Annals of Botany 114: 389– 398, 2014 doi:10.1093/aob/mcu110, available online at www.aob.oxfordjournals.org

PART OF A HIGHLIGHT ON CLONAL PLANT ECOLOGY

Patch size and distance: modelling habitat structure from the perspective of clonal growth Bea´ta Oborny* and Andras G. Hubai Department of Plant Taxonomy, Ecology, and Theoretical Biology, Lora´nd Eo¨tvo¨s University, 1/C Pa´zma´ny Pe´ter stny., Budapest, H-1117, Hungary * For correspondence. E-mail [email protected] Received: 14 January 2014 Returned for revision: 20 March 2014 Accepted: 25 April 2014 Published electronically: 18 June 2014

Key words: Clonal plant ecology, ramet, genet, phenotypic plasticity, physiological integration, foraging, population dynamics, modelling, cellular automata, percolation theory, habitat connectivity, patchy environment.

IN T RO DU C T IO N Clonal plants are capable of vegetative reproduction; thus, the genetic individual (genet) can consist of multiple physiological individuals (ramets). The degree of physiological integration between adult ramets varies between species (Pitelka and Ashmun, 1985; Jo´nsdo´ttir and Watson, 1997), and can also vary within species, between genotypes [e.g. in Ranunculus reptans (van Kleunen et al., 2000) and in Fragaria chiloensis (Alpert et al., 2003)]. In spite of the differences in integration between established, adult ramets, all clonal plants share the trait that juvenile ramets are subsidized by parents. The parent–offspring pair is an elementary unit in which integration is inevitable, at least for some period of time. In a patchy habitat, the parent and the offspring can be under different environmental conditions, for example in terms of light/shade or high/low nutrient concentration. Several experiments have examined parent–offspring ramet pairs in contrasting habitats (e.g. Alpert and Mooney, 1986). These studies show that plants are capable of sensing resource differences, and can respond morphologically

and/or physiologically. The response is vital when the resource is limiting ramet survival, thus, the parent and the offspring are in direct conflict. In some species, allocation into the offspring decreases when the parent is in a good, resource-rich patch, and its stolon (or other organ of vegetative reproduction) grows into a resource-poor region. Thus, the plant’s biomass tends to remain within the good patch, avoiding the bad region. Alternatively, the parent can go on with the subsidy, moving out its resource into the bad region. We call these growth responses avoiding vs. entering, respectively. Lolium perenne provides an example for the avoiding growth response. 14C tracing revealed that when the amount of light became limited, unshaded tillers abandoned the export of photoassimilates to shaded tillers (Ong and Marshall, 1979, cited by Pitelka and Ashmun, 1985). Lycopodium annotinum in the tundra was observed to keep its biomass in favourable soil patches at the expense of ramet death in unfavourable patches (Callaghan et al., 1986). Entering has also been observed in many species. For example, Ambrosia psilostachya can occupy extremely saline

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† Background and Aims This study considers the spatial structure of patchy habitats from the perspective of plants that forage for resources by clonal growth. Modelling is used in order to compare two basic strategies, which differ in the response of the plant to a patch boundary. The ‘avoiding plant’ (A) never grows out of a good (resource-rich) patch into a bad (resource-poor) region, because the parent ramet withdraws its subsidy from the offspring. The ‘entering plant’ (E) always crosses the boundary, as the offspring is subsidized at the expense of the parent. In addition to these two extreme scenarios, an intermediate mixed strategy (M) will also be tested. The model is used to compare the efficiency of foraging in various habitats in which the proportion of resource-rich areas ( p) is varied. † Methods A stochastic cellular automata (CA) model is developed in which habitat space is represented by a honeycomb lattice. Each cell within the lattice can accommodate a single ramet, and colonization can occur from a parent ramet’s cell into six neighbouring cells. The CA consists of two layers: the population layer and the habitat. In the population layer, a cell can be empty or occupied by a ramet; in the habitat layer, a cell can be good (resourcerich) or bad (resource-poor). The habitat layer is constant; the population layer changes over time, according to the birth and death of ramets. † Key Results Strategies M and E are primarily limited by patch distance, whereas A is more sensitive to patch size. At a critical threshold of the proportion of resource-rich areas, p ¼ 0.5, the mean patch size increases abruptly. Below the threshold, E is more efficient than A, whilst above the threshold the opposite is true. The mixed strategy (M) is more efficient than either of the pure strategies across a broad range of p values. † Conclusions The model predicts more species/genotypes with the ‘entering’ strategy, E, in habitats where resourcerich patches are scattered, and more plants with the ‘avoiding’ strategy, A, in habitats where the connectivity of resource-rich patches is high. The results suggest that the degree of physiological integration between a parent and an offspring ramet is important even across a very short distance because it can strongly influence the efficiency of foraging.

390

Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth A 1

2

B 1

2

F I G . 1. An example for a parent and offspring ramet in a patchy habitat (A), and its representation in the spatially discrete model (B). Resource-rich (good) patches 1 and 2 are white; the resource-poor (bad) region between patches is coloured. The shading corresponds to the distance from the nearest patch, ranging from z ¼ 1 (light brown) to z ¼ 4 (dark brown). Throughout the manuscript, we use ‘good patch’ and ‘patch’ as synonyms, and distinguish them from the ‘bad region’ between patches. A parent ramet that is within patch 1 gives birth to an offspring into the bad region. The unit distance (z ¼ 1), i.e. the distance between the centres of hexagons, is equal to the parent–offspring distance.

ecology (for a review, see Oborny et al., 2007). It provides a general framework for the investigation of spreading in disordered media (Stauffer and Aharony, 1994). In the present application, the disordered medium is the patchwork of good and bad sites. A plant with strategy A can be present only in good sites (which are ‘open’ in a percolation terminology), and can never enter into bad sites (which are ‘closed’); therefore, the spreading of A can be directly translated into a percolation problem. We test the following hypotheses: (1) the avoiding strategy is inefficient at low values of p, due to challenge I; (2) the entering strategy is also inefficient at low values of p, because of challenge II; and (3) the efficiency of the mixed strategy is intermediate between that of A and E.

