Available at: http://publications.ictp.it IC/2008/046 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS EVOLUTIONARY FORMALISM FROM RANDOM LESLIE MATRICES IN BIOLOGY Manuel O. C´ aceres 1 Centro Atomico Bariloche, Instituto Balseiro, Universidad Nacional de Cuyo, CNEA, CONICET, Bariloche, 8400, Argentina and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and Iris C´ aceres-Saez CRUB, Universidad Nacional del Comahue, 8400, Bariloche, Argentina. Abstract We present a perturbative formalism to deal with linear random matrix difference equations. We generalize the concept of the population growth rate when a Leslie matrix has random elements (i.e., characterizing the disorder in the vital parameters). The dominant eigenvalue of which defines the asymptotic dynamics of the mean value population vector state, is presented as the effective growth rate of a random Leslie model. This eigenvalue is calculated from the largest positive root of a secular polynomial. Analytical (exact and perturbative calculations) results are presented for several models of disorder. A 3x3 numerical example is applied to study the effective growth rate characterizing the long-time dynamics of a population biological case: the Tursiops sp. MIRAMARE – TRIESTE July 2008 1 Senior Associate of ICTP. [email protected]
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Available at: http://publications.ictp.it IC/2008/046
United Nations Educational, Scientific and Cultural Organizationand
International Atomic Energy Agency
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
EVOLUTIONARY FORMALISM FROM RANDOM
LESLIE MATRICES IN BIOLOGY
Manuel O. Caceres1
Centro Atomico Bariloche, Instituto Balseiro, Universidad Nacional de Cuyo,CNEA, CONICET, Bariloche, 8400, Argentina
andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
and
Iris Caceres-SaezCRUB, Universidad Nacional del Comahue, 8400, Bariloche, Argentina.
Abstract
We present a perturbative formalism to deal with linear random matrix difference equations.
We generalize the concept of the population growth rate when a Leslie matrix has random elements
(i.e., characterizing the disorder in the vital parameters). The dominant eigenvalue of which
defines the asymptotic dynamics of the mean value population vector state, is presented as the
effective growth rate of a random Leslie model. This eigenvalue is calculated from the largest
positive root of a secular polynomial. Analytical (exact and perturbative calculations) results
are presented for several models of disorder. A 3x3 numerical example is applied to study the
effective growth rate characterizing the long-time dynamics of a population biological case: the
1.1 Population growth in a time-fluctuating environment
The effects of a randomly fluctuating environment on the population growth have been studied
since a long time ago. These models go back to non-age structured descriptions where the
fluctuating environment may introduce stochastic elements in the mesoscopic net growth rate r.
For example, if the population size is large, n(t) can be treated as a continuous variable, thus a
stochastic continuous model may be well suited to describe the process. In this case assuming
that environmental changes are due to many factors, and are fast compared to the time-scale
of the population growth, it is possible to approximate the randomly fluctuating environment
by a Gaussian white noise. A stochastic form of r may be r = r + σf(t) with 〈f(t)〉 = 0,
〈f(t)f(0)〉 = δ(t), and σ2 measuring the intensity of the fluctuations. Then starting from the
deterministic evolution in the presence of a fluctuating growth rate, the full description of the
stochastic (Markov) problem can be given in terms of the conditional probability density P (n, t |n0, t0) which is governed by a Fokker-Planck equation (van Kampen (1992)).
In order to fix some ideas consider, for example, the logistic model with stochastic changes
in the net growth rate r. If the population size is far from saturation the stochastic evolution
equation is well described by the linear approximation:
dn
dt= (r + σf(t)) n(t). (1)
Therefore it is possible to calculate the variance of the population size var[n(t)] ≡⟨
n(t)2⟩
−〈n(t)〉2
for different models of stochastic sources f(t). In fact, when the noise fluctuation term f(t) is
Gaussian and white the coefficient of variation is given by (Goel et al. (1974))
√
var [n(t)]
〈n(t)〉 =√
(exp (σ2t) − 1). (2)
As t increases, the coefficient of variation is exponentially increasing, and already for a long-time
this linear-approximation cannot describe the mean growth of the population Verhulst’s model.
By introducing a non-linear change of variable (Goel et al. (1974)) in the original full stochastic
logistic equation, a complete analysis can also be done for the case when the noise is non-Gaussian,
for example using the functional technique presented in (Budini et al. (2004)).
A quite different situation appears when the growth rate r is not randomly fluctuating in
time, but has uncertainty due to heterogeneous conditions in the environment. For example,
random conditions can be the result of inferences that the human beings have in the environment,
and thereof indirectly on the vital parameters of a given population problem. Or just random
conditions can be the sampling error in estimating the vital rates. This random (disordered)
situation leads to a much more complex problem than the one posed in equation (1) when dealing
with a fast time-fluctuating environment. In order to show the differences between both problems,
we are going to analyze in the next section a particular random model.
2
1.2 Population growth in a heterogeneous environment
To study of the effects of an heterogeneous environment on the population growth, we begin
introducing the evolution equation for a field single-type population size n(x, t) in a 1−dimension
space. To fix some notation, consider here the logistic model with migration in a heterogeneous
(in space) environment that changes the mesoscopic growth rate r(x) randomly from site to site.
