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

Tursiops sp.

MIRAMARE – TRIESTE

July 2008

1Senior Associate of ICTP. caceres@cab.cnea.gov.ar

1 Introduction

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

moments of n(t), (see Appendix 1 for details)

〈n(t)m〉 = n(0)m 〈exp [m (r + β) t]〉= n(0)m exp (mrt)Gβ (−imt) , (4)

where Gβ (k) is the Fourier transform of the probability P(β). In order to remark the difference

of this result with the one coming from a fast time-fluctuating environment (the noisy model (1)),

we show here five particular cases considering different statistics for the random variable β.

I) In the case when the uncertainty in the mesoscopic growth rate r = r + β is uniformly

distributed in the interval β ∈ [0, F ], the coefficient of variation of the population n(t) will be:

√

var [n(t)]

〈n(t)〉 =

√

tF (exp (2tF ) − 1)

2 (exp (tF ) − 1)2− 1. (5)

II) In the case when the uncertainty in the mesoscopic growth rate r = r+β is Gaussian with

dispersion θ2 and mean-value zero, the coefficient of variation of the population n(t) will be:

√

var [n(t)]

〈n(t)〉 =√

(exp (θ2t2) − 1). (6)

III) In the case when the number β is a binary random variable with values ±∆, mean-value

zero and dispersion ∆2, the coefficient of variation of the population n(t) will be:

√

var [n(t)]

〈n(t)〉 = tanh (t∆) . (7)

3

IV) In the case when β = p/τ and p is an integer random number characterized by a Poisson

probability with mean-value 〈p〉, the coefficient of variation of the population n(t) will be:

√

var [n(t)]

〈n(t)〉 = exp

[

〈p〉(

et/τ − 1)2]

− 1. (8)

V) In the case when the number β is Gamma distributed with mean-value b/c and dispersion

b/c2, the coefficient of variation of the population n(t) will be:

√

var [n(t)]

〈n(t)〉 =

√

(

(c − t)2

(c − 2t)c

)b

− 1; t < c/2. (9)

From these results we see the huge difference that the coefficient of variation has (as a function

of time) depending on the statistics used to characterize the uncertainty in the parameter r. For

example, using a uniform distribution a long-time we get ∝√

t, using a Gaussian statistics we

get ∝ exp(t2), using a binary probability we get a saturation value ∼ 1, using Poisson statistics

we get ∝ exp(et), and using Gamma distribution we get a power-law behavior! From a similar

analysis (see Appendix I), in general, we can prove that the long-time behavior of the mean-

value 〈n(t)〉 in a random problem, has not an exponential growth with the mean growth rate

〈r〉 = r + 1

2

⟨

β2⟩

, in contrast to the result obtained from the (white) noisy model (1). We have

shown this fundamental result using the simplest scalar model (3). Nevertheless, a single-type

population growth system in a heterogeneous environment is a field problem, thus a rigorous

proof considering the full lattice-field ni(t) ought to be done using the matrix structure of the

system. One of the goals of the paper will be to introduce an approach to tackle this type of

problem.

Due to the fact that in this paper we are only interested in the population dynamics of

age-structured colonies, we now turn to the problem of the description of linear (time-discrete)

projection matrix models where the vital parameters may have sampling errors in their values.

This problem will be studied in terms of quite general statistics for the random elements appearing

in the projection matrices (Caswell (1983)). On the other hand, it should be noted that enlarged

projection matrices can also be used to represent a crude migration process between two or more

population collectivities (Caswell (1978)); in this case the introduction of disordered elements in

the transition rates between spatial locations have a clear meaning in terms of an environmental

disordered problem.

1.3 The age-structured population growth problem

Projection matrix is an increasingly popular tool for modeling population dynamics. Since the

pioneer work of Leslie (1945) to tackle ecology problems, population projection matrices have

been applied to a wide array of demographic problems (vegetative propagation, predator-prey

interaction, competition, etc., for a recent review see van Groenendael et al. (1988)). Time

fluctuating combinations of projection matrices have been used to simulate the time variability

4

of the environment. These studies have shown the dramatic effects of the stochastic variations on

the asymptotic properties of projection matrices, as well as the need to modify the concepts of

population growth rate (Cohen (1979)), and the accuracy in predicting the fate of a population

in a stochastic environment (Tuljapurkar (1982)).

More recently, rigorous results concerning the asymptotic behavior of a certain class of random

(disordered) matrices that arise in evolutionary systems have been tackled from a mathematical

point of view (Cohen (1986), Arnold et al. (1994)). In these examples variability is an important

ingredient to be considered, therefore we immediately arrive to a fundamental question: how

can the dynamics of an age-structured population -in the presence of randomness in the vital

parameters- be described? That is, can a constructive analysis, in terms of general properties

for the random elements of the projection matrices, be done in order to find the behavior of

the mean-value population vector? These and other related problems will be the subject of the

present work.

In the previous sections we have shown some exact results for the evolution of a scalar model,

emphasizing the difference that arises when solving a stochastically perturbed model (1), against

a randomly perturbed one (3). In Appendix 1 we will present the motivation of using a ran-

dom matrix to describe the analysis of a single-type population growth model in a heterogeneous

environment. This motivation also applies to Leslie (age-structures) matrices, because in many

occasions the matrix elements have uncertainties related to human inferences and/or heteroge-

neous environmental patterns which may be connected with only some of the age-structured vital

parameters. For example, the survival parameter for the youngest group (calves) may strongly

depend on its location in space (Mann et al. (1999)).

