Growth and extinction of populations
in a randomly varying environments
Carlos A. Braumann (braumann@uevora.pt)
Universidade de Évora – Portugal
Departamento de Matemática www.dmat.uevora.pt
Outline
Population growth in a random environment can be modeled using stochastic differential equations (SDE).
Instead of considering specific models, we will study a general model in what concerns extinction and existence and existence of stationary
densities. That model is a generalization of previously studied specific models.
Which stochastic calculus is more appropriate: Itô or Stratonovich? Extension to general harvesting models
Further generalization to density-dependent noise intensities. Time to extinction
Deterministic model
)) ( ( ) ( ) ( dt g N t t dN t N = 1known
is
0
0
= N
0>
N )
(
Examples Malthusian g(N) = r Logistic g(N) = r (1-N/K) Gompertz g(N) = r ln (K/N) ………N(t) Population size at time t > 0
g(N) (per capita) growth rate (when population size is N)
Randomly fluctuating environment
noise
white
standard
)
(t
ε
(
(
)
)
(
)
)
(
)
(
K
t
N
K
N
r
K
N
t
r
dt
t
dN
t
N
σε
σ
⎟
⎠
ε
⎞
⎜
⎝
⎛ −
+
⎟
⎠
⎞
⎜
⎝
⎛ −
=
⎟
⎠
⎞
⎜
⎝
⎛ −
+
=
1
1
1
1
) ( ) ( ) ( K t N r dt t dN t N ⎟⎠+σε
⎞ ⎜ ⎝ ⎛ − = 1 1Effect of environmental random fluctuations on the growth rate What has appeared in the literature
• Additive noise Add noise to the growth rate of a specific model
Example: logistic model
• Add noise to a parameter of a specific model
Example: Add noise to the r in the logistic model
Randomly fluctuating environment
Several specific models (specific functions g(N)) have been proposed in the literature starting with Levins (1969) and, for the case of harvesting, with Beddington and May (1977).
Question: Are the properties obtained model specific or real properties of the population?
Our work:
• See if you can obtain properties for general models (arbitrary functions
g(N) satisfying only biologically determined assumptions and some mild
technical assumptions). We seek properties on extinction or non-extinction and on existence of stationary densities.
• We consider realistic noise intensities
•First: additive noise (constant noise intensities)
•Later: density-dependent noise intensities that are positive for positive population sizes
General SDE model
with constant noise intensity
)
(
))
(
(
)
(
)
(
dt
g
N
t
t
t
dN
t
N
=
+
σε
1
)
(
))
(
(
))
(
(
)
(
t
G
N
t
dt
V
N
t
dW
t
dN
=
+
Assumptions on• continuously differentiable strictly decreasing • the limit • • G(0+) = 0
)
,
(
)
,
(
:
)
(
⋅
0
+∞
a
−∞
+∞
g
infinite)
be
(may
0
0
+)
:
=
lim
↓0(
)
≠
(
g
N
g
N 0 < +∞) ( gknown
is
0
0
= N
0>
N )
(
process
Wiener
standard
0 ∫=
ts
ds
t
W
(
)
ε
(
)
g(N) “average” growth rate
G(N)=g(N)N total “average” growth rate
σ >0 noise intensity (constant) V(N)=σN total noise intensity
Stochastic integration
)
(
))
(
(
))
(
(
)
(
t
G
N
t
dt
V
N
t
dW
t
dN
=
+
N )
(
0
= N
0>
0
is
known
∫+
∫+
=
N
tG
N
s
ds
tV
N
s
dW
s
t
N
(
)
0 0(
(
))
0(
(
))
(
)
Decompositions with diameter converging to 0
Intermediate points
The Riemann-Stieltjes sums
have m.s. limits that depend on the choice of the intermediate points.
)
,
,
(
, , , ,L
1
2
L
0
=
t
0n<
t
1n<
<
t
n−1n<
t
n n=
t
n
=
]
,
[
, , ,n i n i n i∈
t
−1t
τ
(
)
∑ =−
− n i i n i n i nt
W
t
W
N
V
1 1)
(
)
(
))
(
(
τ
, , ,Stochastic integration
Itô integral (non-anticipative choice τi,n=ti -1,n )
Nice probabilistic properties. Does not follow ordinary calculus rules. Itô chain rule for Y(t) = h(t,N(t)) with h(t,x) of class C1,2
Stratonovich integral
We will use Stratonovich calculus.
