Pristine Landscape In Elementary Mathematics Pdf

Abstract

The ecological consequences of habitat loss and fragmentation have been intensively studied on a broad, landscape-wide scale, but have less been investigated on the finer scale of individual habitat patches, especially when considering dynamic turnovers in the habitability of sites. We study changes to individual patches from the perspective of the inhabitant organisms requiring a minimum area for survival. With patches given by contiguous assemblages of discrete habitat sites, the removal of a single site necessarily causes one of the following three elementary local events in the affected patch: splitting into two or more pieces, shrinkage without splitting, or complete disappearance. We investigate the probabilities of these events and the effective size of the habitat removed by them from the population's living area as the habitat landscape gradually transitions from pristine to totally destroyed. On this basis, we report the following findings. First, we distinguish four transitions delimiting five main phases of landscape degradation: (1) when there is only a little habitat loss, the most frequent event is the shrinkage of the spanning patch; (2) with more habitat loss, splitting becomes significant; (3) splitting peaks; (4) the remaining patches shrink; and (5) finally, they gradually disappear. Second, organisms that require large patches are especially sensitive to phase 3. This phase emerges at a value of habitat loss that is well above the percolation threshold. Third, the effective habitat loss caused by the removal of a single habitat site can be several times higher than the actual habitat loss. For organisms requiring only small patches, this amplification of losses is highest during phase 4 of the landscape degradation, whereas for organisms requiring large patches, it peaks during phase 3.

Introduction

In today's world, habitat fragmentation is a mostly anthropogenic process that threatens whole ecosystems^1,2,3,4,5. Mitigating the consequences of, or altogether avoiding, habitat fragmentation is one of the main focuses of conservation ecology^6,7,8,9. Theoretical landscape ecology can aid such conservation efforts¹⁰ by supplying reliable measures of habitat fragmentation^11,12 and by analysing models of population and metapopulations to predict the consequences of habitat fragmentation^{13,14,15,16,17,18}. In particular, neutral landscape models (NLMs)^19,20 have been introduced to provide a standard to which real landscape patterns can be compared.

The simplest NLM is the percolation map²⁰ (Fig. 1). In a percolation map, habitat sites and non-habitat sites are randomly distributed according to the fraction p of habitat sites, with q = 1 −p denoting the fraction of non-habitat sites. Such models are being extensively studied by physicists, within the field of percolation theory²¹. Percolation theory shows that in an infinite landscape of randomly distributed habitat and non-habitat sites (physicists sometimes refer to these as open and closed sites, respectively), an infinite cluster (also known as giant component or spanning patch) of habitat sites exists above a critical fraction p _c of habitat sites, called the percolation threshold, whereas below that fraction no such cluster exists. Consequently, the probability that a randomly chosen habitat site belongs to the infinite cluster is zero below the percolation threshold and positive above it. The threshold-like transition from connected to fragmented landscapes observed in percolation maps is instilling caution regarding the dangers of such drastic transitions occurring also in real landscapes^20,22,23.

In recent years, NLMs have proliferated, becoming more and more sophisticated (for a review of landscape-generating algorithms, see²⁴). For example, percolation maps have been generalized to incorporate more than two landscape-cover types^13,25 or a gradient across which p varies from one end of the landscape to the other^26,27,28. Also, as real landscapes generally show aggregated patterns, aggregation has been introduced to NLMs. For example, Gustafson and Parker²⁹ randomly placed rectilinear clumps with random edge lengths onto a grid to create an aggregated pattern. Hiebler³⁰ proposed an iterative method to produce landscapes with given pair-correlation probabilities (see also³¹). In hierarchical random landscapes³², the probability of assigning a patch to one of two landscape-cover types is first applied at larger spatial scales before being applied to the remaining habitat at smaller spatial scales, an approach that can be generalized to multiple landscape-cover types^33,34. Realistically looking mosaic landscapes can also be generated by tessellation methods^35,36, by fractal methods^37,38,39,40, or by modifying/distorting existing landscape patterns to modify their spatial characteristics⁴¹. Many of the aforementioned methods are implemented in a readily available software package⁴². Finally, NLMs have been extended to three dimensions, to model soil layers or forest canopies⁴³.

The spatially explicit modelling of ecological processes^44,45 has become the focus of many ecological studies⁴⁶. In the past three decades, an increasing number of population models have incorporated environmental heterogeneity^{15,25,30,31,39,47,48,49,50,51,52,53,54,55,56,57,58}: the resultant body of literature has underscored how NLMs can aid the development of increasingly insightful and realistic landscape models. Compared to NLMs, landscape patterns or patch structures used in experiments are usually simpler, such as a chessboard mosaic of resource-rich and resource-poor patches^59,60,61 or patterns based on percolation maps^62,63,64. To apply the insights derived from such simple experimental settings to more complex landscapes, suitable models are required, as experimental settings with more complex landscapes^65,66 are rare.

An important development in NLM approaches has been the incorporation of temporal fluctuations into models^{23,49,52,53,67,68,69,70,71,72,73} and experiments⁷⁴. In particular, Hagen-Zanker and Lajoie⁶⁹ have proposed a neutral model of landscape change, introducing a novel method for generating a landscape with multiple landscape-cover types and aggregation. Dynamic neutral landscape models (DNLMs), in general, eschew pre-defined spatial and temporal structures, to study the emergence of these structures from interactions among the components.

The DNLM we employ in the present study can arguably be considered as a minimalistic DNLM, because the process for changing the landscape is very simple and does not induce any spatial or temporal correlations among the habitability of the sites. Consequently, the habitat sites, initially being distributed randomly (as in NLMs), persist being free from any spatial or temporal correlations.

Even this simple representation of a dynamic landscape of habitable sites produces non-trivial emergent phenomena. Many interesting features have already been revealed before, especially by percolation theory elucidating the size distribution of habitable patches (percolation clusters) and the global connectivity of such patches (see²³ for a review). In general, percolation theory has focused on global features, applying at the scale of the whole landscape, while assuming a static landscape. Here, we extend this view to dynamic landscapes and to the finer scale of individual patches.

This mesoscale is intermediate between the global scale (of the landscape as a whole) and the local scale (of individual habitat sites). From the perspective of an organism inhabiting the landscape, this intermediate scale of habitable patches is particularly important. When non-habitat sites are difficult to traverse for such an organism, it may be confined to a single patch for its whole life (concrete examples concerning plants are discussed by⁵⁷). Thus, any event changing the size of the patch the organism is inhabiting is crucial for the organism.