It should be remarked that even when a Leslie population dynamics described by a time-

fluctuating matrix looks similar to the one with a random (disordered) matrix, they do represent

very different problems, and mathematically speaking the methods to solve them are quite differ-

ent. In the stochastic case the important hypothesis to tackle a Markov problem is the short-scale

correlation of the temporal fluctuations, while in the random case (disorder problem) the time-

correlation is infinite (then the problem is non-Markovian), but the space-correlation should be

modeled with some fast decay in order to be able to work out the mathematical problem. The

same difference occurs in solid state physics between transport in random media (Alexander et al.

(1981)) and transport in dynamically disordered lattice (Harrison et al. (1985)), or diffusion-

influenced reaction where the reactivity of the species fluctuates in time (Szabo et al. (1982)),

as in external fluctuations in master equations (Hernandez et al. (1989), Budde et al. (1988)),

etc.

In this work we investigate and generalize the concept of growth rate in order to analyze the

population dynamics in the presence of random elements (uncertainties in the vital parameters).

We introduce an alternative analysis for the asymptotic long-time behavior of the population

vector state, by introducing a Green function technique associated to the linear matrix problem

5

(van Kampen (1992), Caceres (2003), Caceres (2004)). By using a Tauberian theorem for

power series (Hardy (1949)), our method provides a constructive way of analysis to get the

spectrum of the mean-value solution. This method is equivalent to the usual eigenvalue analysis

when it is applied to a non-random Leslie matrix. However, using the Tauberian theorem, it is

straightforward to tackle the random matrix case. We develop a statistical technique to calculate

the asymptotic behavior of the mean-value of the Green function. We show that this mean-value

function is associated to the asymptotic long-time population’s vector state. Then, we introduce

a projector operator technique to get the mean-value of the Green function as a perturbation

series of Terwiel’s cumulants (Terwiel (1974)). This allows us to define the effective growth rate

as the dominant pole of the mean-value Green function. Interestingly, the present definition of

effective growth rate agrees with the recently introduced concept of a positive Perron-Frobenius’

eigenvalue for a random Leslie matrix (Arnold et al. (1994)).

The paper is outlined as follows. In section 2, in the context of a linear matrix dynamics, we

introduce a Tauberian theorem to characterize the population growth for the non-random case. In

section 3, we characterize the mean-value of the Green function of the problem as a perturbation

in series on Terwiel’s cumulants, we also present a detailed analysis of different random models

appearing in Leslie’s matrices. In section 4 we study some exact results. In section 5 we present

a perturbation analysis for the random survival case for a matrix of dimension 3× 3. In section 6

we apply our approach to tackle a biological example, the Tursiops sp., and give some numerical

results. In section 7, we present the conclusions and future extensions of our program. Appendices

are left for presenting details of calculations.

2 Asymptotic analysis of Leslie’s dynamics in the ordered case

2.1 General remarks on age-structured populations

In the pioneer work of Leslie the specific structure of the projection matrices M were based

on age intervals of the same duration as the time step in the model (Leslie (1945)). Then

the age-specific fecundity (fertility parameters) fj were placed in the first row, and age-specific

survival probabilities pj on the subdiagonal, and zeros elsewhere. However, when the demographic

properties of individuals class (subgroups) are not closely related to age, alternative classifications

are needed. The categories into which individuals are classified should be defined in such a

way that transitions between categories are as unambiguous as possible (van Groenendael et al.

(1988)). Thus uncertainty in the vital parameters play a fundamental role in the description of

the system and the problem that we have to face is to learn how to handle a matrix random

model. The sampling error in estimating the vital rates in the definition of a Leslie matrix is an

important ingredient to be considered in order to improve the population dynamics description.

6

2.2 The Green function and Tauberian asymptotic approach

In this section we will be concerned in the analysis of the stability of the population dynamics.

Here we use a Tauberian theorem to study the stability in the Leslie model because it is suitable

when disorder is present.

Consider a m×m Leslie matrix where all its elements are sure quantities (in general we know

that fj ≥ 0, and pj ∈ [0, 1] because these last ones are probabilities)

M =

f1 f2 f3 · · · · · · fm

p1 0 0 0 0 00 p2 0 0 0 00 0 p3 0 0 00 0 0 · · · 0 00 0 0 0 pm−1 · · ·

. (10)

Due to the particular structure of this matrix, it is possible to apply the Perron-Frobenius

theorem and realize (if it is non reducible) the existence of a non-degenerated positive eigenvalue

λ1 fulfilling that λ1 ≥| λj | for all j = 2, 3, 4, · · ·m. This particular eigenvalue, λ1, is associated

to a positive eigenvector Ψ1 (the stable population). Thus, it is simple to prove that the stability

of the population dynamics is controlled asymptotically by the behavior λn1 . If λ1 < 1 the stable

population declines at a constant rate λ1. On the other hand if λ1 > 1 the stable population

Ψ1 grows at a constant rate λ1. For the ordered case, Perron-Frobenius analysis is powerful

to calculate the asymptotic behavior of the vector state. Here, we are going to introduce an

alternative approach in order to study the long time behavior n → ∞ of the population vector

state. Our approach will be a useful technique to calculate the asymptotic behavior of the mean

population vector in the case when the Leslie matrix has random elements, this will be shown in

the next section.

Consider the linear matrix dynamics (difference equations) written in the form:

Xn+1 = M · Xn, (11)

where Xn is a state vector of dimension m characterizing the population at the step time n.