THE MODEL General description

We use a stochastic CA model. This is a frequently applied modelling tool for the study of population dynamics in space (Cza´ra´n, 1998), including ramet populations (for a review, see Oborny et al., 2012). In our CA, space is represented by a honeycomb lattice. Each lattice cell can accommodate a single ramet. Colonization can occur from a parent ramet’s site into six neighbouring sites (Fig. 1B).

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soils, provided that a part of the clone is under relatively good conditions (e.g. in grass hummocks or on slopes bordering salt pans). The ramets under good conditions provide water for the stressed ramets, at the expense of their own biomass production (Salzman and Parker, 1985). Phragmites australis was also observed to enter from a favourable to an unfavourable area, from high to low marsh, subsidized through physiological integration (Amsberry et al., 2000; see more examples in Alpert, 1996; Pennings and Callaway, 2000; Wang et al., 2008). Several avoiding and entering responses, specifically to patchiness caused by the presence of competitors, have been reviewed by Novoplansky (2009), and Herben and Novoplansky (2010). In general, avoiding facilitates the exploitation of the current patch; entering permits the exploration of new patches. This dilemma, exploitation vs. exploration, is at the core of studies on plant foraging (Oborny and Cain, 1997). Strategies of foraging for patchy resources were modelled first in animals, and, since the 1980s, several models of plant foraging have been published (e.g. Sutherland and Stillman, 1988; for a review, see Oborny et al., 2012). Our model can be interpreted as a minimal model of foraging. The response to the environment does not imply any morphological plasticity within the ramet or stolon; it is based on a simple, binary response of whole ramets (survival/death). The unit which responds consists of the smallest number of ramets (a pair) that can meet a local difference in the environment (rich/poor in a resource). We use cellular automata (CA) simulations for producing various patterns of patches, and observing the growth of ramet populations within and between patches (Fig. 1). We vary the proportion of good cells ( p) within the lattice, and compare the performance of three strategies at each value of p. The ‘avoiding’ strategy (A) always avoids the bad region; it remains within the good patch. The ‘entering’ strategy (E) always enters. In addition to these two pure strategies, we test a ‘mixed’ strategy (M), that randomly chooses between avoiding or entering. Each simulation starts by placing a seed into a randomly selected good site, initiating a new genet. Then we let the ramet population grow according to the rule (A/E/M), and reach a steady state. Finally, we evaluate the probability of survival of the genet, and its efficiency of foraging. (Note that the evolutionary unit is the genet, which carries the strategy.) A plant that grows in a patchy habitat typically meets the following challenges. (I) The size of resource patches may be too small. This challenge is particularly severe for A, which cannot leave the patch where its first ramet was initiated. (II) The distance between patches may be too large. This challenge concerns E and M, which can grow out of the parental patch, but only to a limited distance, because growth is subsidized from the parental resource. We study the importance of these challenges in various habitats, in a broad range of p. For challenge I, we investigate the mean patch size, and the probability of confinement within a small patch (when the patch size is 1, 2, 3 . . . units). For challenge II, we estimate the mean patch distance, and the probability of escape from confinement by traversing a small distance (1, 2, 3 . . . units). We investigate these structural properties from the view of a growing plant, borrowing some basic concepts from percolation theory. Percolation theory originates from physics, and has gained several applications in natural sciences (Sahimi, 1994), including

Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth The CA consists of two layers: the population layer and the habitat. In the population layer, a site can be empty or occupied by a ramet; in the habitat layer, a site can be good (resource-rich) or bad (resource-poor). The habitat layer is constant; the population layer is changing over time, according to the birth and death of ramets.

1 2 3

In the lattice, p proportion of the cells is good, and 1 – p is bad. Good and bad cells are distributed randomly (Fig. 3). Although this is a simple system, many of its characteristics are non-trivial. Spatial structures emerge through neighbourhood contacts. Adjacent good sites form patches (‘clusters’ in the percolation

Entering

Avoiding

F I G . 2. Transition rules. In each pair of cells, the left/right cell is the site of the parent/offspring, respectively. Birth and death depend on site qualities. White/ grey represents a good/bad site. Occupancy by a ramet is denoted by a green hexagon. A dead ramet is indicated by a red cross. The transition rules were constructed in a way which ensures that the number of surviving ramets after the transition is always equal to the number of good sites, i.e. to the carrying capacity in the two-cell system. Rule 1: if both the parent and the offspring are in a good site, then both survive. Rule 2: if both ramets are in bad sites, then both die. Rule 3: if the parent’s site is bad, but the offspring’s is good, then the parent dies and the offspring survives. Rules 1 –3 are deterministic. Rule 4 is stochastic: the probability of entering is e, the probability of avoiding the bad site is 1 – e. We compared three strategies: A (e ¼ 0), E (e ¼ 1) and M (e ¼ 0.5).

p = 0·01

p = 0·1

p = 0·3

p = 0·5

F I G . 3. The habitat at various values of p. In the left column, those sites that belong to the largest good patch are red, other good sites are white and bad sites are grey. In the right column, good sites are white and bad sites are coloured. The colour code is the same as in Fig. 1. These are examples in relatively small lattices (20 × 20 cells); the simulations were run in larger ones (100 × 100 or 400 × 400 cells). To avoid edge effects, the boundaries are wrapped around, i.e. cells in the opposite edges of the lattice (upper and lower; left and right) are connected.

