acta oecologica 33 (2008) 280–290

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Original article Stochastic approach to determine spatial patterns of community on a desert island

Crystian Sadiel Venegas-Barrera1, Enrique Morales-Bojo´rquez*, Gustavo Arnaud

Centro de Investigaciones Biolo´gicas del Noroeste (CIBNOR), Mar Bermejo 195, Col. Playa Palo de Santa Rita, La Paz, B.C.S. 23090, Mexico article info abstract

Article history: One of the principal sources of error in identifying spatial arrangements is autocorrelation, Received 23 May 2007 since nearby points in space tend to have more similar values than would be expected by Accepted 30 November 2007 random change. When a Markovian approach is used, spatial arrangements can be mea- Published online 18 April 2008 sured as a transition probability between occupied and empty spaces in samples that are spatially dependent. We applied a model that incorporates first-order Markov chains to an- Keywords: alyse spatial arrangement of numerical dominance, richness, and abundance on a lizard Spatial dependence community at different spatial and temporal scales. We hypothesized that if a spatial de- Markov models pendence on abundance and richness exists in a diurnal desert community, then the Mar- Numerical dominance kov chains can predict the spatial arrangement. We found that each pair of values was richness dependent only on its immediate predecessor segment. In this sense, we found interge- Spatial arrangement neric differences at temporal and spatial scales of recurrence estimates. Also, in desert Nearby samples scrub, species show higher spatial aggregation and had lower species richness than at the island level; the inverse pattern occurred on rocky hillsides. At the species level, Uta stansburiana is the most abundant species in desert scrub, while Sauromalus slevini is the most abundant species on rocky hillsides. This report attempts to understand, using Mar- kovian spatial models, the effect of nearby samples on local abundance and richness on different scales and over several seasons. ª 2008 Elsevier Masson SAS. All rights reserved.

1. Introduction segregation, habitat selection, and territoriality (Legendre, 1993; Bevers and Flather, 1999). Patterns based on spatial ar- The spatial arrangement of several species in a community is rangements are a first approximation for analysing the effect not homogeneous because limiting factors that affect abun- of biotic, that is, inter- or intraspecific interactions and abiotic dance and distribution change spatially and species tend to environmental variations, such as, temperature, pH, topogra- respond differentially to environmental heterogeneity. Gener- phy, and soil on species. One of the principal sources of error ally, species predominate in habitats where conditions are in identifying spatial arrangements is autocorrelation, since suitable, and are rare in unfavourable habitats (Kneitel nearby points tend to have more similar values than would and Chase, 2004). Therefore, spatial elements play a funda- be expected by random change (Lichstein et al., 2002). When mental role in most ecological processes, including spatial a Markovian approach is used, spatial arrangements can be

* Corresponding author. Tel.: þ52 612 123 8484x3351; fax: þ52 612 125 3625. E-mail address: [email protected] (E. Morales-Bojo´ rquez). 1 Present address: Laboratorio de Ecologı´a Evolutiva, Centro de Investigaciones en Recursos Bio´ ticos, Universidad Auto´ noma del Estado de Me´xico, Instituto Literario 100, 50000, Toluca, Mexico. 1146-609X/$ – see front matter ª 2008 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.actao.2008.01.002 acta oecologica 33 (2008) 280–290 281