(
)
∑ ∫ = − − +∞ →−
=
n i i n i n i n n tV
N
s
dW
s
V
N
t
W
t
W
t
1 1 1 0(
(
))
(
)
l.i.m.
(
(
,))
(
,)
(
,)
) ( ) ( ) , ( ) ( ) , ( ) , ( t dW N V x N t h dt N G x N t h t N t h dY ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ( )V (N) x t,N h 2 2 2 2 1 ∂ ∂ +(
)
∑ ∫ = − − +∞ →⎟
⎠
−
⎞
⎜
⎝
⎛
+
=
n i i n i n n i n i n tV
N
s
dW
s
V
N
t
V
N
t
W
t
W
t
1 1 1 02
l.i.m.
(
(
))
(
(
))
(
)
(
)
)
(
))
(
(
)
, , , ,S
(
General growth model
with constant noise intensity
The solution exists and is unique up to an explosion time The solution is a homogeneous diffusion process with
Diffusion coefficient
Drift coefficient
Note: With Itô calculus
(
g
N
t
)
N
dt
dN
S
)
(
)
(
)
(
=
+
σε
2 2 2x
x
V
x
b
(
)
:
=
(
)
=
σ
(
g
x
)
x
dx
x
db
x
x
g
x
a
2
4
1
2/
)
(
)
(
)
(
:
)
(
=
+
=
+
σ
x
x
g
x
a
(
)
:
=
(
)
General growth model
with constant noise intensity
Scale density Scale function Speed density Speed function(
g
N
t
)
N
dt
dN
S
)
(
)
(
)
(
=
+
σε
arbitrary)
0
2
2
0 2 0 0 0⎟⎟
⎠
>
⎞
⎜⎜
⎝
⎛
−
=
⎟
⎠
⎞
⎜
⎝
⎛−
=
∫ ∫d
y
V
G
N
V
y
V
d
b
a
N
s
yN yN(
)
(
)
(
exp
)
(
)
(
)
(
)
(
exp
:
)
(
θ
θ
θ
θ
θ
θ
arbitrary)
0
0 0>
=
∫s
z
dz
x
N
S
N x(
)
(
)
(
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
=
∫N yd
V
G
N
V
y
V
N
b
N
s
N
m
0 2 02
1
1
θ
θ
θ
)
(
)
(
exp
)
(
)
(
)
(
)
(
:
)
(
arbitrary)
0
0 0>
=
∫m
z
dz
x
N
M
(
)
xN(
)
(
[
]
)
(
)
(
)
(
)
(
)
(
a
S
b
S
a
S
x
S
x
N
T
T
x
u
b
N
a
a b−
−
=
=
<
=
+∞
<
<
<
<
0 00
P
General growth model
with constant noise intensity
(
g
N
t
)
N
dt
dN
S
)
(
)
(
)
(
=
+
σε
Boundary N=0 is non-attractiveif there is a right-neighborhood R=]0,y[ of zero such that, for any 0<x<n∈R,
Tz - first passage time by z
Necessary and sufficient condition
This implies (Karlin and Taylor 1981) non-extinction a.s. Similarly, for non-attractiveness of the boundary
With our assumptions we prove that:
The boundary is non-attractive (which implies non-explosion, i.e., existence and uniqueness of the solution for all times).
The boundary N = 0 is attractive if g(0+) < 0 and non-attractive if g(0+) > 0.
[
0]
0 P 0+ ≤T N = x = T n ( ) z z T T 0 0+ = lim ↓ +∞ = N −∞ = +) (0 S +∞ = NGeneral growth model
with constant noise intensity
When both boundaries are non-attractive andthe process is ergodic and there is a stationary density given by
With our assumptions, we prove that happens when g(0+)>0.
CONCLUSIONS:
When g(0+) < 0, extinction occurs a.s.
When g(0+) > 0, there is a zero probability of extinction and there is a stationary density
(the mode of which approximately coincides with the deterministic equilibrium when the noise intensity is small).
(
g
N
t
)
N
dt
dN
S
)
(
)
(
)
(
=
+
σε
,
)
(
)
,
(
0
+∞
=
∫0+∞m
z
dz
<
+∞
M
).
(
)
,
(
)
(
)
(
<
<
+∞
+∞
=
x
M
x
m
x
p
0
0
General growth model
with constant noise intensity
What happens if we use Itô calculus?
