Proceedings of the 14th International Conference on

Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering Costa Ballena, Rota, Cádiz (Spain) July 3rd-7th, 2014

CMMSE 2014 VOLUME II Editor: J. Vigo-Aguiar Associate Editors I. P. Hamilton, J. Medina, P. Schwerdtfeger, W. Sprößig, M. Demiralp, E. Venturino, V.V. Kozlov, P. Oliveira

Contents: Volume II A logic-based approach to compute a direct basis from implications. Cordero P., Enciso M., Mora A., Ojeda-Aciego M., Rodríguez-Lorenzo E. ........................ 331 A new parametric class of iterative methods for solving nonlinear systems. Cordero A., Feng L., Magreñán A.A., Torregrosa J.R. ....................................................... 340 A class of bi-parametric families of iterative methods for nonlinear systems. Cordero A., Maimó J.G., Torregrosa J.R., Vassileva M.P. ................................................. 350 On generalization of the variants of Newton's method for solving nonlinear equations. Cordero A., Torregrosa J.R. ............................................................................................... 364 Adjoint triples versus extended-order algebras. Cornejo M.E., Medina J., Ramírez-Poussa E. .................................................................... 375 An study for the Microwave Heating of a Half-Space through Lie symmetries and conservation laws. de la Rosa R., Gandarias M.L., de los Santos M. .............................................................. 385 Error analysis in the reconstruction of a convolution kernel in a semilinear parabolic problem. De Staelen R.H., Slodicka M. ............................................................................................ 396 A hybrid algorithm for the split generalized equilibrium and the system of variational inequality problems. Deepho J., Kumam W. ...................................................................................................... 399 A mathematical model of a single population with habitat fragmentation in progress. Del-Valle R., Córdova-Lepe F. ........................................................................................... 429 Weighted Tridiagonal Matrix Enhanced Multivariance Products Representation of Finite Interval Data. Demiralp E. ....................................................................................................................... 441 Tridiagonal Matrix Enhanced Multivariance Product Representation (TMEMPR) for Matrix Decomposition. Demiralp E., Demiralp M. ................................................................................................. 446

Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2014 3–7July, 2014.

A mathematical model of a single population with habitat fragmentation in progress Rodrigo Del-Valle1 and Fernando C´ ordova-Lepe2,1 1 2

Facultad de Ciencias B´ asicas, Universidad Cat´ olica del Maule

Facultad de Ciencias B´ asicas, Universidad Metropolitana de Ciencias de la Educaci´ on emails: [email protected], [email protected]

Abstract For a single species, we considered that the habitat of its population, during certain interval of time, is being fragmented by the emergence of an interior border, which divide it in two disjoint patches. Assuming logistic laws for times before and after the fragmentation interval, we connect past with future to study the abundance on the fragments while they appear. The above is got it by the postulating and analyzing of a natural differential system that rules the abundances of the transition. Key words: Habitat fragmentation Population abundance Logistic growth

1

Introduction

Fragmentation of ecosystems are often evaluated as a progressive phenomenon directly linked to the transforming human action. Furthermore, it is considered having a very high impact as a factor of loss of biodiversity and abundance of populations, for an example in native forest fragmentation (see [1]). At a level of habitat, fragmentation is a phenomenon of transformation of the landscape, that is, a habitat that was a continuous medium after a time becomes two or more fragments. In the literature, we recognize at least two meanings to the concept of fragmentation: First: The fragmentation can be understood as the process in which at least one of the patches which form the habitat has a division that is accompanied with reduction in the size of the sum of the emerging patches and / or a gradual isolation of these. The above generally occurs within what is called a hostile matrix (see [3]). The work, at the theoretical or practical level, with mentioned concept has an important drawback, since this concept does