In the case when r =constant the evolution equation for the field n(x, t) is the Fisher equation
(Fisher (1937)), see Appendix 1. In the particular case when the population size is far from the
saturation value and if we introduce a discrete-space and a lattice-Laplace operator, we arrive
to a linear evolution equation for the field population in the lattice: ni(t), see equation (3)
in the Appendix 1. From this equation it is simple to realize that we are now forced to treat a
matrix problem. Related to the mentioned mathematical system, consider now the scalar random
evolution equationdn(t)
dt= (r + β)n(t), (3)
where β is an arbitrary random variable characterized by its probability P(β). Even when this
equation looks similar to (1), its mathematical meaning is quite different because (3) corresponds
to the case when the noise in (1) has an infinite correlation in time. From (3) we get for the
Note that by construction 〈B〉 = 0, and H is a sure Leslie’s matrix with elements given by:
f∗j = fj − 〈αj〉 , p∗j = pj − 〈βj〉 . (22)
In order to calculate the average of the Green function 〈G(z)〉 we need to find its evolution equa-
tion, this can be done by using a projector operator technique, see for example (Hernandez et al.
(1990b), Caceres et al. (1997)). The average of G(z), i.e. averaging over the random variables
9
αj , βj, can formally be carried out introducing the projector operator P that averages over the
disorder, and its complement projector Q ≡ (1 − P), i.e.:
〈G(z)〉 = PG(z), G(z) = PG(z) + QG(z).
Using this projector technique a close exact evolution equation can be obtained. Applying
the operator P to Eq. (20) we obtain
1
z[PG(z) − 1] = HPG(z) + PBPG(z) + PBQG(z). (23)
Also, applying the operator Q to Eq. (20) we obtain
1
zQG(z) = HQG(z) + QBPG(z) + QBQG(z) (24)
A formal solution of Eq. (24) can be obtained using the non-disordered Green matrix:
G0 ≡[
1
z1− H
]−1
. (25)
Applying G0(z) to Eq.(24) and using the definition given in Eq.(25), results in
QG(z) = G0 [QBPG(z) + QBQG(z)] . (26)
This equation can iteratively be solved for QG(z),
QG(z) =
∞∑
k=0
[
G0QB]k PG(z). (27)
Putting this formal solution in Eq. (23) we find a close exact equation for the average of the
Green function PG(z),
PG(z) − 1 = z
[
HPG(z) + PBPG(z) + PB
∞∑
k=0
[
G0QB]k PG(z)
]
. (28)
This equation can be rewritten in a more friendly way
〈G(z)〉 =
[
1 − z
(
H +
⟨ ∞∑
k=0
[
BG0Q]k
B
⟩)]−1
. (29)
Here we can see the non-trivial structure that the average Green function obtained as a conse-
quence of its evolution in time.
We remark that even in the case when the random Leslie matrix M is of dimension m,
the number of z−poles in 〈G(z)〉 will depend on the numbers of non-null contributions from the
series expansion appearing in (29). From this solution we can easily demonstrate that the “naive”
approximation: 〈G(z)〉 ≃ [1− zH]−1 corresponds to neglecting all “cumulant contributions” with
k ≥ 1. As a matter of fact, each cumulant represents a particular structure of correlation that
we need to evaluate carefully.
10
Remark. The important task is to calculate the different k−contributions from the object
⟨ ∞∑
k=0
[
BG0Q]k
B
⟩
, (30)
as a function of z for a given model of disorder. In fact, we will prove that the operator (30)
can be studied in terms of statistical objects called Terwiel’s cumulants, that will be defined later
(Terwiel (1974)), see Appendix 3. In particular, if the intensity of the random variables αj , βjcan be considered as a small parameter, we can analyze the behavior of the dominant pole of the
averaged Green function (29), order-by-order to any contribution that comes from the different
k in Eq. (30). By virtue of the Tauberian theorem the long-time behavior of the averaged Green
function will be dominated by the smallest strictly positive root z1 of
det
∣
∣
∣
∣
∣
1− z
(
H +
⟨ ∞∑
k=0
[
BG0Q]k
B
⟩)∣
∣
∣
∣
∣
= 0. (31)
Remark. We conclude that the stability of the mean-value population vector state shall be
characterized as
limn→∞
〈Xn〉 ∼(
1
z1
)n
. (32)
This formula generalizes (15) in the case when the dynamics are characterized by a random Leslie
matrix. Note that if the pole z1 were degenerated we still can apply the Tauberian theorem and,
of course, a different asymptotic behavior for the growth of the averaged population vector state
would be obtained. In Appendix 4 we present an example of stability analysis for a particular
random survival model in a general m × m Leslie matrix.