CONCLUSIONS:
When g(0+) < σ2/2, extinction occurs a.s.
When g(0+) > σ2/2, there is a zero probability of extinction and there is a stationary density
So, we can have extinction even when the “average” growth rate at low densities is positive.
Which calculus is right? Recipes
(
g
N
t
)
N
dt
dN
I
)
(
)
(
)
(
=
+
σε
Resolution of the controversy
for constant noise intensity
(
)
N
t
N
g
dt
dN
)
(
)
(
+
σε
=
Deterministic model (σ =0)(per capita) growth rate R(x) when population size is x at time t
Stochastic models
Arithmetic average growth rate Ra (x) when population size is x at time t
)
(
)
(
lim
:
)
(
g
x
t
x
t
t
N
x
dt
dN
x
x
R
t x N=
Δ
−
Δ
+
=
⎟
⎠
⎞
⎜
⎝
⎛
=
↓ Δ = 01
1
(
g
N
t
)
N
dt
dN
I
)
i(
)
(
)
(
=
+
σε
(
g
N
t
)
N
dt
dN
S
)
s(
)
(
)
(
=
+
σε
⎩
⎨
⎧
+
=
=
Δ
−
Δ
+
=
↓ Δ2
Stratonovi
ch
Itô
1
E
1
2 0(
)
/
)
(
)
(
)]
(
[
lim
:
)
(
,σ
x
g
x
g
x
a
x
t
x
t
t
N
x
x
R
s i x t t aResolution of the controversy
for constant noise intensity
Geometric average growth rate Rg (x) when population size is x at time t
CONCLUSION (Braumann 2007a)
g(x) means two different “average” growth rates under the two calculi.
It is the arithmetic average growth rate when we use Itô calculus.
It is the geometric average growth rate when we use Stratonovich calculus. Taking into account the difference between the two averages, the two calculi completely coincide.
In both, we have extinction or stationary density according to whether the geometric average growth rate at low densities Rg(0+) is negative or positive.
(
)
⎩
⎨
⎧
−
=
Δ
−
Δ
+
=
↓ ΔStratonovi
ch
Itô
2
E
exp
1
2 0(
)
/
)
(
)]
(
[ln
lim
:
)
(
,x
g
x
g
t
x
t
t
N
x
x
R
s i x t t gσ
Harvesting models
with constant noise intensity
)
(
)
(
)
(
)
(
g
N
t
h
N
dt
dN
N
S
1
=
+
σε
−
h(N) harvesting effort (when population size is N)
H(N)= h(N)N yield (total harvesting rate) q(N)=g(N)−h(N) net growth rate
Assumptions on
• continuously differentiable non-negative
• the limit can be weakened
• H(0+) = 0
)
,
[
)
,
(
:
)
(
⋅
0
+∞
a
0
+∞
h
infinite)
be
(may
0
is
and
exists
0
+)
:
=
lim
↓0(
)
≠
(
q
N
q
N CONCLUSIONS (Braumann 1999b)When q(0+) < 0, extinction occurs a.s.
When q(0+) > 0, there is 0 probability of extinction and there is a stationary density
(the mode of which approximately coincides with the deterministic equilibrium when the noise intensity is small).
Itô and Stratonovich: Braumann (2007c).
General growth model
with density-dependent noise intensity Assumptions on :• strictly positive twice continuously differentiable • V(0+)=0, where V(N)=σ (N)N
(A) for some x0>0;
(B) for some y0>0.
(C) |σ(N)/g(N)| is bounded in a right neighborhood of 0. (D) |σ(N)/g(N)| is bounded in a neighborhood of +∞.
If noise intensity is bounded, it satisfies (A), (B), (C) and (D).
)
,
(
)
,
(
:
)
(
⋅
0
+∞
a
0
+∞
σ
∫ +0 = +∞ 0 1 x dN N N ) ( σ ∫+∞ = +∞ 0 1 y σ(N)N dN)
(N
σ
General growth model
with density-dependent noise intensityCONCLUSION (Braumann 2007b)
g(x) means two different “average” growth rates under the two calculi.
It is the arithmetic average growth rate when we use Itô calculus. It is the φ−average growth rate when we use Stratonovich calculus
(coincides with the geometric average when N approaches 0).
Taking into account the difference between the two averages, the two calculi completely coincide.