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Modelling a process of fragmentation

not allow to separate and calibrate the different concurrent effects that can be distinguished: the division, the habitat loss and the isolation. Although in nature, the division of a patch (as a process of emergence of edge, i.e., border) usually occurs in conjunction with some loss of habitat and also with continued isolation, some authors recognize that this mixture, as a concept unifying factors, is a weakness for the study of fragmentation (see [5]), it is also likely that these contributing factors have independent effects on habitat richness and abundance of populations (see [2]). Second: The appearance of fragments as a simple division in an ideal sense, is other conceptual possibility. Then, exaggerating the reality, we can consider by fragmentation as the splitting process by the development of a border or edge of codimension one, which is not a priori a habitat loss. This means that in a habitat formed by one planar patch, emerges a border which is a curve or separatrix which divides the patch at least in other two. This option, which favorese the analysis, will be the approach assumed by the present article. We also consider that when this fragmentation has been set, there is no possibility of migration between patches, that is, it generates a total isolation. It is recognized the need to further study the effects of fragmentation processes in the above second meaning. In [4], it is shown that experimental studies with this sense of fragmentation per se are quite few. From a total of 120 works found (period Jan. 2000 Nov. 2003 in ISI Web of Knowledge) related to experimental studies of fragmentation, only five papers separate the effects of fragmentation of those effects of habitat loss. Restricting ourselves to study only the theoretical implications of fragmentation per se by mathematical modeling, this paper aims to contribute in this direction. Moreover, some authors suggest that modeling gives better compression expectations when it is taken at the landscape level, analizing one or a small group of species, rather than the entire ecosystem. We limit ourselves to study the consequences on the abundance of a single population in its habitat that has a fragmentation in two patches and only during the interval of time that this process is completed. The literature shows many works centered in try to understand the effect of the habitat fragmentation on abundance of the populations. A simple model that works with a single species, modeled by a logistic law of growth can be found in [6]. Their work resolve the following question “Does habitat fragmentation influence the abundance of organisms in the absence of habitat loss?” It also will be our main research question, but centered in the dynamic during the interval time that the process late. The paper will be organized as follows: Firstly, in Section 2, we represent the fragmentation transition by considering an nonautonomouos differential system. Secondly, in Section 3, we present the implications on the abundance of the total and partial population of the fragmentation process by means of an analytical study of the introduced model. We finish with Section 4, giving some numerical simulations to illustrate the behavior possibilities.

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´ rdova-Lepe Rodrigo Del-Valle, Fernando Co

2

The mathematical model of fragmentation

Let us consider that the population is undergoing a process of fragmentation during a time interval [ti , tf ], ti < tf . This population is located in a habitat represented in mathematical terms by a region (open, connected and non-empty) bounded Ω of �2 . To represent the habitat division into two patches and the appearance of edge, we assume that Ω is the union of three sets: two disjoint regiones A and B, and also a common border Γ = ∂A ∩ ∂B, which is an arcconnected set. Moreover, we assume that Γ is the graph of a regular and simple curve α : (ti , tf ) → Ω such that α(ti −) and α(tf +) exist. Denoting Γ(t) = {α(s) : s ∈ (ti , t)} with t ∈ [ti , tf ], we have that, at any instant t of the process, the habitat is represented by the set Ω(t) = Ω \ Γ(t). Notice that Ω(ti ) = Ω and Ω(tf ) = A∪B, because Γ(ti ) = φ and Γ(tf ) = Γ respectively. Then, at t = ti the population is in a unique patch Ω, but at the final instant, t = tf , the population is distributed in two isolated patches A and B. In set-theoretic terms, the loss of original habitat to a time t ∈ (ti , tf ) is Γ(t). As this is a set of measure (area) null, we will say, by abuse of language, we will say that we have a process of fragmentation by arise of edge, but without habitat loss. Our interest is in connecting population dynamics that existed before the onset of fragmentation (t < ti ), with that which occurs after completion of the fragmentation, (t > tf ). To simplify this scenario, we assume a before and after fragmentation of the population modeled by classical logistic equations. To work with population densities, we will assume that the sets Ω, A and B, and those considered hereafter, are quantified by a nonnegative measure (area) m(·). Notice that, m(Ω) = m(Ω(t)) = m(A) + m(B), for each t ∈ �. A unique patch (t < ti ): If we represent by xA (t) and xB (t) the abundances of the population in the zones A and B respectively at time t, and we assume that there are no individuals in areas of zero measure, then we will consider that the total population, xA (t) + xB (t), is given by the logistic equation: � � xA + xB � , (1) (xA + xB ) = r(xA + xB ) 1 − K where r denotes the intrinsec rate of growth and K the carrying capacity of Ω. Two patches (t > tf ): The zones A and B are now phisically divided and there is not migration between them (isolation). So they form two closed patches. However, there was not loss of habitat in the process. So we have two ordinary differential equation governing the growth one for any patch. This is,  � �  x˙ A (t) = rxA (t) 1 − xA (t) , KA � � (2)  x˙ B (t) = rxB (t) 1 − xB (t) . KB c �CMMSE

