Mathematical Biosciences 181 (2003) 165–176 www.elsevier.com/locate/mbs
Habitat destruction, habitat restoration and eigenvector–eigenvalue relations Otso Ovaskainen
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Department of Ecology and Systematics, Metapopulation Research Group, University of Helsinki, P.O. Box 65, Viikinkaari 1, FIN-00014, Helsinki, Finland Received 5 October 2001; received in revised form 10 April 2002; accepted 25 July 2002
Abstract According to metapopulation theory, the capacity of a habitat patch network to support the persistence of a species is measured by the metapopulation capacity of the patch network. Mathematically, metapopulation capacity is given by the leading eigenvalue kM of an appropriately constructed non-negative n n matrix M, where n is the number of habitat patches. Both habitat destruction (in the sense of destruction of entire patches) and habitat deterioration (in the sense of partial destruction of patches) lower the metapopulation capacity of the patch network. The effect of gradual habitat deterioration is given by the derivative of kM with respect to patch attributes and may be straightforwardly evaluated by sensitivity analysis. In contrast, destruction of entire patches leads to a rank modification of matrix M, the effect of which on kM may be derived from eigenvector–eigenvalue relations. Eigenvector–eigenvalue relations have previously been analyzed only for symmetric matrices, which restricts their use in biological applications. In this paper I generalize some of the previous results by deriving eigenvector–eigenvalue relations for general non-symmetric matrices. In addition to the exact eigenvector–eigenvalue relations, I also derive eigenvalue perturbation formulae for rank-one modifications. These results lead to simple and intuitive approximation formulae, which may be used e.g. to assess the contribution of particular habitat patches to the metapopulation capacity of the landscape. The mathematical results presented are not restricted to the metapopulation context, but they should find a number of useful applications in biology, engineering and other applied sciences, where the removal (or addition) of matrix rows and columns often corresponds in a natural manner to decreasing (or increasing) the degrees of freedom of the focal system. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: Eigenvalue–eigenvector relation; Rank-one modification; Metapopulation dynamics; Habitat destruction; Habitat restoration; Patch value
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Tel.: +358-91 915 7735; fax: +358-91 915 7694. E-mail address: otso.ovaskainen@helsinki.fi (O. Ovaskainen).
0025-5564/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 0 2 ) 0 0 1 5 0 - 5
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1. Introduction One of the greatest challenges in the current development of biological theory is to account for various kinds of heterogeneities. Individuals are not identical, populations are not homogeneously mixed, and the world is not a regular lattice of identical sites. Often heterogeneity is of discrete nature and is most naturally described through a matrix. Consequently, matrix analysis has become an integral part of biological theory. A number of fundamental biological quantities have been found to appear as eigenvalues, examples including the growth rate of an age-structured population [1], the basic reproduction ratio [2] of an infectious disease, measures of community behaviour [3], the fitness of an individual [4] and the effective size of a subdivided population in population genetics [5]. Often one is interested not only in the absolute value of the quantity represented by the eigenvalue, but also in its sensitivity with respect to relevant model parameters. For example, in addition to knowing the current growth rate of an age-structured population, one would like to know the expected change in the growth rate that would follow from improving the survival of juveniles. As a consequence, the perturbation theory of eigenvalues is being increasingly used in the biological literature. In its most traditional form, eigenvalue perturbation theory is focused on the derivative of an eigenvalue, i.e., the change in an eigenvalue due to a small change in matrix elements. Such perturbation methods are most natural for considering the effect of small and gradual changes. However, in many biological applications, it is often relevant to consider large perturbations. As an example, I will study the following problem in this