Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Stochastic Population Systems
Xuerong Mao FRSE
Department of Statistics and Modelling Science
University of Strathclyde
Glasgow, G1 1XH
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
One of the important problems in life science is the
specification of the stochastic process governing the behaviour
of an underlying quantity. We here use the term underlying
quantity to describe any interested object whose value is
known at present but is liable to change in the future. Typical
examples are
number of cancer cells,
number of HIV infected individuals,
number of people who have the swine flu.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
One of the important problems in life science is the
specification of the stochastic process governing the behaviour
of an underlying quantity. We here use the term underlying
quantity to describe any interested object whose value is
known at present but is liable to change in the future. Typical
examples are
number of cancer cells,
number of HIV infected individuals,
number of people who have the swine flu.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
One of the important problems in life science is the
specification of the stochastic process governing the behaviour
of an underlying quantity. We here use the term underlying
quantity to describe any interested object whose value is
known at present but is liable to change in the future. Typical
examples are
number of cancer cells,
number of HIV infected individuals,
number of people who have the swine flu.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Now suppose that at time t the underlying quantity is x(t). Let
us consider a small subsequent time interval dt, during which
x(t) changes to x(t) + dx(t). (We use the notation d· for the
small change in any quantity over this time interval when we
intend to consider it as an infinitesimal change.) By definition,
the intrinsic growth rate at t is dx(t)/x(t). How might we model
this rate?
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
If, given x(t) at time t, the rate of change is deterministic, say
R = R(x(t), t), then
dx(t)
= R(x(t), t)dt.
x(t)
This gives the ordinary differential equation (ODE)
dx(t)
= R(x(t), t)x(t).
dt
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
However the rate of change is in general not deterministic as it
is often subjective to many factors and uncertainties e.g.
system uncertainty, environmental disturbances. To model the
uncertainty, we may decompose
dx(t)
= deterministic change + random change.
x(t)
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
The deterministic change may be modeled by
R̄dt = R̄(x(t), t)dt
where R̄ = r̄ (x(t), t) is the average rate of change given x(t) at
time t. So
dx(t)
= R̄(x(t), t)dt + random change.
x(t)
How may we model the random change?
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
In general, the random change is affected by many factors
independently. By the well-known central limit theorem this
change can be represented by a normal distribution with mean
zero and and variance V 2 dt, namely
random change = N(0, V 2 dt) = V N(0, dt),
where V = V (x(t), t) is the standard deviation of the rate of
change given x(t) at time t, and N(0, dt) is a normal distribution
with mean zero and and variance dt. Hence
dx(t)
= R̄(x(t), t)dt + V (x(t), t)N(0, dt).
x(t)
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
A convenient way to model N(0, dt) as a process is to use the
Brownian motion B(t) (t ≥ 0) which has the following
properties:
B(0) = 0,
dB(t) = B(t + dt) − B(t) is independent of B(t),
dB(t) follows N(0, dt).
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
The stochastic model can therefore be written as
dx(t)
= R̄(x(t), t)dt + V (x(t), t)dB(t),
x(t)
or
dx(t) = R̄(x(t), t)x(t)dt + V (x(t), t)x(t)dB(t)
which is a stochastic differential equation (SDE).
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Linear Model
Nonlinear Models
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Linear Model
Nonlinear Models
Linear model
If both R̄ and V are constants, say
R̄(x(t), t) = µ,
V (x(t), t) = σ,
then the SDE becomes
dx(t) = µx(t)dt + σx(t)dB(t).
This is the exponential growth model in population.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Linear Model
Nonlinear Models
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Linear Model
Nonlinear Models
Logistic model
If
R̄(x(t), t) = b + ax(t),
V (x(t), t) = σx(t),
then the SDE becomes
dx(t) = x(t) [b + ax(t)]dt + σx(t)dB(t) .
This is the well-known Logistic model in population.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Linear Model
Nonlinear Models
Square root process
If
R̄(x(t), t) = µ,
V (x(t), t) = p
σ
x(t)
,
then the SDE becomes the well-known square root process
p
dx(t) = µx(t)dt + σ x(t)dB(t).
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Linear Model
Nonlinear Models
Mean-reverting model
If
R̄(x(t), t) =
α(µ − x(t))
,
x(t)
σ
V (x(t), t) = p
x(t)
,
then the SDE becomes
dx(t) = α(µ − x(t))dt + σ
p
x(t)dB(t).
This is the mean-reverting square root process in population.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Linear Model
Nonlinear Models
Theta process
If
R̄(x(t), t) = µ,
V (x(t), t) = σ(x(t))θ−1 ,
then the SDE becomes
dx(t) = µx(t)dt + σ(x(t))θ dB(t),
which is known as the theta process.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
In the classical theory of population dynamics, it is assumed
that the grow rate is constant µ. Thus
dx(t)
= µdt,
x(t)
which is often written as the familiar ordinary differential
equation (ODE)
dx(t)
= µx(t).
dt
This linear ODE can be solved exactly to give exponential
growth (or decay) in the population, i.e.
x(t) = x(0)eµt ,
where x(0) is the initial population at time t = 0.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
We observe:
If µ > 0, x(t) → ∞ exponentially, i.e. the population will
grow exponentially fast.