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The habitat

or

4

The plant population

The population consists of ramets of a single genet. The rules of ramet birth and death are summarized in Fig. 2. We apply several simplifying assumptions. To observe the strongest possible effect of resource patchiness, we assume that the contrast between good and bad sites is maximal. A good site can sustain a ramet unconditionally; a bad site can never sustain any ramet, unless the ramet is subsidized from the outside. This is expressed in the transition rules (Fig. 2). We compare two pure strategies, the avoiding (e ¼ 0) and the entering (e ¼ 1) strategy, with a mixed strategy (e ¼ 0.5). The strategies are abbreviated as A, E and M, respectively. M is half-way between A and E, as if the choice was made by tossing an unbiased coin. Utilization of the resource by the plant is consistent throughout the model. One good cell is capable of sustaining one ramet. In every case (from 1 to 4 in Fig. 2), the number of ramets at the end of a transition is equal to the number of good cells in the two-cell system. Thus, case 1 yields two ramets (reproduction); case 2 ends with zero ramets (mortality); and cases 3 and 4 preserve the number of ramets. In case 4, the place of the ramet is optional: remaining in the parent’s site (avoiding) or colonizing a new site (entering). Each elementary step (updating) in the computer simulation starts by selecting a ramet randomly from the existing N ramets. Then we select a random cell from the six neighbours of the parent cell. If that cell is empty, i.e. can be colonized, then we apply the transition rule described in Fig. 2. After the elementary step, time is increased by 1/N. One time unit (Monte Carlo step) consists of N elementary steps. We run the simulations for 5000 time units. This period of time was sufficient for reaching a steady state in the population in every case. At the end, the program records whether the genet is alive, i.e. consists of at least one ramet. If it does, then the efficiency of foraging is also recorded, n(e), which is defined as the proportion of good sites occupied by ramets. e indicates the strategy (A/E/M). n(e) ¼ 1 when the plant occupies every good site. In most cases, n(e) , 1, because some good sites may be unattainable by stepwise spreading (limited dispersal). To investigate the limitations of spreading, we review some results from percolation theory, and complement them with our computer simulations. We construct the habitat layer in a way that is simple, and is compatible with percolation theory.

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Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth

terminology; e.g. 1 and 2 in Fig. 1B). To characterize the patch structure at each value of p, we ask the following questions.

of escape is defined as a cumulative probability, R(b) =

Patch size, si. What is the number of good cells that are accessible

from the initial cell by always stepping from good to good cells? For example, in Fig. 1B, the sizes of patches are s1 ¼ 7, and s2 ¼ 4. Patch size is equal to the maximum area that is available for A, because it cannot leave the patch. Probability of confinement, C(a). To characterize the whole lattice,

ma a G

ra = Note that



(1)

qj

(4)

j=1

This is the probability that the plant can reach another patch by traversing a distance that is not longer than b cells (i.e. di ≤ b). We examine b ¼ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Mean patch distance, D. We define the mean patch distance as

D=



qb b

(5)

b

D characterizes the minimum thickness of the wall between patches. This is a limitation upon the growth of E and M: the plant cannot grow less in a bad region, even if it finds the optimal pathway. S, C(a), D and R(b) provide an organism-centred description of the habitat pattern. The value of each depends on p, the proportion of good cells over the area.

ra = 1. We define a cumulative probability as

a

C(a) =

a 

Computer simulations

rj

(2)

j=1

This is the probability that the size of the patch in which the seed has landed is not larger than a (i.e. si ≤ a). We call this the ‘probability of confinement’ within a. We examine a ¼ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Mean patch size, S. We define the mean patch size as

S=



ra a

(3)

a

S is the expected size of the patch where the seed lands. Patch distance, di. This variable characterizes a between-patch distance. Take patch i, in which the seed has landed. Let di denote the minimum distance between i and the nearest good patch (multiple patches can have the same minimum distance; equality does not influence the data). In Fig. 1B, the minimum distance between the plant’s actual patch (1) and the nearest good patch (2) is d1 ¼ 3, because three steps (new ramets) are needed in bad cells in order to reach a good cell in patch 2. Patch 2 may have another neighbouring patch (not shown in the figure) which is closer than patch 1, so d2 can be different from d1. Patch distances are crucial for plants E and M, because plants with these strategies can traverse only di ¼ 1 (Fig. 2). For convenience, we call the bad region between good patches a ‘wall’. The right panel in Fig. 3 shows the thickness of the wall. Probability of escape, R(b). This variable describes the lattice according to patch distances. We estimate qb, the probability that a seed, landing in a good site, arrives at a patch which has a nearest neighbour at distance b (so that di ¼ b). The probability

Two kinds of simulations were performed: the study of population dynamics (that we abbreviate as PD) and the study of habitat structure (HS). In the PD, we used both layers of the CA (the habitat and the population layer), and followed the development of the system over 5000 time steps. In the HS, we used only the habitat layer at a single time (t ¼ 0), because this layer was constant. The lattice. The lattice consisted of L ¼ 10 000 cells (100 × 100) in the PD, and L ¼ 160 000 cells (400 × 400) in the HS. The boundaries were wrapped around to eliminate edge effects (Fig. 3). Thus, every cell had six nearest neighbours. Input. The PD had two input parameters: p, which was needed for creating the habitat; and e, which described the plant strategy. p ranged from 0.1 to 0.9 in 0.1 steps. In addition, we tested p ¼ 0.01, p ¼ 0.45 and p ¼ 0.55. Using p ¼ 0.01 allowed us to observe the populations at a very low proportion of good sites. Examining p ¼ 0.45 and 0.55 was motivated by the fact that some of the observables changed significantly around p ¼ 0.5. In the HS, the range and resolution of p were the same as those in the PD. Initial conditions. Every repetition started by producing a random

map (with a given L and p). In the PD, a good site was selected randomly for the place of the seed from which clonal growth started. Repetitions. In the PD, we made 200 independent repetitions at every parameter combination ( p, e). In the HS, we used 100 repetitions at every parameter value ( p). Results of the repetitions were averaged. We do not introduce a separate notation for the averages, but adopt the notations that were introduced above (see a summary table in the Appendix). The evaluation of D requires at least two patches. This condition was not satisfied in some runs at p ¼ 0.9. These cases were omitted, and the average was calculated from the remaining runs.