⎩
⎨
⎧
+
+
=
=
ch
Stratonovi
2
Itô
1
2(
)
/
(
)
'
(
)
)
(
)
(
)
(
)
(
x
x
x
x
x
g
x
g
x
a
x
x
R
s i aσ
σ
σ
∫=
x cdz
c
z
z
x
1
fixed
arbitrary
constant)
average
-(
)
(
)
(
σ
φ
φ
(
)
ch
Stratonovi
for
E
1
1 0)
(
))]
(
(
[
lim
:
)
(
,g
x
t
x
t
t
N
x
x
R
t x s tΔ
=
−
Δ
+
=
− ↓ Δφ
φ
φGeneral growth model
with density-dependent noise intensityIn terms of Y, the drift coefficient is, with y=φ(x), Therefore
Apply φ to both sides, expand about y and notice that
to obtain from which the result follows
(
)
ch
Stratonovi
for
E
1
1 0(
)
))]
(
(
[
lim
:
)
(
,g
x
t
x
t
t
N
x
x
R
t x s tΔ
=
−
Δ
+
=
− ↓ Δφ
φ
φ Proof ofUnder Stratonovich calculus, Y=φ(N) satisfies the SDE
)
(
))
(
(
))
(
(
)
(
dt
dW
t
Y
Y
g
dY
S
=
s −+
− 1 1φ
σ
φ
[
]
)) ( ( )) ( ( ) ( lim , y y g y t t Y t s x t t 1 1 0 E 1 − − ↓ Δ Δ + Δ − = σ φ φ[
]
( ). )) ( ( )) ( ( ) ( , t o t y y g y t t Y s x t + Δ = + − Δ + Δ − 1 1 E φ σ φ ) ( / ) ( ) ( x x dx x d dy y d σ φ φ−1 = 1 =(
t,x[Y(t + Δt])
= x + xgs(x)+o(Δt), − E 1 φGeneral growth model
with density-dependent noise intensity)
(N
σ
With the assumptions made, the same conclusions hold:
• When the geometric average growth rate at low densities is negative, extinction occurs a.s.
• When the geometric average growth rate at low densities is positive, there is a zero probability of extinction and there is a stationary density
Time to extinction
)
(
))
(
(
)
(
)
(
)
(
g
N
t
t
dt
t
dN
t
N
S
1
=
+
σε
N )
(
0
=
N
0=
x
>
0
is
known
with the assumptions made on g and g(0+)>0.
There is no “mathematical” extinction and there is a stationary density. What about a population of 0.4 individuals? What about Allee effects? Set extinction threshold a>0. We assume a<N0.
Note: To study pest outbreaks, we could also consider a>N0 “Realistic” extinction occurs if ever N(t) reaches the threshold
Since the process is ergodic it will do it (sooner or later) with probability one.
So, “realistic” extinction occurs a.s.
Time to extinction
{
t
N
t
a
}
T
a=
inf
>
0
:
(
)
=
we call extinction time To the first passage timeTime to extinction
(
)
( )
[
]
[
]
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
σ
σ
d m kV a S S x u d m kV S b S x u x V x V k b V a V x kV x dS x dV x dM d x kV dx x dV x a dx x V d x b x N T x V a S b S a S x S x N T T x u T T T b N a x x b x x g x a k x a k b x k k k k k k k k k ab k a b b a ab ) ( ) ( )) ( ) ( ( )) ( ( ) ( ) ( )) ( ) ( ( ) ( ) ( ) ( ,...) , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( } , min{ ) ( / ) ( ) ( 1 1 0 1 1 2 2 0 0 0 2 2 1 2 2 1 2 1 0 0 2 1 0 2 1 moment order th -k 0 2 − − − − ∫ ∫ − − + − = ≡ = = = = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + + = = − − = = < = = +∞ < < < < = + = E PTime to extinction
We can also obtain an ODE for the Laplace transform
Solving the equation and inverting the Laplace transform, one obtains the p.d.f. of Tab
[
]
0
0
2
1
0=
=
=
−
⎟
⎠
⎞
⎜
⎝
⎛
=
−
=
)
(
)
(
)
(
)
(
)
(
)
(
)
exp(
)
(
b
U
a
U
x
U
x
dS
x
dU
x
dM
d
x
N
T
x
U
ab λ λ λ λ λλ
λ
E
Time to extinction
Since the process is ergodic, if we let , we obtain as limit of Vk(x)
So, we obtain (after some indeterminations are removed)