4 An exact 2 × 2 soluble case
Consider a 2 × 2 Leslie matrix where the fertility of the sub-class 2 has a random element of
the form f2 − α2, then following the previous sections we see that the problem is completely
characterized by defining the matrices:
G0 =
[
1
z1− H
]−1
, H =
(
f1 f∗2
p1 0
)
, B =
(
0 ξ2
0 0
)
, (33)
where ξ2 = 〈α2〉 − α2, f∗2 = f2 − 〈α2〉. From (33) we can calculate the Terwiel operator (30). We
get for every k⟨
[
BG0Q]k
B⟩
=
(
0 gk21
⟨
[ξ2Q]k ξ2
⟩
0 0
)
,
here, as before, gjl are the matrix elements of the ordered Green function G0. Summing all
contributions k we obtain⟨ ∞∑
k=0
[
BG0Q]k
B
⟩
=
(
0∑∞
k=0 gk21
⟨
[ξ2Q]k ξ2
⟩
0 0
)
. (34)
11
Then, we have proved that for this 2 × 2 case and for any statistics of the random variables ξ2,
we only have to calculate the statistical object
⟨
[ξ2Q]k ξ2
⟩
, k = 1, 2, 3, · · · . (35)
As we have remarked before, these are in fact Terwiel’s cumulants, see Appendix 3.
4.1 Binary disorder in the fertility
In order to continue the analysis of our model (33), suppose now that the random variable α2
can only take two discrete values ±∆, i.e.,
α2 =
∆−∆
with probability cwith probability (1 − c)
. (36)
In order to assure that random fertility f2 − α2 is a positive quantity for each sample of the
disorder, we have to assume that 0 ≤ ∆ ≤ f2. From (36) it is simple to see that
⟨
α2q+1
2
⟩
= ∆2q+1 (2c − 1) ;⟨
α2q2
⟩
= ∆2q; q = 1, 2, 3, · · · . (37)
Then, it is also possible to prove that Terwiel’s cumulants of the random variable ξ2 = 〈α2〉 −α2
are⟨
[ξ2Q]k ξ2
⟩
= ∆k+1c (1 − c) (2c − 1)k−1 2k+1, k = 1, 2, 3, · · · . (38)
From this result we get the important conclusion that for a symmetric binary random perturbation
(i.e., with c = 1/2) all Terwiel’s cumulants vanish for k ≥ 2. Then in the symmetric case the
only non-null Terwiel’s cumulant appearing in (35) will be 〈ξ2Qξ2〉 = ∆2. In order to remark the
difference between Terwiel’s cumulant with the simple cumulants, we write here the formula for
the usual cumulants corresponding to the random variable ξ2; using (36) for the symmetric case,
i.e., ξ2 = α2 (when c = 1/2) we get
⟨⟨
ξ2q2
⟩⟩
=−22q−1(22q − 1)Bq
i2qq∆2q, q = 1, 2, 3, · · · ,
where Bq are the Bernoulli numbers: Bq = 1/6, 1/30, 1/42, · · · . This result shows, for the
symmetric binary case, the simplicity of Terwiel’s cumulants against the usual ones.
4.1.1 The symmetric binary case
From all these previous facts we see that for this 2× 2 case we can write the exact solution of the
averaged Green function. From model (33) with a symmetric binary random variable, using the
general expression (29) and noting that f∗2 = f2 we get
〈G(z)〉 =
[
1− z
(
H +
⟨ ∞∑
k=0
[
BG0Q]k
B
⟩)]−1
(39)
=
[
1− z
(
f1 f2 + g21∆2
p1 0
)]−1
,
12
here
g21 =p1z
2
1 − f1z − p1f2z2=
p1
(1/z)2 − (1/z) f1 − p1f2
=p1
(
1z − λ1
) (
1z − λ2
) ,
where λ1,2 are the eigenvalues of the sure matrix H, in the case when c = 1/2 these eigenvalues
coincide with the 2 × 2 non-random Leslie matrix M, see (20) and (33), i.e.,
λ1,2 =1
2
(
f1 ±√
f21 + 4p1f2
)
. (40)
In order to find the dominant pole of 〈G(z)〉 we study (39) introducing the notation z = 1/λ,
then we have to solve the roots of
(
λ2 − λf1 − f2p1
)
=(p1∆)2
(λ2 − λf1 − f2p1).
This equation implies fourth roots (we adopt 0 ≤ ∆ ≤ f2 to assure the positivity of the Perron-
Frobenius eigenvector Ψ1 for each sample of the disorder), then
λ1,2 =1
2
(
f1 ±√
f21 + 4p1 (f2 + ∆)
)
λ3,4 =1
2
(
f1 ±√
f21 + 4p1 (f2 − ∆)
)
.
It is clear now that the largest positive one is
λ1 =1
2
[
f1 +√
f21 + 4p1 (f2 + ∆)
]
. (41)
As we mentioned before this effective eigenvalue is different from the average of λ1.
Remark. The effective finite growth rate of the disordered Leslie model (33) with a symmetric
binary random perturbation α2 is characterized by λ1. This exact result shows, by using the
Tauberian theorem, that the average of the population grows faster than in the ordered case
(without a random element in the fertility f2), i.e.,
limn→∞
〈Xn〉 ∼(
1
z1
)n
=
(
1
2
[
f1 +√
f21 + 4p1 (f2 + ∆)
])n
, (42)
where ∆2 is the dispersion of α2 (see (37)). An equivalent analysis can also be carried out by
putting a random element in the survival parameter p1.