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we consider every possible position where the seed could land to establish a ramet (i.e. every good cell). Let L denote the total number of cells, and G the number of good cells; thus G ¼ pL. For each patch size a, the number of patches that consist of exactly a cells is denoted by ma. The probability that the size of the patch in which the seed has landed is exactly a (i.e. si ¼ a) can be calculated as

b 

Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth Output. The PD yielded the proportion of surviving runs from 200

repetitions, which we abbreviate as u(e), and the efficiency of foraging n(e). The HS provided S, D, C(a) and R(b).

R E S U LT S Habitat structure

A 100

Probability of confinement, C(a) (Fig. 5A). Within the challenge of getting stuck in a patch (challenge I), a particular problem is to get stuck in a small patch. Therefore, we examine the frequency distribution of small patches (from a ¼ 1 to 10). At p ¼ 0.01, the probability of confinement is C(a ¼ 1) ¼ 0.94. This means that 94 out of 100 seeds are expected to land in a solitary, single-cell patch. The remaining six seeds are likely to land in two-cell patches. At p ¼ 0.2, which is still a relatively low value within the range of p, the distribution is different. The seed has a high chance for a larger patch; even si . 10 may occur (see the area above the upper curve in Fig. 5A). At p ¼ 0.6, the amount of small patches is negligible [C(10) ¼ 0.01].

Size

Mean patch size, S (%)

80

60

40

20

All patches Largest patch removed

Probability of escape, R(b) (Fig. 5B). In the same way as C(a) is

0 0

0·2

0·4

0·6

0·8

1

p

B

5 Distance 4

Mean patch distance, D

p ≥ 0.5, the two curves diverge. This indicates that the largest patch is considerably larger than the others. The dotted line shows a sudden increase at p ¼ 0.5, which implies that the connectivity of the good sites changes abruptly in this region. At p . 0.5, most of the good sites belong to the same (the largest) cluster. If all belonged, then the dotted line would be at 100 %, and the solid line would be at 0 %. Our data do not reach these extremes, but are very near at p ¼ 0.7 already. Consequently, the range of p can be divided into two distinct phases, with a relatively narrow range of transition around p ¼ 0.5. In the fragmented phase, at low values of p, the habitat contains numerous, relatively small patches. In the connected phase, a single large patch dominates the area, and the rest of the patches are small. Patches with intermediate size occur in significant numbers only in the vicinity of the transition. Mean patch distance, D (Fig. 4B). At p ¼ 0.01, the mean distance is considerable. The value decreases rapidly with the increase of p. At p ¼ 0.2, the mean distance is only 1.036 cells. At p ¼ 0.3, practically every cell is attainable by one step across the bad region (D ¼ 1.003).

3

2

related to the mean patch size (S), this probability, R(b), is related to the mean patch distance (D). At p ¼ 0.01, we find R(1) ¼ 0.12. Only 12 out of 100 seeds can be expected to land in a patch which has a neighbour at a short distance, b ¼ 1. These thin walls can be penetrated by an E or M plant. At p ¼ 0.1, 76 out of 100 seeds land in a patch with a thin wall [R(1) ¼ 0.76]. The value of p is still low, p ¼ 0.2, when as many as 96 out of 100 seeds are expected to land in a thin-walled patch [R(1) ¼ 0.96]. At p ¼ 0.3, practically every seed lands in a thin-walled patch [R(1) ¼ 0.997], i.e. an escape route is almost always available. Plant population

1

0 0

0·2

0·4

0·6

0·8

1

p F I G . 4. General features of the habitat. (A) Mean patch size as a percentage of the total good area. (B) Mean distance between patches. The dotted line shows all the patches; the solid line is without the largest patch. p is the proportion of good sites. The smallest value shown is p ¼ 0.01. Error bars represent standard deviations. In most cases, the error bar is smaller than the symbol.

Frequency of survival, u(e) (Fig. 6A). None of the strategists survives at p , 0.2. The M strategy becomes viable at the lowest value, p ¼ 0.2, and shows a rapid increase at 0.2 , p , 0.3. The E strategy switches on at p ¼ 0.3, also with an abrupt increase. M is better than E in a broad range of p. The A strategy always survives: it does not move any resource out of the good cells, thus, its persistence in the good area is unlimited. Efficiency of foraging, n(e) (Fig. 6B). The efficiencies also increase abruptly with a gradual increase of p. First M becomes efficient (0.2 , p , 0.3), then E switches on (0.3 , p , 0.4) and, finally, A starts to become efficient (0.45 , p , 0.5). The tipping point,

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Mean patch size, S. Figure 4A shows the mean patch size (S) as a percentage of the total good area (G). In each simulated lattice (i.e. independent repetition), we calculated the mean from all the patches (dotted line); then we omitted the largest patch, and recalculated the mean (solid line). When multiple patches had the same size, only one of them was omitted. Figure 3 illustrates the effect of omitting the largest patch (in a smaller lattice, 20 × 20 cells, instead of 400 × 400 cells, that were used in the simulations). Figure 4A shows that at low values of p, omission of the largest patch does not change the mean significantly; but at

393

394 A

Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth 1

A

Confinement

0·6

0·4

0·2

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Frequency of survival, u(e) (%)