measured as a transition probability between occupied and abundance of species, then we can estimate the average dis- empty spaces in samples that are spatially dependent. tance before we again observe the same species (recurrence) A first-order Markov chain is a stochastic model in which and rate of change between species without the necessity of the future development of a system is dependent on the pres- evaluating the spatial structure of habitats. We predicted ent state of the system and is independent of the way in which high probabilities of replacement between species with simi- that state has developed (Formacion and Saila, 1994). lar habitat requirements as species with different require- In ecology, Markov chains have been used principally in ments. Also, we can compare the numerical dominance and temporal succession of ecological states (Tanner et al., 1996; richness at two spatial scales (island and landscape) to ana- Wootton, 2001a; Hill et al., 2004), time of recovery and restora- lyse the effect of scale on the estimation of spatial arrange- tion of forests (Orlo´ ci and Orlo´ ci, 1988; Hall et al., 1991; Tucker ment. Finally, we calculated the spatial arrangement for and Anand, 2004), patterns of change in parental stock and re- more abundant species on two landscapes. cruitment in fisheries (Roshchild and Mullen, 1985), estimates of bird populations (Wileyto et al., 1994), and anthropogenic impact on marine mammals (Lusseau, 2003). Markov chains 2. Materials and methods spatial models have been used to predict sequences of - laying in butterflies (Root and Kareiva, 1984), ontogenic 2.1. Study area change in habitat preference of cotton rats (Kincaid and Cameron, 1985), movements of Canada geese (Hestbeck Isla Coronados is a volcanic, land bridge island (260801500 N, et al., 1991), and spatial inhibition by allelopathy in plants 111160500 W) located w10 km northeast of Loreto, B.C.S., (Kenkel, 1993). These studies highlight the potential of Markov Mexico and w3 km from the closest shore of the Baja Califor- chains in the study of population dynamics and importance of nia Peninsula (Fig. 1). The climate is hot and arid in summer proximity neighbourhoods in the state of system. However, (mean July temperature 33 C) and warm in winter (mean Jan- rarely has it been used to explore spatial changes in richness, uary temperature 16 C). Summer precipitation comes from abundance, or dominance of species at a community scale, convectional storms on the Peninsula (average 190 mm/year) despite the utility of that approach to analyse spatial depen- (Grismer, 1994). Four types of habitats occur on Isla Coronado; dence between near samples and calculate the average dis- rocky hillsides, desert scrubland, a coastal zone, and a transi- tance between ecological states. tion zone between rocky hillsides and desert scrub (Venegas, The advantages of Markov chains are that: (1) such models 2003). On rocky hillsides, approximately 45% of the surface is are relatively easy to derive from continuous data; (2) the covered with rocks (ranging in diameter from 20 to 100 cm), model does not require deep insight into the mechanisms of 15% is bare soil (includes fallen leaves, soil, and gravel dynamic change; (3) the basic transition matrix summarizes <2 cm), and 40% is covered with vegetation (Jatropha cuneata, essential parameters of dynamic change in a systems in Euphorbia magdalenae, and Simmondsia chinensis). In desert a way that few others models achieve; and (4) a model has scrub, approximately 7% are rocks (<10 cm diameter), 38% is much potential for identifying recent history in dynamic com- bare soil, and 55% is covered with vegetation (principally Fou- munities and population dynamics (Formacion and Saila, quieria diguetii, Gossypium harknessii, Hibiscus denudatus, and 1994). These characteristics are ideal for calculating spatial ar- Jatropha cuneata). The coastal zone has a ground surface that rangement of species. Additionally, Markov chains can be is 5% rock (<10 cm diameter), 65% bare soil, and 30% covered used to estimate the probability of any state of abundance or with vegetation (principally Salicornia virginica, Maytenus phyl- richness occur in the space, assuming that it is dependent lanthoides, and Atriplex barclayana). The transition habitat is on the preceding area. We assumed that: (1) contagious bio- characterized by 30% rocks (5–50 cm diameter), 17% bare logical process that affect local abundance of individual spe- soil, and 53% vegetation (Jatropha cuneata, Hibiscus denudatus, cies are spatially dependent (Legendre, 1993), i.e. conditions Euphorbia magdalenae, and Lycium sp.). of growth, survival, and reproduction tend to be similar at nearby sites, (2) well-selected habitats provide high fitness po- 2.2. Species tential (Railsback et al., 2003), (3) individuals in a population do not show a random distribution, i.e. occurrence of an individ- Isla Coronados is one of four islands in the Gulf of California ual does not affect the presence of others, and (4) species with with a largest number of diurnal lizard species (Grismer, similar adaptations will tend to occur together at the same 2002). Of the ten species in the study area, four are ground- sites (Bell, 2001). In this study, we developed models that in- dwelling species: desert iguana (Dipsosaurus dorsalis), zebra- corporate first-order Markov chains to analyse spatial changes tailed lizard (Callisaurus draconoides), side-blotched lizard (Uta in states of numerical dominance (more abundant species by stanburiana), and orange-throated whiptail (Aspidoscelis hyper- unit area), richness (number of species by unit area), and ythra) and six are rock dwellers: slevin’s chuckwalla (Sauroma- abundance of one lizard community on Isla Coronados in lus slevini), black-tailed brush lizard (Urosaurus nigricaudus), the Gulf of California. We chose diurnal lizard species on central Baja California banded rock lizard (Petrosaurus repens), Isla Coronados because they are conspicuous, abundant, and granite spiny lizard (Sceloporus orcutti), western whiptail represent one of the four islands in Gulf of California with (A. tigris), and Baja California spiny lizard (S. zosteromus) high richness (ten species). We hypothesized that, if a pattern (Grismer, 2002). Diurnal desert lizard species show different of abundance and richness exist in a desert community, then sizes of home range, exhibit varying territoriality, and differ- Markov chains can predict the spatial arrangement. If we ent kinds of foraging behaviour (Pough et al., 2004). The foli- knew the sequences of presence-occurrence, richness, and vores, D. dorsalis and S. slevini, have a small home range, 282 acta oecologica 33 (2008) 280–290

Fig. 1 – Study area on Isla Coronado in the Gulf of California, off the east coast of the Baja California Peninsula. Transects and principal habitats are shown.