Now we show another exact result for the effective finite growth rate, but in the case of having
a symmetric random perturbation α1 in the fertility parameter f1 → f1 − α1. As in (33) the
problem is now defined by considering
G0 =
[
1
z1− H
]−1
, H =
(
f1 f2
p1 0
)
, B =
(
ξ1 00 0
)
, (43)
13
where ξ1 = −α1, f∗1 = f1 adopting a symmetric binary random variable for α1. The exact averaged
Green function now looks like
〈G(z)〉 =
[
1− z
(
H +
⟨ ∞∑
k=0
[
BG0Q]k
B
⟩)]−1
(44)
=
[
1− z
(
f1 + g11∆2 f2
p1 0
)]−1
,
where
g11 =p1/z
(
1z − λ1
) (
1z − λ2
) ,
and as before λ1,2 are the eigenvalues of the sure matrix H, see Eq. (40). From the poles of Eq.
(44) we immediately get that the dominant (smallest positive) pole z1 is (adopting 0 ≤ ∆ ≤ f1)
characterized by the largest positive eigenvalue
λ1 =1
2
[
f1 + ∆ +
√
(f1 + ∆)2 + 4p1f2
]
. (45)
This exact result shows that also from the model (43), the average of the population grows faster
than in the ordered case. In this case the population vector state grows as
limn→∞
〈Xn〉 ∼(
1
z1
)n
=
(
1
2
[
f1 + ∆ +
√
(f1 + ∆)2 + 4p1f2
])n
. (46)
It is important to mention that the convexity of the effective growth rate λ1 (45) as a
function of the random intensity ∆, is different when compared with the previous case (41).
Nevertheless, in both cases the effective eigenvalue λ1 is larger than in the non-random case
λ1 = 12
(
f1 +√
f21 + 4p1f2
)
. In order to quantify this comment we can take the derivative of λ1
with respect to the strength ∆ and evaluate dλ1/d∆ at ∆ = 0. In this form we can measure
the variation of the effective eigenvalue to a small random perturbation and prove that if the
perturbation is symmetric the effective eigenvalue λ1 is always larger than in the non-random
case.
For a symmetric binary random perturbation in the fertility f2, i.e., from (41) we get
λ1 ≃ λ1 +p1
√
f21 + 4p1f2
∆. (47)
But for a symmetric binary random perturbation in the fertility f1, i.e., from (45) we get
λ1 ≃ λ1 +1
2
(
1 +f1
√
f21 + 4p1f2
)
∆. (48)
These simple but interesting results can be of great utility in modeling biological population
growth, for example, using fixed (mean values) Leslie vital parameters, it may occur that λ1 < 1.
Nevertheless considering symmetric fluctuations (sampling error in estimating the vital rates) we
could get λ1 larger than 1, and in this way predict an increasing population.
14
One last remark concerning our 2 × 2 model: suppose now that random elements appear in
both fertilities f1, f2, or simultaneously in the three Leslie vital parameters f1, f2, p1. Then, it is
possible to see that even if we would have used the statistical independent assumption for the set
ξ1, ξ2, η1 the Terwiel operator⟨ ∞∑
k=0
[
BG0Q]k
B
⟩
, (49)
would not cut in the second Terwiel’s cumulant! This is due to the occurrence of a higher
order non–trivial Terwiel’s structure between the different random variables. For example, in the
presence of random elements in both fertilities f1, f2, it is possible to see that apart from the
simplest second order contribution:⟨
BG0QB⟩
, higher order statistical contributions come from
non-null Terwiel’s cumulants like:
〈ξ1Qξ2Qξ1Qξ2〉 ; 〈ξ1Qξ2Qξ2Qξ1〉 ;
〈ξ1Qξ2Qξ2Qξ2Qξ2Qξ1〉 ; 〈ξ1Qξ2Qξ1Qξ2Qξ1Qξ2〉 ; etc.
These cumulants lead to the occurrence of a non-trivial structure in the calculation of the domi-
nant pole of the mean-value Green function.
Remark. Note that even in the case when the random variables ξ1, ξ2 are statistical independent
these cumulants do not cancel. Terwiel’s cumulants can easily be evaluated using diagrams, but
we will leave this discussion for a future contribution, see Appendix 3 for details. In order to
calculate the averaged Green function we have to introduce a criterion to cut the Terwiel cumulant
series. A possible one is to invoke an expansion in the intensity of the random perturbation. For
example, if ∆ is a small parameter it is clear that higher Terwiel’s cumulants are of lower order,
then we can approximate (49) up to some O (∆q) in order to calculate the mean-value Green
function. From this approximated (truncated) function 〈G(z)〉 we can estimate the effective
finite growth rate of the mean-value population vector state. An example in that direction will
be shown in the next sections where we consider a 3×3 Leslie matrix in the presence of uniformly
distributed random variables perturbing all the survival rates.