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The transition (ti < t < tf ): We are counting the population size according to a border that is forming. It is expected that: � x˙ A (t) = xA (t) ft (xA (t), xB (t)), (3) x˙ B (t) = xB (t) gt (xA (t), xB (t)), where, ft (·, ·) and gt (·, ·) could be linear functions of the pair (xA , xB ), for any t ∈ [ti , tf ].

In order to postulate, for any t ∈ [ti , tf ], a form for the functions ft , gt : [0, ∞]2 → �, we follow the classical deduction logistic model x� (·) = r(x(t)) x(t). Here, in each patch the per capita rate of growth is decomposed into the difference r(x) = b(x) − d(x), with the natality b(x) as the mortality d(x), are linearly affected by the density D. Then, b(x) = b0 − b · D and d(x) = d0 + d · D,

(4)

where b0 and d0 are respectively the natality and mortality to very low densities. Moreover, b and d are the decreasing (resp. increasing) of the natality (resp. mortality) per unit density. The density D at the instant t = ti , this is, before fragmentation, and that affects both the patch individuals A and B, is D = (xA (ti ) + xB (ti ))/(m(A) + m(B)), where m(·) is some space mesure, for intance, area. Substituting this value D in (4), we obtain (1), with r = b0 − d0 and K = KA + KB , where KC = m(C)

b0 − d 0 , C ∈ {A, B}. b+d

Now at t = tf , the density D in patch A (resp. B) is xA /m(A) (resp. xB /m(B)). Substituting this density in (4), we obtain the first equation of (2) (resp. the second). These last two paragraphs lead us to wonder about the density at time t intermediate in [ti , tf ]. Note that it is natural to think that A, at each time t, while it is not completely isolated, is an area influenced (for purposes of limiting rates) by the density in a territory Bη , whose measure represents a fraction η(t) of m(B), i.e., we can work with a weighted average density: (5) DA (t) = p · (density in A) + q · (density in Bη ), certain positives p and q, such that p + q = 1. The weights p and q can be assumed representing proportionaly the influencing zones on the rates of A, this is, A and Bη . As we have a mesure and m(A∪Bη ) = m(A)+η ·m(B), we suppose: η · m(B) m(A) and q = . p= m(A) + η · m(B) m(A) + η · m(B) Substituting in (5), we obtain DA (t) =

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If, at the same instant t, the focus is in zone B, then assuming a fraction ξ of A which is influencing B, and assuming the average perspective, we have that the density DB in area B is given by: ξ(t) · xA + xB . DB (t) = ξ(t) · m(A) + m(B) Doing the calculations, by replacing these densities in (4) for each of the areas A and B, we obtain the following differential system:  � �  x� = xA ft (xA , xB ) = rxA 1 − xA +η(t)xB , A KA +η(t)KB � � (6)  x� = xB gt (xA , xB ) = rxB 1 − ξ(t)xA +xB , B ξ(t)KA +KB

where η, ξ : [ti , tf ] → [0, 1] are decreasing functions, such that η(ti ) = ξ(ti ) = 1 and η(tf ) = ξ(tf ) = 0. The functions η(·) and ξ(·) pretend to be a measure of the “connectivity” between patches A and B at time t, t ∈ [ti , tf ]. Hereinafter, for simplicity, we assume that η(·) and ξ(·) are equal. There is an other route to postulate the system (6). It is the case when we are inspired in a simple competition model with variable curriying capacity and competitors for any subpopulation. Notice that equation (1) can be deduced from the additon of the equations of (6), because η(t) = ξ(t) = 1, for each t ≤ ti .