where size tends to increase as the patchiness of food in- the result of availability of resources and reproductive events creases, and defend polygynous territories (Kwiatkowski and that affect the size of the home range; therefore, the distance Sullivan, 2002). species, genus Callisarus, between states of numerical dominance, richness, and abun- Uta, Urosaurus, Sceloporus, and Petrosaurus, commonly sit and dance depend on seasonality. wait to prey on mobile prey and defend a small home range (ambush, Pough et al., 2004). Finally, Teiidae species, Aspido- 2.3. Data collection scelis, are active predators that mostly feed on cryptic or slow-moving prey, have larger home ranges with a high over- Patterned annual activity of desert lizard species varies, which lap with neighbours, but with very little overlap of their core affects the frequency of sightings. For example, Aspidoscelis areas where they obtain most of their food (Eifler and Eifler, hyperythra and A. tigris are more abundant in warm months 1998). We assumed that changes in spatial distribution are (May–July), Dipsosaurus dorsalis and Sauromalus slevini are acta oecologica 33 (2008) 280–290 283

sighted more frequently after summer rainfall (September– The matrix of one-step transition probabilities for all the m October), while Sceloporus spp. and Petrosaurus repens prefer states of the lizard community is given by: months with temperatures below 25 C (November–Decem- Pii Piw Pim ber). We chose May, September, and November to evaluate Piw Pww Pwm P ¼ the frequency of sighting lizard species. We delimited homo- ... geneous areas (landscape units) based on interpretations of Pim Pmw Pmm aerial photographs (scale 1:75,000) and prepared four perma- where Pii ¼ the dominant species in the lizard community nent transects (3 to 9.8 km long and 6 m wide) for censuses from one segment to the next remains Species i and Piw ¼ the of the four habitats in 2004 in four days in May, September, predominant species in the lizard community changes from and November (Fig 1). The 6-m width was close to the limit one segment dominated by Species i to Species w, where of identifying species by sight. Transects were divided into isw. When the matrix is post-multiplied by vector w(f), repre- 25-m segments because GPS resolution was 6–15 m with senting the composition of states at segment s, according to MapInfo 5.0 software (MapInfo, New York). We analysed Equation 1, data at two levels, island and landscape, for dominance, rich- f f ; ness, and abundance of diurnal lizard species. At the island Wð þ 1Þ¼Pwð Þ (1) level, 862 segments (21,745 m) were evaluated during each the resulting vector w(f þ 1) describes the composition of the sampling period, distributed among the four habitats. At the lizard’s community at segment s þ 1. The previous equation landscape level, we evaluated only the two principal land- can be used iteratively to simulate the composition of the scapes, rocky hillsides with 328 segments (8200 m) and desert community over an area. The resulting vector is the steady scrubland with 224 segments (5600 m). We sighted 446 states probability ‘wi’, which means that the probability that in May, 691 in September, and 330 in November. Transects the lizard’s community will be in a particular state is indepen- were sampled from 08:00 to 14:00 h and again from 16:00 to dent of its initial state (Hill et al., 2004). This is a relatively 18:00 h because most of species have bimodal daily activity. simple model, in which probability distribution of the Observations were not taken at 14:00 to 16:00 h during the hot- model-states stabilizes after a given number of model steps test time of day when most species reduce their activity (Tucker and Anand, 2004). This property makes the model (Grismer, 2002). The species and the GPS-defined location of ideal for presenting spatial pattern dominance. every sighting were recorded. We tested three Markovian assumptions of models. (1) Transition probabilities are constant and depend only on pre- vious states. We hypothesized that if each transition probabil- 2.4. Model ity was independent of the previous state, then we would expect ‘n/y’, where ‘y’ is defined like a number of segments be- First-order Markovian models are based on probabilities of tween a pair of states (cells), where ‘n’ is the total number of transition from one state to another after a one-distance segments, and ‘y’ is the total number of cells. The chi-square step (segment), which depends only on the previous state criterion (P < 0.05) was used to test this hypothesis (Roshchild (Formacion and Saila, 1994; Wootton, 2001a; Hill et al., 2004). and Mullen, 1985). (2) Irreducibility implies that every state is We assumed that the population is closed during the month possibly recurrent, that is, having occurred once, there is of the survey, without loss or gain of individuals. Wootton a non-zero probability that it will occur again. (3) When ergo- (2001b) found that the application of second-order Markov dicity, a recurrent state within a finite class of communicating chains showed similar results as first-order Markov chains; states, is not periodic, it means that a class of communicating s therefore, we assumed that the present state of systems states need only contain at least one member i with pii 0. only depend on the previous sample. We employed three kinds of information: (1) numerical dominance, most abun- 2.5. Parameters dant species by segment and ecological states are the species and empty segments; (2) richness or number of species by seg- Replacement Analysis (P) is the probability that a randomly- ment and states are the number of species and empty seg- selected point in a stationary lizard community is replaced ments; and (3) abundance or number of lizards by species by a different species. This probability is the average of the and states are the number of lizards and empty segment. stationary distribution of the probability of replacement, de- For example, for numerical dominance, let the lizard’s com- fined as: munity be sampled at regular intervals, and let there be a re- cord of the state observed at each sampling. P ¼ wið1 Pii PeiÞ=wi; (2) We defined a state ‘i’ at segment ‘s’ of the lizard commu- where wi is the stationary frequency of species i, and e is in an nity to mean that species ‘i’ is the dominate species present empty state (Hill et al., 2004). in the lizard community at segment ‘f’, and let there be ‘m’ Expected turnover segments (E) measure the number of species that dominate the lizard community at one segment segments in which a point changes state (Hill et al., 2004). or another on segments under study (Roshchild and Mullen, The turnover segments of species i is the distance that a point 1985). Let the one-step transition probability ‘piw’ be defined changes to state j, defined as: as: Piw ¼ the probability that the dominant species in the lizard E ¼ d ¼ 1=ð1 P Þ: (3) communities is species w, given that the dominant species in i ii the lizard community during the immediately preceding seg- Recurrence segments (R) is a point in any state that will ment was species i. eventually leave this state and then return to it after some 284 acta oecologica 33 (2008) 280–290

