Theoretical Population Biology 74 (2008) 68–73

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Theoretical Population Biology

journal homepage: www.elsevier.com/locate/tpb

On second order sensitivity for stage-based population projection matrix modelsDominic McCarthy a,∗, Stuart Townley a, Dave Hodgson b

a School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, EX4 4QE, UKb Centre for Ecology and Conservation, School of Biosciences, University of Exeter in Cornwall, Tremough, TR10 9EZ, UK

a r t i c l e i n f o

Article history:Received 11 July 2007Available online 10 May 2008

Keywords:ConvexitySensitivityTransfer functionStructured perturbationsPopulation projection matrices

a b s t r a c t

In this paper we present a simple method for identifying life-history perturbations in populationprojection matrices that yield an accelerating population growth rate. Accelerating growth means thatthe dependence of the growth rate on the perturbation is convex. Convexity, when the second sensitivityof the growth rate is positive, is calculated using a new formula derived from the transfer functionof the perturbed system. This formula is used to explore the relationship between stasis and growthprobabilities from stage-structured population projection matrices.

© 2008 Elsevier Inc. All rights reserved.

1. Introduction

Predictions and strategies based on the analysis of matrixmodels are used extensively across many sciences. Consider thecase of the dominant eigenvalue of a population projection matrix(PPM). If this is less than one then the model predicts that thepopulation declines and may eventually become extinct (Caswell,2001). This prediction from the matrix model is contingent on theaccuracy of the model. A conservation strategy for an endangeredspecies may require intervention in some part of the life cycle ofthemodelled organism and this intervention could bemodelled byperturbing (changing) some matrix entries (Hodgson et al., 2006).The perturbed model could then be used to predict the effect ofthese changes on the dominant eigenvalue and hence the futuremodelled population. In order to achieve a desired conservationstrategy model perturbations need to be biologically realistic andshould strive to meet requirements with the least cost in termsof expended resources. Perturbation analysis of PPM models cantherefore assist in the study of model stability as well as the cost-effective selection of population management strategies (Deineset al., 2007). Due to the particular structure of PPMs (non-negativeand irreducible) we can identify the dominant eigenvalue witheither the spectral radius of the PPM or the asymptotic populationgrowth rate (Horn and Johnson, 1991).

If there is some uncertainty regarding model parameters,and a conservation strategy which requires some manipulationof model parameters is desired, then we ask not only how

∗ Corresponding author.E-mail addresses: D.Mc-Carthy@ex.ac.uk (D. McCarthy),

townley@maths.ex.ac.uk (S. Townley), D.J.Hodgson@exeter.ac.uk (D. Hodgson).

0040-5809/$ – see front matter© 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.tpb.2008.04.008

sensitive is the growth rate of a PPM when the matrix isperturbed, but how sensitive is this sensitivity to perturbations?This second sensitivity is important for several reasons. Equalsuccessive increments or decrements in some vital rate can resultin successively increasing or decreasing changes in the spectralradius and this corresponds to acceleration or deceleration ofthe growth rate (Cohen, 1978). Applications in this paper focusprimarily on accelerating the growth rate although they couldbe applied to reducing the growth rate of invasive species. Thisproperty of second sensitivity is termed the convexity or concavityof the spectral radius (see Fig. 1.1 for an example) and is givenby the second derivative of the spectral radius with respect to aperturbation. In order to increase a population by manipulationof some vital rates, the perturbation giving the largest resultantconvexity is preferable to others resulting in less convexity.Similarly, to reduce an invasive population by manipulation ofsome vital rates the perturbation giving the largest resultantconcavity is preferable to others resulting in less concavity.For example, given a choice of practical perturbations, a smallchange to one perturbation could result in a larger change to thegrowth rate than an equally sized change to another perturbation(Hodgson et al., 2006).

