ecological modelling 202 (2007) 441–453

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When corridors work: Insights from a microecosystem

Martin Hoyle ∗

School of Biology, University of Nottingham, Nottingham NG7 2RD, UK article info abstract

Article history: Evidence for a beneficial effect of corridors on and in Received 14 February 2006 patches is mixed. Even in a single microecosystem of microarthropods living in moss patches Received in revised form connected by a moss corridor, experiments have had different results (positive and neutral). 2 November 2006 This paper attempts to provide an explanatory framework for understanding these results. I Accepted 7 November 2006 developed a stochastic individual-based model of the moss—microarthropod microecosys- Published on line 12 December 2006 tem. Some of the movement parameter values were estimated from two manipulation experiments. Assuming mortality independent of the season, and assuming the corridors Keywords: merely increase migration rates between patches, only a very weak beneficial effect of corri- Conservation dors was possible in simulations. Incorporating a seasonal pattern to mortality caused some simulated populations to die out, which were then occasionally rescued by migrants from Individual-based model the adjacent patch. Corridors were slightly beneficial if there was little or no immigration from the surrounding matrix. In contrast, corridors were very beneficial in simulations that Microarthropod incorporated lower emigration to the matrix when a corridor was present, even for moder- ate levels of immigration from the matrix. Thus corridors may reduce the chance of species extinction in patches even when the lifespan of the individuals is long relative to the time- scale in question. The beneficial effect in this case can act via two possible mechanisms: seasonal mortality imposing brief periods of high vulnerability to extinction, and the pres- ence of a corridor reducing the rate of emigration to the matrix by encouraging movement along the corridor. Either one or both of these mechanisms may have operated in the study whereby corridors had a beneficial effect, but not in the study whereby corridors did not have a beneficial effect. This work demonstrates that corridor effectiveness is dependent on the species and landscape in question, and that it is important to understand the mechanisms by which corridors function. © 2006 Elsevier B.V. All rights reserved.

1. Introduction Corridors may function by reducing the negative impacts of demographic and environmental stochasticity via the rescue Habitat loss and isolation pose perhaps the most serious effect (Brown and Kodric-Brown, 1977). Furthermore, they may threat to biological diversity (Kareiva et al., 1993). Habitat cor- increase the size of the effective gene pool, thereby reducing ridors have been proposed as a means of reducing the negative inbreeding depression. However, there has been no research effects of habitat isolation (Diamond, 1975) by increasing ani- on the impact of corridors on emigration from habitat patches mal movement rates between habitat patches (Saunders and (Hudgens and Haddad, 2003): they will be beneficial if they Hobbs, 1991). However, corridors are controversial (Mann and direct migrants through corridors. On the other hand, cor- Plummer, 1995). ridors may increase the chance of extinction by increasing

∗ Current address: School of Biosciences, Hatherly Laboratories, University of Exeter, Prince of Wales Road, Exeter, Devon EX4 4PS, UK. Tel.: +44 1392 263796; fax: +44 1392 263700. E-mail address: [email protected]. 0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.11.008 442 ecological modelling 202 (2007) 441–453