number of segments (Hill et al., 2004). A recurrence segment of Dipsosaurus dorsalis after sighting a rock-dwelling Sauromalus state i is the average distance elapsing between points leaving slevini in the previous segment is 0.0. The same pattern was state i and returning to it, defined as: observed for several other species, including lower values than those estimated for transition of species of the same R ¼ qi ¼ð1 wiÞ=wið1 PiiÞ: (4) group. We observed that transition probabilities among We analysed three spatial levels. (a) Island, which describes ground-dwellers are greater than among rock-dwellers (Table spatial segregation between species and richness in all seg- 2). We used a simple assumption where our expected number ments evaluated during the three months of sampling. We of segments by cell in all matrices was 7.09. This means that expected high probabilities of replacement between species every cell could have the same value. When we analysed with similar habitat requirements as species with different re- this assumption with a chi-square test, there was a significant quirements. We calculated the steady-state probabilities of difference (c2 ¼ 173, d.f. ¼ 120, P < 0.001). Consequently, we numerical dominance and richness for the three months, concluded that the number of transitions by cell was not uni- since the species had different annual activity cycles (Grismer, formly distributed. 2002). We expected seasonal variations. To test this hypothe- In Table 2, the steady-state probabilities of seasonal var- sis, we used the G test at P < 0.05 (Sokal and Rohlf, 1981). (b) iations in numerical dominance and richness of lizard com- Landscape, which describes the properties of species with munities are shown. For example, these probabilities mean similar habitat requirements. We analysed only the species that in summer Uta stansburiana could be dominant with in the rocky hillside and flatland desert scrubland landscapes a probability of 0.17, whereas Sceloporus zosteromus could in September because the largest number of lizards occur at be dominant with a probability of 0.001. Changes in the this time. We expected lower rates of replacement, turnover, abundance of lizards and the space occupied by them de- and distance of recurrence than at the island level. We used pend on the season. The estimate of steady-state probabili- the G test at P < 0.05. (c) Species, which describes the spatial ties for numerical dominance at the island level is lower in pattern of the most abundant species at the landscape level. May than in September (Gtest ¼ 42.8, d.f. ¼ 10, P < 0.001) The estimates illustrate a potential use of Markov chains in and the inverse pattern occurred from September to Novem- spatial arrangement studies because each species responds ber (Gtest ¼ 790, d.f. ¼ 10, P < 0.001) (Table 2). During Septem- differently to habitat and will show differences as distance be- ber, more segments were occupied, principally by tween lizards. Dipsosaurus dorsalis and Sauromalus slevini,andground- dwellers were more abundant, showing the expected lower turnover and recurrence segments than for rock-dwellers 3. Results (Table 3). The richness steady-state probabilities of occupied states (more than one species) in September are higher than 3.1. Island level during spring and autumn (Gtest ¼ 25.7, d.f. ¼ 3, P ¼ 0.004); the opposite tendency occurred in the empty state The matrix of transition probabilities of numerical dominance (Gtest ¼ 175, d.f. ¼ 3, P < 0.001) (Table 2). The steady-state showed spatial segregation between ground-dwelling probabilities showed seasonal variations in numerical dom- and rock-dwelling species because the transition probabilities inance and richness of the lizard community (Table 2). tend to have lower values between species with different September had the highest level of richness; numerical niche requirements (Table 1). For example, the probability dominance was most dynamic between seasons. The transi- of finding a segment dominated by a ground-dwelling tion probabilities are shown in Fig. 2a.