≤ 10 ≤9 ≤8 ≤7 ≤6 ≤5 ≤4 ≤3 ≤2 =1

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A M E

60

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0·2

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p

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p

B

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Escape

≤ 10 ≤9 ≤8 ≤7 ≤6 ≤5 ≤4 ≤3 ≤2 =1

0·6

0·4

0·2

0

Efficiency of foraging, n(e) (%)

Probability of escape, R(b)

Distance, b 0·8

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0 0

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p F I G . 5. Probabilities of (A) confinement and (B) escape. The habitat is described from the view of a ramet that is in a random site in a good patch. C(a) is the probability that the patch is not larger than a. R(b) is the probability that another good patch is available within distance b. The horizontal axis is the same as in Fig. 4. Error bars represent standard deviations.

where the change is the most abrupt, is at p ¼ 0.5. The turning point, from which A becomes more efficient than E, is at 0.5 , p , 0.55. At p ≥ 0.7, all the strategies have 100 % efficiency. Standard deviations in and near the tipping point are larger in A than in E or M. DISCUSSION Percolation across a patchy habitat

Clonal growth typically constrains the plant to a stepwise exploration and exploitation of resource patches. This can be a considerable limitation compared with seed dispersal, in which jumps are possible (see an example in Hieracium pilosella, modelled by Winkler and Sto¨cklin, 2002; and a general model by

0

0·2

0·4 p

F I G . 6. Performance of the strategies in various habitat types. The horizontal axis is the same as in Fig. 4. (A) Frequency of survival in 200 independent trials. (B) Efficiency of foraging for the resource. In (A), the points represent individual data, thus the standard deviation cannot be defined; in (B), standard deviations are shown by error bars. At some data points, the error bar is smaller than the symbol.

Winkler and Fischer, 2002). Those studies that address the evolution of asexual vs. sexual reproduction often mention strict dispersal limitation as a disadvantage of the former. Our results show that step-by-step spreading is seriously limiting at low values of p, but not at high values. We propose that percolation theory provides a plausible explanation for the phenomenon. Percolation theory is a general framework for the study of spreading in disordered media (Stauffer and Aharony, 1994), in the present case, clonal growth across a patchy habitat. The theory has already been applied in ecology (see reviews by Sole´ and Bascompte, 2006; Oborny et al, 2007), but only some results have been utilized from the vast literature of the topic that is available in physics. Mainly the results that are related to the probability of percolation have been applied in

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Probability of confinement, C(a)

Size, a 0·8

Survival

Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth

K/( p − pc )b

(6)

where / denotes ‘proportional to’. Theoretical considerations suggest that b ¼ 5/36 in all two-dimensional lattices (Stauffer and Aharony, 1994). The scaling law has been found to be general: the value of b does not depend on the geometry of the lattice (which is honeycomb in our case), the size of the neighbourhood (which is six cells) and other local details of the model. This broad generality (‘universality’ in the terminology of physics) makes percolation theory suitable in a large variety of models (see Sahimi, 1994, regarding diverse applications in physics, chemistry and other natural sciences). The value of pc, on the other hand, is non-universal: it depends on the details of the model, e.g. on the geometry of the lattice. We chose the honeycomb lattice because it does not distort Euclidean distances. Therefore, the threshold is exactly at 50 % good and 50 % bad cells, pc ¼ 0.5. Equation (6) implies that the percolation threshold is sharp: surpassing pc, the probability of percolation rises abruptly (as a second-order phase transition). Thus, the range of p can be divided into two distinct phases. We call p , pc the ‘fragmented phase’, and p . pc the ‘connected phase’ (Oborny et al., 1997).

The mean patch size (S) also changes abruptly at pc. As p increases, and approaches the threshold ( p  pc), the mean patch size goes to infinity, S / ( pc − p)−g

(7)

This is also a scaling law with a universal exponent, g ¼ 43/18 {Stauffer and Aharony, 1994; our definition of S [eqn (3)] is applicable only in finite lattices, because it is based on the number of patches; the definition Stauffer and Aharony use is compatible with ours, but is applicable to infinite lattices, too, as it is based on patch densities}. Surpassing the percolation threshold ( p . pc), an infinite patch appears, thus, the mean becomes infinite. When the infinite patch is omitted, and the mean is recalculated only for the finite patches, we obtain a power law with the same exponent, S / ( p − pc )−g

(8)

Equations (7) and (8) express that the mean patch size increases rapidly near the threshold, and declines rapidly after surpassing the threshold, as the infinite patch contains most of the good sites (see an illustration in Fig. 3). Our results are in agreement with the theory (Fig. 4A). At low values of p, the mean patch size is very small, compared with the total good area (S/G is close to 0 %). This is the fragmented phase. At high p, the mean is almost as large as the total good area (S/G is nearly 100 %), because the largest patch dominates the field. This is the connected phase. The transition between the phases is not as sharp as suggested by the theory, because the lattice is finite (see chapter 4 in Stauffer and Aharony, 1994, regarding finite-size effects). There is a narrow transition zone between the two phases, rather than a single point of transition. The tipping point is at p ¼ 0.5, the predicted value. In agreement with eqns (7) and (8), the mean patch size without the largest patch (the lower curve in Fig. 4A) also peaks at p ¼ 0.5. The increase of standard deviations toward the threshold is also predicable from percolation theory. In an infinite lattice, the standard deviation goes to infinity as p  pc (Stauffer and Aharony, 1994). Equations (7) and (8) are applicable only in a narrow range around pc, the so-called critical region. In a honeycomb lattice in two dimensions, the critical region is approx. 0.45 ≤ p ≤ 0.55 (cf. Hoshen et al., 1979). Outside this region, the parameters describing the shape of S( p) are non-universal and thus less interesting from the view of physics. Consequently, a lot of information is available about a narrow range, and much less is known about a broad range of p. From the view of ecologists, the whole 0 ≤ p ≤ 1 range is relevant, because any value may occur in a natural environment. Our simulations (Figs 4 and 5) complement the scope of percolation theory. Crossing obstacles