genetic uniformity, and by synchronising population dynam- ics (Earn et al., 2000). Furthermore they may increase the incidence of disease, the movement of , and in general (Hess, 1994). Corridors may act as sinks (Amarasekare, 1998; Hudgens and Haddad, 2003) since the quality of the habitat may be poor due to the edge effect, and since individuals may wander unproductively in corridors. Unfortunately therefore, it is difficult to generalise about the value of corridors (Debinski and Holt, 2000). Instead the effectiveness will probably be specific to the species and land- scape in question. Other factors will include the scale of the system relative to the size and dispersal of the species (Berggren et al., 2002), patch isolation and the demographic parameters of the species both in and out of the habitat. The time-scale and of the major threat of local extinction (Hudgens and Haddad, 2003), such as sustained population decline or boom-bust cycles, will also be important. It is therefore important to understand the mechanisms by which corridors may be beneficial, perhaps in conjunction with a with estimates of rates of movement, birth and death. For example corridors may boost the patch-to- patch migration rate (Haddad, 1999; Tewksbury et al., 2002), perhaps by organisms following corridor edges (Levey et al., 2005), and corridors may increase the rate of immigration from the surrounding matrix by acting as drift-fences, where organ- isms are intercepted from the matrix and then directed into connected patches (Fried et al., 2005). Fig. 1 – The effect of corridor connectivity on the abundance Studies of corridor effectiveness are either in the form of five common mite species in moss landscapes after 6 of controlled experiments (e.g. Burkey, 1997; Schmiegelow et months. (a) In the positive study (Gilbert et al., 1998; al., 1997; Gilbert et al., 1998; Gonzalez et al., 1998; Haddad, Gonzalez et al., 1998), there were significantly more 1999; Collinge, 2000; Berggren et al., 2002; Forbes and Chase, individuals in the complete- compared to the 2002; Mabry and Barrett, 2002; Tewksbury et al., 2002; Hoyle broken-corridor treatment for mite species A (circles) and B and Gilbert, 2004), or correlational (e.g. Saunders and Hobbs, (triangles) (t20 = 2.32, P=0.03; t20 = 2.38, P = 0.03, respectively) 1991; Mech and Hallett, 2001; Selonen and Hanski, 2003). Two (species descriptions are not available), and significantly reviews of the literature (Beier and Noss, 1998; Debinski and more individuals in the mainland compared to the Holt, 2000) concluded that most studies support the utility of complete-corridor treatment for mite species A (t20 = 2.11, corridors, and further studies have agreed (e.g. Gilbert et al., P = 0.05), but not for species B (t20 = 1.57, P > 0.05). (b) By 1998; Gonzalez et al., 1998; Mech and Hallett, 2001; Haddad et contrast, in the neutral study (Hoyle and Gilbert, 2004) al., 2003). On the other hand, some studies suggest that cor- where there were two patches per treatment, there were no ridors may have neutral or even negative effects on species significant differences in abundance among the treatments persistence (Burkey, 1997; Collinge, 2000; Forbes and Chase, (Cryptostigmata mite morphospecies 9 (circles), 4 (triangles) 2002; Hoyle and Gilbert, 2004), perhaps because dispersers act and 8 (squares), see Appendix A for morphospecies to synchronise the (Burkey, 1997; Earn et descriptions). Quasi-Poisson errors were used in an al., 2000). analysis of deviance, with contrasts. Error bars represent There are great logistic difficulties in designing replicated one standard error. experiments to test whether corridors increase the rate of migration between habitat patches in the field (Inglis and Underwood, 1992). Hence, a microecosystem approach can tion. Broken corridors of the same total area did not have be a very practical solution, although extrapolation to com- this effect. The abundance of most species was higher in con- munities on a larger scale should be treated with care. The nected patches (Fig. 1a). Henceforth, I refer to this experiment moss—microarthropod microecosystem is a rare example of a as the “Positive Study” and the greater species richness and field-based microecosystem of wildlife corridors. A large com- abundance in the complete- compared to the broken-corridor munity of microarthropods of different trophic levels live in patches as the “corridor effect”. The mechanism behind the a small patch of moss and are easily extracted for censusing. corridor effect was not investigated, but the authors suggested Furthermore, the effects of fragmentation on the microarthro- that corridors facilitated the dispersal of microarthropods pod populations can occur in as little as 6 months (Gilbert et between the habitat patches, maintaining the distribution and al., 1998; Gonzalez et al., 1998). abundance of species through the rescue effect, and that the Gilbert et al. (1998) and Gonzalez et al. (1998) found that scale of fragmentation and dispersal distance of the organisms connecting patches of moss habitat with moss corridors for were likely to be appropriate for the metapopulation concept 6 months slowed the rate of microarthropod species extinc- to apply. By contrast, Hoyle and Gilbert (2004) did not find ecological modelling 202 (2007) 441–453 443