The remainder of this paper contains a brief introduction topopulation projection models and structured perturbations vialinear systems theory. We then introduce eigenvalue sensitivityusing the transfer function of the perturbed system and derivea formula for the second order sensitivity using this transferfunction. This formula gives a simple method for determiningwhich perturbation structures allow convex or concave behaviourof the growth rate of the perturbed matrix. We link trade-offperturbation structures with certain matrix regions to find cost-efficient ways to implement a conservation strategy.

D. McCarthy et al. / Theoretical Population Biology 74 (2008) 68–73 69

Fig. 1.1. Illustration of an eigenvalue changing in a convex or concave manner. Perturbations to a PPM with the dominant eigenvalue plotted against the perturbation. Theleft figure shows a perturbation to a diagonal element and the dominant eigenvalue varies in a convex manner. A line joining any two points on a convex curve will beabove the curve. The right figure shows a perturbation to a subdiagonal element and the dominant eigenvalue varies in a concave manner. A line joining any two points ona concave curve will be below the curve.

2. Population projection models

The dynamics of many population projection models can berepresented by the equation

x(t + 1) = Ax(t). (2.1)

Here, t ∈ {0, 1, 2, . . .} denotes the year or reproductive interval,x(t) is an n × 1-dimensional age- or stage-structured populationprofile and A is the n × n matrix of rates of transition betweenstages of the life cycle. The matrix A is the PPM. This model islinear, discrete, time-invariant and autonomous. Time invarianceimplies that the matrix parameters are constant over time andautonomous implies there are no external inputs. The long-termor asymptotic dynamics of the population are captured by thedominant eigenvalue and associated left and right eigenvectorof A. The dominant eigenvalue (i.e. of largest magnitude) of Apredicts the asymptotic rate of geometric increase or decline of thepopulation (Caswell, 2001). We can write a simple perturbation tothe PPM in the form

A → A + pS. (2.2)

Here S is a sparse matrix with a fixed structure and p is a scalarwhich measures the magnitude of changes to A. We can oftendecompose S into a pair of vectors, D and E and rewrite (2.2) as

A → A + pDE. (2.3)

In this case p is a scalar representing the magnitude of theperturbation, D is a vector of dimension n × 1 and E is a vectorof dimension 1 × n. These vectors represent the structure of theperturbation. This perturbation has one free parameter, p, and arank one structure, D E. For a rank-k perturbation where k rowsor columns are perturbed, D and E have structure n × k and k × n.Here we only consider rank-one perturbations. Formore details onstructured perturbations in a general setting, see, e.g., Doyle (1982)and Hinrichsen and Pritchard (1990) or for more on structuredperturbations in a PPM setting see Hodgson et al. (2006). Thevector product D E evaluates to an n × n matrix which picks outthe elements of the matrix A to perturb. Trade-off perturbationsoffset a change to a matrix element (or elements) with a changeto some other matrix element (or elements) and hence impliessome correlation between matrix entries. Correlated vital ratesare common (Doak et al., 1994). Although there are a number ofinteresting trade-off perturbations to consider, in this paper wefocus on perturbing only stasis and growth rates, we do not perturb

fecundities, see Caswell (1996) for trade-off changes relating tofecundity. If a single element representing a probability of stasis orgrowth in a matrix is changed then the mortality for that stage islikely to change by the same amount but in the opposite direction.This is because the sum of mortality and transition probabilitiescannot exceed one. Alternatively, it is not unreasonable to assumethat if there are N individuals in stage j at some time and weincrease the proportion of individuals going to stage j+ 1 then theprobability of an individual staying in stage j is reduced. We canaugment the dynamical system (2.1) with an input and output:

x(t + 1) = Ax(t) + Du(t)y(t) = Ex(t). (2.4)

In dynamical systems theoryu(t) is an input to the systemeffectingchanges given by the structure of Dwhile E captures the measuredoutput of the system. However, in this case the vectors D and E arenot actual control and output measurement vectors but describethe perturbation of the system matrix, A. This is the feedbackinterpretation of the perturbed system (Hinrichsen and Pritchard,1990) and represents amapping from the canonical formof a linearsystem to a perturbed linear system. By taking z-transforms thetransfer function of this system, parameterised by the scalar z, canbe written (Farina and Rinaldi, 2000)