If µ < 0, x(t) → 0 exponentially, that is the population will
become extinct.
If µ = 0, x(t) = x(0) for all t, namely the population is
stationary.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
We observe:
If µ > 0, x(t) → ∞ exponentially, i.e. the population will
grow exponentially fast.
If µ < 0, x(t) → 0 exponentially, that is the population will
become extinct.
If µ = 0, x(t) = x(0) for all t, namely the population is
stationary.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
We observe:
If µ > 0, x(t) → ∞ exponentially, i.e. the population will
grow exponentially fast.
If µ < 0, x(t) → 0 exponentially, that is the population will
become extinct.
If µ = 0, x(t) = x(0) for all t, namely the population is
stationary.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
However, if we take the uncertainty into account as explained
before, we may have
dx(t)
= rdt + σdB(t).
x(t)
This is often written as the linear SDE
dx(t) = µx(t)dt + σx(t)dB(t).
It has the explicit solution
h
i
x(t) = x(0) exp (µ − 0.5σ 2 )t + σB(t) .
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Recall the properties of the Brownian motion
B(t)
=1
2t log log t
lim sup p
t→∞
and
lim inf p
t→∞
a.s.
B(t)
= −1 a.s.
2t log log t
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If µ > 0.5σ 2 , x(t) → ∞ exponentially with probability 1, i.e.
the population will grow exponentially fast.
If µ < 0.5σ 2 , x(t) → 0 exponentially with probability 1, that
is the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ 2 where
the population will grow exponentially in the corresponding
ODE model but it will become extinct in the SDE model.
This reveals the important feature - noise may make a
population extinct.
If µ = 0.5σ 2 , lim supt→∞ x(t) = ∞ while lim inft→∞ x(t) = 0
with probability 1.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If µ > 0.5σ 2 , x(t) → ∞ exponentially with probability 1, i.e.
the population will grow exponentially fast.
If µ < 0.5σ 2 , x(t) → 0 exponentially with probability 1, that
is the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ 2 where
the population will grow exponentially in the corresponding
ODE model but it will become extinct in the SDE model.
This reveals the important feature - noise may make a
population extinct.
If µ = 0.5σ 2 , lim supt→∞ x(t) = ∞ while lim inft→∞ x(t) = 0
with probability 1.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If µ > 0.5σ 2 , x(t) → ∞ exponentially with probability 1, i.e.
the population will grow exponentially fast.
If µ < 0.5σ 2 , x(t) → 0 exponentially with probability 1, that
is the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ 2 where
the population will grow exponentially in the corresponding
ODE model but it will become extinct in the SDE model.
This reveals the important feature - noise may make a
population extinct.
If µ = 0.5σ 2 , lim supt→∞ x(t) = ∞ while lim inft→∞ x(t) = 0
with probability 1.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If µ > 0.5σ 2 , x(t) → ∞ exponentially with probability 1, i.e.
the population will grow exponentially fast.
If µ < 0.5σ 2 , x(t) → 0 exponentially with probability 1, that
is the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ 2 where
the population will grow exponentially in the corresponding
ODE model but it will become extinct in the SDE model.
This reveals the important feature - noise may make a
population extinct.
If µ = 0.5σ 2 , lim supt→∞ x(t) = ∞ while lim inft→∞ x(t) = 0
with probability 1.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If µ > 0.5σ 2 , x(t) → ∞ exponentially with probability 1, i.e.
the population will grow exponentially fast.
If µ < 0.5σ 2 , x(t) → 0 exponentially with probability 1, that
is the population will become extinct.
In particular, this includes the case of 0 < µ < 0.5σ 2 where
the population will grow exponentially in the corresponding
ODE model but it will become extinct in the SDE model.
This reveals the important feature - noise may make a
population extinct.
If µ = 0.5σ 2 , lim supt→∞ x(t) = ∞ while lim inft→∞ x(t) = 0
with probability 1.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
The logistic model for single-species population dynamics is
given by the ODE
dx(t)
= x(t)[b + ax(t)].
dt
Mao
Stochastic Population Systems
(3.1)
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If a < 0 and b > 0, the equation has the global solution
x(t) =
b
−a +
e−bt (b
+ ax0 )/x0
(t ≥ 0) ,
which is not only positive and bounded but also
lim x(t) =
t→∞
b
.
|a|
If a > 0, whilst retaining b > 0, then the equation has only
the local solution
b
x(t) =
(0 ≤ t < T ) ,
−bt
−a + e (b + ax0 )/x0
which explodes to infinity at the finite time
ax0
1
T = − log
.
b
b + ax0
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If a < 0 and b > 0, the equation has the global solution
x(t) =
b
−a +
e−bt (b
+ ax0 )/x0
(t ≥ 0) ,
which is not only positive and bounded but also
lim x(t) =
t→∞
b
.
|a|
If a > 0, whilst retaining b > 0, then the equation has only
the local solution
b
x(t) =
(0 ≤ t < T ) ,
−bt
−a + e (b + ax0 )/x0
which explodes to infinity at the finite time
ax0
1
T = − log
.
b
b + ax0
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Once again, the growth rate b here is not a constant but a
stochastic process. Therefore, bdt should be replaced by
bdt + N(0, v 2 dt) = bdt + vN(0, dt) = bdt + vdB(t),
where v 2 is the variance of the noise intensity. Hence the ODE
evolves to an SDE
dx(t) = x(t) [b + ax(t)]dt + vdB(t) .