Each component j of the population vector, Xn(j), represents the number of individual in each

recognized category j. The recurrence relation (11) can be solved by using a generating function

technique. We define the generating function G(z) associated to the state vector Xn by

G(z) =∞∑

n=0

zn Xn, | z |< 1, (12)

7

then multiplying (11) by zn and summing over all n we get

∞∑

n=0

zn Xn+1 =1

z

∞∑

n=0

zn+1 Xn+1

=1

z

∞∑

n=1

zn Xn

=1

z(G(z) − X0)

= M ·∞∑

n=0

zn Xn = M · G(z).

From this equation we can solve G(z) and get the following expression for the generating function

(vector G(z))

G(z) − z M · G(z) = X0. (13)

Introducing the (m×m) identity matrix 1 we can define an associated Green function matrix

to Eq. (11) in the form

G(z) = [1 − z M]−1 . (14)

It is now clear that the Green function G(z) is a matrix of dimension (m×m), and the dynamics

information of the recurrence relation (11) is contained in the poles of the G(z). In the non-

random case these poles are completely equivalent to the eigenvalues of the matrix M. The

solution of (11) can be obtained by using the z−inversion technique. Nevertheless, what is more

important here is the asymptotic value of Xn for large n, this behavior can be obtained from a

Tauberian theorem for power series (Hardy (1949)).

If the matrix M is irreducible, the matrix G(z) will have a simple pole of the form (z1 − z),

and each element of the Green function G(z) will have, in the limit z → z1, the dominant

diverging form: G(z) ∼ (z1 − z)−1, then applying Tauberian’s theorem we get asymptotically for

large n that (see Appendix 2 Eq. (24))

Xn ∼(

1

z1

)n

= λn1 , (15)

here z1 is the smallest positive pole of G(z). This is the expected result in the ordered case. For the

disordered case, the average of the Green function 〈G(z)〉 characterizes the asymptotic behavior

of the mean-value vector state 〈Xn〉. The new dominant pole z1 (the smallest positive one, which

of course is not the average of z1) characterizes the effective rate at which the population grows

in a random Leslie model.

8

3 Calculating the average of the Green function

Consider a Leslie matrix as in (10) but with random elements. Noting that all the elements of

M must be positives we adopt the following notation:

fj → fj − αj with (fj − αj) ≥ 0 (16)

pj → pj − βj with 1 ≥ (pj − βj) > 0, (17)

where the quantities αj , βj represent the random elements in a general Leslie dynamics. In

principle we are going to work out the problem for arbitrary random variables αj , βj, with the

only restriction that the support of these random variables must fulfill conditions (16) and (17)

for each sample of the disorder, see Appendix 1 for some models of random variables (the use of

the statistical independence assumption of the set αj , βj will be analyzed in the next sections).

Therefore, in what follows we do not need to specify any specific distribution for these random

variables. Using the definition (16) and (17) we can rewrite vital parameters in the form

fj − αj = fj − 〈αj〉 + 〈αj〉 − αj ≡ f∗j + ξj (18)

pj − βj = pj − 〈βj〉 + 〈βj〉 − βj ≡ p∗j + ηj, (19)

where 〈αj〉 and 〈βj〉 are mean values, thus it is clear that f∗j ≡ fj − 〈αj〉 and p∗j ≡ pj − 〈βj〉 are

sure positive numbers, and ξj , ηj are random numbers with mean-value zero. Using these facts

we can write the random equation for the Green function (14) as:

1

zG(z) − 1 = (H + B) ·G(z), (20)

where we have defined H + B ≡ M. Here H is a sure Leslie’s matrix and B a random matrix

(not necessarily with positive elements) but with the particular structure:

B =

ξ1 ξ2 ξ3 · · · · · · ξm

η1 0 0 0 0 00 η2 0 0 0 00 0 η3 0 0 00 0 0 · · · 0 00 0 0 0 ηm−1 · · ·

. (21)

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.

〈η1Qη2〉 = 〈η1η2〉 (54)

〈η1Qη2Qη3Qη4〉 = 〈η1η2η3η4〉 − 〈η1η2〉 〈η3η4〉〈η1Qη2Qη3Qη4Qη5Qη6〉 = 〈η1η2η3η4η5η6〉 − 〈η1η2〉 〈η3η4η5η6〉

− 〈η1η2η3η4〉 〈η5η6〉 + 〈η1η2〉 〈η3η4〉 〈η5η6〉 .

16

In analogy with the calculation up to O(

B2)

given in Eq. (32) (Appendix 4), for a general

m × m matrix, we get

⟨

BG0QB⟩

=

0 0 0F 2

1

12g12 0 0

0F 2

2

12g23 0

, (55)

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

〈ηlQηmQηmQηl〉 = 〈ηlQηlQηmQηm〉 = 〈ηlQηmQηlQηm〉= (∆1∆2)

2 , l 6= m = 1, 2.

Then from (59) and (60) we get that the polynomial is

0 = 1 − z2f2

(

p1 + ∆21g12 + ∆2

1∆22g13g23g22

)

(78)

−z3f3

(

p1 + ∆21g12 + ∆2

1∆22g13g23g22

) (

p2 + ∆22g23 + ∆2

1∆22g13g12g22

)

−z(

∆21∆

22g22g

213

)

− z2f3

(

∆21∆

22g13g

222

)

+ O(

B6)

,

where gjm are given as before from (56). Solving numerically this equation we found twelve roots

and the largest positive one (λ = 1/z) gives

λ1 = 1.8366 + O(

B6)

. (79)

It is interesting to emphasize the good convergency of our Terwiel cumulant method to get the

effective eigenvalue for a random Leslie problem.