Percolation implies that the hypothetical organism never crosses any bad region, but goes around it. To understand the opportunities of a clonal plant, we examine an alternative, when the organism can penetrate into the bad region by 1 unit, as the E and M strategists do. We define the ‘transit threshold’, pt, as the value of p at which the mean distance becomes statistically indistinguishable from 1. This threshold is far from the percolation

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an ecological context. Here we extend the scope to the sizes of patches. The theory, in its simplest form, assumes complete disorder, i.e. no correlation. Among lattice models in ecology, the counterpart of this system is the so-called random map, which is frequently used in landscape ecology as a neutral landscape model (e.g. Gardner et al., 1987; Gustafson and Parker, 1992; Andre´n, 1994; With et al., 1997; Turner et al., 2001, chapter 6). We used the same system, but we modelled fine-scale phenomena. The spatial unit (the lattice constant) was the typical parent–offspring distance, which is of the magnitude of centimetres in most clonal species. The random map is a seemingly simple system, but many of its characteristics are non-trivial. Spatial structures emerge through neighbourhood contacts, as adjacent good sites form patches (Fig. 1). One of the exciting questions, that is routinely studied by percolation theory, is about the probability of percolation (K). We present the question in the context of clonal growth. Consider a seed that lands in a randomly selected good cell. What is the probability that the cell is part of a patch within which it is possible to reach the edge of the lattice without getting stuck, i.e. without being surrounded by a set of bad cells? The answer depends on the size of the lattice; therefore, K is defined for infinite lattices. K is the probability that it is possible to spread infinitely far from the point of origin over infinite time. In practice, K is studied by increasing the lattice size, and extrapolating for an infinite lattice. This quantity has been thoroughly studied in statistical physics (Stauffer and Aharony, 1994). Some key results are directly related to clonal growth. The probability of percolation shows a critical transition at a particular value of p, the percolation threshold ( pc). Below the threshold ( p , pc), K ¼ 0; therefore, spreading within a patch is necessarily limited to a finite distance. Above the threshold ( p . pc), K . 0; thus, unlimited spreading becomes possible, depending on the landing site of the seed. Close to the threshold, in the so-called critical region, K( p) can be described by the following scaling law,

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Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth 0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9

I. Size

S, C(1)

pc

II. Distance D, R(1)

pt

III. Plant

M

u(e), n(e)

1

E

A

threshold, within the range 0.2 , pt , 0.3. Although p ¼ 0.3 is still a relatively low proportion of good sites, the mean distance is already close to 1 unit (D ¼ 1.003; Fig. 4B), and the probability of escape is nearly 1 [R(1) ¼ 0.997; Fig. 5B].

Assessment of the initial hypotheses Habitat types

The transit threshold ( pt) and the percolation threshold ( pc) divide the p axis into three regions, which represent qualitatively different habitat types (Fig. 7). Below the transit threshold ( p , pt), some of the patches are surrounded by thick walls, that cannot be crossed by a single step. Between the thresholds ( pt ≤ p ≤ pc), all the walls are thin, but they fragment the patches. Above the threshold ( pc , p), the walls break up; thus, the good portion of the habitat is connected. The efficiency of foraging related to habitat types

With the increase of p, the efficiency of foraging increased rapidly within distinct, narrow ranges for each strategy, in the following order: M, E and A (green, blue and red in Fig. 7, respectively). In M and E, the range of transition in efficiency and in survival coincided (compare Fig. 6A and B), indicating that efficient foraging is crucial for survival. (For a comparison, strategy A is not limited by this factor, because it never moves out any resource from a good patch; see its unlimited survival in Fig. 6A.) The mixed strategy M proved to be more successful in foraging than any of the pure strategies. Its limit of survival was at the transit threshold, pt. This is not evident, since the minimum distance between patches was low already at a lower value of p (D ¼ 1.3 at p ¼ 0.1). The results suggest that the probability of escape must be extremely high for survival (nearly 1), when the point(s) of escape, i.e. the exact places where the two patches are near, must be found by the a trial-and-error of ramet birth and death.

Hypothesis 1 was that the avoiding strategy is inefficient at low values of p, and this was confirmed. Challenge I, getting stuck in a patch, was a serious limitation for A in half of the possible range of p, up to the percolation threshold. Hypothesis 2 was that the entering strategy is also inefficient at low values of p, and this too was confirmed. Challenge II, being unable to reach the next patch, did limit E and M. In particular, the probability of escape had to be high for successful foraging. Hypothesis 3 was that the efficiency of the mixed strategy is intermediate between that of A and E, but this was rejected. Foraging efficiency of the mixed strategy (M) was not intermediate, but better than either of them in a broad range of p. O UT LOO K The model predicts more species (or genotypes within species) with strategy E in those habitats where the resource patches are scattered, and more plants with strategy A in those places where the connectivity of good areas is high. Pure avoiding or entering is probably rare in nature: we should rather expect mixed strategies with high vs. low e, respectively. A further step from the present study is to run the simulations at various values of e. Patch contrast is also important. At present, we assumed maximal contrast between the good vs. bad sites. Experiments have indicated that contrast can significantly influence the performance of a growing clone (e.g. Wijesinghe and Hutchings, 1999). A model on physiological integration suggests that keeping the same total amount of resource, but lowering the contrast, decreases the advantage of resource translocation (Kun and

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F I G . 7. Summary of the results. Rows show the main output variables that characterize habitat patches (I and II) or plants (III). The value of p varies from 0 to 1 in a resolution that was used in the simulations (0.1, with three additional values: 0.01, 0.45 and 0.55). Shading indicates a range of p within which the corresponding output variables change abruptly. (I) Patch size shows a rapid increase near pc. The vertical dotted line marks the theoretical value of the percolation threshold ( pc ¼ 0.5). (II) Patch distance decreases rapidly at 0.01 , p , 0.1. The transit threshold ( pt) is above that range. (III) Plant strategies switch from a non-viable to a viable state within distinct ranges, in the following order: M, E and A (green, blue and red, respectively). Within the region marked by purple, foraging efficiency turns from n(A) , n(E) to n(A) . n(E), i.e. A becomes more successful than E.