paper. Consider a fragmented landscape consisting of a discrete set of n habitat patches. According to metapopulation theory, the capacity of the patch network to support the persistence of a species is given by the metapopulation capacity kM of the fragmented landscape, defined as the leading eigenvalue of an appropriate Ôlandscape matrixÕ M, which describes how the spatial structure of the habitat patch network (e.g. patch areas and interpatch distances) affects the colonization–extinction dynamics of the species in the patch network [6]. A possible management scenario in such a landscape is the destruction of one (or some) of the habitat patches or, in the positive case, the restoration of a new habitat patch into a particular location. In order to evaluate the likely consequences of such management options, one should be able to estimate the change in the eigenvalue kM due to a rank modification of matrix M. For example, the destruction of a habitat patch corresponds to the deletion of a matrix row and a matrix column. Due to the singular nature of such a perturbation, approaches based on eigenvalue derivatives may not be applicable and eigenvalue–eigenvector relations should be used instead. Eigenvector–eigenvalue relations have been previously analyzed in the mathematical literature only for symmetric matrices, which restricts their applicability in many biological situations. The main contribution of this paper is given in Section 2, where I generalize some of the previous results by deriving eigenvector–eigenvalue relations for non-symmetric matrices. As a corollary to the exact eigenvalue–eigenvector relations, I derive approximation formulae for eigenvalue perturbations due to rank-one modifications. In particular, I show that if the dth row and the dth column of matrix A are deleted, the change in a simple eigenvalue k is given as k k ky d xd ;
ð1Þ
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where k is an eigenvalue of the modified matrix, and x and y are the right and left eigenvectors corresponding to the eigenvalue k, respectively. I also give an analogous formula for the change in a simple eigenvalue due to addition of a new row and a new column to the matrix. After presenting the mathematical theory in Section 2, I apply the eigenvector–eigenvalue relations in Section 3 to study the effects of habitat destruction and habitat restoration in the metapopulation context.
2. Eigenvector–eigenvalue relations Eigenvalue-eigenvector relations for rank-one modifications have previously been presented for symmetric matrices [7–9]. The principal result is that if A is a symmetric matrix and A is obtained by deleting the last row and the last column from A, it holds that Qn1 j¼1 ðki kj Þ ðiÞ 2 Q ½xn ¼ n ; ð2Þ j¼1;j6¼i ðki kj Þ n1 ðiÞ is the normalized where fkj gnj¼1 and fk j gj¼1 are the eigenvalues of A and A , respectively, and x eigenvector corresponding to a simple eigenvalue ki . I start by showing that a formula analogous to Eq. (2) holds for non-symmetric matrices as well. Although in most biological applications the matrix will be non-negative, and the eigenvalue of interest will be the leading eigenvalue, I state the result for simple eigenvalues of complex valued matrices, as the proof is not more difficult for the general case. I also give a formula analogous to Eq. (2), which holds for the addition of a new row and a new column to matrix A.
Theorem 2.1. Let A 2 Cnn , and let A 2 Cðn1Þðn1Þ be the matrix obtained by deleting the dth row and dth column from A, and let Aþ be the matrix obtained by adding a new row and a new column to A, A aC þ A ¼ H ; aR a n1 þ nþ1 where the superscript H denotes conjugate transpose. Let fki gni¼1 , fk i gi¼1 and fki gi¼1 be the eigenvalues of A, A and Aþ , respectively. Let xðiÞ and yðiÞ be the right and left eigenvectors corresponding to a simple eigenvalue ki , normalized as yðiÞH xðiÞ ¼ 1. Then Qn1 j¼1 ðki kj Þ ðiÞ ðiÞ y d xd ¼ Qn ; ð3Þ j¼1;j6¼i ðki kj Þ
ðy
ðiÞH
aC ÞðaHR xðiÞ Þ
Qnþ1 þ j¼1 ðki kj Þ ¼ Qn : j¼1;j6¼i ðki kj Þ
ð4Þ
Proof. The proof follows closely the proofs of Lemma 2.1 and Theorem 2.1 in [7]. Let YH ; X 2 Cnn be such that J ¼ YH AX is a Jordan decomposition of A, and let B ¼ ðA kIÞ1 for 1 k 6¼ ki , where i ¼ 1; . . . ; n. It is easy to see that B ¼ XðJ kIÞ YH , and thus
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bkl ¼
ðjÞ ðjÞ n X xk y l ; kj k j¼1
ð5Þ
where bkl is the ðk; lÞ element of B. As ki is a simple eigenvalue, it follows that ðiÞ ðiÞ
lim ðki kÞbkl ¼ xk y l :
k!ki
ð6Þ