Remark. The large value of λ1 = 1.8366, compared with the non-random one λ1 = 1.5, teaches

us that even when the randomness of the survival parameters has mean-value zero, fluctuations

can drive the system to large value (sample) excursions of the Perron-Frobenius eigenvalue. Re-

member that in the present binary case the survival parameters can only take discrete values:

p1 = 0.75 ± 0.25 and p2 = 0.6 ± 0.3. This example shows that a correct study of the fluctuations

leads to very important consequences when applying to random population problems.

For this symmetric case it is interesting to present here a table showing the different values of

λ1 = λ1(f1, f2, pj) for the four possible realizations of disorder, and compare these values with

λ1. Using 〈βj〉 = 0 in (71), and from the effective value (79), up to O(

B6)

, we write

λ1 = λ∗1 = 1.5 (80)

λ1 = 1.8366 + · · ·,

24

and for each realization of the disorder we get the significative table

∥

∥

∥

∥

∥

∥

p1 + ∆1

p2 + ∆2

p1 + ∆1

p2 − ∆2

p1 − ∆1

p2 + ∆2

p1 − ∆1

p2 − ∆2

λ1 (f1, f2, pj) 1.8519 1.4372 1.4313 1.0899

∥

∥

∥

∥

∥

∥

.

To end this comment note that due to the fact that all realizations are equiprobable, the arithmetic

average gives 〈λ1〉 = 1

4

∑

pj λ1 (f1, f2, pj) = 1.4525, which is below the sure value. As we

commented before this arithmetic average is not representative of the dynamics of the mean-

value population vector state 〈Xn〉.A more extended analysis considering higher order corrections, and the possibility of handling

non-statistically independent random variables in the vital parameters of a m×m Leslie matrix,

can also be solved within the present theoretical framework, these subjects are under investigation

and will be presented elsewhere.

7 Conclusions

The main concern of this paper was to relate the characteristics of disorder (sampling error in

estimating the vital rates) appearing in a Leslie matrix M with the dynamics of the population.

The focus was on the effects that fluctuations have on the dominant eigenvalue λ1 (the largest

positive one) associated to the mean-value Green function of the random matrix problem. A

general approach, to get this effective eigenvalue, was described. We calculate the dynamics

of the mean-value population vector state under the assumption that the random variables,

appearing in the Leslie matrix, are described with arbitrary distributions. The problem was

reduced to the calculation of the smallest positive root z1 of the secular polynomial appearing in

the general expression for the mean-value Green function 〈G(z)〉. This non-trivial polynomial can

be obtained order by order in terms of a diagrammatic technique built with Terwiel’s cumulants,

which have carefully been identified in the present work. By understanding how this smallest

positive root z1 = 1/λ1 depends on the model of disorder one can link the asymptotic population

dynamics with the statistical properties of the errors in the vital parameters. Particular examples

(using the statistical independent assumption) were presented using uniformly distributed, and

binary random variables affecting the survival and the fertility parameters in Leslie matrices of

dimensions 2×2 and 3×3. It was shown that the effective growth rate λ1 has a non-trivial response

to the perturbation. In particular, it was proved that if the random variables are symmetric, the

effective positive eigenvalue is enlarged with respect to the mean-value growth rate. On the other

hand, if the fluctuations (random variables) always reduce the vital parameters of the model, the

effective eigenvalue is larger than the naive approximation associated to the mean-value of the

Leslie matrix 〈M〉 = H. This result teaches us that fluctuations increase the final effective value

growth rate. In particular we have worked out an analytical approximation to measure this fact

in a 3 × 3 model: λ1 ≃ λ∗1 + ǫ, showing that ǫ is a positive quantity. In addition, we have proved

25

this fact from the analysis of an exact 2×2 case, and also showed this behavior numerically when

applied to a particular 3 × 3 biological case.

Our theory also provides a means to determine precise criteria by which the extinction of a

population can be affected by the randomness of the environment affecting the vital parameters

of the species in study. Works in that direction are in progress.

Acknowledgments

This work was done within the framework of the Associateship Scheme of the Abdus Salam

International Centre for Theoretical Physics, Trieste, Italy. MOC is thankful for the grants from

SECTyP, Uni. Nac. Cuyo., and PIP 5063 (2005) CONICET, Argentina.

1 Appendix: Population growth in a random environment

An interacting population diffusion system is generally characterized by a source term and a

flux transport mechanism. The source term in an ecological context could represent the birth-

death process and the transport can be emulated by a diffusion term. For example, consider

the logistic population growth model with a linear reproduction rate r and a carrying capacity

of the environment K. The resulting equation with constant diffusion coefficient D is known as

the Fisher equation (Fisher (1937)), who proposed the one-dimensional case as a model for the

spread of an advantageous gene in a population:

∂n(x, t)

∂t= rn(x, t)

(

1 − n(x, t)

K

)

+ D∂2n(x, t)

∂x2; x ∈ (+∞,−∞) . (2)

When K → ∞; i.e., either the population is far from saturation or the supply of food is unlimited,

Fisher’s equation can easily be studied analytically. The case when the reproduction rate r(x) is

heterogenous corresponds to a diffusion problem in a random multiplicative media, this equation

is equivalent to a polymer in a random potential (Tao (1988)). Also, it models chemical reactions,

biological multiplication, and the evolution of species (Ebeling et al. (1984)). The quantities of

interest are either the averaged Green function (over the disorder) taken in the initial point

〈G (x, t/x, 0)〉, or the total contribution N (t) =∫

〈G (x, t)〉 dx. The case when the random

reproduction rate r(x) is Gaussian has been the subject of much research, and both quantities

present the same exponential behavior ∼ exp(

constant t3)

and differ only in a factor (Tao (1988),

Guyer et al. (1990), Gross et al. (1983), Valle et al. (1991)).