The E strategy was similar to M, but more sensitive to the probability of escape, because of the relatively higher number of trials (by definition). Our simulated plants could not sense the presence of a next good patch; the offspring was initiated in a random direction. We predict that near the transit threshold, considerable selection acts upon the plant’s ability to sense the advantageous direction of growth. In the simulations, the step length was short (1 unit) and the contrast between good and bad patches was maximal. Therefore, missteps were strictly punished: whenever a mother ramet placed an offspring into a bad site, and then the next offspring could not find any good site, then the mother and the offspring died (see the rules in Fig. 2). We applied the highest contrast to observe the maximal effect of habitat patchiness. Permitting more steps or increasing the step length in the bad region would certainly diminish the effect of dispersal limitation, and consequently would increase the efficiency of foraging. Further numerical experiments with lower contrasts and/or variable step length would provide more insight into the system, and would increase the realism of the model (see Outlook section, below). All the strategies became successful in foraging in the middle range, pt ≤ p ≤ pc. The percolation threshold, pc, was the exact place where strategy A switched on. Surpassing the percolation threshold, A became more successful than E (row III in Fig. 7). In summary, A was dependent on patch size (row I in Fig. 7), whereas E and M were primarily limited by patch distance (row II in Fig. 7).

Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth

per se, without any plasticity in the step length, can significantly influence the shape of a growing clone (e.g. Oborny et al., 2001; Herben 2004; Crowley et al., 2005). A study by Oborny and Kun (2003) adds that not only broadly integrated clones can forage successfully, but also those in which the genet becomes fragmented into small groups of ramets. The present study reduces the group size to the minimum: two ramets. Pseudoannual species (e.g., Trientalis europaea) may provide real-life examples for such a minimal system, because a short ramet life span prevents the connection of multiple ramet generations (Piqueras and Klimesˇ, 1998). Nevertheless, this is not a model of any particular plant species, but a general minimal model, to which more complex clonal growth strategies can be related. On the basis of this minimal model, we suggest an organism-centred description of the habitat pattern. We propose that percolation theory provides insight into the challenges met by a plant that grows within and between resource patches. ACK NOW LED GE MENTS This work was supported by the National Science Fund of Hungary [OTKA, K 109215]. L IT E R AT U R E CI T E D Alpert P, Mooney HA. 1986. Resource sharing among ramets in the clonal herb, Fragaria chiloensis. Oecologia 70: 227– 233. Alpert P. 1996. Does clonal growth increase plant performance in natural communities? In: Oborny B, Podani J, eds. Clonality in plant communities. Special Features in Vegetation Science 11, Uppsala: Opulus Press, 11– 16. Alpert P, Stuefer JF. 1997. Division of labour in clonal plants. In: van Groenendael J, de Kroon H, eds. Clonal growth in plants: regulation and function. The Hague: SPB Academic Publishers, 137– 154. Alpert P, Holzapfel C, Slominski C. 2003. Differences in performance between genotypes of Fragaria chiloensis with different degrees of resource sharing. Journal of Ecology 91: 27–35. Amsberry L, Baker MA, Ewanchuk PJ, Bertness MD. 2000. Clonal integration and the expansion of Phragmites australis. Ecological Applications 10: 1110–1118. Andre´n H. 1994. Effect of habitat fragmentation on birds and mammals in landscapes with different proportion of suitable habitat: a review. Oikos 71: 355–366. Bell AD. 1991. Plant form. An illustrated guide to flowering plant morphology. Oxford: Oxford University Press. Callaghan TV, Headley AD, Svensson BM, Lixian L, Lee JA, Lindley DK. 1986. Modular growth and function in the vascular cryptogam Lycopodium annotinum. Proceedings of the Royal Society B: Biological Sciences 228: 195–206. Crowley PH, Stieha CR, McLetchie DN. 2005. Overgrowth competition, fragmentation and sex ratio dynamics: a spatially explicit, sub-individual-based model. Journal of Theoretical Biology 233: 25– 42. Cza´ra´n T. 1998. Spatiotemporal models of population and community dynamics. New York: Chapman and Hall. Gardner RH, Milne BT, Turner MG, O’Neill RV. 1987. Natural models for the analysis of broad-scale landscape pattern. Landscape Ecology 1: 19– 28. Gustafson EJ, Parker GR. 1992. Relationships between landcover proportions and indices of landscape spatial pattern. Landscape Ecology 7: 101 –110. Herben T. 2004. Physiological integration affects growth form and competitive ability in clonal plants. Evolutionary Ecology 18: 493– 520. Herben T, Novoplansky A. 2010. Fight or flight: plastic behavior under selfgenerated heterogeneity. Evolutionary Ecology 24: 1521– 1536. Herben T, Suzuki J. 2002. A simulation study of the effects of architectural constrains and resource translocation on population structure and competition in clonal plant. Evolutionary Ecology 15: 403– 423. Hoshen J, Stauffer D, Bishop GH, Harrison RJ, Quinn GD. 1979. Monte Carlo experiments on cluster size distribution in percolation. Journal of Physics A: Mathematical and General 12: 1285– 1307.