5 Application to a 3 × 3 random survival model
In this section we are going to consider a 3× 3 Leslie matrix with statistical independent random
elements. In particular we assume that the uncertainties are located in the survival parameters
p1 and p2, then we use a 3 × 3 random perturbation matrix B like in Appendix 4. As in the
previous 2 × 2 example, from (31) we see that we need to study the Terwiel operator
⟨ ∞∑
k=0
[
BG0Q]k
B
⟩
, (50)
15
now associated to the matrices:
G0 =
[
1
z1 −H
]−1
, H =
0 f2 f3
p∗1 0 00 p∗2 0
, B =
0 0 0η1 0 00 η2 0
, (51)
where p∗j ≡ pj − 〈βj〉 and ηj ≡ 〈βj〉 − βj . Given any statistics for the random variables βj,from (51) we could analyze the exact behavior of the averaged Green function 〈G(z)〉 if we were
able to sum all Terwiel’s cumulants appearing in (50). Note that if we were interested in random
variables βj which were not statistical independent, we need to know the joint probability of
the set βj.
5.1 Uniformly distributed disorder in the survival parameters
In order to clarify this example, let us analyze the particular situation when the distribution of
the random variables βj are uniform in the interval [0, Fj ]. But, of course, we can use any other
distribution in our approach
P(βj) = 1/Fj , with 0 < Fj < pj. (52)
Note that we have excluded the situation when Fj = pj because, in that case, the random survival
parameter pj − βj would have a finite probability to be null, this situation is very extreme from
a biological point of view and will not be analyzed here. From (52) it is simple to calculate the
moments of βj , and so all the moments associated to the random variables ηj = 〈βj〉 − βj
⟨
βqj
⟩
=1
Fj
∫ Fj
0
βqj dβj =
F qj
q + 1, q = 1, 2, · · · (53)
⟨
η2qj
⟩
≡⟨
(〈βj〉 − βj)2q⟩
=1
2q + 1
(
Fj
2
)2q
,⟨
η2q+1
j
⟩
= 0.
From these results it is now clear that higher moments are less important.
As we mentioned before, the calculation of Terwiel’s cumulants are not so straightforward,
but using the partition property mentioned in Appendix 3 and the fact that⟨
η2q+1
j
⟩
= 0, from
(53), we can prove that Terwiel’s cumulants simplify considerably. In the present case, i.e., using
the uniform distribution (52), odd moments are null then also their odd Terwiel’s cumulants, this
fact simplifies even more the calculation of (50), see Appendix 3 for details.
glm are the elements of the ordered Green function G0, see (51), i.e.,
G0 =
1/z −f2 −f3
−p∗1 1/z 00 −p∗2 1/z
−1
(56)
=1
(
1z − λ∗
1
) (
1z − λ∗
2
) (
1z − λ∗
3
)
1/z2 (f3p∗2 + f2/z) f3/z
p∗1/z 1/z2 f3p∗1
p∗1p∗2 p∗2/z
(
−f2p∗1 + 1/z2
)
,
where λ∗j are the eigenvalues of the sure matrix H appearing in (51). Denoting Θ ≡ f3p
∗1p
∗2 =
detH 6= 0 and Ω ≡ f2p∗1 we can write
λ∗1 =
(
2
3
)1/3
Ω(
9Θ +√
3√
27Θ2 − 4Ω3
)−1/3
(57)
+1
21/332/3
(
9Θ +√
3√
27Θ2 − 4Ω3
)1/3
λ∗2 =
−Ω(
1 + i√
3)
21/332/3
(
9Θ +√
3√
27Θ2 − 4Ω3
)−1/3
−Ω(
1 − i√
3)
2 21/332/3
(
9Θ +√
3√
27Θ2 − 4Ω3
)1/3
,
with λ∗3 the complex conjugated of λ∗
2. Note that here we are preserving the notation p∗j because
〈βj〉 6= 0.
Now we calculate the next contribution for the 3 × 3 model given in (51). From (54) we get
that the only non-null components of (50) up to O(
B4)
are
⟨
BG0QBG0QBG0QB⟩∣
∣
21= g3
12 〈η1Qη1Qη1Qη1〉 (58)
+g13g23g22 〈η1Qη2Qη2Qη1〉⟨
BG0QBG0QBG0QB⟩∣
∣
31= g2
22g13 〈η2Qη1Qη2Qη1〉⟨
BG0QBG0QBG0QB⟩∣
∣
22= g22g
213 〈η1Qη2Qη1Qη2〉
⟨
BG0QBG0QBG0QB⟩∣
∣
32= g3
23 〈η2Qη2Qη2Qη2〉+g22g12g13 〈η2Qη1Qη1Qη2〉 .
We remark that this general result is supported only by the fact that the random variables
ηj are statistical independent (partition property of Terwiel’s cumulant) and that⟨
η2q+1
j
⟩
= 0.
We see that up to O(
B4)
we only have to calculate a few fourth-order Terwiel’s cumulants of
the forms:
〈η1Qη1Qη1Qη1〉 ; 〈η1Qη2Qη2Qη1〉 ; 〈η2Qη1Qη2Qη1〉 ; etc.
17
In general, the Terwiel cumulants that appear in (58) belong to a class that can easily be drawn
using diagrams (see Hernandez et al. (1989)). The Terwiel cumulants that we need to evaluate
in (58) can just be obtained from (54). For the particular case of the uniform distribution, see
(53), we get for Terwiel cumulants, at the same point,
〈ηjQηjQηjQηj〉 =⟨
η4j
⟩
−⟨
η2j
⟩ ⟨
η2j
⟩
=1
4 + 1
(
Fj
2
)4
−(
1
2 + 1
(
Fj
2
)2)2
=F 4
j
180, j = 1, 2,
and for a couple of points we have
〈ηnQηjQηjQηn〉 = 〈ηjQηnQηjQηn〉 =⟨
η2j
⟩ ⟨
η2n
⟩
=1
144F 2
j F 2n , j 6= n = 1, 2.