3 3.1

Main results Equal densities

Theorem: Let us consider that (xA (·), xB (·)) is a solution of (6) such that for t = ti we have xA (ti )/KA = xB (ti )/KB . Then for each t ∈ [ti , tf ], we have: (a) xA (t)/KA = xB (t)/KB . (b) xA (t) + xB (t) satisfy (1). Remark: The equality (a) indicates that when the regions A and B present at t = ti same densities, these persist over time as long as the fragmentation. In this case, with respect to the total abundance, the fragmentation makes no difference. Proof (a): Indeed, considering the integral versions of the equations of the system (6), we have � t |α(t) − β(t)| ≤ |α(ti ) − β(ti )| + r |∆(s)|ds, t ∈ [ti , tf ], (7) ti

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where α(·) = xA (·)/KA , β(·) = xB (·)/KB and � � � � α + ηβ(KB /KA ) ηα(KA /KB ) + β ∆=α 1− −β 1− (8) 1 + η(KB /KA ) η(KA /KB ) + 1 Notice that, � � α(1 + λη) + β(1 + λ−1 η) , ∆ = (α − β) 1 − (1 + ηλ)(1 + ηλ−1 ) where λ = KA /KB . Let us define M = max{KA + KB , xA (ti ), xB (ti )}. Then xA (t), xB (t) ≤ M for each t ≥ ti . Indeed, if there exists t1 > ti with xA (t1 ) > M ≥ xA (ti ), then there exists t2 ∈ (ti , t1 ] such that xA (t2 ) > M and x�A (t2 ) > 0. However, xA (t2 ) > KA + KB implies xA (t2 ) + ηxB (t2 ) (1 − η(t2 ))KB + η(t2 )xB (t2 ) 0 such that Then α(·) and β(·) are bounded functions. Therefore, there exists M ˆ |∆(s)| ≤ M |α(s) − β(s)|, for each s > ti . Using (7) and Gronwall Inequality, we obtain: 1−

ˆ

|α(t) − β(t)| ≤ |α(ti ) − β(ti )|eM (t−ti ) ,

t ≥ ti .

Since α(ti ) = β(ti ), we have α(t) = β(t), for each t ≥ ti , this is, (a) is proved. Proof (b): Notice that by (a), (xA + ηxB )/(KA + ηKB ) = xA /KA and (ηxA + xB )/(ηKA + KB ) = xB /KB . Then x�C = r xC (1 − xC /KC ), for any C ∈ {A, B}, on [ti , tf ]. Therefore, � � x A + xB (xA + xB )� = r(xA + xB ) 1 − + rω, KA + KB where � � 2 xA x2B (xA + xB )2 . − + ω= K A + KB KA KB Using xA /KA = xB /KB , it is straightforward to conclude that ω = 0.

3.2

Equilibria

It is clear that the function t → (xA (t), xB (t)) = (0, 0) is a constant solution. Nevertheless, if (xA (·), xB (·)) is a constant solution such that xA (t)xB (t) �= 0, for any t ∈ [0, 1] (for simplicity, in what follows we will identify [ti , tf ] with [0, 1]), then from (6) we have: � η(t)(KB − xB (0)) = xA (0) − KA , (9) η(t)(xA (0) − KA ) = KB − xB (0).

From (9), we have that xA (0) = KA if only if xB (0) = KB . For other hand, if xB (0) �= KB or xA (0) �= KA , then combining both identities in (9), η 2 (t) = 1, for any t ∈ [0, 1], i.e., which is a contradiction. Then, the function (xA (·), xB (·)) equals to (KA , KB ), is the unique non trivial constant solution.

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3.3

Growth

In order to determine some properties of the trayectories defined by (2)–(6), we divided the states set in six zones according: Z(1): xA ≤ KA and xB ≤ KB ; Z(2): xA ≥ KA and xB ≥ KB ; Z(3): xA ≤ KA , xB > KB and xA + xB < KA + KB ; Z(4): xB ≤ KB , xA > KA and xA + xB > KA + KB ; Z(5): xA < KA , xB > KB and xA + xB > KA + KB ; and Z(6): xB < KB , xA > KA and xA + xB < KA + KB . See Figure (2). Given a point (xA , xB ) in some Z(i), i = 1, · · · , 6, at some time t ∈ [0, 1], we present in the table (10) an overview of the signs of x�A (t) and x�B (t) at the same instant.

Figure 1: The states set associated to system (3)–(6) divided in six zones according the signs of xA − KA , xB − KB and (xA + xB ) − (KA + KB ).