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Oborny, 2003). Therefore, we hypothesize that lowering the contrast in the present model would give an advantage to A over E in a broader range of p. It would be promising to make further simulations, varying the birth and death rates of ramets in good vs. bad sites, in order to set the cost of integration within a broader range. Some experiments suggest that the cost is not necessarily high [e.g. in Fragaria chiloensis (Alpert and Mooney, 1986) and in Potentilla anserina (Yu et al., 2002); see a review in Song et al., 2013]. In the present model, the lattice constant (i.e. the distance between neighbouring cells) was defined as the distance between parent and offspring. There are two kinds of deviation from this assumption: the lattice constant may be larger or smaller than the parent– offspring distance. In the first case, more generally, the habitat quality is autocorrelated on the scale of spacer length. We chose the uncorrelated random map in order to keep the percolation problem simple. However, the basic percolation model can be extended to some types of autocorrelated lattices. More than two states can also be introduced (i.e. ranks of resource content, instead of good/ bad). A model by Mendelson (1999) is an example for such a habitat. In that system, the characteristic scaling laws, and thus the framework of percolation theory, remain applicable. However, this is not necessarily true for every autocorrelated pattern, especially when long-range correlations occur. In the second case, larger steps from the parent’s cell have to be considered. Figure 5 shows the habitat pattern analysed at larger distances (up to 10 units). This could be complemented with simulating plant growth with larger parent– offspring distance (spacer length). The consideration of morphological plasticity would also be important. Not all clonal plants are plastic in terms of spacer length (see, for example, Alpinia speciosa; Bell, 1991), but many experiments have demonstrated plasticity in this trait, and its importance in foraging [e.g. in Glechoma hederacea (Slade and Hutchings, 1987); for a review see de Kroon and Hutchings, 1995]. Computer models have confirmed the importance in a broad range of potential habitats (Sutherland and Stillman, 1988; Oborny, 1994; for a review, see Oborny et al. 2012). Our present model does not assume any plasticity in the spacer length, but it implicitly assumes plasticity in the branching probability, through the environment dependence of ramet birth. Explicit introduction of plasticity into the model is a promising direction of research. Several studies have demonstrated the richness of processes that unfold from the interaction between plasticity and integration, both in experiments (Alpert and Stuefer, 1997; Hutchings and Wijesinghe, 1997; Wijesinghe and Whigham, 2001; Ye et al. 2006; de Kroon et al., 2009) and in models (Herben and Suzuki, 2002; Mony et al., 2011). In this paper, we presented a minimal model of integration. We assumed that only parent – offspring pairs were connected, and the connection influenced only the establishment of the offspring. This simple interaction between ramets is widespread in almost all clonal plant species. Only those species in which the offspring becomes detached from the parent before finding its place (e.g. dispersal by bulbils in Dentaria bulbifera) are exceptions. Our results show that even this minimal extent of integration is sufficient for successful foraging in a broad range of habitat types. This study fits into a logical sequence of previous models on integration. Several studies have demonstrated that integration

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Oborny & Hubai — Modelling habitat structure from the perspective of clonal growth Stauffer D, Aharony A. 1994. Introduction to percolation theory, revised 2nd edn. London: CRC Press. Sutherland WJ, Stillman RA. 1988. The foraging tactics of plants. Oikos 52: 239– 244. Turner M, Gardner RH, O’Neill RV. 2001. Landscape ecology in theory and practice: pattern and process. Berlin: Springer. Wang N, Yu F-H, Li P-X, et al. 2008. Clonal integration affects growth, photosynthetic efficiency and biomass allocation, but not the competitive ability, of the alien invasive Alternanthera philoxeroides under severe stress. Annals of Botany 101: 671– 678. Wijesinghe DK, Hutchings MJ. 1999. The effect of environmental heterogeneity on the performance of Glechoma hederacea: the interactions between patch contrast and patch scale. Journal of Ecology 87: 860–872. Wijesinghe DK, Whigham DF. 2001. Nutrient foraging in woodland herbs: a comparison of three species of Uvularia (Liliaceae) with contrasting belowground morphologies. American Journal of Botany 88: 1071– 1079. Winkler E, Fischer M. 2002. The role of vegetative spread and seed dispersal for optimal life histories of clonal plants: a simulation study. Evolutionary Ecology 15: 281 –301. Winkler E, Sto¨cklin J. 2002. Sexual and vegetative reproduction of Hieracium pilosella L. under competition and disturbance: a grid-based simulation model. Annals of Botany 89: 525– 536. With KA, Gardner RH, Turner MG. 1997. Landscape connectivity and population distribution in heterogeneous environments. Oikos 78: 151–169. Ye X-H, Yu F-H, Dong M. 2006. A trade-off between guerrilla and phalanx growth forms in Leymus secalinus under different nutrient supplies. Annals of Botany 98: 187 –191. Yu F-H, Chen Y, Dong M. 2002. Clonal integration enhances survival and performance of Potentilla anserina, suffering from partial sand burial on Ordos plateau, China. Evolutionary Ecology 15: 303– 318.

A P P E N DI X: S U M M ARY O F T HE VARI AB L E S Values of the input variables are always given by the user; the output values are always observed in the simulations. At the other variables, we note the origin of the values in parentheses. Many of the variables depend on p; we do not indicate that dependence (see the text).

Input p e Output si S C(a) di D R(b) u(e) n(e) Others L G ma ra qb pc pt

Proportion of good sites in the lattice Probability of entering into a bad site Size of patch i Mean patch size Probability of confinement within patch size a Minimum distance between patch i and its nearest neighbour Mean patch distance Probability of an escape route to distance b Survival rate of plants with strategy e Efficiency of foraging with strategy e Number of cells in the lattice (given by the user) Number of good cells in the lattice (calculated) Number of patches with size a (observed) Probability of size a (calculated) Probability of distance b (calculated) Percolation threshold (observed) Transit threshold (observed)

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