Let ed denote the dth unit vector with edi ¼ ddi , and let ðA kIÞd denote the matrix A kI with its dth column replaced by the unit vector ed . By Eq. (6) and by CramerÕs rule, det½ðA kIÞd ; detðA kIÞ
ð7Þ
Qn1 j¼1 ðk kj Þ ¼ lim ðk ki Þ Qn ; k!ki j¼1 ðk kj Þ
ð8Þ
ðiÞ ðiÞ
xd y d ¼ lim ðki kÞ k!ki
ðiÞ ðiÞ xd y d
from which Eq. (3) follows. Eq. (4) follows in a similar fashion. I will next derive two eigenvalue perturbation formulae approximating the change in a simple eigenvalue due to rank-one modifications. The formulae will appear as corollaries to Theorem e Þ the matching distance between the eigenvalues of A and A e , defined 2.1. I will denote by mdðA; A by e Þ ¼ minfmax jk~pðiÞ ki jg; mdðA; A p
i
ð9Þ
where p is taken over all permutations of f1; 2; . . . ; ng. Corollary 2.1. Let A 2 Cnn , and let A 2 Cnn be the matrix obtained from A by setting the elements in the dth row and dth column to zero. Let k be a simple eigenvalue of A with corresponding right and left eigenvectors x and y (normalized as yH x ¼ 1). Let d be the minimum distance between k and the other eigenvalues of A, and let m ¼ mdðA; A Þ: Then, if m < d, A has a unique eigenvalue k for which ð10Þ k k ¼ ky d xd ð1 þ Þ; where n1 d ðn 1Þm þ Oðm2 Þ: jj 6 1¼ ð11Þ dm d Proof. Let ki and k i denote the eigenvalues of A and A , arranged so that k1 ¼ k and ki matches ki in the sense of Eq. (15). By Theorem 2.1, k k ¼ ky d xd
n Y k kj : k k j j¼2
The claim follows as ðk kj Þ=ðk k j Þ ¼ 1 þ j , where jj j < m=ðd mÞ.
ð12Þ
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e ; Aþ 2 Cðnþ1Þðnþ1Þ be the matrices Corollary 2.2. Let A 2 Cnn , and let A A aC A 0 þ e A¼ ; A ¼ H : aR a 0 0 Let k 6¼ 0 be a simple eigenvalue of A with corresponding right and left eigenvectors x and y (normalized as yH x ¼ 1). Let d be the minimum distance between k and the other eigenvalues of A augmented by zero, and let m ¼ mdðA; Aþ Þ: Then, if m < d, Aþ has an unique eigenvalue kþ for which 1 kþ k ¼ ðyðiÞH aC ÞðaHR xðiÞ Þð1 þ Þ; k where
n d nm þ Oðm2 Þ: jj 6 1¼ dm d Proof. Proof is similar to that of Corollary 2.1.
ð13Þ
ð14Þ
In order to use Corollaries 2.1 and 2.2, one should be able to say that the matching distance e ¼ Aþ ) is sufficiently small. It is well-known that this is guaranteed e Þ (where A e ¼ A or A mdðA; A e A is sufficiently small. In the general case, it holds that [10] if the matrix E ¼ A e Þ 6 ð2n 1ÞðkAk þ k A e k Þ11=n kEk1=n : mdðA; A 2 2 2
ð15Þ
If A is diagonalizable, the bound can be improved to [10] e Þ 6 ð2n 1ÞkX 1 EX k; mdðA; A
ð16Þ where X AX ¼ K is a diagonalization of A and k k denotes any consistent matrix norm such that kdiagða1 ; . . . ; an Þk ¼ maxi jai j. For example, k k may be any matrix norm subordinate to an absolute vector norm [10]. 1
3. Habitat destruction and habitat restoration In this section I will apply the results derived in Section 2 to analyze the consequences of habitat destruction and habitat restoration in the metapopulation context. Consider a species inhabiting a highly fragmented landscape consisting of a discrete set of n habitat patches (for a thorough discussion of the ecology of such species see [11]). According to the metapopulation concept, the species may persist in the network if local extinctions are compensated for by recolonizations of empty habitat patches. For the purpose of illustration, I model the colonization–extinction dynamics of the species with a relatively simple metapopulation model, the spatially realistic version of the Levins model [6], which belongs to the larger family of patch occupancy models [12]. The spatially realistic Levins model is a deterministic continuoustime model, which gives the rate of change in the probability pi of patch i being occupied as [6,13]
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dpi ¼ Ci ð1 pi Þ Ei pi ; ð17Þ dt where Ci and Ei are the colonization and extinction rates of patch i, respectively. Let Ai denote the area of patch i and dij the distance between patches i and j. I assume that P Ci ¼ cAfi im j6¼i Afj em f ðdij Þpj and that Ei ¼ e=Afi ex . Here e and c are extinction and colonization rate parameters, and fim , fem and fex describe the scaling of immigration, emigration and extinction rates by patch area, respectively. The justification of the functional form for Ci is based on the reasoning that the colonization rate of an empty patch increases with increasing size of the patch, with increasing number of potential source patches (occupied patches in the vicinity of the empty patch), with increasing sizes of potential source patches, and with decreasing distances to potential source patches. The