Important elementary events on the patch scale have been reviewed before, by Forman⁷⁵ (pp. 407), Jaeger⁷⁶, Akçakaya⁷⁷, and Didham et al.⁷⁸. Here we adopt this event-based view and develop it through a quantitative analysis of the events. Specifically, we characterize the relative event frequencies in dependence on the sizes of the patches that are involved. Following Akçakaya⁷⁷, we consider changes in both directions: habitat loss and habitat gain. Accordingly, we systematically distinguish and analyse six elementary local events that can affect a patch (Fig. 2):

Patch formation is the appearance of a habitat site within a non-habitat area,

Patch loss is the event occurring when a habitat patch ceases to exist by the removal of its last site,

Patch enlargement is the addition of a habitat site to an existing habitat patch,

Patch shrinkage is the removal of a habitat site from an existing habitat patch,

Patch coalescence is the joining of two habitat patches that were previously isolated, and

Patch splitting is the division of a habitat patch into two or more isolated ones.

The event-based description of landscape dynamics enabled by the definition of these elementary events offers a general framework for studying patch dynamics in changing landscapes. Here, we apply this general framework to a simple DNLM, a random map with random local fluctuations. We also assume that the landscape-level habitat pattern is in a steady state when habitat loss occurs. We investigate the consequences of a single local habitat loss for the affected patch and analogously study the consequences of a single local habitat gain (see the corresponding loss/gain events in Fig. 2).

Our analysis underscores that landscape fragmentation is not only a problem resulting from the loss of connections between habitat patches, but also a problem related to a minimum feasible patch size. The habitat patches arising from dynamical habitat fragmentation may be smaller than the minimal area needed for sustaining a viable local population of the focal organism. Therefore, by removing a single unit of habitat, a potentially much larger area may be lost from the organism's perspective. We estimate the magnitude of this kind of loss, and predict the level q of habitat loss at which it is the most perilous.

Methods

The probabilities of the elementary events, as defined above, cannot be obtained analytically for a given landscape, as this would require the full enumeration of all patches together with their sizes and shapes. The number of possible patch shapes or configurations grows exponentially with patch size^79,80,81,82. An enumeration of patch configurations has been achieved only up to a patch size of 45 sites⁸³, later pushing this limit to 56 sites⁸⁴. This poses a problem, since percolation maps without habitat patches larger than 56 sites are either very small or have very low values of p.

Therefore, we use Monte Carlo simulations to obtain the functions describing how the probabilities of the elementary events depend on the habitat cover p, by recording all events taking place on a percolation map with random fluctuations in site habitability. For this purpose, we use lattices consisting of N = 100 × 100 = 10,000 sites with periodic boundaries (i.e., the lattice is wrapped around a torus). Initially, pN randomly chosen sites are habitat sites and qN sites are non-habitat sites.

In our model, we assume a constant proportion q = 1 −p of non-habitat sites. Thus, when a randomly chosen habitat site is turned into a non-habitat site, at the same time a randomly chosen non-habitat site is turned into a habitat site. For each value of q, we record 6 · 10⁶ elementary events. The record of an event consists of the type of the event and of the sizes of all the patches affected by the event. Since the total proportion of habitat sites does not change, the probabilities of patch loss and patch formation are equal, the probabilities of patch enlargement and patch shrinkage are equal, and the probabilities of patch splitting and patch coalescence are equal (see the pairs of opposing processes in Fig. 2). For this reason, it suffices to show results for the three elementary local events of patch loss, patch shrinkage, and patch splitting.

The probability of patch loss can be calculated analytically, assuming an infinite lattice. This enables us to compare our numerical results to an analytical baseline. Patch loss occurs when the local configuration matches the case shown in Fig. 2a. The probability of encountering such a local configuration at a random site of a hexagonal lattice (also referred to as a honeycomb lattice or equilateral triangular lattice) is p(1 −p)⁶, whereas it is p(1 −p)⁴ on a square lattice (considering the von Neumann neighbourhood, i.e., the four nearest neighbours). This example illustrates that the probabilities of elementary events are sensitive to the geometry of the considered lattice. Also the percolation threshold, which is analytically known for simple lattice geometries²¹, varies with the geometry of the considered lattice: p _c = 0.5 (q _c = 0.5) for the hexagonal lattice and p _c = 0.592746 (q _c = 0.407254) for the square lattice with a four-site neighbourhood. The threshold value is exact for the hexagonal lattice and is numerically estimated for the square lattice. In line with these differences, we must expect that the geometry of the lattice affects patch dynamics and the probabilities of the elementary events. For this reason, we include both the hexagonal lattice and the square lattice in our investigations.

For the hexagonal lattice, our numerical investigations are based on values of p ranging from 0.05 to 0.95 with 0.05 increments and from 0.45 to 0.55 with 0.01 increments; in addition, the values 0.57, 0.63, and 0.67 were also considered. For the square lattice, our numerical investigations are based on values of p ranging from 0.05 to 0.95 with 0.05 increments and from 0.45 to 0.80 with 0.01 increments.

Results

In this section, we identify three kinds of landscape transitions that are important in addition to the traditionally recognized percolation transition. The resultant four transitions naturally subdivide the process of habitat loss into five phases. Finally, we show how the effective habitat loss experienced by organisms having different requirements for minimal patch sizes increases with the actual habitat loss.

Three impacts of habitat loss

The functions describing how the probabilities of patch loss, patch shrinkage, and patch splitting change with the level of habitat loss q turn out to have similar shapes for the hexagonal lattice and the square lattice (Fig. 3a,b). The numerically obtained probabilities of patch loss perfectly match the aforementioned analytical predictions, increasing monotonically as the level of habitat loss q is raised. The probability of patch splitting reaches its maximum at a value of q that is higher than the percolation threshold (q >q _c). For the hexagonal lattice, patch splitting is most probable when ca. 60% of the habitat is lost (Fig. 3a), while for the square lattice this peak occurs at ca. 50% of habitat loss (Fig. 3b). From there up to ca. 80% of habitat loss, the probability of patch shrinkage remains relatively constant, independent of the two considered lattice geometries (Fig. 3a,b).