Table 1 – Estimated transition matrix P for numerical dominance at insular level in summer. Ddor, Dipsosaurus dorsalis; Usta, Uta stansburiana; Ahyp, Aspidoscelis hyperythra; Cdra, Callisaurus draconoides; Ssle, Sauromalus slevini; Unig, Urosaurus nigricaudus; Atig, Aspidoscelis tigris; Prep, Petrosaurus repens; Sorc, Sceloporus orcutti; and Szos, S. zosteromus. From State To

Ground-dwelling Rock-dwelling

Empty Ddor Usta Ahyp Cdra Ssle Unig Atig Prep Sorc Szos

Empty 0.663 0.059 0.129 0.061 0.0 0.041 0.039 0.0 0.002 0.004 0.002 Ddor 0.353 0.318 0.235 0.094 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Usta 0.452 0.075 0.377 0.075 0.007 0.0 0.007 0.0 0.0 0.007 0.0 Ahyp 0.364 0.309 0.073 0.2 0.0 0.018 0.018 0.018 0.0 0.0 0.0 Cdra 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Ssle 0.636 0.0 0.0 0.03 0.0 0.212 0.091 0.0 0.03 0.0 0.0 Unig 0.667 0.03 0.03 0.0 0.0 0.061 0.212 0.0 0.0 0.0 0.0 Atig 0.556 0.0 0.0 0.0 0.0 0.333 0.111 0.0 0.0 0.0 0.0 Prep 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.0 0.0 Sorc 0.667 0.0 0.333 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Szos 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 acta oecologica 33 (2008) 280–290 285

we found two ground-dwellers (Uta stansburiana and Aspido- Table 2 – Steady-state probabilities of numerical dominance and richness at the insular level during the scelis hyperythra) with low probabilities of occurrence (Table 4). three sampling periods. Abbreviations of species names When we compared the steady-state probability of numer- are presented in Table 1. ical dominance by species at the island level versus rocky hill- State May Sep. Nov. sides, we found the steady-state probability of Petrosaurus repens, Aspidoscelis tigris, Sceloporus orcutti, and S. zosteromus Numerical dominance Empty 0.621 0.569 0.693 on rocky hillsides was greater than at the island level (G ¼ 130, d.f. ¼ 8, P < 0.001) (Table 4). The expected turnover Ground-dwelling Usta 0.162 0.17 0.175 test 2 Ddor 0.067 0.103 0.001 distance (c ¼ 16, d.f. ¼ 8, P ¼ 0.04), recurrence segment 2 Ahyp 0.076 0.073 0.025 (c ¼ 200, d.f. ¼ 8, P < 0.001), and distance in meters Cdra 0.001 0.001 0.001 (c2 ¼ 400, d.f. ¼ 8, P < 0.001) on rocky hillsides were signifi- cantly different than at the island level. However, the Rock-dwelling Unig 0.038 0.037 0.065 expected turnover segments are similar between rocky hill- Ssle 0.017 0.036 0.009 sides and the island level (c2 ¼ 1.64, d.f. ¼ 8, P ¼ 0.99). On rocky Atig 0.014 0.004 0.01 hillsides, there is a greater probability of sighting a rock- Prep 0.001 0.002 0.007 Sorc 0.001 0.003 0.009 dwelling species than at the island level and an inverse Szos 0.001 0.001 0.003 pattern occurs for ground-dwelling species (Table 4). In the rocky hillside landscape, there was a greater proba- Richness Empty 0.624 0.572 0.759 bility (0.72) of an empty state than at the island level (0.56) One 0.319 0.326 0.203 (Gtest ¼ 45, d.f. ¼ 3, P < 0.000) (Tables 2 and 4, Fig. 2c). The Two 0.052 0.088 0.035 states of richness on rocky hillsides show no difference at Three 0.005 0.014 0.002 the island level on expected segment turnover (c2 ¼ 0.7, d.f. ¼ 3, P ¼ 0.83), whereas the turnover distance expected (c2 ¼ 31, d.f. ¼ 3, P < 0.001), recurrence segments (c2 ¼ 14, 3.2. Landscape level d.f. ¼ 3, P < 0.002), and distance (c2 ¼ 30, d.f. ¼ 3, P < 0.001) were different. 3.2.1. Rocky hillsides During summer, rocky hillsides were characterized by numer- 3.2.2. Desert scrubland ical dominance of rock-dwelling species (Sauromalus slevini During the summer, the desert scrub landscape is character- and Urosaurus nigricaudus) and a lower position for Sceloporus ized by numerical dominance of ground-dwelling species, zosteromus (Fig. 2b). There were many empty segments and principally Uta stansburiana, and less of Aspidoscelis hyperythra a higher replacement probability between species (from 0 to (Fig. 2d). There was a lower probability of empty segments 1) than at the island level. This matched the expected segment (0.46) and a moderate replacement probability between spe- turnover and estimations of recurrence segments. The same cies (0.23 to 0.41). The same pattern was found for expected pattern was observed in the richness index. In this landscape, segment turnover, estimation of recurrence segments, and

Table 3 – Expected probability of replacement, expected turnover, and recurrence segments at the insular level during September. Numbers in parentheses show estimated distance in meters. Abbreviations of species names are presented in Table 1. Replacement by Expected segments Recurrence other species (P) turnover (m) segments (m)

Numerical dominance Empty – 2.97 (74) 2.25 (56)

Ground-dwelling Ddor 0.33 1.47 (36) 12.80 (319) Usta 0.17 1.6 (40) 7.81 (195) Ahyp 0.44 1.25 (31) 15.90 (396) Cdra 1.00 1.00 (25) 856 (21401)

