = potential growth rate, = maximum growth rate, when
population size is effectively zero.
• Usually
U
ll jjustt called
ll d r, nott r0.
– When population size is effectively zero, effects of limiting
factors are negligible.
– As population size N increases, the parenthetical term
decreases toward zero.
Logistic model for limited population growth
• Logistic model:
• To solve:
Separate
variables and
integrate:
Evaluate
integrals:
Take
exponentials
ti l
of both sides:
Rearrange
ea a ge
and simplify:
dN
 N
 rN 1  
dt
 K
(instantaneous
rate of population
change at time t)
KdN
 N  K  N    r dt
1

1
  N   K  N   dN   r dt


log e N  log e  K  N   rt  c
log e N  log e  K  N   rt  log e N  0   log e  K  N  0  
N  0  e rt
N

K  N K  N  0
N t  
K
 K
  rt
1 
 1 e
 N  0 
(
(also
called the
Verhulst-Pearl
equation, ~1840)
Logistic model for limited population growth
• Logistic model:
dN
 N
 rN 1  
dt
 K
• Integrated time-dependent form: N  t  
K
 K
  rt
 1 e
1 
 N  0 
K
• Alternate equivalent form: N  t  
 r t h
1

e
(reparameterized)
– Parameter h is the time at which the population
reaches half of the carrying capacity: N  h   12 K
= ‘half-saturation’ constant.
– Assumes that initial p
population
p
size N(0)
( ) is less than
the carrying capacity K.
Logistic model for limited population growth
• Properties of the logistic growth curve:
(1) The increase of population size with time follows a
sigmoidal or S
S-shaped
shaped curve
curve. N=0 and N=K are
asymptotes.
Logistic model for limited population growth
• Properties of the logistic growth curve:
(2) The growth rate increases with population size to a
peak rate of r K/4 when the population reaches half
of carrying capacity. This occurs at a point of
inflexion on the logistic curve, where the slope is
greatest:
Logistic model for limited population growth
• Properties of the logistic growth curve:
(3) Unlike the case of exponential growth, the logarithm
of population size log(N(t)) increases at a decreasing
rate:
Logistic model for limited population growth
• Properties of the logistic growth curve:
(4) The log-growth rate decreases linearly with
population size
size, with slope –rr / K :
Logistic model for limited population growth
• Properties
P
ti off the
th logistic
l i ti growth
th curve:
(5) Shape of the curve depends on rate constant, r.
For practical purposes, the growth curve covers the
time span h ± 5/r .
h – 5/r
h
h + 5/r
Logistic model for limited population growth
• E
Ex: G
Growth
th off a population
l ti off yeastt cells
ll (Carlson,
(C l
1913):
– Introduced nutrients and a small number of yeast cells
cells.
– Number of cells estimated at hourly intervals for 18 hrs.
– Logistic
g
model fitted to data by
y nonlinear regression.
g
Logistic model for limited population growth
• Case in which initial population size greater than the
carrying capacity:
– Logistic
g
model typically
yp
y depicts
p
p
population
p
that increases
steadily toward carrying capacity, K.
• Never exceeds it.
– IIn theory,
th
could
ld ‘‘seed’
d’ an iinitial
iti l population
l ti size
i th
thatt
exceeds carrying capacity.
• Population
opu at o ssize
e tthen
e dec
declines
es steadily
stead y to K.
Logistic model has a stable
equilibrium at N(t)=K because
population
l ti returns
t
th
there ffrom
any displacement either above
or below.
A second equilibrium at N(t) = 0
is ‘trivial’, and is unstable.