Summing up all the contributions to O(
B4)
we get
⟨
[
BG0Q]3
B⟩
(59)
=
0 0 0(
〈1111〉T g312 + 〈1221〉T g13g23g22
)
〈1212〉T g22g213 0
〈2121〉T g13g222
(
〈2222〉T g323 + 〈2112〉T g13g12g22
)
0
,
where we have used an obvious short notation for Terwiel’s cumulants 〈nlpq〉T ≡ 〈ηnQηlQηpQηq〉.It is interesting to note that if we had used symmetric binary statistical independent random
variables βj, there would not have been a great simplification in the expression (59). Non-null
Terweil’s cumulants of the form
〈ηnQηjQηjQηn〉 , 〈ηnQηjQηnQηj〉 , j 6= n = 1, 2
will always appear, and then there would not be a great simplification in getting an analytical
formula for the pole z1. This is the reason why we introduce in this section an example using
uniform distributed random variables. If we want to analyze the behavior of the effective growth
rate up to O(
B4)
we can find numerically the smallest positive root of
det∣
∣1 − z(
H +⟨
BG0QB⟩
+⟨
BG0QBG0QBG0QB⟩)∣
∣ = 0, (60)
18
which in the present case leads to the analysis of the roots of the following polynomial
0 = 1 − z2f2
(
p∗1 +F 2
1
12g12 +
1
180F 4
1 g312 +
1
144F 2
1 F 22 g13g23g22
)
(61)
−z3f3
(
p∗1 +F 2
1
12g12 +
1
180F 4
1 g312 +
1
144F 2
1 F 22 g13g23g22
)
×(
p∗2 +F 2
2
12g23 +
1
180F 4
2 g323 +
1
144F 2
1 F 22 g13g12g22
)
−z
(
1
144F 2
1 F 22 g22g
213
)
− z2f3
(
1
144F 2
1 F 22 g13g
222
)
+z3f3
(
1
144F 2
1 F 22 g22g
213
)(
1
144F 2
1 F 22 g13g
222
)
,
the elements gjl are functions of z, see (56).
Note. If we use only one symmetric binary random variable β affecting both survival Leslie
parameters p1, p2. Higher order Terwiel’s cumulant vanishes (as can be seen from (38)), thus (55)
would be the only contribution to the averaged Green function (this case would be a sort of global
disorder model, which may be of interest in biology for some particular cases).
5.1.1 Analytic approximation for the effective growth rate 1/z1
Even when expression (61) looks very complicated it is still possible to get an analytical formula
for the smallest positive root z1, if we introduce a simple perturbation analysis. Using, as in the
previous 2 × 2 example, the transformation λ = 1/z, it is possible to study the largest positive
eigenvalue λ1 by introducing a perturbation around the value λ∗1; here λ∗
1 6= λ1 because λ∗1 is the
eigenvalue of H considering that 〈βl〉 6= 0, see (51). Note that the value of the growth rate in the
non-random case λ1 can be read from λ∗1 by replacing p∗j → pj in (57).
We define a small quantity ǫ in the form:
λ1 = λ∗1 + ǫ + · · · . (62)
In principle ǫ is positive or negative indicating that the effective eigenvalue λ1 could be larger or
smaller than λ∗1. The polynomial given in (61) simplifies considerably if we keep only contributions
up to O(
F 2j
)
, then we get
(
λ − λ∗1
)2 (
λ − λ∗2
)2 (
λ − λ∗3
)2
=F 2
1 f22
12
(
λ − λ+
)(
λ − λ−)
, (63)
where λ∗j , j = 1, 2, 3 are given in (57) and
λ± =f3p
∗2
f2
(
−1 ± ip∗1F2
p∗2F1
)
. (64)
In order to find an analytical solution for the largest positive root of (63) we assume that if F 2j
are small quantities, then the value λ1 is not so different from λ∗1 and in this form we can solve
(63) using the small perturbation ǫ introduced in Eq. (62). Therefore from (63) we get
ǫ2 (λ∗1 + ǫ − λ∗
2)2 (λ∗
1 + ǫ − λ∗3)
2 =F 2
1 f22
12(λ∗
1 + ǫ − λ+) (λ∗1 + ǫ − λ−) . (65)
19
We can solve this equation consistently up to the order ǫ. Thus we get the non-trivial result
ǫ ≃ F1f2√12
|λ∗1 − λ+|
|λ∗1 − λ∗
2|2> 0. (66)
If ǫ is a small quantity this analytical formula gives the result we were looking for. Note that
depending on the magnitude of ǫ against λ∗1 − λ1, the effective growth rate
λ1 ≃ λ∗1 +
F1f2√12
|λ∗1 − λ+|
|λ∗1 − λ∗
2|2+ · · · , (67)
will be much more smaller or not than the non-random growth rate λ1. This formula shows an
explicit non-trivial dependence between the fluctuations in the survival parameter p1 and the
magnitude of the fertility f2 (see model (51) with the probability distribution (52))
We remark that Eq. (63) is exact up to O(
B2)
. If we need to evaluate the effective growth
rate with more accuracy, we can solve numerically the roots of the polynomial (61).