Rock-dwelling Unig 0.12 1.27 (31) 32.80 (820) Ssle 0.15 1.27 (31) 34.40 (860) Atig 0.44 1.00 (25) 276 (6900) Sorc 0.33 1.00 (25) 286 (7138) Prep 1.00 1.00 (25) 446 (11138) Szos 1.00 1.00 (25) 860 (21401)

Richness Empty – 2.97 (74) 2.22 (55) One 0.11 1.64 (40) 3.40 (84) Two 0.47 1.22 (30) 12.60 (315) Three 0.83 1.20 (30) 85.40 (2134) 286 acta oecologica 33 (2008) 280–290

Fig. 2 – Diagram of transition probabilities of: (a) richness at the insular level, (b) numerical dominance; and (c) richness in rocky hillsides, (d) numerical dominance, and (e) richness in desert scrub.

richness index. In this landscape, we did not find rock- selected five species (Table 5). Table 5 shows that recurrence dwelling species (Table 4). for empty segments is, on average, between 1.2 to 1.8 seg- Steady-state probabilities for numerical dominance in des- ments (from 30 m to 45 m) for all species. When a species is ert scrub differ significantly from the island level (Gtest ¼ 15, sighted in a segment, we expect to sight it again within 4.7 d.f. ¼ 3, P ¼ 0.001). In desert scrubland, Uta stansburiana had to 27.7 segments (117 to 692 m), depending on the species. a steady-state probability of 0.23 as a dominant species and The state with four lizards per segment varied from 111 to Aspidoscelis hyperytrhus has a steady-state probability of 0.11. 115 (2787 to 2890 m). Some species were absent from this Comparing expected segments and distance turnover, we state, with different combinations of this description applying did not find significant statistical differences between desert to states, as shown in Table 5. According to these results, Uta scrubland and the island level (c2 ¼ 0.54, d.f. ¼ 3, P ¼ 0.96; stansburiana had the most individuals by segment and the c2 ¼ 7.9, d.f. ¼ 3, P ¼ 0.09). We found lower expected recur- shortest recurrence segment in the desert scrublands. On rence segments and distance for richness states within desert rocky hillsides, Sauromalus slevini had more members by seg- scrubland than at the island level (c2 ¼ 7.9, d.f. ¼ 3, P ¼ 0.04; ment and Urosaurus nigricaudus had the shortest recurrence c2 ¼ 80, d.f. ¼ 8, P ¼ 0.001). In this landscape, there were less segments and distances. empty segments than at the island level (Gtest ¼ 8.6, d.f. ¼ 3, P ¼ 0.035, see Tables 2 and 4, Fig. 2e). Number of species by segment in desert scrubland were no different than at the is- 4. Discussion land level, as in the case for expected segment turnover and distance (c2 ¼ 0.27, d.f. ¼ 3, P ¼ 0.96; c2 ¼ 5.16, d.f. ¼ 3, Markov chains predict the most probable state of a system and P ¼ 0.16). However, recurrence segments and distance were it depends only on the previous state of system before an different (c2 ¼ 8, d.f. ¼ 3, P ¼ 0.04; c2 ¼ 30, d.f. ¼ 3, P ¼ 0.001). elapsed interval of time or space (Formacion and Saila, 1994). In temporal models, Markov chains can predict behav- 3.3. Species level iour, dominance, or ecological succession, while spatial models have been used to explain movements of . In This analysis was made for the most abundant species in both this sense, Hestbeck et al., 1991 noted that Markov chains landscapes. They were defined according to estimates of are not strictly appropriate for projected equilibrium regional steady-state probabilities that are >0.095; the value among abundance of Canada geese; however, Markov chains are use- species ranged from 0.003 to 0.235 (Table 4); therefore, we ful for the study of population dynamics. Root and Kareiva acta oecologica 33 (2008) 280–290 287

Table 4 – Expected probability during September of steady states, replacement, expected turnover, and recurrence segments for numerical dominance and richness for numerical dominance and richness at the landscape level. Numbers in parentheses indicate estimated distance in meters. Abbreviations of species names are presented in Table 1. Landscape Model State Steady Replacement by Expected segments Recurrence states other species (P) turnover (m) segments (m)

Rocky hillside Numerical dominance Empty 0.720 – 4.18 (104) 1.62 (40) Ssle 0.097 0.63 1.28 (32) 11.90 (297) Unig 0.095 0.68 1.29 (32) 12.30 (307) Ahyp 0.024 0.75 1.00 (25) 40.30 (1007) Atig 0.024 0.63 1.00 (25) 40.30 (1007) Usta 0.024 0.33 2.00 (50) 80.30 (2007) Prep 0.006 0.00 1.00 (25) 164 (4100) Sorc 0.006 1.00 1.00 (25) 164 (4100) Szos 0.003 0.00 1.00 (25) 330 (8250) Richness Empty 0.720 – 4.18 (104) 1.63 (40) One 0.240 0.04 1.44 (36) 4.53 (113) Two 0.034 0.50 1.11 (27) 31.50 (787) Three 0.006 1.00 1.00 (25) 154 (3850)