G(z) = E(zI − A)−1D. (2.5)

This function gives the relationship between the input and outputof the system. We map the perturbation structure vectors to thisformula where E and D represents the uncertainty structure. Bywriting the inverse of (zI − A) as a polynomial in z the transferfunction can be described by the rational polynomial

G(z) =n(z)d(z)

. (2.6)

In this case the denominator d(z) is the characteristic polynomialof the matrix A and is fixed by A. The numerator n(z) is thepolynomial corresponding to a combination ofminors of (zI−A)−1.Using the Sherman-Morrison-Woodbury formula (Golub and VanLoan, 1996) we combine the above to obtain:

λ is an eigenvalue of A + pDE if, and only if,

1 = p G(λ) = pn(λ)

d(λ). (2.7)

This equation shows clearly how λ (the growth rate) is relatedto the perturbation pDE. In the next section we look at the

70 D. McCarthy et al. / Theoretical Population Biology 74 (2008) 68–73

sensitivities of the growth rate to perturbations. For a morecomprehensive discussion of transfer function applications tomatrix population models see Hodgson and Townley (2004);Hodgson et al. (2006).

3. Eigenvalue sensitivity

The sensitivities of the eigenvalues of a PPM to parameterperturbations are important tools in ecology (Caswell, 2001).There are different ways to calculate these sensitivities but inbroad terms they either explicitly involve the eigenvectors of thesystem matrix or they do not. We give formulae, based on thesystem transfer function, for the first order sensitivity and thesecond order sensitivity without using eigenvectors. Eigenvectorscan change dramatically for only small perturbations to matrixentries (Horn and Johnson, 1991) and thus a formula that doesnot use eigenvectors is potentially more stable than one that does.The formula for the first derivative has been in use for sometime. By first order sensitivity we mean the partial derivative ofthe growth rate with respect to (w.r.t.) the matrix entries. Bysecond order sensitivity we mean the second partial derivativeof the growth rate w.r.t. the matrix entries. The structure of ourformulae yield additional insight into system dynamics and arealso computationally elegant and efficient.

The sensitivity of a simple eigenvalue to the perturbation of asingle element, aij, in a matrix is often given by the formula (Hornand Johnson, 1991)

∂λ

∂aij= viwj, (3.8)

where w and v are right and left eigenvectors of A correspondingto the simple (unique) eigenvalue λ and are normalised so thatvTw = 1, with vi the ith component of v and wj the jth componentof w. This formula defines the sensitivity explicitly in terms of theproducts of the eigenvector components. Suppose that thematrixAis perturbed, A → A+pDE.We require the derivative of the growthrate w.r.t the parameter p. This is the sensitivity of the eigenvalueλ of A in the direction DE. The eigenvalue λ and parameter p arerelated by (2.7). When λ → λmax then p → 0 and by applicationof l’Hopital’s rule (2.7) becomes

1 = limp→0

pn(λ)

d(λ)=

n(λ)∂λ∂p d′(λ)

. (3.9)

If λ(A) is a simple eigenvalue then d′(λ) 6= 0. and (3.9) becomes

∂λ

∂p=

n(λ(A))

d′(λ(A)). (3.10)

Choosing D = ei and E = eTj as ith and jth standard basis vectors,then (3.10) reduces to (3.8).

Although (3.8), an eigenvector-based derivative and (3.10), atransfer function based derivative both calculate sensitivity w.r.t.a change in the matrix A, there are some interesting differencesbetween them. As presented the eigenvector- based derivativeformula calculates the sensitivity w.r.t. a change in one parameter,whereas the transfer function based derivative formula calculatesthe sensitivity w.r.t. a change in as many parameters that theperturbation can affect. The calculus chain rule is not requiredto calculate the sensitivity for some rank-one perturbations (e.g.perturbations described by formula (2.3)).