the corridor effect (Fig. 1b) in a similar study. Henceforth, this 10, and patches i =1,2(Renshaw, 1991): experiment is referred to as the “neutral study”. Again, the mechanism of corridor function was not investigated. U − ln( i) , N gives the time to the next birth Here, I aim to account for these contradictory results. This ˇ i Ni(t) paper is organised as follows. I introduce a mathematical U − ln( i+2) , population model (based on Renshaw, 1991) used to analyse N time to the next death ı i Ni(t) the empirical data. The model assumes that a pair of popu- ln(Ui+ ) lations are subject to density-dependent births and deaths, − 4 time to the next migration from patch i, Ni N t and are connected by migration across a corridor. Next I i( ) describe two manipulation experiments designed to estimate U − ln( i+6) , N time to the next emigration the movement parameters of the microarthropods. I then use ε i Ni(t) the parameterized model to present two possible explanations ln(Ui+ ) for the corridor effect of the positive study, and to suggest why − 8 time to the next immigration. Ni the effect was not found in the neutral study.

The next simulated event is the one corresponding to the min- 2. Materials and methods imum of these times. The populations in the two patches are then recomputed—either increasing or decreasing by one unit, 2.1. Population model or remaining unchanged. The algorithm is then repeated: 10 random numbers are generated, the time to the next event is The model chosen to describe the population dynamics of computed, and the populations adjusted, and so on until the the microarthropods, based on Renshaw (1991), suits the data maximum time of 6 months (the length of the positive and practically obtainable on the microecosystem and the ques- neutral studies) is reached. tions being addressed in this paper. The model (coded in Visual The model assumes the following. First, the time to the Basic for Applications, Appendix A) is “structured individual- next event is exponentially distributed. The death process based”: the future abundance of a single microarthropod imposes the simplifying assumption that the mortality rate species in each of two moss patches is simulated, assuming is independent of age. The duration of migration between the following density-dependent processes: birth and death, patches is assumed to be zero (tested in the second manipu- migration between the two patches, emigration from the lation experiment). Since inter-specific data were patches into the surrounding matrix, and immigration from unavailable, the model is of a single species (see Section the matrix into the patches. The deterministic form of the 4). Species interactions are allowed for only indirectly by model is given by a pair of coupled non-linear differential the existence of an equilibrium abundance for each species. equations, one for each patch: Microarthropods are assumed to be reproductively active immediately after birth. dNi(t) = ˇNi t N t − ıNi t N t − Ni t N t + Nj t N t t ( ) i( ) ( ) i( ) ( ) i( ) ( ) j( ) d 2.2. Microarthropod movement rates N N + i (t) − ε i (t)Ni(t) (1) There is very little literature concerning microarthropod where Ni(t) is the abundance of a microarthropod species in movement rates (Norton, 1994). Dispersal of Cryptostigmata one patch at time t and Nj(t) is the abundance in the other mites may be due to seeking of food or favourable oviposi- patch. In patch i: tion sites by gravid females (Norton, 1994) and is probably restricted primarily to adults, since they are better equipped Ni birth rate : ˇ (t) = ˇ0 − ˇ1Ni(t), to deal with danger and desiccation (Norton, 1994) N than immatures. Berthet (1964) measured movement of Cryp- death rate : ı i (t) = ı + ı Ni(t), 0 1 tostigmata mites in soil of two to four centimetres per day. Ni between-patch migration rate : (t) = 0 + 1Ni(t), Ojala and Huhta (2001) found lower dispersal rates of Collem- bola (0.5–10 cm per week) than for Cryptostigmata mites εNi t = ε + ε N t , emigration rate to matrix : ( ) 0 1 i( ) (1–20 cm per week) in soil. Unfortunately, it is not possible Ni to observe the microarthropods move, since they are usually immigration rate from matrix : (t) = 0 − 1Ni(t), (2) buried deep in the moss. Rates of microarthropod migration across the corridor and immigration across the bare rock were where ˇ0, ˇ1, ı0, ı1, 0, 1, ε0, ε1, 0, 1 are constants assumed obtained indirectly from the first and second manipulation equal for both patches. It was assumed that 0 and 1 would be greater when the patches were connected with a moss corri- experiments, respectively. dor. For biological reasonableness, all parameters were subject to a minimum of zero. The intrinsic capacity of increase is 2.2.1. First manipulation experiment r = ˇ0 − ı0. The first manipulation experiment was designed (1) to test A Poisson process for the stochastic version of the deter- whether the rate of migration (0, 1) between patches is ministic equations is assumed (Renshaw, 1991). The time to greater with a connecting corridor than across bare rock, (2) the next event for each of the ten processes in Eq. (2) is dis- to estimate the density-independent corridor migration rate ... tributed exponentially. For random numbers 0 < Uj <1,j =1, , ( 0) and (3) test for density-dependent corridor migration ( 1). 444 ecological modelling 202 (2007) 441–453