The following summary is very useful in order to compare our theoretical predictions. For a
sure 3 × 3 Leslie matrix M, the growth rate is (from Eq. (57))
λ1 = λ1(Θ,Ω); with Θ ≡ f3p1p2, Ω ≡ f2p1.
Consider now perturbations in all the survival parameters pj → pj − βj , with 〈βj〉 6= 0. Then,
noting that p∗j ≡ pj − 〈βj〉 we get from Eqs. (53), (57) and (64) the significative table
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
Naiveapprox.
H =
0 f2 f3
p∗1 0 00 p∗2 0
→
λ∗1 = λ∗
1(Θ,Ω) withΘ ≡ f3p
∗1p
∗2, Ω ≡ f2p
∗1
To O(
B4)
dominant pole
solve numericallyz1 from Eq.(61)
→ λ1 = λ1(f3, f2, p∗1, p
∗2)
To O(
B2)
analytic approx.
λ1 ≃ λ∗1 + ǫ + O
(
B4)
→ ǫ ≃ F1f2√12
|λ∗
1−λ+|
|λ∗
1−λ∗
2|2+ · · ·
To O(
B2)
non statist.independ.assump.
Use Eq. (31) Ap-4 &Eq.(60) up to O
(
B2)
solve numerically z1
→ λ1 = λ1(f3, f2, p∗1, p
∗2)
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
.
(68)
We see that the naive approximation λ∗1 = λ∗
1(Θ,Ω) is far away from the more accurate description
given in terms of our effective growth rate λ1. In fact, from this table it is easy to see that if
〈βl〉 = 0, ∀l, the naive approximation gives the same value as the sure Leslie matrix. Nevertheless,
our theory predicts that symmetric fluctuations will lead to new critical scenarios.
Note. If we had used two statistical independent binary symmetric random variables ±∆1,±∆2perturbing the survival parameters p1, p2 (with 0 < pj − ∆j and pj + ∆j ≤ 1), ǫ would have
been
ǫ ≃ ∆1f2
|λ1 − λ+||λ1 − λ2|2
,
20
then, the final expression for the effective (analytic approximation) growth rate up to O(
B2)
would be
λ1 ≃ λ1 + ∆1f2
|λ1 − λ+||λ1 − λ2|2
, (69)
where
λ+ =f3p2
f2
(
−1 + ip1∆2
p2∆1
)
. (70)
Note that in formula (69) we have written λj because if the random variables are symmetric we
get p∗j = pj. Expression (69) is considerably much more complex when compared to the 2 × 2
case with only one symmetric random variable, as was presented in the examples (47) and (48)
perturbing the fertility parameters f1 and f2, respectively. We note that the complexity in (69)
is due to the occurrence of two random variables, and of course, due to the larger dimensionality
of Leslie’s matrix.
6 Random Leslie numerical example
Consider a closed, single-sex population model with three age classes (calves, juveniles and adults).
Here we describe an example based on heuristics, but not biologically implausible, numbers. As
a matter of fact, we got the numbers for the vital parameters from a recent study on female
reproductive success in bottlenose dolphins (Tursiops sp.), see Mann et al. (1999). In that work
the authors examined whether factors affecting predation or food availability, water depth, and
group size, were related to female reproductive success; also calf survivorship from birth to age 3
were analyzed. Infanticide, female visibility and distribution of prey and predator may also alter
the survival parameters. From those works, and from similar female reproductive researches, see
for example Berta et al. (2005), Rayen (2005), it is not difficult to realize the error introduced
in estimating vital parameters for a given specie in study, that is why we are going to consider
the effect of disorder in the handling of those numbers. Reproduction is moderately seasonal,
and survivorship strongly depends on calf age, showing a stable value between the age of 2 and
3. Modeling the fertility parameters as sure values, we are going to analyze a case when only the
survival parameters are uncertain. Therefore from Mann et al. (1999), we can estimate that our
3 × 3 population dynamics model is subject to a Leslie matrix characterized by the elements
M =
0 1 53
4− β1 0 00 3
5− β2 0
, (71)
where βj are random numbers. From our approach we can consider many possibilities for modeling
these random numbers.
21
6.1 Non-symmetric uniformly distributed disorder
An interesting possibility will be to consider that the survival parameters are always randomly
reduced by environmental circumstances, then we can assume that β1 is uniformly distributed in
the interval 0 < β1 < F1 = 9/20. On the other hand, because the survival for the juvenile (the
second age class) are not well known it is reasonable to assume that β2 (independently form β1)
runs from values similar to calves at age 3, and are uniformly distributed in the interval 0 < β2 <
F2 = 3/10. Using Eq. (53),⟨
βqj
⟩
= F qj /(q+1), it is simple to calculate the variance associated to
these random variables. In this case we get the small dispersions:⟨
η21
⟩
=⟨
β21
⟩
−〈β1〉2 = 0.016875
and⟨
η22
⟩
=⟨
β22
⟩
− 〈β2〉2 = 0.0075.