Four transitions caused by habitat loss

We observe the probabilities of the elementary events during the degradation from a pristine landscape (q = 0) to the disappearance of all habitat (q = 1). Based on the results shown in Fig. 3, we propose to divide the process of habitat loss into five phases, each corresponding to a certain interval of habitat loss. The boundaries between adjacent phases are determined by four major transitions in landscape structure and dynamics occurring during the course of landscape degradation:

Transition 1 (occurring at a level of habitat loss of ca. 20%). Appearance of isolated patches and a very slowly increasing incidence of patch splitting. This transition can be defined to occur at the level of habitat loss q at which the probability of patch shrinkage falls below 99%.

Transition 2 (occurring at a level of habitat loss of ca. 45%). Transition from a fully connected to a fragmented landscape. This transition happens at the percolation threshold, which is defined in, and well known from, the literature on percolation processes²¹.

Transition 3 (occurring at a level of habitat loss of ca. 55%). Switch from an increasing to a decreasing probability of patch splitting. This transition is thus defined to occur at the level of habitat loss q at which the patch-splitting probability peaks.

Transition 4 (occurring at a level of habitat loss of ca. 85%). Sudden decline in the probability of patch shrinkage. This transition happens when the probability of patch loss starts to dominate, and we can define it to occur at the habitat loss q at which the probability of patch loss surpasses the probabilities of patch shrinkage and patch splitting.

The four values of habitat loss q listed above are just rough indications, mentioned here only for the sake of approximate orientation. Naturally, the exact values depend on the details of the model, and in particular on the considered lattice geometry (Table 1).

Table 1 Five phases of habitat loss.

Full size table

Five phases of habitat loss

The aforementioned four transitions naturally divide a landscape's degradation process into five phases (Table 1; see Fig. 1 for illustration):

Phase 1 (occurring at levels of habitat loss from 0% to ca. 20%; Fig. 1a). The landscape is dominated by a single large habitat patch, known from percolation theory as a spanning cluster, which spans across the entire landscape. By far, the most frequent event during this phase is the shrinkage of the spanning patch (Fig. 3a,b). In contrast, patch splitting and patch loss are rare events, as they happen only to the non-spanning smaller patches (Fig. 3a,b). Throughout this phase, all three event probabilities remain almost constant (Fig. 3a,b).

Phase 2 (occurring at levels of habitat loss from ca. 20% to ca. 45%; Fig. 1b). The spanning patch still exists, but the number and size of the non-spanning smaller patches become significant. Close to the percolation threshold at this phase's upper boundary, the spanning patch is filamental, which means that the random removal of a single habitat site in the considered lattices of 100 × 100 sites results in its splitting into two habitat patches in about 1/4 of the cases (Fig. 3a,b). In this phase, patch shrinkage and patch splitting co-occur at appreciable frequencies, while patch loss remains rare (Fig. 3a,b). The sensitivities of the frequencies of patch shrinkage and patch splitting to habitat loss (slopes in Fig. 3a,b) both increase toward the percolation threshold, where they reach their peaks.

Phase 3 (occurring at levels of habitat loss from ca. 45% to ca. 55%; Fig. 1c). This phase is similar to the previous one in terms of the occurrence of the elementary events, while being characterized by a key difference in landscape structure: as habitat loss q exceeds the percolation threshold q _c, the landscape is fragmented. In the theoretical limit of infinite landscape size, no spanning patch would exist anymore; however, for finite landscapes, we may still find a spanning patch when q −q _c is small, but the probability of such configurations rapidly declines with an increase of q −q _c and/or of landscape size²¹. This fragmentation implies that it is not possible for a population to spread across the landscape without being obstructed by non-habitat sites, which poses a serious burden on its persistence (see²³ for a review). In this phase, the frequencies of patch shrinkage and patch splitting continue to decrease and increase, respectively, while the frequency of patch loss also increases, but still remains at low levels (Fig. 3a,b). The sensitivities of the frequencies of patch shrinkage and patch splitting to habitat loss (slopes in Fig. 3a,b) both decrease above the percolation threshold, where they had reached their peaks (Fig. 3a,b).

Phase 4 (occurring at levels of habitat loss from ca. 55% to ca. 85%; Fig. 1d). In this phase, the frequency of patch shrinkage is surprisingly unresponsive to the further increase of habitat loss, exhibiting a broad plateau (Fig. 3a,b). After having reached its peak at the transition between Phases 3 and 4, the frequency of patch splitting declines with the further increase of habitat loss, and patch loss is gaining importance, gradually becoming as frequent as patch shrinkage (Fig. 3a,b). These effects are caused by decreasing patch sizes.

Phase 5 (occurring at levels of habitat loss from ca. 85% to 100%; Fig. 1e). In this final phase of landscape degradation, the frequency of patch loss continues to increase super-linearly with further habitat loss, exceeding the frequencies of patch shrinkage and patch splitting, with the latter both continuing to decline with further habitat loss (Fig. 3a,b). Populations inhabiting these kinds of landscapes are confined to live primarily in small habitat patches, which are frequently lost. Therefore, sufficient dispersal between waxing and waning habitat patches is crucial for a population's persistence.

The aforementioned phases can be observed on a hexagonal lattice (Fig. 3a) and on a square lattice with von Neumann neighbourhood (Fig. 3b). In the hexagonal lattice, each phase starts later, i.e., at a higher level of habitat loss (Table 1). The reason for this is the higher number of connections per site (degree): each site is connected to six others, as opposed to four others in the square lattice with von Neumann neighbourhood. Therefore, more sites need to be removed from a hexagonal lattice for causing a habitat patch to split, or for isolating a habitat site completely, making it less vulnerable to patch loss.

Effective habitat loss

To assess the severity of impact of landscape degradation on a population, we assume that sustaining a viable local population requires a habitat patch with a minimum of m habitat sites. When a single isolated habitat site is sufficient for sustaining a local population (m = 1), then every site removal decreases the population's living area by exactly one site. In contrast, when m = 2, then fragmentation of a habitat patch of size 3 creates two fragments, each of size 1, which effectively removes three habitat sites from the population's living area. In this manner, every elementary local event is evaluated according to the effective habitat loss it causes. Some site removals cause no loss of habitat, as they affect patches that are already too small to sustain a local population. These habitat patches are part of the landscape; they are just lost from the population's living area.