One formula to calculate the second derivative of one eigen-value λ1 is Caswell (2001)

∂2λ1

∂aij∂akl= v̄(1)

i w(1)l

∑m6=1

v̄(m)k w(m)

j

λ1 − λm+ v̄(1)

k w(1)j

∑m6=1

v̄(m)i w(m)

l

λ1 − λm. (3.11)

Here v̄i and wj are i and j eigenvector components of the leftand right eigenvectors, v̄ is the complex conjugate of the vector vand λm is the m’th eigenvalue. This formula expresses the secondderivative as a linear combination of eigenvectors exceptingthose vectors associated with the concerned eigenvalue and thusrequires all the eigenvectors and eigenvalues of a matrix A.

There are other eigenvector-based second sensitivity formulaewhich do not require all the eigenvectors. e.g. see Nam et al. (2000)where only one pair of eigenvectors are required. Another methodis derived from the characteristic equation of the associatedeigenvalue problem see Prells and Friswell (2000). There are alsomethods which use the group inverse of the M-matrix, see Deutchand Neumann (1984). The method presented here is simpler thanany of those referred to above.

To obtain an expression for the second derivative using thesystem transfer function instead of eigenvectors we differentiate(2.7) twice to obtain

d′(λ)∂2λ

∂p2+

∂λ

∂p

{d′′(λ)

∂λ

∂p

}− 2n′(λ)

∂λ

∂p

− p{n′(λ)

∂2λ

∂p2+

∂λ

∂pn′′(λ)

∂λ

∂p

}= 0. (3.12)

Taking the limit as p → 0 and dividing the result by d′(λ) gives

∂2λ

∂p2+

(∂λ

∂p

)2 d′′(λ)

d′(λ)− 2

n′(λ)n(λ)

d′(λ)2= 0. (3.13)

Rearranging (3.13) gives

∂2λ

∂p2=

(∂λ

∂p

)2 (2n′(λ)

n(λ)−

d′′(λ)

d′(λ)

). (3.14)

Structurally, this formula has a positive term (the sensitivitysquared) and the difference of the two terms between brackets.Of these two terms between brackets the rightmost one d′′(λ)

d′(λ)is

fixed by A, while the left-hand term is an explicit function of aperturbation to the matrix. The constant term d′′(λ)

d′(λ)is a ratio of

derivatives of the characteristic polynomial of the matrix A andbehaves as a convexity/concavity threshold. This fixed term isalways non-negative. This sign of ∂2λ

∂p2indicates how the growth

rate, λmax, changes with respect to the perturbation p. While thefirst derivative gives the local slope of λmax, the second derivativeshows how this slope changes. If a number of perturbations whichhave zero sensitivity locally (where p → 0) are selected thenthese perturbations could have quite different scales of convexityor concavity as the perturbation ranges through admissible values.The larger the value of the second derivative the larger the changein the growth rate per change in the perturbed matrix element(s).Conservation efforts which require intervention in a life-cycle andhence alteration of the vital rates in a matrix, could use convexityand concavity measurements to assist the ranking of differentpotential strategies.

For amore compact version of Eq. (3.14) we can use the identityf ′(x)f (x) = (loge f (x))′ and rewrite this equation as

∂2λ

∂p2=

(∂λ

∂p

)2 (loge

n2

d′

)′

. (3.15)

The sign of the second derivative is now given by the slopeof loge

n2d′ and hence by whether the log of n2

d′ is increasingor decreasing. See Nussbaum (1986) for a more mathematicaltreatment of the log convexity of the growth rate.

D. McCarthy et al. / Theoretical Population Biology 74 (2008) 68–73 71

Let λ denote λ(A), the unperturbed eigenvalue. Using formulasfor first and second sensitivity and a Taylor expansion wecan write

λ(A + pDE) = λ +∂λ

∂pp +

12

∂2λ

∂p2p2 + higher order terms.