function f describes the dispersal kernel, which gives the effect of distance on colonization success. For example, [6] made the phenomenological assumption that f ðdÞ ¼ ead with 1=a giving the average migration distance. More generally, the function f may be derived from submodels (such as correlated random walks) for the migration phase. The functional form for the extinction rate Ei is based on the reasoning that large patches tend to have large population sizes, and they consequently have a small extinction risk. For further discussion and justification of the functional forms of the colonization and extinction processes see [11,12]. The above assumptions lead to a Ôlandscape matrixÕ M with elements
f þf f Ai ex im Aj em f ðdij Þ for j 6¼ i; ð18Þ mij ¼ 0 for j ¼ i: The element ðc=eÞmij gives the contribution that patch j makes to the colonization rate of patch i when patch i is empty, multiplied by the expected lifetime of patch i when the patch is occupied. As the fraction of time that patch i is occupied is determined by its colonization and extinction rates, ðc=eÞmij may be viewed to measure the fraction of time that patch i would be occupied if the only source of immigrants would be patch j. The metapopulation capacity kM of the habitat patch network is defined as the leading eigenvalue of matrix M. Defining d ¼ e=c, the threshold condition for persistence (in the sense of the existence of a stable non-trivial equilibrium state of Eq. (17)) is given as [6,12] ð19Þ kM > d: In Eq. (19), the metapopulation capacity kM is a landscape index measuring the capacity of the habitat patch network to support the long-term persistence of a species, whereas the species parameter d sets the threshold value for the persistence of the particular species. The parameter d is independent of the landscape, 1=d measuring the expected number of colonization events that a local population inhabiting a unit size patch would cause in its lifetime to an empty patch of unit size located at distance d with f ðdÞ ¼ 1. Metapopulation capacity is analogous to the basic reproduction ratio R0 , which has been used in epidemiology to describe the number of new infections that a single infectious individual is expected to give raise to [2]. The two quantities are related by R0 ¼ kM =d, and thus the well-known threshold condition R0 > 1 is equivalent with the condition given by Eq. (19). The reason for separating the species parameter d in the metapopulation context is that Eq. (19) allows one to analyse the metapopulation capacity of a landscape even for species for which the parameter d is not known.
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While metapopulation capacity characterizes the capacity of the entire network to support the persistence of a species, it would often be desirable to be able to assess the values of individual habitat patches. The ÔvalueÕ of a habitat patch is not a well-defined quantity as such, as a multitude of measures can be justified. For example, one might consider the contribution of a habitat patch to the expected size of the metapopulation, or the contribution of a habitat patch to the expected time to metapopulation extinction [14,15]. I will concentrate here on a simple measure that is appropriate for the case of rare species, which is Vi , the contribution of a habitat patch i to the metapopulation capacity of the landscape. More precisely, Vi is defined as the decrease in metapopulation capacity due to the destruction of patch i. As the destruction of patch i corresponds to the deletion of a row and a column from matrix M, and as the leading eigenvalue of an irreducible non-negative matrix is simple, the theory developed in Section 2 applies immediately. Most importantly, Corollary 2.1 gives the approximation formula ð20Þ Vi Vei ¼ kM y i xi ; where x and y are the right and the left leading eigenvectors corresponding to the eigenvalue kM . Similarly, one may apply Corollary 2.2 to consider the value of a new patch that would be added to the network, by analyzing the increase in metapopulation capacity due to the addition of a new row and a new column to matrix M. For the purpose of illustration, consider the hypothetical network of 30 habitat patches shown in Fig. 1. Fig. 2 compares the approximations Vei for all the 30 patches with the exact values. The approximations are very close to the exact values, the largest relative deviations occurring for the patches that have the largest contribution to the metapopulation capacity of the patch network, as expected from Corollary 2.1.