Z(i) x�A x�B t∈

(1) + + [0, 1]

(2) [0, 1]

(3) + + [0, t1 ]

(3) + [t1 , 1]

(4) + [0, t1 ]

(4) [t1 , 1]

(5) [0, t2 ]

(5) + [t2 , 1]

(6) + + [0, t2 ]

(6) + [t2 , 1]

(10)

Notice that reordering the terms of (6), the signs of ft (xA , xB ) and gt (xA , xB ) are given respectively by the signs of the function that follows: F (xA , xB ) = (KA − xA ) + η(KB − xB ) and G(xA , xB ) = η(KA − xA ) + (KB − xB ).

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If (xA (·), xB (·)) is in Z(1), then KA − xA > 0 and KB − xB > 0 and is clear that F (xA , xB ) and G(xA , xB ) are positive numbers. In the Z(2) the argument is similar for obtaining that F (xA , xB ) and G(xA , xB ) are negative numbers. Now, if the point (xA , xB ) is in Z(3), then xA < KA , xB > KB and (KA + KB ) − (xA + xB ) > 0. Noting that F (xA , xB ) is equal to [(KA + KB ) − (xA + xB )] + (xB − KB )(1 − η), it is concluded that x�A (t) is positive. In order to determine the sign of x�B , notice that λ = (xB − KB )/(KA − xA ) is a positive number less that one. So, there exists t1 ∈]0, 1[ such that: If t < t1 (resp. t > t1 ), then η > λ (resp. η < λ). Since, G(xA , xB ) = (η−λ)(KA −xA ), we have x�B > 0 (resp. ). Since in Z(4) we have that F (xA , xB ) equal to −η[(xA +xB )−(KA +KB )]−(1−η)(xA − KA )(1−η), a negative number, then x�A < 0. For other hand, λ = (KB −xB )/(xA −KA ) < 1 and G(xA , xB ) = (λ − η)(xA − KA ). Notice that t < t1 (resp. >) implies η > λ (resp. ). Considering q = (KA −xA )/(xB −KB ), in Z(5), we can express F (xA , xB ) = (KB −xB ) and G(xA , xB ) equals to −[(xA + xB ) − (KA + KB )] − (1 − η)[KA − xA ]. Then x�B < 0 and x�A < 0 or x�A > 0 depending if q < η or q > η, this is, t < t2 or t > t2 , where η(t2 ) = q. Finally, in Z(6) it is convenient to express function F and G by (KB − xB )(η − q) and (KB − xB )(1 − ηq). Then x�B > 0 and x�A > 0 (resp. ). Notice that if (xA , xB ) satisfies xA + xB = KA + KB (i.e., is on the antidiagonal line), then: F = −(1 − η)(KB − xB ) = (1 − η)(KA − xA ) = G. So that, in the border of: i) (3) and (5), F > 0 and G < 0, this is, x�A > 0 and x�B > 0. ii) (4) and (6), F < 0 and G > 0, this is, x�A > 0 and x�B > 0. To know in which direction the vector (x�A , x�B ) indicates, we need to do the comparison � � � K � xA � xA ηKA + KB (xA /KA ) η KBA + 1 � �= = . � x� � xB KA + ηKB B (xB /KB ) 1 + η K B K

(11)

A

This expression also has the following formulation: � � � � xA � ˜ − η), � � = 1 + x B KB − x A KA ( λ � x� � xB (KA + ηKB ) B

(12)

˜ = (xA KB − xB KA )/(xB KB − xA KA )). where λ

Over the line xA + xB = KA + KB , we have the possibilities that follows:

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If KA = KB , then |x�A /x�B | = xA /xB . So that, in the common border of Z(3) and Z(5) we have |x�A | < |x�B |. Similarly, on the border of Z(4) and Z(6) the inequality is |x�A | > |x�B |. If KA > KB , we have two cases. When xA > KA and xB < KB , in whose case, by (11), it is clear that |x�A | > |x�B |. The case C: [xA < KA , xB > KB ] contains other two ˜ > 1, then possibilities: [C: xA < xB ] and [C: xA > xB ]. In case [C: xA < xB ], we have λ � � by (12) it is concluded |xA | > |xB |. If KA < KB , there are also two cases. Firstly, D: [xA > KA , xB < KB ], that will be ˜ > 1, open in the subcases [D: xA < xB ] and [D: xA > xB ]. In case [D: xA < xB ] we have λ then by (12) it is concluded |x�A | > |x�B |. Nevertheless, the case xA < KA and xB > KB , by (11), implies |x�A | < |x�B |. In cases [C: xA > xB ] or [D: xA > xB ], it can be proved that λ < 1, so that there ˜ So if t < t˜ (resp. >), we have η > λ ˜ (resp. ).