+∞
↑
b
( )
[
T
N
x
]
x
V
k(
)
=
E
a k 0=
(
θ
θ
θ
)
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξkV
m
d
d
s
d
m
kV
a
S
S
d
m
kV
a
S
x
S
x
V
x a k k x a k x k ∫ ∫ ∫ ∫ ∞ + − − − ∞ +=
−
+
−
=
)
(
)
(
)
(
)
(
)
(
))
(
)
(
(
)
(
)
(
))
(
)
(
(
)
(
1 1 12
2
2
Application
Gompertz model with additive noise
0
0
0
0
with
1
0>
>
>
>
+
=
t
r
K
N
N
K
r
dt
t
dN
t
N
S
(
)
ln
(
)
,
,
,
)
(
)
(
σε
σ
( )
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
=
=
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
=
>
+∞
=
≡
=
+ 2 2 2 2 2 0 2 21
choosing
2
0
0
and
s
assumption
the
all
satisfy
K
x
r
Kx
x
m
K
x
r
x
K
x
s
K
y
x
x
b
x
x
K
r
x
a
g
N
N
K
r
N
g
ln
exp
)
(
ln
exp
)
(
)
(
ln
)
(
)
(
)
(
ln
)
(
σ
σ
σ
σ
σ
σ
σ
Application
Extinction has 0 probability of occurring and there is a stationary density proportional to m(n) with support n>0, the mode of which is Kexp(σ2/(2r)).
Change of variable
y = ln (n/K)
Y(t) = ln ( N(t) /K ) has stationary density proportional to m(n) dn/dy, which one
immediately sees to be Gaussian with mean 0 and variance σ2/(2r).
We can obtain the transient p.d.f. of Y(t)
Y(t) satisfies the SDE
(
)
∫=
+
=
=
+
+
−
=
t rs rt rt rt rtK
N
Y
s
dW
e
S
Y
e
t
Y
t
dW
e
Ydt
re
dY
e
S
t
dW
rYdt
dY
S
0 0 0 0(
)
(
)
with
ln
/
)
(
)
(
)
(
)
(
)
(
σ
σ
σ
Application
(
)
.
,
,
)
(
)
(
+∞
→
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
→
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
+
=
∫ ∫ − − − − −t
r
e
r
ds
e
e
e
Y
s
dW
e
e
e
Y
t
Y
t rs rt rt rt t rs rt rtas
2
0
Gaussian
1
2
Gaussian
2 0 2 2 2 2 2 0 0 0σ
σ
σ
σ
[
]
( )
[
]
K
x
r
K
a
r
dtds
dze
e
z
e
r
x
N
T
x
V
dt
e
t
r
x
N
T
x
V
t z s t s a t aln
ln
))
(
(
)
(
))
(
(
)
(
σ
γ
σ
α
π
π
γ α α γ α=
=
Φ
−
=
=
=
Φ
−
=
=
=
− ∞ + ∫ ∫ ∫ ∫and
with
2
1
8
2
1
2
2 2 2 2 2 0 2 2 0 1E
E
Gráfico de r E[Ta] como função de N0/a. Aqui R=r /σ2 e d=a/K.
Gráfico de r DP[Ta] como função de N0/a. Aqui R=r /σ2 e d=a/K.
Case of logistic model
If we use instead the logistic model with additive noise
0 0 0 0 with 1 1 0 > > > > + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = t r K N K N r dt t dN t N S ( ) ( ) , , , ) ( ) (
σε
σ
( )
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+
−
=
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+
−
−
=
=
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛ −
=
>
=
≡
⎟
⎠
⎞
⎜
⎝
⎛ −
=
+K
x
K
x
r
Kx
x
m
K
x
K
x
r
x
K
x
s
K
y
x
x
b
x
K
x
r
x
a
r
g
N
K
N
r
N
g
1
2
1
1
2
choosing
2
1
0
0
and
s
assumption
the
all
satisfy
1
2 2 2 0 2 2ln
exp
)
(
ln
exp
)
(
)
(
)
(
)
(
)
(
)
(
σ
σ
σ
σ
σ
σ
σ
Case of logistic model
[
]
( )
[
]
x
K
r
a
K
r
dtds
e
t
dy
e
y
y
r
e
s
x
N
T
x
V
dt
e
y
y
r
x
N
T
x
V
t r y r s t s t a y r a 2 2 2 1 2 2 1 2 4 0 2 2 1 2 2 2 0 12
and
2
with
2
8
2
2
2 2 2 2σ
γ
σ
α
σ
σ
σ
σ
σ σ γ α σ α σ γ α=
=
⎟
⎠
⎞
⎜
⎝
⎛
Γ
=
=
=
⎟
⎠
⎞
⎜
⎝
⎛
Γ
=
=
=
− − − ∞ + − − − − ∫ ∫ ∫ ∫'
'
,
)
(
,
)
(
/ / ' ' ' / / ' 'E
E
x1090
x1090
Application
of Gompertz additive noise model
to Gause´s 1934 data on Paramecia caudatum
0)
set to
figure
(in
day
per
096
0
137
0
cm
per
s
individual
2
23
2
58
day
per
15
0
49
0
intervals
confidence
95%
e
Approximat
2 3.