Desert scrub Numerical dominance Empty 0.460 – 2.20 (55) 2.58 (64) Usta 0.235 0.23 1.410 (35) 4.57 (114) Ddor 0.190 0.34 1.66 (41) 6.99 (174) Ahyp 0.115 0.41 1.23 (30) 9.71 (242) Richness Empty 0.570 – 2.10 (74) 2.07 (51) One 0.330 0.11 1.60 (40) 2.58 (64) Two 0.090 0.48 1.20 (30) 11.50 (287) Three 0.010 1.00 1.00 (25) 233 (5833)

(1984) suggested that a flight sequence of cabbage butterflies probabilities, steady-state probabilities, replacement analysis, was Markovian. Kenkel (1993) demonstrated that individual expected turnover, and recurrence show different useful plant performance is determined in part by proximity of information of the characteristics of spatial arrangement of neighbourhood, i.e. spatially dependent. Finally, Kincaid and the lizard community. These include: (1) a view of the possible Cameron (1985) found that movements of cotton rats Sismodus interactions between species and their environment without hispidus between habitats follow a Markovian pattern and deeper evaluation (in the form of transition probabilities), were dependent on age. Results in this study are similar to re- the interactions occurring in areas where growth, survival, sults reported in seasonal modelsdpreceding samples can and reproduction are suitable for maintaining population dy- predict the state of the system (Wootton, 2001a,b; Hill et al., namics in the absence of immigration (Pulliam, 2000)dalso, 2004). dominance and spatial recurrence change, depending on avail- Transition probabilities of numerical dominance and rich- ability of resources and habitat use where some species tend to ness at the island, landscape, and species levels can be used to be dominant in certain areas and rare in other; (2) an index compress data into a few predictive quantities. Transition of dominance and richness (steady-states probabilities);

Table 5 – Recurrence segments for the most abundant species in desert scrub and rocky hillside landscapes. –, not recorded. Numbers in parentheses indicate estimated distance in meters. Abbreviations of species names are presented in Table 1. State Recurrence segments (m)

Desert scrub Rocky hillside

Usta Ddor Ahyp Unig Ssle

Number of lizards/segment 0 1.6 (40) 1.7 (45) 1.6 (40) 1.2 (30) 1.8 (45) 1 4.7 (117) 8.5 (212) 9.7 (242) 10.5 (262) 27.7 (692) 2 7.5 (187) 18.4 (460) 36.5 (912) 41.7 (1043) 116 (2900) 3 58.7 (1467) 43.2 (1080) 74.0 (1850) – 176 (4400) 4 115.6 (2890) – 111.5 (2787) – – Number expected for cell 9.5 14.6 9 19 11.1 d.f. 24 15 24 8 15 c2 81 75 81 60 73 P 0.00001 0.00001 0.00001 0.00001 0.00001 288 acta oecologica 33 (2008) 280–290