Figure 2: The arrows in zone (1) and (2) indicate the direction of movement of trayectories in the zone. For instance, the arrows forming an “L” in Zone (3) indicate a first direction on a time interval [0, t1 ] and the other direction the complementary time, [t1 , 1]. Notice that (1) and (2) are repulsor zones. Starting in a particular zone, a trajectory can be remain in this one or continue to another according to the following combinations: (1) → (3) → (5), (1) → (6) → (4), (2) → (5) → (3) or (2) → (4) → (6).

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4

Numerical Simulations

Let us consider an interval of fragmentation [ti , tf ] = [20, 70] a subset of a horizon time [0, T ] = [0, 100]. We take parameters: b0 d0 b d m(A) m(B) 2/10 1/10 2/100 1/100 60 40

(13)

Notice that with this parameters: r = 1/10, KA = 600 and KB = 400. Considering a linear connectivity η(t) = ξ(t) = (tf − t)/(tf − ti ), t ∈ [ti , tf ], and initial conditions: xA (ti ) xB (ti ) xA (ti ) xB (ti ) , S1 40 60 S2 50 10 we have the behavior showed in figures (3) and (4) for simulations S1 and S2 respectively, where continuous (resp. dashed) line is the situation with (resp. without) fragmentation. Total biomass vs time K 1000

X(t)

800 600 400 200 0

0

10

20

30

40

50 t−time

60

70

80

90

600 KA

700

800

900

Planar Trayectory KA+KB 1000

KB

xB(t)

800 600 400 200 0

0

100

200

300

400

500 xA(t)

1000 KA+KB

Figure 3: With equal initial densities the total abundances are always the same.

Acknowledgment: This work was partially financed by the Fondecyt 1120218.

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Total biomass vs time K 1000

X(t)

800 600 400 200 0

0

10

20

30

40

50 t−time

60

70

80

90

600 KA

700

800

900

Planar Trayectory KA+KB 1000

KB

xB(t)

800 600 400 200 0

0

100

200

300

400

500 xA(t)

1000 KA+KB

Figure 4: With independence of function of connectivity, we conjecture that different initial densities imply a total abundance with fragmentation below the unfragmented case.

References [1] BUSTAMANTE, R.O., GREZ, A.A. & SIMONETTI, J.A. Efectos de la fragmentaci´ on del bosque maulino sobre la abundancia y la diversidad de las especies nativa, in Grez AA, Siminetti JA & Bustamante R.O. (eds) Biodiversidad en ambientes fragmentados de Chile: patrones y procesos a diferentes escalas, Editorial Universitaria (2006) 83–97. [2] FAHRIG, L., Effects of habitat fragmentation on biodiversity. Annual Review of Ecology, Evolutions and Systematics 34 (2003) 487–515. [3] FORMAN, R.T.T. Land mosaic: the ecology of landscape and regions. Cambridge University Press, Cambridge (1995). [4] GREZ, A.A. & BUSTAMANTE-SANCHEZ, M.A. Aproximaciones experimentales en estudios de fragmentaci´ on, in Grez AA, Siminetti JA & Bustamante RO (eds) Biodiversidad en ambientes fragmentados de Chile: patrones y procesos a diferentes escalas, Editorial Universitaria (2006) 17–40.

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[5] HURRISON, S. & BRUNA, E. Habitat fragmentation and large–scale conservation: what do we know for sure?. Ecpgraphy 22 (1999) 225–232. [6] HERBERNER, K.W., TAVENER, S.J. & HOBSS, N.T. The distinct effects of habitat fragmentation on population size. Theor. Ecol. 5 (2012) 73–82.

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