Fig. 1. A hypothetical habitat patch network of 30 patches used in Figs. 2–4. The patches are randomly located within a 5 5 square, patch areas being lognormally distributed with mean 1 and standard deviation 1. The patch indicated by an arrow is further analyzed in Fig. 3. The contour lines depict the value of the function gðzÞ in Eq. (24). Moving one contour line away from the core of the network corresponds to the reduction of the value of gðzÞ by one half.
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Fig. 2. The accuracy of the approximation formula (20). The dots depict the true and approximate contributions of individual habitat patches to the metapopulation capacity of the patch network shown in Fig. 1. The line shows identity. The parameter values fex ¼ 0:8, fim ¼ 0:5, fem ¼ 0:5 and f ðdÞ ¼ ed are used also in Figs. 3 and 4.
The approximation formulae (Corollaries 2.1 and 2.2) relate in an interesting way to the traditional eigenvalue perturbation theory. It is well known that if matrix A is perturbed to a matrix e as A e ¼ A þ E, a simple eigenvalue k of A is perturbed to an eigenvalue k~ of A e as [10,16] A k~ ¼ k þ yH Ex þ OðkEk2 Þ; ð21Þ H where x and y are the right and the left eigenvectors (normalized as y x ¼ 1) corresponding to k. Comparison of Eqs. (21) and (10) reveals that the metapopulation capacity of the habitat patch network behaves non-linearly with respect to gradual habitat deterioration. This is illustrated in Fig. 3, where I consider a perturbation obtained by multiplying the area of a single patch by 1 s, where 0 < s < 1 represents the loss of patch area. Applying Eq. (21) for small s and extrapolating to s ¼ 1 would lead to ð22Þ Vi fkM y i xi ; where f ¼ fim þ fem þ fex is called the patch area scaling factor of the model [15]. The Eq. (22) is, in the general case, in disagreement with Eq. (20). The problem arises because the error term 2 OðkEk Þ in Eq. (21) is of the same magnitude as the first order term ðy H ExÞ=y H x for s ¼ 1. In Fig. 3a, f ¼ 1:8 > 1, and consequently habitat deterioration is most detrimental in the early stage of destruction when the patch loosing area is still large. On the contrary, in Fig. 3b, where f ¼ 0:5 < 1, habitat deterioration is most detrimental at the stage when most of the patch has already been destroyed. Corollaries 2.1 and 2.2 together with Eqs. (15) and (16) give explicit upper bounds for the error in the approximations, the convergence rate ¼ OðkEkÞ being optimal for the case of diagonalizable matrices. A caveat to the error bounds is that the quantitative bounds are likely to largely
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Fig. 3. The effect of gradual deterioration (loss of area) of the patch indicated by an arrow in Fig. 1. The continuous curves depict the change in the metapopulation capacity kM of the patch network due to deterioration of the patch obtained by multiplying the patch area by the factor 1 s. Dot A depicts the metapopulation capacity of the original patch network, and dot B gives the metapopulation capacity of the patch network from which the patch has been removed. Dots C and D are approximations of dot B, dot C being based on the approximation formula (20), and dot D being based on linear extrapolation of the derivative as given by Eq. (22). In panel (a), the parameter values are as in Fig. 2, whereas in panel (b) they have been modified to fex ¼ 0:3, fim ¼ 0:1, fem ¼ 0:1.