.
.
.
.
.
±
±
±
σ
K
r
Estimation for
Gompertz additive noise model
Assume we have observations in a single trajectory at times
t0 = 0< t1 < t2 < ...< tk
N0, N1, N2 , ... , Nk com Ni=N(ti ) Y0, Y1, Y2 , ... , Yk com Yi=Y(ti )=ln(Ni/K)
n0=N0 , n1 , n2 , ..., nk concrete observations y0, y1, y2 , ... , yk com yi=ln(ni/K)
K
n
y
K
n
y
n
Y
Y
y
y
p
n
N
N
n
n
p
K
e
y
y
p
n
n
p
i i Y i i i i i i y Y i i i i*
ln
*
ln
*
)
*
(
*
)
*
(
)
*
(
)
*
(
=
=
=
=
=
− − − − − − −and
with
given that
of
p.d.f.
transition
given that
of
p.d.f.
transition
1 1 1 1 1 1Estimation for
Gompertz additive noise
model
and so
from which one obtains
From the Markov property, we obtain the log-likelihood function
From
Y
(
t
)
=
Y
0e
−rt+
σ
e
−rt ∫0te
rsdW
(
s
)
one obtains ∑ ∑ = − − − − − ==
=
k i y i i Y i i i i i i k i kK
e
y
y
p
n
n
p
n
n
K
r
L
i 1 1 1 1 1 1 1,...,
)
(
)
(
)
,
,
(
σ
1 1 1with
− − Δ − −+
Δ
=
−
=
i r rt ∫tt− rs i i i iY
e
e
e
dW
s
t
t
Y
i i i iσ
(
)
(
)
(
(
)
)
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
−
−
−
−
=
Δ − Δ − Δ − − i i i r r r Y i ie
r
e
y
y
e
r
y
y
p
2 2 2 2 2 11
2
2
1
2
2
1
σ
σ
π
*
exp
*)
(
Estimation for
Gompertz additive noise model
(
)
(
)
∑ = − Δ − Δ − Δ −⎟⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
−
+
−
+
−
−
+
−
=
k i r i r i r i k i i ie
K
n
e
K
n
r
e
K
n
K
r
k
n
n
K
r
L
1 2 2 1 2 2 11
1
2
1
2
2
2
ln
ln
ln
ln
ln
ln
ln
ln
)
,...,
,
,
(
σ
σ
π
σ
Maximizing we obtain the ML estimators
).
,
,
(
)
,
,
(
)
ˆ
,
ˆ
,
ˆ
(
)
ˆ
,
ˆ
,
ˆ
(
θ
1θ
2θ
3=
r
K
σ
of
θ
1θ
2θ
3=
r
K
σ
intervals conf. approx. of n computatio allowing variance and mean ith Gaussian w asympt. is 1 2 3 2 1 3 2 1 k j i j i L K r K r ,..., , ) , , ( ) , , ( ) ˆ , ˆ , ˆ ( ) ˆ , ˆ , ˆ ( = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ − = =θ
θ
σ
θ
θ
θ
σ
θ
θ
θ
EPrediction for
Gompertz additive noise model
For prediction of population size at a future time t > tk it is better to work with the Gaussian random variable Y(t).
A good predictor is
[
] [
]
)).
(
ˆ
exp(
ˆ
)
(
ˆ
ˆ
ln
ˆ
ˆ
)
(
ˆ
,...,
)
(
ˆ
)
(
ˆ
( )t
Y
K
t
N
K
N
Y
e
Y
Y
t
Y
Y
Y
t
Y
t
Y
k k t t r k k k k=
=
=
=
=
−which
from
with
1E
E
Conclusions
•We have studied general models of population growth in random environments so that properties obtained are not model specific. We have first considered constant noise intensity and then allowed density-dependent noise intensity.