In order to carry out a similar program, as done in the present paper, i.e., taking into account

the influence of a random environment, we consider now the discrete version of (2). Therefore, in

the case when the reproduction rate is heterogenous in the space and taking the limit of K → ∞,

from (2) we arrive to a discrete random evolution equation of the form

dni(t)

dt= rini(t) + [ni+1(t) + ni−1(t) − 2ni(t)] . (3)

26

Here we have associated a lattice diffusion coefficient equal to one. Defining the vector P =

(· · · , ni, ni+1, ni+2, · · · ), this last equation can be written in a matrix form:

dP

dt= [H + B] · P, (4)

where H is a sure tridiagonal matrix representing the discrete Laplacian operator, and B a random

diagonal matrix representing the influence of the environment in the heterogeneous reproduction

rate.

Here we argue that a similar method, as presented in the present paper, can also be applied to

study the mean-value of the vector P. Nevertheless, because we are interested in an age-structured

population model we have focused our program in random Leslie matrices, representing either

the influence of environmental disorder and/or uncertainties in the definition of the values of the

associated vital parameters.

Related to equation (3) is the (scalar) simplest random multiplicative evolution problem

dn(t)

dt= (r + β)n(t), (5)

where β is an arbitrary random variable characterized by its probability P(β). If we are interested

in the time-evolution of the m−moments of n(t), we take from the solution of (5) the mean-value

of n(t)m = n(0)m exp [m (r + β) t] over the probability P(β). Then, we arrive at the following

formula for the moments

〈n(t)m〉 = n(0)m 〈exp [m (r + β) t]〉= n(0)m exp (mrt) 〈exp [mtβ]〉= n(0)m exp (mrt)Gβ (−imt) , (6)

where Gβ (k) is the Fourier transform of the probability P(β), i.e., its characteristic function

(van Kampen (1992), Caceres (2003)),

Gβ (k) =

∫

DP(β)eikβdβ, β ∈ D.

1.1 Case when β is uniformly distributed

The Fourier transform of the uniform probability distribution with positive support is

Gβ (k) =

∫ F

0

eikβ

Fdβ =

exp (ikF ) − 1

ikF. (7)

From (6) and (7) we get for the first and second moments of n(t)

〈n(t)〉 = n(0) exp (rt)

(

exp (tF ) − 1

tF

)

(8)

⟨

n(t)2⟩

= n(0)2 exp (2rt)

(

exp (2tF ) − 1

2tF

)

. (9)

27

1.2 Case when β is Gaussian

The Fourier transform of a centered Gaussian probability distribution with square mean value⟨

β2⟩

= θ2 is

Gβ (k) =

∫ +∞

−∞

exp(

−β2

2θ2

)

√2πθ2

eikβdβ = exp

(−θ2

2k2

)

. (10)

From (6) and (10) we get for the first and second moments of n(t)

〈n(t)〉 = n(0) exp (rt) exp

(

θ2

2t2)

(11)

⟨

n(t)2⟩

= n(0)2 exp (2rt) exp(

2θ2t2)

. (12)

Note that the behavior of 〈n(t)〉 is bounded from above by the result reported in the case of

diffusion in a random Gaussian multiplying 1−dimensional medium ∝ exp(

t3)

(Tao (1988),

Guyer et al. (1990), Gross et al. (1983)).

1.3 Case when β is binary and symmetric

The Fourier transform of a symmetric probability associated to a binary random variable β = ±∆

is

Gβ (k) =∑

β

.eikβ (δβ,+∆ + δβ,−∆) = cos k∆ (13)

In this case from (6) and (13) the first and second moments of n(t) are

〈n(t)〉 = n(0) exp (rt) cosh (t∆) (14)⟨

n(t)2⟩

= n(0)2 exp (2rt) cosh (2t∆) (15)

1.4 Case when β is a Poisson number

When β = p/τ and p is an integer random number characterized by a Poisson probability with

mean-value 〈p〉, the Fourier transform is

Gβ (k) =∞∑

p=0

exp (ikp/τ) 〈p〉p exp (−〈p〉)p!

= exp [−〈p〉 (1 − exp (ik/τ))] . (16)

From (6) and (16) the first and second moments of n(t) are

〈n(t)〉 = n(0) exp (rt) exp [−〈p〉 (1 − exp (t/τ))] (17)⟨

n(t)2⟩

= n(0)2 exp (2rt) exp [−〈p〉 (1 − exp (2t/τ))] (18)

1.5 Case when β is Gamma distributed

Consider now the quite ubiquitous Gamma distribution:

P(β) =cb

Γ(b)βb−1e−cβ; β ∈ [0,∞], b, c > 0. (19)

28

If b = n is an integer, the Erlang density results. For b = n/2 and c = 1/2 this density is denoted

by χ2(n) and is called the Chi-square with n degrees of freedom. In general, its Fourier transform

will be

Gβ (k) =

∫ ∞

0

P(β)eikβdβ =cb

(c − ik)b; Im[k] + c > 0. (20)

Then, from (4) and (20) the first and second moments of n(t) are

〈n(t)〉 = n(0) exp (rt)cb

(c − t)b; (c − t) > 0 (21)