For any given total level of habitat loss q, we determine the amplification factor of effective habitat loss by averaging the effective habitat loss (i.e., the loss of a population's living area) over a large number of elementary events (6 · 10⁶), each involving an actual habitat loss of one habitat site. This factor starts to exceed 1 appreciably when the level of habitat loss q exceeds roughly 30% (Fig. 3c,d), which is well below the percolation threshold and roughly coincides with the transition from Phase 1 to Phase 2. The amplification factor of effective habitat loss peaks at levels of habitat loss well above the percolation threshold, close to the level of habitat loss at which the probability of patch splitting is maximal (compare Fig. 3a–d). This peak shifts toward lower levels of habitat loss q as the minimal patch area needed for a viable local population increases (Fig. 3c,d). After reaching its peak, the amplification factor of effective habitat loss strongly declines and typically drops below 1 in Phase 4 (Fig. 3c,d), which means that by removing a single habitat site on average less than one habitat site is lost from the population's living area. This is because in this phase most habitat patches are already too small for sustaining a local population, so that the further removal of a habitat site often causes no further loss from a population's living area. Naturally, this effect is stronger and starts earlier, i.e., at lower levels of habitat loss q, for populations requiring habitat patches of larger minimal size m.

Discussion

Here we have presented a dynamical view of habitat loss by observing the probabilities of elementary local events on the spatial mesoscale of individual habitat patches. Our framework for studying changing landscapes allows us to identify three additional landscape transitions beside the one associated with the well-known percolation threshold. The four corresponding thresholds demarcate five distinct phases of habitat loss. Phase 1 begins from a pristine habitat and is characterized by the shrinkage of contiguous habitat area. In Phase 2, detached habitat patches begin to appear, even though a spanning patch still exists. The percolation threshold marks the transition to Phase 3, in which the effective habitat loss peaks, and at the end of which the frequency of patch splitting reaches its peak. It is important to emphasize that – although the connectivity of habitat sites plummets at the percolation threshold – our results demonstrate that the severity and frequency of patch splitting peaks at higher levels of habitat loss. It is in Phase 3 that organisms requiring a larger number of habitat sites for sustaining a viable local population experience a peak in effective habitat loss. Phase 4 commences with the peak in the probability of patch splitting and is characterized by a nearly constant probability of patch shrinkage. Finally, in Phase 5, habitat loss is so severe that the landscape mainly consists of isolated habitat sites, which then disappear one-by-one as habitat cover decreases.

The same process can alternatively be described from the opposite direction, which is a trend toward habitat gain. For example, when sites with arable land are created, the corresponding habitat patches may become suitable for weeds or other agricultural pests, increasing their population sizes, as well as their probabilities of spreading across the landscape.

Dynamical Neutral Landscape Models (DNLMs) serve as natural benchmarks and references to which data observed on real landscapes can be compared. High correlations between a landscape index and a response variable measured on artificial landscapes have often been confirmed by measurements on real landscapes¹⁶. Real landscapes differ from percolation maps in at least two important characteristics. (1) Non-habitat patches can be traversed up to some distance^85,86,87, and thus, habitat patches can still be functionally connected¹⁴ in spite of their physical fragmentation. (2) Real landscapes are, in general, not as randomly structured as percolation maps, but instead exhibit some degree of aggregation^34,88,89,90. When habitat sites are aggregated, patch splitting is less likely, and thus occurs at higher levels of habitat loss. This is corroborated, for example, by observations on birds and mammals⁸⁵, which have led to the conclusion that a landscape becomes significantly fragmented for these species only when 60–70% of the habitat is destroyed. Habitat fragmentation on a percolation map can thus be considered as a worst-case scenario.

Our numerical investigations in the present study have been carried out on a percolation map, since this is the first²⁰ and best-studied NLM, and also the most widely used in spatially explicit models^{13,15,30,67,68,91,92,93,94}. Percolation maps are often applied to study habitat fragmentation because of the critical behaviour they exhibit^13,20,21,22. In contrast, habitat change in other models of spatial ecology has mostly been defined in terms of disturbance regimes^{13,15,91,95,96,97,98,99,100}. Up to now, NLMs with temporal changes have scarcely been considered (for exceptions, see^{23,24,48,56,69,73}). The DNLM framework could be extended in a variety of ways. First, the dynamical properties of NLMs other than percolation maps could be similarly investigated. This perspective could also be applied to landscape models defined in continuous space (rather than on a lattice, as in our present study): In those contexts, it would then be interesting to examine how the dynamical landscape properties are changing when scanning through ranges of distances for defining whether nearby locations are regarded as being connected or not. The frequency distribution of events on other landscapes is expected to be different. In particular, on landscapes with spatially aggregated patterns of habitat sites, we expect patch splitting to dominate at higher levels of habitat loss. Second, the dynamics of populations can be described in greater mechanistic detail. For example, the costly – but in general not impossible – movement of individuals through non-habitat sites could be included (for an example of such a model, see⁵⁷). The explicit study of population dynamics within the patches, including the study of extinction times, is also a promising direction for future research (see²³ about connecting habitat dynamics and population dynamics in a percolation-theoretical framework). Third, in our present study, we have removed habitat sites randomly, according to a spatially uniform probability distribution. Future research could consider more elaborate models of habitat loss, reflecting real-word processes such as agricultural land-use expansion or other anthropogenic environmental changes. In this context, an examination of contagious habitat loss¹⁰¹ would be of particular interest.

The importance of the percolation threshold for understanding landscape dynamics and habitat fragmentation has been recognized in the literature from the early days of landscape ecology²⁰ (see reviews of relevant studies in^58,102). Here we have suggested considering three additional thresholds that are fundamentally related to understanding the dynamics of habitat patches and the impacts of habitat loss on different organisms that require habitat patches of a certain minimal size for their local survival. Altogether, the four thresholds we have defined, examined, and discussed here distinguish five characteristic phases of habitat loss. We believe that the distinction between these phases according to a landscape's dynamical properties, focussing on the patch-scale events, is crucial for the protection of species subject to habitat loss and fragmentation. In particular, important qualitative changes in the dynamics of habitat patches can be revealed by studying the frequencies of the elementary local events we have investigated here. Habitat changes thus becoming detectable will often foreshadow critical landscape transitions to occur, which allows scientists and managers to take action before disastrous and irreversible damage is done.

Data Availability

The datasets generated and analysed for the current study are available from the corresponding author on reasonable request.

References

1.
Foley, J. A. et al. Global consequences of land use. Science 309, 570–574, https://doi.org/10.1126/science.1111772 (2005).

ADS CAS Article PubMed Google Scholar
2.
Fahrig, L. Effects of habitat fragmentation on biodiversity. Annu. Rev. Ecol. Evol. Syst. 34, 487–515, https://doi.org/10.1146/annurev.ecolsys.34.011802.132419 (2003).