overestimate the true errors. Fig. 4 gives an example, panel (a) showing the error bars given by Corollary 2.1 together with Eq. (16) for the example in Fig. 2. The error bars become narrower only when the relative contributions of the habitat patches to the metapopulation capacity are negligible (Fig. 4b). Corollary 2.2 allows us to analyze the effect of habitat restoration, defined here as a creation of a new patch in a particular location. Consider again a network of n patches, and let a new patch of area A be added to the network in a location with distances di (i ¼ 1; . . . ; n) from the existing patches. Utilizing Corollary 2.2, the increase in metapopulation capacity due to the addition of the new patch is given by ! ! n n X Af X fex þfim fem þ y i Ai f ðdi Þ xi Ai f ðdi Þ ; ð23Þ kM kM kM i¼1 i¼1
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Fig. 4. The error bounds given by Corollary 2.1 together with Eq. (16). Panel (a) shows the error bars for the example shown in Fig. 2. In panel (b), the structure of the habitat patch network has been modified so that the patch areas are lognormally distributed with mean 1 and standard deviation 4. The line shows identity.
where the accuracy of the approximation increases with decreasing size of the new patch. A convenient feature of Eq. (23) is that it separates the effects of area and location of the patch to be added to the network. Letting z ¼ ðzx ; zy Þ denote the ðx; yÞ-coordinates of the new patch, Eq. (23) suggests that the value of the location z is given by ! ! n n X X fex þfim fem y i Ai f ðdi Þ xi Ai f ðdi Þ ; ð24Þ gðzÞ ¼ i¼1
i¼1
where the right-hand side depends on the location z through the distances di . The function gðzÞ reveals the core of the habitat patch network, as illustrated by the contour lines in Fig. 1. By Eq. (23), the effect of the area of the new patch to be added to the network scales to the power of f. As expected, the more sensitive the basic ingredients of metapopulation dynamics (local extinction, immigration and emigration) are to patch area, the more important is the area of the patch to be restored.
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4. Discussion The main contribution of this paper is to generalize eigenvalue–eigenvector relations for nonsymmetric matrices. While the exact eigenvalue–eigenvector relations given by Theorem 2.1 are mainly of theoretical interest, the approximations given by Corollaries 2.1 and 2.2 can be readily applied to practical problems. As illustrated by Fig. 2, the approximations may be fairly accurate for a number of potential applications, while the quantitative error bounds may often be far too pessimistic (Fig. 4). However, one should note that the error bounds apply to the general case, whereas Fig. 4 relates to the approximation of the leading eigenvalue of a non-negative matrix, which is certainly a special case. In order to demonstrate the applicability of eigenvalue–eigenvector relations in population biological theory, I have applied Corollaries 2.1 and 2.2 to examine the effects of habitat destruction and habitat restoration in the metapopulation context. This problem has been studied earlier mainly through simulation studies (see e.g. [17,18]), where it is possible to include a much more detailed description of the ecology of the species, but at the same time the analysis is restricted to comparing numerical results for a set of parameter values. Analytical results from simpler models such as the one studied in this paper enhance general understanding and bring conceptual clarity to the interpretation of simulation results. For example, it would have been hard to demonstrate with a numerical study the separation of the contributions of patch area and patch location to metapopulation capacity (Eq. (23) and the contour lines in Fig. 1). Similarly, it would be hard to see with numerical studies only the effect that f has on the non-linear relationships between gradual habitat deterioration and the metapopulation capacity of a patch network (Eqs. (20) and (22) and Fig. 3). As the mathematical theory presented in Section 2 is fairly general, I expect that its applicability is not restricted to the metapopulation context. As an example, one potential application of these results might be found in epidemiology, where the basic reproduction ratio R0 is given by the leading eigenvalue of a matrix describing the spatial or social heterogeneity in the host population [2]. Vaccination scenarios effectively correspond to removal of susceptibles, their effect being thus measured by the decrease in R0 due to rank modifications of the relevant matrix. Another application could be found in the study of food webs (see e.g. [3,19]), where the effect of extinction or introduction of species on community stability may be measured in the change of the eigenvalues of the community matrix due to rank modifications. Indeed, as the removal (or addition) of matrix rows and columns often corresponds in a natural way to decreasing (or increasing) the degrees of freedom of the focal system, and as eigenvalues are of major importance in characterizing the behaviour of systems described through matrices, useful applications of the theory presented here should be found also in other fields of biology, engineering and other applied sciences.
Acknowledgements I thank Ilkka Hanski and Marko Huhtanen for their valuable comments. This study was supported by the Academy of Finland (grant number 50165 and the Finnish Centre of Excellence Programme 2000–2005, grant number 44887).
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