⟨

n(t)2⟩

= n(0)2 exp (2rt)cb

(c − 2t)b; (c − 2t) > 0 (22)

2 Appendix: The Tauberian approach

The Tauberian theorem is as follows (Hardy (1949)). Let U(y) be defined by

U(y) =

∞∑

n=0

an exp (−ny) (23)

where an > 0. Let U(y) have the asymptotic form, as y → 0,

U(y) ∼ ϕ(y−1) = y−γ L(y−1),

where L(x) is a slowly varying function, and xγL(x) is a positive increasing function of x for

sufficiently large x. Then as n → ∞

a0 + a1 + a2 + · · · + an ∼ ϕ(n)

Γ(γ + 1),

where Γ(γ + 1) is the gamma function. If the an are monotonic and ϕ(x) is differentiable, it

follows that

an ∼ dϕ(x)/dx

Γ(γ + 1)

∣

∣

∣

∣

x=n

.

This is the important result that we use to study the asymptotic behavior of Xn for large n.

In order to apply the Tauberian theorem to our problem, we introduce the change of variable

z → z1e−y, where z1 = 1/λ1, then from the generating function (Section 2, Eq. (12))

G(z) =

∞∑

n=0

znXn, | z |< 1,

we get

G(z = z1e−y) =

∞∑

n=0

exp(−yn) zn1 Xn.

29

Thus, we can associate from (23) that an = zn1 Xn. On the other hand, if the Leslie matrix M is

irreducible, the matrix G(z) (Section 2, Eq. (14)) will have a simple pole of the form (z1 − z),

thus the Green function G(z) will have in the limit z → z1 the dominant diverging form

G(z) → 1

(z1 − z)G,

where the m × m matrix G remains finite in the limit z → z1. Using U(y) = G(z = z1e−y) ∼

(z1 − z1e−y)

−1, and in the limit of y → 0 we get

U(y) ∼ ϕ(y−1) ∼ y−γ

z1

, where γ = 1, L(x) = 1.

For n → ∞, and using the Tauberian theorem we get asymptotically that an ∼ ϕ′

(n)/

Γ(2) ∼ 1.

Going back to the old variable we obtain in the limit n → ∞

Xn ∼(

1

z1

)n

= λn1 , (24)

which is the asymptotic behavior of the vector state in the ordered case.

Remark. The Tauberian approach teaches us that if we want to tackle the random case we

should first calculate the average of the Green function 〈G(z)〉, then from its poles we can infer

which is the asymptotic behavior of the average of the vector state 〈Xn〉. The smallest positive

pole z1 = 1/λ1 characterizes the rate at which the population grows in a random Leslie model.

3 Appendix: Terwiel’s cumulants

The calculations of Terwiel’s cumulants are not so complex (Terwiel (1974)). Here we recall

some general properties of these cumulants. Consider the general situation when we have a set

different random variables ξj, a Terwiel cumulant of order q can be written in terms of the

moments of the variables ξj by using the following formula

〈ξ1Qξ2Qξ3Q· · · ξq−1Qξq〉 (25)

=

q−1∑

r=0

(−1)r∑

1≤l1≤···≤lr≤q

〈ξ1 · · · ξl1〉 〈ξl1+1 · · · ξl2〉 · · · 〈ξlr+1 · · · ξq〉 ,

where as before Q is the projection operator (1 − P) . Explicit examples of this formulae are:

〈ξ1Qξ2〉 = 〈ξ1ξ2〉 − 〈ξ1〉 〈ξ2〉 (26)

〈ξ1Qξ2Qξ3〉 = 〈ξ1ξ2ξ3〉 − 〈ξ1〉 〈ξ2ξ3〉 − 〈ξ1ξ2〉 〈ξ3〉 + 〈ξ1〉 〈ξ2〉 〈ξ3〉〈ξ1Qξ2Qξ3Qξ4〉 = 〈ξ1ξ2ξ3ξ4〉 − 〈ξ1〉 〈ξ2ξ3ξ4〉 − 〈ξ1ξ2〉 〈ξ3ξ4〉

− 〈ξ1ξ2ξ3〉 〈ξ4〉 + 〈ξ1〉 〈ξ2〉 〈ξ3ξ4〉+ 〈ξ1ξ2〉 〈ξ3〉 〈ξ4〉 + 〈ξ1〉 〈ξ2ξ3〉 〈ξ4〉 − 〈ξ1〉 〈ξ2〉 〈ξ3〉 〈ξ4〉 .

30

In the particular case when the random variables ξj have zero mean-value, Terwiel’s cumulants

simplify notably, for example:

〈ξ1Qξ2〉 = 〈ξ1ξ2〉 (27)

〈ξ1Qξ2Qξ3〉 = 〈ξ1ξ2ξ3〉〈ξ1Qξ2Qξ3Qξ4〉 = 〈ξ1ξ2ξ3ξ4〉 − 〈ξ1ξ2〉 〈ξ3ξ4〉

〈ξ1Qξ2Qξ3Qξ4Qξ5〉 = 〈ξ1ξ2ξ3ξ4ξ5〉 − 〈ξ1ξ2〉 〈ξ3ξ4ξ5〉− 〈ξ1ξ2ξ3〉 〈ξ4ξ5〉

〈ξ1Qξ2Qξ3Qξ4Qξ5Qξ6〉 = 〈ξ1ξ2ξ3ξ4ξ5ξ6〉 − 〈ξ1ξ2〉 〈ξ3ξ4ξ5ξ6〉− 〈ξ1ξ2ξ3〉 〈ξ4ξ5ξ6〉 − 〈ξ1ξ2ξ3ξ4〉 〈ξ5ξ6〉+ 〈ξ1ξ2〉 〈ξ3ξ4〉 〈ξ5ξ6〉 .