(3.2)
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
The variance may or may not depend on the state x(t). We first
consider the latter, say
v = σx(t).
Then the SDE (3.2) becomes
dx(t) = x(t) [b + ax(t)]dt + σx(t)dB(t) .
How is this SDE different from its corresponding ODE?
Mao
Stochastic Population Systems
(3.3)
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Significant Difference between ODE (3.1) and SDE
(3.3)
ODE (3.1): The solution explodes to infinity at a finite time
if a > 0 and b > 0.
SDE (3.3): With probability one, the solution will no longer
explode in a finite time, even in the case when a > 0 and
b > 0, as long as σ 6= 0. Moreover, the stochastic
population system is persistent.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Significant Difference between ODE (3.1) and SDE
(3.3)
ODE (3.1): The solution explodes to infinity at a finite time
if a > 0 and b > 0.
SDE (3.3): With probability one, the solution will no longer
explode in a finite time, even in the case when a > 0 and
b > 0, as long as σ 6= 0. Moreover, the stochastic
population system is persistent.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Example
dx(t)
= x(t)[1 + x(t)],
dt
has the solution
x(t) =
t ≥ 0, x(0) = x0 > 0
1
−1 +
e−t (1
+ x0 )/x0
(0 ≤ t < T ) ,
which explodes to infinity at the finite time
1 + x0
T = log
.
x0
However, the SDE
dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)]
will never explode as long as σ 6= 0.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
200
(a)
150
100
50
0
0
2
4
6
8
10
80
(b)
60
40
20
0
0
2
4
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6
8
Stochastic Population Systems
10
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Note on the graphs:
In graph (a) the solid curve shows a stochastic trajectory
generated by the Euler scheme for time step ∆t = 10−7 and
σ = 0.25 for a one-dimensional system (3.3) with a = b = 1.
The corresponding deterministic trajectory is shown by the
dot-dashed curve. In Graph (b) σ = 1.0.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Key Point:
When a > 0 and ε = 0 the solution explodes at the finite time
t = T ; whilst conversely, no matter how small ε > 0, the
solution will not explode in a finite time. In other words,
noise may suppress explosion.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
If the variance is independent of the state x(t), namely
v = σ = const., then the SDE (3.2) becomes
dx(t) = x(t) [b + ax(t)]dt + σdB(t) .
How is this SDE different from others?
Mao
Stochastic Population Systems
(3.4)
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
For this SDE, we require a < 0 in order to have no
explosion.
If b < 0.5σ 2 then limt→∞ x(t) = 0 with probability 1,
namely the noise makes the population extinct.
If b > 0.5σ 2 , the population is persistent and
1
lim
T →∞ T
Z
T
x(t)dt =
0
Mao
b − 0.5σ 2
|a|
a.s.
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
For this SDE, we require a < 0 in order to have no
explosion.
If b < 0.5σ 2 then limt→∞ x(t) = 0 with probability 1,
namely the noise makes the population extinct.
If b > 0.5σ 2 , the population is persistent and
1
lim
T →∞ T
Z
T
x(t)dt =
0
Mao
b − 0.5σ 2
|a|
a.s.
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
For this SDE, we require a < 0 in order to have no
explosion.
If b < 0.5σ 2 then limt→∞ x(t) = 0 with probability 1,
namely the noise makes the population extinct.
If b > 0.5σ 2 , the population is persistent and
1
lim
T →∞ T
Z
T
x(t)dt =
0
Mao
b − 0.5σ 2
|a|
a.s.
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Outline
1
Stochastic Modelling
2
Well-known Models
Linear Model
Nonlinear Models
3
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Noise changes the behaviour of population systems
significantly.
Noise may suppress the potential population explosion.
Noise may make the population extinct.
Noise may make the population persistent.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Noise changes the behaviour of population systems
significantly.
Noise may suppress the potential population explosion.
Noise may make the population extinct.
Noise may make the population persistent.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Noise changes the behaviour of population systems
significantly.
Noise may suppress the potential population explosion.
Noise may make the population extinct.
Noise may make the population persistent.
Mao
Stochastic Population Systems
Stochastic Modelling
Well-known Models
Stochastic verse Deterministic
Exponential growth model
Logistic Model
Summary
Noise changes the behaviour of population systems
significantly.
Noise may suppress the potential population explosion.
Noise may make the population extinct.
Noise may make the population persistent.
Mao
Stochastic Population Systems