all experimental blocks, as

Ct= − Dt= 1 1 (3) Bt=0.5

where At=1, Bt=1, Ct=1, Dt=1 are the average abundances of any given species in patches A, B, C and D, respectively, at the end of the 1 month experiment, and Fig. 2 – A single replicate of the first manipulation experiment. Each replicate consisted of one 10 cm diameter B ≈ . B + B moss patch containing a complete of t=0.5 0 5( t=1 t=0) and microarthropods (patch B, black) plus three patches cleared Bt=0 ≈ (Bt=1 − Dt=1) + (Ct=1 − Dt=1) + (At=1 − Dt=1) (4) of microarthropods (patches A, C and D, white) and one cleared 7 cm long moss “corridor”. The moss was laid on a are the estimated abundances in patch B half way through and bed of gravel in two seed trays, which were widely at the start of the experiment, respectively. separated. Migration from patch B is assumed to occur If migration is density-dependent then: across the corridor and gravel (complete arrows), and immigration from the surrounding environment (broken dC(t) ∝ B(t)(1 + bB(t)), (5) arrows). dt

where b is a constant.

Hence, assuming little change in B over time, Ct=1 is approx- Each of six replicates consisted of four moss patches imately proportional to Bt = 0.5 (1 + bBt=0.5), a quadratic in Bt=0.5. (mostly mixtures of Homalothecium sericeum, Brachythecium Therefore, to test for density-dependent migration, Ct=1 was rutabulum and Hypnum lacunosum var. lacunosum or Isothecium modelled as a quadratic function of Bt=0.5. Poisson errors were myosuroides (Brid.) var. myosuroides, as in the neutral study) assumed, again corrected for overdispersion. and one moss corridor (Fig. 2). The moss was collected from The most important finding was that there were always the site of the neutral study (see Hoyle and Gilbert, 2004). more individuals in the connected patch (C) compared to the One patch contained microarthropods, while the remaining disconnected patch (A) (Table 1) (binomial test: P < 0.0005). patches were cleared of the fauna, by being placed in Tullgren This suggests that microarthropods move more readily across funnels for 48 h. The moss patches were then chosen at ran- the moss corridor than across the gravel, i.e. that 0 is dom and placed in the arrangement of Fig. 2 in the grounds greater when a corridor is present. Significance may not have of the University of Nottingham, and covered with netting to been achieved for some species due to a lack of statistical retain moisture. After 1 month, in July 2001, the moss (which power. The number of individuals in patch A was some- was still alive) was removed and placed in Tullgren funnels times higher and sometimes lower than the number in the for 48 h (which preliminary experiments had shown extracts completely isolated patch (D) (Table 1), but with no overall 98% of all individuals; Hoyle, unpublished data). All emerg- trend, suggesting that the gravel was a significant barrier to ing microarthropods were collecting in an ethanol/glycerol movement. mixture, and then sorted into the same morphospecies (with Surprisingly, there was only occasionally evidence that the advice from experts, see Acknowledgements) as in the neutral abundance in patch C depended on the abundance in the study. The microarthropods are commonly found in moss, and microarthropod-rich patch (B) (Table 1). Nonetheless, in com- vary in size from approximately 0.1–1 mm. mon with much theory (e.g. Renshaw, 1991), migration was The first test examined whether the abundance of