These formulae are general for any kind of random distribution. Possible random variables useful

to characterize the disorder in a Leslie matrix are shown in Appendix 1, where we present the

characteristic function for several examples.

To end these remarks, note that Terwiel’s cumulants preserve the order of the random variables

ξj. There is another very important property of any Terwiel cumulant

〈ξ1Qξ2Q· · ·QξkQξk+1 · · · Qξm〉 ,

if it is possible to split it into two sets ξ1ξ2 · · · ξk and ξk+1 · · · ξm without altering the order

of the ξ′s in such a way that the variables in one of the sets are statistical independent of those in

the other set, the cumulant vanishes (this is call the partition property of Terwiel’s cumulants).

Terwiel’s cumulants are different from the simple cumulants that naturally appear in a Taylor

expansion of the logarithm of the characteristic function of a random variable (van Kampen

(1992), Caceres (2003)).

4 Stability, up to 2nd order, in a random Leslie matrix model

Here we apply the general formula:

det

∣

∣

∣

∣

∣

1− z

(

H +

⟨ ∞∑

k=0

[

BG0Q]k

B

⟩)∣

∣

∣

∣

∣

= 0, (28)

to calculate the dominant pole z1. Then, from the asymptotic behavior:

limn→∞

〈Xn〉 ∼ (1/z1)n ,

we can analyze the mean-value population stability in a concrete biological case. We chose here

a particular model of disorder in an arbitrary m×m Leslie matrix. Following the notation given

31

in section 3, we define a random survival model. Then the matrix B will have the particular

structure:

B =

0 0 0 0 · · · 0η1 0 0 0 0 00 η2 0 0 0 00 0 η3 0 0 00 0 0 · · · 0 00 0 0 0 ηm−1 · · ·

, (29)

with 〈ηl〉 = 0. The random fertility model can also be worked in an analogous way. From this

matrix B, we consider now the cumulant structure given in Eq. (28); using that 〈B〉 = 0 the first

non-null contribution in (28) is of the form

⟨

BG0QB⟩

. (30)

This cumulant is of O(

B2)

in the random perturbation and has the structure of a second Terwiel

cumulant (Terwiel (1974)). This particular Terwiel structure comes from the time evolution of

the perturbed Green function.

There are some special cases that can be solved in an exact way, but in general we have

to invoke a perturbation approach to keep only a few cumulants in order to arrive to some

analytical calculation. In solid state physics this technique is the starting point to introduce a self-

consistent approximation to tackle the problem of transport in random media (Hernandez et al.

(1990a), Hernandez et al. (1990b), Pury et al. (2002)). A self-consistent approximation is a

good technique to tackle enlarged Leslie’s matrices with transitions rates between spatial locations

(Caswell (1978)), this will be the subject of a future work.

In general, using the definition of G0 ≡[

1z1 − H

]−1in terms of the sure m ×m Leslie matrix

H and using (29), we get up to O(

B2)

⟨

BG0QB⟩

(31)

=

0 0 0 0 · · · 0〈η1Qη1〉 g12 〈η1Qη2〉 g13 〈η1Qη3〉 g14 · · · · · · 0〈η2Qη1〉 g22 〈η2Qη2〉 g23 〈η2Qη3〉 g24 · · · · · · 0〈η3Qη1〉 g32 〈η3Qη2〉 g33 〈η3Qη3〉 g34 · · · · · · 0

· · · 〈η4Qη2〉 g43 〈η4Qη3〉 g44 · · · · · · 0· · · · · · 〈η5Qη3〉 g54 · · · 〈ηm−1Qηm−1〉 gm−1,m · · ·

.

This expression is the exact contribution considering all the correlations up to second order. Here

gjl are the matrix elements of the ordered Green function G0, this formula can easily be handled

in a computer. Thus we see that our approach is not restricted to the assumption of statistical

independent random perturbations. Nevertheless in order to arrive at some analytical expression

the independent assumption will be used in the rest of the paper.

A great analytical simplification arises if we consider that all the random variables are statis-

32

tical independent, in this case and using that the set ηj has mean-value zero, we get

⟨

BG0QB⟩

(32)

=

0 0 0 0 · · · 0〈η1Qη1〉 g12 0 0 · · · · · · 0

0 〈η2Qη2〉 g23 0 · · · · · · 00 0 〈η3Qη3〉 g34 · · · · · · 00 0 0 · · · · · · 00 0 0 · · · 〈ηm−1Qηm−1〉 gm−1,m · · ·

.

In this case it is now clear that up to O(

B2)

the only statistical objects that we need to calculate

are the second Terwiel cumulants:

〈ηjQηj〉 , j = 1, 2, · · · ,m − 1.

These numbers depend on the statistical properties that we chose for the set of random variables

ηj. In total analogy, if we want to study the perturbation up to O(

B3)

we have to calculate

the Terwiel operator:⟨

BG0QBG0QB⟩

.

This object looks much more complex, but if we use the statistical independence assumption the

corresponding expression can also be handled analytically.

Remark. Up to O(

B2)

in the random perturbation and by virtue of the Tauberian theorem,

the long-time behavior of the averaged Green function will be dominated by the smallest positive

root z1 of

det∣

∣1 − z(

H +⟨

BG0QB⟩)∣

∣ = 0. (33)

Following Boyce (1977) we can call λ1 = 1/z1 the effective finite growth rate in a disordered

Leslie’s population model.