Rescaling p by sensitivity, i.e. p̃ =∂λ∂p p and using (3.14) we obtain

λ(A + pDE) = λ + p̃ +12

(2n′(λ)

n(λ)−

d′′(λ)

d′(λ)

)p̃2

+ higher order terms.

This formula emphasizes the role of the second sensitivity inshaping the dependence of λ(A + pDE) on p.

The new formula has a form that lends itself to moreinteresting interpretations of the dynamics of convexity orconcavity of the growth rate than existing formulae, which aremore algorithmic in their representation. There is a clear interplaybetween the perturbed and unperturbed system dynamics, withthe unperturbed part representing a barrier or threshold. Moreinterpretation of these polynomials could yield additional insights.Furthermore, the role of the sensitivity as a scaling factor isrevealed. As the sensitivity is squared (and thus positive) it doesnot have a role in determiningwhether the dynamics are convex orconcave, it reduces or magnifies the convexity or concavity whichis given by the relationship between the polynomials representingperturbed and unperturbed components.

4. Constructing the perturbations

If a number of distinct matrix elements are perturbedsimultaneously, will the second sensitivity of the growth rate (w.r.tthe growth rate) of this disturbed matrix be convex or concave?More specifically, can wemanipulate the perturbation tomake thesecond sensitivity convex or concave? With some constraints onthe perturbation structure, the answer is yes. We do this by firstequating the second sensitivity to zero and then obtaining fixedperturbations. We then parameterise these fixed perturbations.Changes in this parameter then give the desired behaviour to thesecond sensitivity.

The growth rate of a PPM, as a non-negative irreducible matrix,is a convex function of diagonal elements (Cohen, 1978) andis typically (but not always) a concave function of off-diagonalelements. Perturbing a single element is not illuminating in thecontext of using formula (3.14) because the result will either beconvex or concave behaviour. In order to utilise the structure offormula (3.14) we consider perturbations which simultaneouslyalter diagonal and off-diagonal elements and these perturbationsmap naturally to life-history trade-off perturbations. We seekperturbations such that the second sensitivity is zero because theseperturbations partition regions of convex or concave behaviour ofthe growth rate. There are a number of cases in which the secondderivative could be zero: the first derivative could be zero, the twoterms between the brackets of Eq. (3.14) could be equal, or both ofthese cases may occur, although not necessarily with biologicallyadmissible perturbations. To implement this for a single matrixcolumn we parameterise a perturbation structure vector D witha parameter, a, so that, Da = D1 + aD2 where we decomposeD into two vectors, a fixed D1 and one parameterised by aD2. Inthis case

G(z) = E(zI − A)−1(D1 + aD2) = E(zI − A)−1D1

+ aE(zI − A)−1D2 =n1

d+ a

n2

d.

The first and second sensitivities are then given by

∂λ

∂p=

n1(λ) + an2(λ)

d′(λ). (4.16)

∂2λ

∂p2=

(∂λ

∂p

)2 (2n′

1(λ) + 2an′

2(λ)

n1(λ) + an2(λ)− K

)(4.17)

where K is the constant given by d′′(λ)

d′(λ). The second sensitivity is

zero if the first sensitivity is zero in which case using (4.16)

a = af = −n1(λ)

n2(λ). (4.18)

According to (4.17) the second sensitivity is also zero when2n′

1(λ)+2an′2(λ)

n1(λ)+an2(λ)= K , i.e.

a = as = −Kn1(λ) − 2n′

1(λ)

Kn2(λ) − 2n′

2(λ). (4.19)

Here af and as represent the possibly distinct values of theparameter a that equates first or second sensitivity to zero. Wedefine the maximum convexity S2max as

maxa

S2 = S2max = maxa

(∂2λ

∂p2

).

The value of the parameter awhich corresponds to S2max is amax.

5. Applications

To illustrate the utility of the above methodology we considertwo examples. For comparative purposes we look at two stage-structured PPMs, one representing a declining population and onerepresenting a growing population. For the declining populationwe use a PPM (Doak et al., 1994) representing the desert tortoise,Gompherus agassizii, with medium fecundity.