Article Google Scholar
3.
Fischer, J. & Lindenmayer, D. B. Landscape modification and habitat fragmentation: a synthesis. Global Ecol. Biogeogr. 16, 265–280, https://doi.org/10.1111/j.1466-8238.2007.00287 (2007).

Article Google Scholar
4.
Haddad, N. M. et al. Habitat fragmentation and its lasting impact on Earth's ecosystems. Science Advances 1, e1500052, https://doi.org/10.1126/sciadv.1500052 (2015).

ADS Article PubMed PubMed Central Google Scholar
5.
Pfeifer, M. et al. Creation of forest edges has a global impact on forest vertebrates. Nature 551, 187, https://doi.org/10.1038/nature24457 (2017).

ADS CAS Article PubMed PubMed Central Google Scholar
6.
Primack, R. B. Essentials of Conservation Biology (Sinauer Associates, 1998).
7.
Lindenmayer, D. B. & Fischer, J. Tackling the habitat fragmentation panchreston. Trends Ecol. Evol. 22, 127–132, https://doi.org/10.1016/j.tree.2006.11.006 (2007).

Article PubMed Google Scholar
8.
Fazey, I., Fischer, J. & Lindenmayer, D. B. What do conservation biologists publish? Biol. Conserv. 124, 63–73, https://doi.org/10.1016/j.biocon.2005.01.013 (2005).

Article Google Scholar
9.
Fardila, D., Kelly, L. T., Moore, J. L. & McCarthy, M. A. A systematic review reveals changes in where and how we have studied habitat loss and fragmentation over 20 years. Biol. Conserv. 212, 130–138, https://doi.org/10.1016/j.biocon.2017.04.031 (2017).

Article Google Scholar
10.
Hansson, L. & Angelstam, P. Landscape ecology as a theoretical basis for nature conservation. Landscape Ecol. 5, 191–201 (1991).

Article Google Scholar
11.
Hargis, C. D., Bissonette, J. A. & David, J. L. The behaviour of landscape metrics commonly used in the study of habitat fragmentation. Landscape Ecol. 13, 167–186 (1998).

Article Google Scholar
12.
Tischendorf, L. & Fahrig, L. How should we measure landscape connectivity? Landscape Ecol. 15, 633–641 (2000).

Article Google Scholar
13.
With, K. A. & Crist, T. O. Critical threshold in species responses to landscape structure. Ecology 76, 2446–2459 (1995).

Article Google Scholar
14.
Andrén, H. Population responses to habitat fragmentation: statistical power and the random sample hypothesis. Oikos 235–242 (1996).

Article Google Scholar
15.
Neuhauser, C. Habitat destruction and competitive coexistence in spatially explicit models with local interaction. J. Theor. Biol. 193, 445–463 (1998).

CAS Article Google Scholar
16.
Tischendorf, L. Can landscape indices predict ecological processes consistently? Landscape Ecol. 16, 235–254 (2001).

Article Google Scholar
17.
Söngerath, D. & Schröder, B. Population dynamics and habitat connectivity affecting the spatial spread of populations – a simulation study. Landscape Ecol 17, 57–70 (2002).

Article Google Scholar
18.
Solé, R. V. & Bascompte, J. Habitat loss and extinction thresholds. In Self-Organization in Complex Ecosystems (eds Solé, R. V. & Bascompte, J.) 171–214 (Princeton University Press, 2006).
19.
With, K. A. & King, A. W. The use and misuse of neutral landscape models in ecology. Oikos 79, 219–229 (1997).

Article Google Scholar
20.
Gardner, R. H., Milne, B. T., Turner, M. G. & O'Neill, R. V. Neutral models for the analysis of broad-scale landscape pattern. Landscape Ecol. 1, 19–28 (1987).

Article Google Scholar
21.
Stauffer, D. & Aharony, A. Introduction to Percolation Theory. Revised Second Edn (Taylor and Francis, 1994).
22.
Boswell, G. P., Britton, N. F. & Franks, N. R. Habitat fragmentation, percolation theory and the conservation of a keystone species. Proc. R. Soc. London, Ser. B 265, 1921–1925 (1998).

Article Google Scholar
23.
Oborny, B., Szabó, G. & Meszéna, G. In Scaling Biodiversity (eds Storch, D., Marquet, P. A. & Brown, J. H.) 409–440 (Cambridge University Press, 2007).
24.
Kun, Á. Generation of temporally and spatially heterogeneous landscapes for models of population dynamics. Appl. Ecol. Environ. Res. 4, 73–84 (2007).

Article Google Scholar
25.
Stoddard, S. T. Continuous versus binary representations of landscape heterogeneity in spatially-explicit models of mobile populations. Ecol. Model. 221, 2409–2414, https://doi.org/10.1016/j.ecolmodel.2010.06.024 (2010).

Article Google Scholar
26.
Keitt, T. H. & Johnson, A. R. Spatial heterogeneity and anomalous kinetics: emergent patterns in diffusion-limited predator-prey interaction. J. Theor. Biol. 171, 127–139 (1995).

Article Google Scholar
27.
Milne, B. T., Keitt, T. H., Hatfield, C. A., David, J. & Hraber, P. T. Detection of critical densities associated with pinion-juniper woodland ecotones. Ecology 77, 805–821 (1996).

Article Google Scholar
28.
Gastner, M., Oborny, B., Pruessner, G. & Zimmermann, D. Transition from connected to fragmented vegetation across an environmental gradient: scaling laws in ecotone geometry. Am. Nat. 174, 23–39 (2009).

Article Google Scholar
29.
Gustafson, E. J. & Parker, G. R. Relationships between landcover proportions and indices of landscape spatial pattern. Landscape Ecol. 7, 101–110 (1992).

Article Google Scholar
30.
Hiebeler, D. Populations on fragmented landscapes with spatially structured heterogeneities: landscape generation and local dispersal. Ecology 81, 1629–1641 (2000).

Article Google Scholar
31.
Huth, G., Lesne, A., Munoz, F. & Pitard, E. Correlated percolation models of structured habitat in ecology. Physica A: Statistical Mechanics and its Applications 416, 290–308, https://doi.org/10.1016/j.physa.2014.08.006 (2014).

ADS Article Google Scholar
32.
O'Neill, R. V., Gardner, R. H. & Turner, M. G. A hierarchical neutral model for landscape analysis. Landscape Ecol. 7, 55–61 (1992).