References

Alexander S., J. Bernasconi, W.R. Schneider & R. Orbach, (1981). Excitation dynamics in random

one-dimensional systems, Rev. Mod. Phys. 53, 175-198.

Arnold L., V.M. Gundlach and L. Demetrius, (1994). Evolutionary Formalism for Products of

Positive Random Matrices, The Annals of Applied Probability, Vol. 4, 859-901.

Berta A., J.L. Sumich, K.M. Kovacs, P.A. Folkens & P.J. Adam (2005). Marine Mammals: Evo-

lutionary Biology (Second Edition), Elsevier, Berlin.

Boyce M.S., (1977). Population growth with stochastic fluctuations in the life table, Theor. Popul.

Biolog. 12, 366-373.

33

Budde C.E. & M.O. Caceres, (1988). Diffusion in presence of external anomalous noise, Phys.

Rev. Lett., 60, 2712-2714.

Budini A.A. & M.O. Caceres, (2004). Functional characterization of linear delay Langevin equa-

tions, Phys. Rev. E., 70, 046104-12.

Caswell H., (1978). A general formula for the sensitivity of population growth rate to changes in

life history parameters, Theor. Popul. Biolog. 14, 215-230.

Caswell H., (1983). Lectures on mathematics. The Life Science 18, 171-233.

Caceres M.O., H. Matsuda, T. Odagaki, D.P. Prato & W. Lamberti, (1997). Theory of diffusion

in finite random media with a dynamic boundary condition, Phys. Rev. B, 56, 5897-5908.

Caceres M.O., (2003). (in Spanich) Elementos de estadistica de no equilibrio y sus aplicaciones

al transporte en medios desordenados, Reverte S.A., Barcelona.

Caceres M.O., (2004). From Chandrasekhar to the Stochastic Transport Theory, in: Trends in

Statistical Physics, Vol. 4, 85-122.

Cohen J.E., (1979). Comparative statistic and stochastic dynamics of age-structured populations,

Theor. Popul. Biolog. 16, 159-171.

Cohen J.E., (1986). Products of Random Matrices and Related Topics in Mathematics and Science

-A Bibliography. Contemp. Math. 50 199-213.

Ebeling W., Engel A., Esser B., and Feistel R., Diffusion and Reaction in a Random Media and

Models of Evolution Procesess (1984) J. Stat. Phys., 37, 369-384.

Fisher R.A., (1937). The wave of advance of advantageous gene. Ann. Eugenics 7, 353-369.

Goel N.S. and Richter-Dyn N, (1974). Stochastic Models in Biology, Academic Press, New York.

Gross E.P., (1983). Partition Function of a Particle Subject to Gaussian noise, J. Stat. Phys. 33,

107-132.

Guyer. R.A. and J. Machta, (1990). Comment on: Exact Solution for Diffusion in an Random

Potential Phys. Rev. Lett. 64, 494.

Hardy G.H., (1949). Divergent series, Oxford University Press, Oxford.

Harrison A. & R. Zwanzig, (1985). Transport on a dynamically disordered lattice, Phys. Rev. A,

32, 1072-1075.

Hernandez Garcia E., L. Pesquera, Rodrigues M. & M. San Miguel, (1989). Random walk in

dynamically disordered chains, J. Stat. Phys. 55, 1027-052.

34

Hernandez Garcia E., M.A. Rodrigues, L. Pesquera & M. San Migue, (1990a). Transport proper-

ties for random walks in disordered one dimensional media: perturbative calculations around

the effective-medium approximation, Phys. Rev. B, 42, 10653-10672.

Hernandez Garcia E. & M.O. Caceres, (1990b). First passage time statistics in disordered media,

Phys. Rev. B, 42, 4503-4518.

Leslie P.H., (1945). On the use of matrices in certain population mathematics. Biometrikra 33,

part III, 183-212.

Mann J., R.C. Connor, L.M. Barre & M.R. Heithaus, (1999). Female reproductive success in

bottlenose dolphins (Tursiops sp.): life history, habitat, provisioning, and group-size effects,

Behavioral Ecology 11, 210-219.

Pury P.A. & M.O. Caceres, (2002). Survival and residence times in disordered chains with bias,

Phys. Rev. E, 66, 21112(01)-21112(13).

Rayen T.J., (2005). Marine Mammal Research: Conservation beyond crisis, The Johns Hopkins

University Press, New York.

Szabo A., D. Shoup, S. Northrup & J. Mc Cammon, (1982). Stochastically gated diffusion-

influenced reactions, J. Chem. Phys., 77, 4484-4493.

Tao R, (1988). Exact Solution for Diffusion in an Random Potential, Phys. Rev. Lett., 61, 2405-

2408.

Terwiel R.H., (1974). Projection operator method applied to stochastic linear differential equa-

tions, Physica A 74, 248-252.

Tuljapurkar S.D., (1982). Population dynamics in variable environments II, correlated environ-

ments sensitivity analysis and dynamics, Theor. Popul. Biolog. 21, 114-140.

Valle A., Rodriguez M.A., and Pesquera L., (1991). Diffusion in a Random Multiplying Medium:

Exact Bounds and Simulations. Phys. Rev. A 43, 2070-2073.

van Groenendael J., H. de Kroon & H. Caswell, (1988). Projection matrices in population biology.

TREE ( 3), 264-269.

van Kampen N.G., (1992). Stochastic processes in physics and chemistry, North Holland, Ams-

terdam.

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