Atort

=

0 0 0 0 0 1.3 1.98 2.570.716 0.567 0 0 0 0 0 0

0 0.149 0.567 0 0 0 0 00 0 0.149 0.604 0 0 0 00 0 0 0.235 0.560 0 0 00 0 0 0 0.225 0.678 0 00 0 0 0 0 0.249 0.851 00 0 0 0 0 0 0.016 0.86

.

(5.20)

For the increasing population we use a PPM (Peters, 1990)representing a tropical fruit, Grias peruviana.

Afruit =

0 0 0 0 0 16 34 59 970.52 0.1 0 0 0 0 0 0 00 0.02 0.52 0 0 0 0 0 00 0 0.16 0.57 0 0 0 0 00 0 0 0.3 0.91 0 0 0 00 0 0 0 0.05 0.93 0 0 00 0 0 0 0 0.05 0.95 0 00 0 0 0 0 0 0.03 0.94 00 0 0 0 0 0 0 0.04 0.98

. (5.21)

To examine the effect of convexity and concavity of thegrowth rate with life-history trade-off perturbations, we perturbeach column of each of the matrices above where both tran-sition probabilities are non-zero; we do not perturb the firstor last columns. The relative increase/decrease on the convex-ity/concavity of lambda is found by varying the parameter awhich

72 D. McCarthy et al. / Theoretical Population Biology 74 (2008) 68–73

represents the relative perturbation of an off-diagonal to a diago-nal. When a = 0 only stasis is perturbed. The range of variationof the parameter a for the perturbation vectors is given by solvingEqs. (4.18) and (4.19).

5.1. Results and discussion

Tables 1 and 2 summarise the second sensitivity valuesresulting from perturbing the matrices. The changes to thesubdiagonal element to set the second sensitivity to zero are given(af and as) aswell as the value of themaximumconvexity and valueof the parameter that results in the maximum convexity (S2max andamax).

With reference to Tables 1 and 2, comparison of the convexity ofthe growth rate for both matrices shows an interesting difference.The desert tortoise matrix shows relatively constant maximumconvexity, approximately 0.3 to 0.5, until the penultimate columnis perturbed when the maximum convexity increases by almostan order of magnitude to a value of approximately 2. In thecase of the tropical fruit the convexity shows low convexity,approximately 0 to 0.15, until the fifth column is perturbed.The convexity then increases to a relatively large value. Theremaining perturbed columns (excepting the penultimate column)also show large convexity. Thematrix for the increasing populationhas many more stages with this larger convexity comparedto the matrix for the declining population. The tropical fruitmatrix therefore has more scope for changes to attain the largeconvexity.

Qualitatively the data for the desert tortoise and the tropicalfruit share some common characteristics (see Tables 1 and 2).For some trade-off perturbations the maximum convexity isachieved near or at the point when the parameter, a, is zero.It is interesting that in many cases even small changes to thesubdiagonal probability reduces the convexity. Thus the matrixappears to be optimised in the sense that irrespective of whetherthe subdiagonal element is increased or decreased, relative to thediagonal element, the convexity of the spectral radius decreasesand so no changes to thematrix are needed tomaximise convexity.This is shown in graphs for perturbations to column 2 for thedesert tortoise and column 2 for the tropical fruit. However, insome cases the parameter, a, needs to be increased or decreasedfor maximum convexity, e.g. perturbations to columns 4, 5 and 7for the desert tortoise and columns 3, 4, 5 and 8 for the tropicalfruit.