Article Google Scholar
33.
Johnson, G. D., Myers, W. L. & Patil, G. P. Stochastic generating models for simulating hierarchically structured multi-cover landscapes. Landscape Ecol. 14, 413–421 (1999).

Article Google Scholar
34.
Turner, M. G. Spatial simulation of landscape changes in Georgia: A comparison of 3 transition models. Landscape Ecol. 1, 29–36 (1987).

Article Google Scholar
35.
Gaucherel, C. Neutral models for polygonal landscapes with linear networks. Ecol. Model. 219, 39–48, https://doi.org/10.1016/j.ecolmodel.2008.07.028 (2008).

Article Google Scholar
36.
Gaucherel, C., Fleury, D., Auclair, D. & Dreyfus, P. Neutral models for patchy landscapes. Ecol. Model. 197, 159–170, https://doi.org/10.1016/j.ecolmodel.2006.02.044 (2006).

Article Google Scholar
37.
With, K. A., Gardner, R. H. & Turner, M. G. Landscape connectivity and population distribution in heterogeneous environments. Oikos 78, 151–169 (1997).

Article Google Scholar
38.
Palmer, M. W. The coexistence of species in fractal landscapes. Am. Nat. 139, 375–397 (1992).

Article Google Scholar
39.
Hovestadt, T., Messer, S. & Poethke, H. J. Evolution of reduced dispersal mortality and "fat-tailed" dispersal kernels in autocorrelated landscapes. Proc. R. Soc. London, Ser. B 268, 385–391 (2001).

CAS Article Google Scholar
40.
Kashian, D. M., Sosin, J. R., Huber, P. W., Tucker, M. M. & Dombrowski, J. A neutral modeling approach for designing spatially heterogeneous jack pine plantations in northern Lower Michigan, USA. Landscape Ecol. 32, 1117–1131, https://doi.org/10.1007/s10980-017-0514-y (2017).

Article Google Scholar
41.
van Strien, M. J., Slager, C. T. J., de Vries, B. & Grêt-Regamey, A. An improved neutral landscape model for recreating real landscapes and generating landscape series for spatial ecological simulations. Ecology and Evolution 6, 3808–3821, https://doi.org/10.1002/ece3.2145 (2016).

Article PubMed PubMed Central Google Scholar
42.
Etherington, T. R., Holland, E. P. & O'Sullivan, D. NLMpy: a python software package for the creation of neutral landscape models within a general numerical framework. Methods in Ecology and Evolution 6, 164–168, https://doi.org/10.1111/2041-210X.12308 (2015).

Article Google Scholar
43.
Kirkpatrick, L. A. & Weishampel, J. F. Quantifying spatial structure of volumetric neutral models. Ecol. Model. 186, 312–325, https://doi.org/10.1016/j.ecolmodel.2005.01.056 (2005).

Article Google Scholar
44.
Czárán, T. Spatiotemporal Models of Population and Community Dynamics (Chapman and Hall, 1998).
45.
Tilman, D. & Kareiva, P. Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (Princeton University Press, 1997).
46.
Dieckmann, U., Law, R. & Metz, J. A. J. The Geometry of Ecological Interactions: Simplifying Spatial Complexity (Cambridge University Press, 2000).
47.
Bascompte, J. & Solé, R. V. Habitat fragmentation and extinction threshold in spatially explicit models. J. Anim. Ecol. 65, 465–473 (1996).

Article Google Scholar
48.
Szabó, G., Gergely, H. & Oborny, B. Generalized contact process on random environments. Phys. Rev. E: Stat. Phys. Plasmas, Fluids 65, 066111 (2002).

Article Google Scholar
49.
Kun, Á. & Oborny, B. Survival and competition of clonal plant populations in spatially and temporally heterogeneous habitats. Community Ecology 4, 1–20 (2003).

Article Google Scholar
50.
Ezard, T. H. G. & Travis, J. M. J. The impact of habitat loss and fragmentation on genetic drift and fixation time. Oikos 114, 367–375, https://doi.org/10.1111/j.2006.0030-1299.14778.x (2006).

Article Google Scholar
51.
Le Ber, F. et al. Neutral modelling of agricultural landscapes by tessellation methods – Application for gene flow simulation. Ecol. Model. 220, 3536–3545, https://doi.org/10.1016/j.ecolmodel.2009.06.019 (2009).

Article Google Scholar
52.
Wimberly, M. C. Species dynamics in disturbed landscapes: when does a shifting habitat mosaic enhance connectivity? Landscape Ecol. 21, 35–46, https://doi.org/10.1007/s10980-005-7757-8 (2006).

Article Google Scholar
53.
Schrott, G. R., With, K. A. & King, A. W. On the importance of landscape history for assessing extinction risk. Ecol. Appl. 15, 493–506 (2005).

Article Google Scholar
54.
Graves, T. A. et al. The influence of landscape characteristics and home-range size on the quantification of landscape-genetics relationships. Landscape Ecol. 27, 253–266, https://doi.org/10.1007/s10980-011-9701-4 (2012).

Article Google Scholar
55.
Dytham, C. The effect of habitat destruction pattern on species persistence: a cellular model. Oikos 74, 340–344 (1995).

Article Google Scholar
56.
Gaucherel, C. Self-organization of patchy landscapes: hidden optimization of ecological processes. Journal of Ecosystem & Ecography 1, 105, https://doi.org/10.4172/2157-7625.1000105 (2011).

Article Google Scholar
57.
Oborny, B. & Hubai, A. G. Patch size and distance: modelling habitat structure from the perspective of clonal growth. Ann. Bot. 114, 389–398, https://doi.org/10.1093/aob/mcu110 (2014).

Article PubMed PubMed Central Google Scholar
58.
Oborny, B., Benedek, V., Englert, P., Gulyás, M. & Hubai, A. G. The plant in the labyrinth: Adaptive growth and branching in heterogeneous environments. J. Theor. Biol. 412, 146–153, https://doi.org/10.1016/j.jtbi.2016.10.015 (2017).

MathSciNet Article PubMed MATH Google Scholar
59.
Fransen, B., de Kroon, H. & Berendse, F. Soil nutrient heterogeneity alters competition between perennial grass species. Ecology 82, 2534–2546 (2001).

Article Google Scholar
60.
Wijesinghe, D. K. & Hutchings, M. J. The effects of spatial scale of environmental heterogeneity on the growth of a clonal plant: an experimental study with Glechoma hederacea. J. Ecol. 85, 17–28 (1997).