•For the general model considered, we have shown that “mathematical” extinction occurs if the geometric average growth rate at low population
densities is negative. If it is positive, “mathematical” extinction does not occur and there is a stationary density.
•”Realistic” extinction (population dropping to a positive low extinction threshold) allways occurs and one can obtain explicit expressions for the moments of the extinction time (the extinction time pd.f. can also be obtained numerically). We have applied to the Gompertz model with additive noise and obtain graphs of the mean and standard deviation of the extinction time. The same ideas can apply to high threshold crossing times (study of pest outbreaks).
•For the same specific model we have illustrated using real data the issues of parameter estimation and prediction.
Conclusions
•We have also resolved the controversy on whether to use Itô or Stratonovich calculus, which was a major obstacle to the use of these models.
Indeed, we have shown that it was due to the implicit wrong assumption that the deterministic term of the SDE meant the same average growth rate under the two calculi. We have shown that it means two different averages and that,
taking into account the difference between them, the two calculi give completely coincidental results.
References
Levins, R. (1969). The effect of random variations of different types on population growth.
Proc. Natl. Acad. Sci. USA 62: 1062-1065.
Braumann, C. A. (2007). Itô versus Stratonovich calculus in random population growth.
Math. Biosc. 206: 81-107.
Braumann, C. A. (1999). Applications of stochastic differential equations to population
growth. In Proceedings of the Ninth International Colloquium on Differential Equations (D. Bainov,ed.), VSP, Utrecht, p. 47-52.
Braumann, C. A. (2001). Crescimento de populações em ambiente aleatório:
generalização a intensidades de ruído dependentes da densidade da população. In A
Estatística em Movimento (M. Neves, J. Cadima, M. J. Martins and F. Rosado, eds.),
Edições SPE, Lisboa, p. 119-128.
Braumann, C. A. (1995). Threshold crossing probabilities for population growth models in
random environments. J. Biological Systems 3: 505-517.
Carlos, C. and Braumann, C. A. (2005). Tempos de extinção para populações em
ambiente aleatório. In Estatística Jubilar (C. A. Braumann, P. Infante, M. M. Oliveira, R. Alpízar-Jara, and F. Rosado, eds.). Edições SPE, Lisboa, p. 133-142.
Carlos, C. and Braumann, C. A. (2006). Tempos de extinção para populações em
ambiente aleatório e cálculos de Itô e Stratonovich. In Ciência Estatística (L. Canto e Castro, E. G. Martins, C. Rocha, M. F.Oliveira, M. M. Leal, and F. Rosado, eds.). Edições SPE, Lisboa, p. 229-238.
Braumann, C. A. (1999). Variable effort fishing models in random environments, Math.
Biosc. 156: 1-19.
Braumann, C. A. (2002). Variable effort harvesting models in random environments:
References (cont.)
Braumann, C. A. (2007). Harvesting in a random environment: It\^o or Stratonovich
calculus. J. Theor. Biol. 244: 424-432.
Beddington, J. R. and May, R. M. (1977). Harvesting natural populations in a randomly
fluctuating environment. Science 197: 463-465.
Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications. Wiley, New
York.
Braumann, C. A. (1983). Population growth in random environments. Bull. Math. Biol. 45:
635-641.
Braumann, C. A. (2007). Population growth in a random environment: which stochastic
calculus? Bull. International Statistical Institute LVI
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic
Press, New York.
Ghiman, I. I. and Skorohod, A. V. (1991). Stochastic Differential Equations. Springer,
Berlin.
Lungu, E. and Oksendal, B. (1997). Optimal harvesting from a population in a stochastic
crowded environment. Math. Biosc. 145: 47-75.
Alvarez, L. H. R. and Shepp, L. A. (1997). Optimal harvesting in stochastically fluctuating
populations. J. Math. Biol. 37: 155-177.
Alvarez, L. H. R. (2000). On the option interpretation of rational harvesting planning. J.
Math. Biol. 40: 383-405.
Braumann, C. A. (1985). Stochastic differential equation models of fisheries in an uncertain
world: extinction probabilities, optimal fishing effort, and parameter estimation. In
Mathematics in Biology and Medicine (Capasso, V., Grosso, E., and Paveri-Fontana, L. S.,