Evolutionary interpretation of matrix population models tendsto focus on age-based models (Charlesworth, 1994). If we

Table 1Perturbation to growth/stasis rates to the desert tortoise matrix

Column 2 3 4 5 6 7

S2max 0.298 0.298 0.354 0.289 0.517 2.123amax 0.085 0.085 0.14 0.13 0.04 −0.215af −0.38 −0.38 −0.665 −0.565 −1.095 −0.855as 0.555 0.555 0.945 0.825 1.185 0.42

Table 2Perturbation to growth/stasis rates to the tropical fruit matrix

Column 2 3 4 5 6 7 8

S2max 0.039 0.114 0.14 1.747 2.272 2.764 1.753amax 0.005 0.18 0.365 0.155 0.045 −0.045 0.145af −0.025 −0.32 −0.67 −0.455 −0.69 −0.68 −0.8as 0.02 0.675 1.4 0.765 0.775 0.585 1.1

interpret the second-sensitivity as a measure of non-linearselection (Phillips and Arnold, 1989; Caswell, 2001) then thevalues of S2max in Tables 1 and 2 indicate that correlationalselection (i.e. the correlation between stasis and growth) ismaximised within the fertile stages of a life cycle and only smallchanges are required to the growth rates to maximise convexity.This may indicate that these age- based models representoptimal life histories with regard to the stasis and growthprobabilities. This observation deserves further study in a moreextensive sample of projection matrices, parameterised for realpopulations.

6. Conclusions

We have derived a new formula to calculate the secondsensitivity of the growth rate of population projection matrices.This second sensitivity can be interpreted as a measure ofaccelerating the growth rate. The formula is simple in structureand relatively easy to implement, particularly for structuredperturbations where convexity can be calculated when allelements in a row or column are perturbed by applying the sameformula that is used for perturbing a single element. The transferfunction, n

d , and its derivatives are easily obtained with MATLABby calling the functions ss2tf (state-space to transfer function) andpolyder (polynomial derivative).

It has been shown for the population matrices above thatconvexity of the growth rate is maximised at the fertile stages ofthe life cycle and that the transition rates for early stages of a lifecycle are optimised in that they represent themaximum convexityavailable. There is more scope to alter the transition rates at thefertile life cycle stages to maximise convexity. The value of themaximum convexity changes by roughly an order of magnitudefrom early to later life cycle stages. The sensitivity of the growthrate is relatively small when the convexity ismaximum. In general,for most stages we do not get much of an enhanced convexity bymoving from perturbing only stasis to perturbing both stasis andgrowth probabilities.

Convexity, viewed as a measure of selection strength betweenstasis and growth rates, may be useful in consideration of selectionpressures in evolutionary demography (Jessica et al., 2007). Asevolution may occur on contemporary timescales (Stockwell et al.,2003) theremay also be evolutionary implications of the convexityfor conservation decisions.

The structure of the formula also reveals some aspects ofthe second sensitivity in a way that existing formulae do not.Inspection of Eq. (3.14) shows that the square of the sensitivityscales the convexity and does not affect the sign of the secondderivative. However, the overall effect of this scaling does notappear to be large (see Fig. 5.2). Convex or concave behaviour ofthe growth rate is determined by the interplay of the constant,non-negative threshold term d′′

d′ and the variable term 2 n′

n . Thesetwo terms represent the perturbed and unperturbed systems.The behaviour of the rational function n′

n plays a major role inthe convexity or concavity of the growth rate of a perturbedsystem.

D. McCarthy et al. / Theoretical Population Biology 74 (2008) 68–73 73

Fig. 5.2. Life-cycle trade-off perturbations to desert tortoise and tropical fruitmatrices. In each case the parameter a is varied and the first and second sensitivity is calculated.The horizontal axis shows the variation in the parameter a. The vertical axis showsmagnitude of the first and second sensitivity of the growth rate. The dashed line representsthe variation in first order sensitivity while the solid line represents the variation in second order sensitivity. Plots (a) and (b) represent perturbations to columns 2 and 7 ofthe desert tortoise matrix and plots (c) and (d) represent perturbations to columns 2 and 8 of the tropical fruit matrix. ei denotes the standard ith basis vector whose entriesare zero except for a one in the ith position. eT denotes the vector transpose.

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