Note that the random Leslie matrix (71) can show for each sample of the disorder a continuous
variability in the behavior of the population dynamics, running from extinction to grow rapidly
depending on the values of the set of random variables βj. Therefore it is extremely important
to know whether the average over the disorder will predict an extinction or not in the population.
In the previous sections we have shown that the answers to this question can be measured by
calculating the effective eigenvalue λ1. From our expansion, up to O(B2), we can use formula
(67) to calculate analytically λ1.
From (71) we get the eigenvalues of the associated non-random Leslie matrix
λ1 = 1.5 (72)
λ2,3 = −0.75 ± i0.968246.
Using that 〈βj〉 = Fj/2 we calculate p∗j and so the eigenvalues of H = 〈M〉, using (57) we write
λ∗1 = 1.2215 (73)
λ∗2,3 = −0.6107 ± i0.7707.
From a physical point of view λ1 is the non-random value of the finite growth rate, and λ∗1 the
first naive correction, just considering the substration of the mean-values 〈βj〉 to each survival
parameters pj, see the table (68).
Using the values of p∗j and fj in (64) we obtain
λ± =1
4(−9 ± i7) ,
From (66) we get
ǫ ≃ F1f2√12
|λ∗1 − λ+|
|λ∗1 − λ∗
2|2= 0.184497.
Thus, the analytical effective growth rate gives
λ1 = λ∗1 + ǫ + O
(
ǫ2)
≃ 1.40.
22
Remark. It is interesting to compare the non-random value λ1 = 1.5 against the naive
expectation value λ∗1 = 1.2215, and the effective growth rate λ1 ≃ 1.40. From these results we
see that even when for each sample of the disorder the vital parameters are reduced, fluctuations
enlarge the value of λ1 with respect to the trivial average λ∗1. The message from this result is
that from mean-value environmental vital parameter values it could happen that we get λ∗1 <
1, therefore “predicting” the extinction of the population. However, the important point is
that taking into account the average over the fluctuations (i.e., our mean-value Green function
technique) it may result that the fluctuations drive the effective growth rate λ1 to a value larger
than 1, therefore restoring the expectation for a growing (stable) mean-value population vector
state.
6.2 Symmetric discrete disorder (analysis of the different cumulant contribu-tions)
Another interesting possibility to analyze here is when the survival parameters fluctuates sym-
metrically around some specific values. Therefore we can assume that βj are binary random
variables with mean-value zero. As we mention in the previous sections this situation can also
be tackled with our approach using (69) to calculate analytically λ1. Just in order to show the
quantitative difference with the previous analysis we assume here that the values are ∆1 = 0.25
and ∆2 = 0.3, and for the non-random vital parameters pj, fj we use the same values as before,
see (71) (with this value of ∆j we always fulfil for each sample of the disorder the condition
0 < pj − βj ≤ 1). Note that for symmetric binary fluctuations⟨
β2qj
⟩
= ∆2q, then it is simple
to see that the dispersions associated to these random variables are:⟨
η21
⟩
=⟨
β21
⟩
= 0.0625 and⟨
η22
⟩
=⟨
β22
⟩
= 0.09, which, in fact, are much more larger than in the previous uniformly dis-
tributed case. The values of λj are as given before in (72), and the values of λ± are now from
(70)
λ± = −3 ± i9
2. (74)
If we want to use the analytical approximation Eq. (69) for the effective eigenvalue λ1 we first
evaluate ǫ
ǫ = ∆1f2
|λ1 − λ+||λ1 − λ2|2
=3
8√
2= 0.26515.
This ǫ is not really a small number so our analytical expression for λ1 should be handled with
care. Just in order to see how good this approximation is let us write λ1, from (69) we get
λ1 ≃ λ1 + ǫ = 1.76 + O(
ǫ2)
. (75)
We want to compare this analytical result with the numerical evaluation of λ1. Up to O(
B2)
the value for the effective growth rate λ1 can be found by solving a polynomial analogous to (61)
but for binary random variables, using that 〈ηjQηj〉 = ∆2j the polynomial reads
0 = 1 − z2f2
(
p1 + ∆21g12
)
− z3f3
(
p1 + ∆21g12
) (
p2 + ∆22g23
)
, (76)
23
where gjm are given from (56) changing p∗j → pj and λ∗j → λj (because here 〈βj〉 = 0).
This equation neglects terms of O(
B4)
, which in the present case means a maximum error of the
order max(⟨
η4j
⟩)
≃ 0.008. Solving numerically the roots of this equation we find six roots, and
the largest positive one is
λ1 = 1.814 + O(
B4)
. (77)
This result shows that our analytical approximation (75) is quite good even in this disfavored
case when ǫ is not too small.
On the other hand, up to O(
B6)
the value for the effective growth rate can be found by
solving the corresponding secular polynomial but taking into account all fourth-order Terwiel
cumulants. From (58) this polynomial can easily be written noting that 〈ηlQηjQηkQηm〉 are
almost all nulls. The only non-null fourth-order Terwiel’s cumulants are