Article Google Scholar
61.
Wijesinghe, D. K. & Hutchings, M. J. The effect of environmental heterogeneity on the performance of Glechoma hederacea: the interactions between patch contrast and patch scale. J. Ecol. 87, 860–872 (1999).

Article Google Scholar
62.
Wijesinghe, D. K. & Handel, S. N. Advantages of clonal growth in heterogeneous habitats: an experiment with Potentilla simplex. J. Ecol. 82, 495–502 (1994).

Article Google Scholar
63.
Wiens, J. A., Schooley, R. L. & Weeks, D. Jr. Patchy landscapes and animal movements: do beetles percolate? Oikos 78, 257–264 (1997).

Article Google Scholar
64.
McIntyre, N. E. & Wiens, J. A. Interaction between habitat abundance and configuration: experimental validation of some predictions from percolation theory. Oikos 88, 129–137 (1999).

Article Google Scholar
65.
With, K. A., Pavuk, D. M., Worchuck, J. L., Oates, R. K. & Fisher, J. L. Threshold effects of landscape structure on biological control in agroecosystems. Ecol. Appl. 12, 52–65, https://doi.org/10.1890/1051-0761(2002)012[0052:teolso]2.0.co;2 (2002).

Article Google Scholar
66.
With, K. & Pavuk, D. Habitat area trumps fragmentation effects on arthropods in an experimental landscape system. Landscape Ecol. 26, 1035–1048, https://doi.org/10.1007/s10980-011-9627-x (2011).

Article Google Scholar
67.
Magyar, G., Kun, A., Oborny, B. & Stuefer, J. F. Importance of plasticity and decision-making strategies for plant resource acquisition in spatio-temporally variable environments. New Phytol. 174, 182–193, https://doi.org/10.1111/j.1469-8137.2007.01969.x (2007).

Article PubMed Google Scholar
68.
Kun, Á., Oborny, B. & Dieckmann, U. Intermediate landscape disturbance maximizes metapopulation density. Landscape Ecol. 24, 1341–1350, https://doi.org/10.1007/s10980-009-9386-0 (2009).

Article Google Scholar
69.
Hagen-Zanker, A. & Lajoie, G. Neutral models of landscape change as benchmarks in the assessment of model performance. Landscape Urban Plann. 86, 284–296 (2008).

Article Google Scholar
70.
Hiebeler, D. E., Houle, J., Drummond, F., Bilodeau, P. & Merckens, J. Locally dispersing populations in heterogeneous dynamic landscapes with spatiotemporal correlations. I. Block disturbance. J. Theor. Biol. 407, 212–224, https://doi.org/10.1016/j.jtbi.2016.07.031 (2016).

MathSciNet Article PubMed MATH Google Scholar
71.
Hiebeler, D. E. & Hill, J. L. Locally dispersing populations in heterogeneous dynamic landscapes with spatiotemporal correlations. II. Habitat driven by voter dynamics. J. Theor. Biol. 407, 81–89, https://doi.org/10.1016/j.jtbi.2016.07.033 (2016).

Article PubMed MATH Google Scholar
72.
Hiebeler, D. E. & Morin, B. R. The effect of static and dynamic spatially structured disturbances on a locally dispersing population. J. Theor. Biol. 246, 136–144, https://doi.org/10.1016/j.jtbi.2006.12.024 (2007).

MathSciNet Article PubMed Google Scholar
73.
Keymer, J. E., Marquet, P. A., Velasco-Hernández, J. X. & Levin, S. A. Extinction threshold and metapopulation persistence in dynamic landscapes. Am. Nat. 156, 478–494 (2000).

Article Google Scholar
74.
Fransen, B., Blijjenberg, J. & De Kroon, H. Root morphological and physiological plasticity of perennial grass species and the exploitation of spatial and temporal heterogeneous nutrient patches. Plant Soil 211, 179–189 (1999).

CAS Article Google Scholar
75.
Forman, R. T. T. Land Mosaics: The Ecology of Landscapes and Regions (Cambridge University Press, 1995).
76.
Jaeger, J. A. G. Landscape division, splitting index, and effective mesh size: new measures of landscape fragmentation. Landscape Ecol. 15, 115–130 (2000).

Article Google Scholar
77.
Akçakaya, H. R. E. Linking population-level risk assessment with landscape and habitat models. Sci. Total Environ. 274, 283–291, https://doi.org/10.1016/S0048-9697(01)00750-1 (2001).

ADS Article PubMed Google Scholar
78.
Didham, R. K., Kapos, V. & Ewers, R. M. Rethinking the conceptual foundations of habitat fragmentation research. Oikos 121, 161–170, https://doi.org/10.1111/j.1600-0706.2011.20273.x (2012).

Article Google Scholar
79.
Sykes, M. F. & Glen, M. Percolation processes in two dimensions. I. Low-density series expansion. J. Phys. A: Math. Gen 9, 87–95 (1976).

ADS Article Google Scholar
80.
Margolina, A., Djordjevic, Z. V., Stauffer, D. & Stanley, H. E. Corrections to scaling for branched polymers and gels. Phys. Rev. B 28, 1652–1654 (1983).

ADS CAS Article Google Scholar
81.
Mertens, S. Lattice animals: A fast enumeration algorithm and new perimeter polynomials. J. Stat. Phys. 58, 1095–1108 (1990).

ADS MathSciNet Article Google Scholar
82.
Mertens, S. & Lautenbacher, M. E. Counting lattice animals: A parallel attack. J. Stat. Phys. 66, 669–678 (1992).

ADS MathSciNet Article Google Scholar
83.
Jensen, I. Enumeration of lattice animals and trees. J. Stat. Phys 102, 865–881 (2001).

ADS MathSciNet Article Google Scholar
84.
Jensen, I. Series for lattice animals or polyominoes, http://www.ms.unimelb.edu.au/~iwan/animals/series/square.site.ser (2004).
85.
Andrén, H. Effect of habitat fragmentation on birds and mammals in landscapes with different proportion of suitable habitat: a review. Oikos 71, 355–366 (1994).

Article Google Scholar
86.
Wiens, J. A. In Dispersal (eds Clobert, J., Danchin, E., Dhondt, A. A. & Nichols, J. D.) 96–109 (Oxford University Press, 2001).
87.
Bowne, D. R. & Bowers, M. A. Interpatch movements in spatially structured populations: a literature review. Landscape Ecol. 19, 1–20 (2004).