(3) importance of seasonal activity and scale in the spatial dispo- At the species level in desert scrubland, Aspidoscelis hyper- sition of species; (4) an estimate of spatial change in dominance ythus had the highest number of recurrence segments for their and richness (replacement and turnover segments); and (5) states of abundance, possibly because this species actively observation of abundance of individual species (recurrence forages (Pianka, 1982) and defends a mobile territory from segments). Also, since we include four different kinds of habi- other conspecifics (Eifler and Eifler, 1998). Uta stansburiana is tats, we can analyse effect of transition probabilities between a passive, general predator, using broad spatial resources by pairs of species among diverse environmental conditions. sitting and waiting (Grismer, 2002), which could explain its The results of the island and landscape models provide high abundance and low recurrence in this landscape. On some interesting insights into the importance of seasonal the rocky hillside, Sauromalus slevini is the rock-dwelling spe- change in the spatial arrangement of lizard species. Four cies with more lizards per segment, possibly because the main results were found. (1) In contrast to sedentary species males protect their polygynous territories (Kwiatkowski and (Hill et al., 2004), season influenced the estimates of parame- Sullivan, 2002). In general, inter- and intraspecific differences ters of numerical dominance and richness that can be in abundance depend on spatially-fixed environmental varia- explained by interspecific differences in periods of activity, tions and seasonal environmental variations (Ives and thermal preferences, or foraging behaviour. For example, Dip- Klopfer, 1997; Pulliam, 2000). In lizards, abundance autocorre- sosaurus dorsalis is active in warmer months and reduces its lation can result from social organization (Glinsky and Krekor- activity at temperatures below 25 C(Muth, 1980)asdoPetro- ian, 1985; Eifler and Eifler, 1998), foraging mode (Pianka, 1982), saurus repens and Sceloporus orcutti (Grismer, 1994). (2) There mating systems (Sinervo et al., 2001; Kwiatkowski and Sulli- were higher transition probabilities within a habitat group van, 2002), or food resources (Vitt, 1991). (ground-dwelling or rock-dwelling) than between species. Our estimates of first-order Markovian chains are domi- This spatial segregation could reflect choice of different mi- nated by empty segments and these probably play a funda- croclimates, reflecting physiological tolerance or choice of dif- mental role in inter- and intraspecific interactions. Gilpin ferent vegetation (Pianka, 1982; Vitt, 1991). (3) Since steady- and Hanski, 1991 proposed that unoccupied, but habitable, state probabilities, expected turnover, and recurrence seg- areas may also be important in the dynamics of the spatial ments were lower at the island than the landscape level, it is structure of populations. In this sense, populations may be- necessary to select the appropriate scale for evaluation in sur- come locally extinct, even in perfectly suitable habitats, and vey methods to obtain a more realistic estimate of abundance the delay between extinction and re-colonization should leave and richness of species (Collins and Glenn, 1995; Bradbury some fraction of suitable habitats unoccupied at any given et al., 2001). In this case, the island level was used to contrast time (Thomas and Kunin, 1999). Patches and travelling waves species with different niche requirements and the landscape can be produced when populations are influenced by densities level to analyse species with similar niche requirement. (4) of neighbouring populations (Sinervo et al., 2001), where spa- We found lower probabilities of steady states for rock-dwell- tial-temporal patterns occur because local dispersal of indi- ing species at the island level and rocky hillsides than species viduals link the dynamics of adjacent areas (Morris, 1990; in the desert scrubland. Johnson, 2000). We found that Uta stansburiana will probability become the Legendre (1993) explained that spatial structuring is an im- dominant species at the island level and in the desert scrub- portant component of ecosystems, where heterogeneity is land and Urosaurus nigricaudus will continue to be the domi- functional, and not the result of some random, noise-generat- nant species on rocky hillsides. In contrast, Sceloporus ing process. Because the spatial pattern in the desert lizard zosteromus and Petrosaurus repens have the lowest probabilities community is variable, with spatial patterns that change sea- to become dominant species at the island level and in the sonally (Vitt, 1991; Case, 2002; Grismer, 2002), we quantified rocky hillside landscape. Our results seem to be consistent the probability that a random sighting of a lizard at an insular with observations about generalist species tending to achieve and landscape level in different periods of year. To predict dominance (Formacion and Saila, 1994; Brown et al., 1995). spatial behaviour in biological communities, it was necessary According to Hanski and Gyllenberg, 1997, generalist species, to quantify the complete distribution of probabilities that such as U. stansburiana, or species using broad resources are describe movements and sightings along all possible land- common and widely distributed; whereas specialist species, scapes and seasons (Bradbury et al., 2001). If the dynamics of such as P. repens, are narrowly distributed. movement are even moderately complex, it is not possible We found that three species co-exist in high abundance to describe complete sight probability of movement in conve- and low recurrence in segments of the desert scrubland, while nient, closed forms, such as those found by other researchers eight species are present in low abundance and high recur- (Yoccoz et al., 2001; MacKenzie et al., 2003). We need to know rence in segments of rocky hillside. This result is consistent not only what physical boundaries delineate lizard species, with the models of He and Legendre (2002) and Kneitel and but also to show how rapidly the components of a species Chase (2004) that higher species richness in a sampling area mix change within those boundaries. occurs if the species are spatially more regularly distributed In conclusion, we could establish a model where each pair in the community, while high spatial aggregation of individ- of values was dependent only upon its immediate predecessor uals of a species result in lower species richness in a study segment. According to Liebhold and Gurevitch, 2002, spatial area. When environmental factors or resources are spatially dependence of ecological data has typically been a problem variable, different species find suitable microhabitats in dif- that can reduce a field observer’s capacity to understand the ferent localities and co-exist regionally (Morris, 1990; He and spatial pattern of the population or community. This effort Legendre, 2002). illustrates how first-order Markovian spatial chains may be acta oecologica 33 (2008) 280–290 289

used to examine spatial dependence of abundance and rich- Hill, M.F., Witman, J.D., Caswell, H., 2004. Markov chain analysis ness at different scales and seasons. However, understanding of succession in rocky subtidal community. Am. Nat. 164, how neighbouring elements affect each other or how they af- E46–E61. Ives, A.R., Klopfer, E.D., 1997. Spatial variation in abundance created fect a process is quite different from classical ecological by stochastic temporal variation. Ecology 78, 1907–1913. concerns for the structure and function of discrete communi- Johnson, M.P., 2000. Scale of density dependence as an alternative ties, populations, or ecosystems (Pickett and Cadenasso, 1995). to local dispersal in spatial ecology. J. Anim. Ecol. 69, 536–540. Kenkel, N.C., 1993. Modeling Markovian dependence in populations of Aralia nudicaulis. Ecology 74, 1700–1708. Kincaid, M.B., Cameron, G.N., 1985. 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