Article Google Scholar
88.
Matlack, G. R. & Leu, N. A. Persistence of dispersal-limited species in structured dynamic landscapes. Ecosystems 10, 1287–1298 (2007).

Article Google Scholar
89.
Neel, M. C., McGarigal, K. & Cushman, S. A. Behavior of class-level landscape metrics across gradients of class aggregation and area. Landscape Ecol. 19, 435–455, https://doi.org/10.1023/B:LAND.0000030521.19856.cb (2004).

Article Google Scholar
90.
Ovaskainen, O., Sato, K., Bascompte, J. & Hanski, I. Metapopulation models for extinction threshold in spatially correlated landscapes. J. Theor. Biol. 215, 95–108, https://doi.org/10.1006/jtbi.2001.2502 (2002).

Article PubMed Google Scholar
91.
Turner, M. G., Gardner, R. H., Dale, V. H. & O'Neill, R. V. Predicting the spread of disturbance across heterogeneous landscapes. Oikos 55, 121–129 (1989).

Article Google Scholar
92.
Oborny, B. & Kun, Á. Fragmentation of clones: how does it influence dispersal and competitive ability? Evol. Ecol. 15, 319–346 (2002).

Article Google Scholar
93.
Oborny, B., Czárán, T. & Kun, Á. Exploration of resource patches by clonal growth: a spatial model on the effect of transport between modules. Ecol. Model. 141, 151–169 (2001).

Article Google Scholar
94.
Oborny, B., Kun, Á., Czárán, T. & Bokros, S. The effect of clonal integration on plant competition for mosaic habitat space. Ecology 81, 3291–3304 (2000).

Article Google Scholar
95.
Inghe, O. Genet and ramet survivorship under different mortality regimes – A cellular automata model. J. Theor. Biol. 138, 257–270 (1989).

MathSciNet Article Google Scholar
96.
Franklin, J. F. & Forman, R. T. T. Creating landscape pattern by forest cutting: Ecological consequences and principles. Landscape Ecol. 1, 5–18 (1987).

Article Google Scholar
97.
Savage, M., Sawhill, B. & Askenazi, M. Community dynamics: what happens when we rerun the tape. J. Theor. Biol. 205, 515–526 (2000).

CAS Article Google Scholar
98.
Lavorel, S., O'Neill, R. V. & Gardner, R. H. Spatio-temporal dispersal strategies and annual plant species coexistence in a structured landscape. Oikos 71, 75–88 (1994).

Article Google Scholar
99.
Winkler, E. & Fischer, M. The role of vegetative spread and seed dispersal for optimal life histories of clonal plants: a simulation study. Evol. Ecol. 15, 281–301 (2002).

Article Google Scholar
100.
Campos, P. R. A., Neto, E. D. C., Oliveira, V. M. D. & Gomes, M. A. F. Neutral communities in fragmented landscapes. Oikos 121, 1737–1748 (2012).

Article Google Scholar
101.
Boakes, E. H., Mace, G. M., McGowan, P. J. K. & Fuller, R. A. Extreme contagion in global habitat clearance. Proc. R. Soc. London, Ser. B 277, 1081–1085, https://doi.org/10.1098/rspb.2009.1771 (2010).

Article Google Scholar
102.
Solé, R. V., Bartumeus, F. & Gamarra, J. P. G. Gap percolation in rainforests. Oikos 110, 177–185 (2005).

Article Google Scholar

Download references

Acknowledgements

We thank Ferenc Jordán, Gabriella Magyar, and Péter Mandl for their helpful comments on earlier versions of this manuscript. Péter Englert kindly allowed us to use his graphical modelling tool PLA-MAZE for producing the pictures in Fig. 1. We are also very grateful to Géza Meszéna and György Szabó for enjoyable discussions about our project. This work was supported by the Hungarian National Research Fund and the National Research, Development and Innovation Office (NKFIH) under the grant numbers T29789, T35009, K109215, K124438, K100299, K119347, GINOP-2.3.2-15-2016-00057, GINOP-2.3.2-15-2016-00019, and NWO-OTKA N34028. Á.K. is grateful for support from the European Science Foundation (TBA/01) for his participation in the Young Scientists Summer Program of the International Institute for Applied Systems Analysis in Laxenburg, Austria, and for support for a János Bolyai Research Fellowship from the Hungarian Academy of Sciences. B.O. acknowledges support by the International Program of the Santa Fe Institute, USA. U.D. gratefully acknowledges support by the Sixth Framework Program of the European Commission, the European Science Foundation, the Austrian Science Fund, the Austrian Ministry of Science and Research, and the Vienna Science and Technology Fund.

Author information

Affiliations

Evolution and Ecology Program, International Institute of Applied System Analysis, Schlossplatz 1, A-2361, Laxenburg, Austria

Ádám Kun & Ulf Dieckmann
Parmenides Center for the Conceptual Foundations of Science, Parmenides Foundation, Kirchplatz 1, D-82049, Pullach, Germany

Ádám Kun
Department of Plant Systematics, Ecology and Theoretical Biology, Loránd Eötvös University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

Ádám Kun & Beáta Oborny
MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

Ádám Kun
Evolutionary Systems Research Group, MTA Centre for Ecological Research, Klebelsberg Kuno u. 3., H-8237, Tihany, Hungary

Ádám Kun
GINOP Sustainable Ecosystems Group, MTA Centre for Ecological Research, Klebelsberg Kuno u. 3., H-8237, Tihany, Hungary

Beáta Oborny
Department of Evolutionary Studies of Biosystems, The Graduate University for Advanced Studies (Sokendai), Hayama, Kanagawa, 240-0193, Japan

Ulf Dieckmann

Contributions

Á.K., B.O. and U.D. jointly devised the study and wrote the manuscript. Á.K. implemented the numerical investigations and analysed the data. All authors reviewed and approved the manuscript.

Corresponding author

Correspondence to Ádám Kun.

Ethics declarations

Competing Interests

The authors declare no competing interests.

Additional information

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kun, Á., Oborny, B. & Dieckmann, U. Five main phases of landscape degradation revealed by a dynamic mesoscale model analysing the splitting, shrinking, and disappearing of habitat patches. Sci Rep 9, 11149 (2019). https://doi.org/10.1038/s41598-019-47497-7

Download citation

Received: 13 August 2018
Accepted: 17 July 2019
Published: 31 July 2019
DOI : https://doi.org/10.1038/s41598-019-47497-7

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.