each expressed as a rate per individual. The density-independent species varied among the cleared patches. Generalised lin- migration rate ( 0) was estimated to lie in the range 0.05–0.21 ear models with Poisson errors corrected for overdispersion, for Cryptostigmata mites (Table 2). There was little evidence with statistical block as a factor, were performed in “R” (Ihaka for density-dependent migration ( 1), hence 1 was set to zero. and Gentleman, 1996). Two non-orthogonal contrasts were used: (1) between patches C and A, to test whether there was 2.2.2. Second manipulation experiment greater movement across the corridor than across the gravel; The second manipulation experiment was designed to esti- (2) between patches A and D, to test for migration across the mate immigration rates from the surrounding environment, gravel. The second test, for density-dependent migration, pro- and migration rates across corridors of various lengths and ceeded as follows. It was assumed that microarthropods in all widths at three times of the year, at the same site as the neu- patches had arrived from the surrounding environment. Addi- tral study in the peak district, and in the same valley as the tionally, the fauna in patch A (Fig. 2) had migrated across the positive study. gravel from patch B, fauna in patch C had migrated across the corridor from patch B, and fauna in patch B were those orig- 2.2.2.1. Methods. For the first part (from March to April 2002), inally present, minus those that had emigrated to patches A within each of nine blocks, eleven treatments (Fig. 3, Table 3) and C. As a first approximation, double-migrations and mor- were randomly allocated on the top of the dry-stone wall of the tality over the relatively short duration of the experiment were neutral study in the peak district, in the same valley as used in ignored. The approximate migration rate per species across the positive study. The source moss was cut from the mainland the corridor over 1 month was then calculated, averaged over moss of the same dry-stone wall, and was randomly allocated ecological modelling 202 (2007) 441–453 445 ? 2 B ∝ C oups in the (negative (negative (negative *** *** *** = 0.29, n.s. = 0.13, n.s. = 0.01, n.s. = 0.01, n.s. = 0.50, n.s. = 0.43, n.s. = 3.84, n.s. = 0.70, n.s. = 5.40, n.s. = 553 = 637 = 489 = 2.73, n.s. 1,3 1,3 1,3 1,3 1,3 1,3 1,3 1,3 1,3 1,3 1,3 1,3 1,3 density-dependence) density-dependence) density-dependence) F F F F F F F F F F F F F abundance in Density-dependence: ? * B ∝ C = 0.36, n.s. = 6.14, n.s. = 0.06, n.s. = 0.13, n.s. = 1.00, n.s. = 0.03, n.s. = 0.53, n.s. = 0.05, n.s. = 0.44, n.s. = 1.73, n.s. = 0.72, n.s. = 7.94 = 0.21, n.s. in 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 1,4 F F F F F F F F F F F F F (C>B) (C>B) (B>C) *** *** * 1.35, n.s. 1.93, n.s. 0.39, n.s. 1.53, n.s. 5.57 4.68 − − − − − − = +1.38, n.s. = 0.00, n.s. = +0.87, n.s. = = +2.44 = +2.15, n.s. = = 0.00, n.s. = = = = = +1.50, n.s. 10 10 10 10 10 10 10 10 10 10 10 10 10 t t t t t t t t t t t t t (A>B) (A>B) (A > B) (A>B) (A>B) (A>B) ** *** *** ** * * = +6.36 = +0.35, n.s. = +2.94 = +1.71, n.s. = +6.20 = +3.14 = +2.10 = +0.78, n.s. = +1.32, n.s. = +1.72, n.s. = +1.06, n.s. = +0.08, n.s. = +2.30 10 10 10 10 10 10 10 10 10 10 10 10 10 t t t t t t t t t t t t Difference in abundance between patches Abundance C > A (one-tailed) A vs. D (two-tailed) t ) 0 rate ( Corridor migration . a Appendix A 12 0.35 0.07 5689101219 0.12 0.13 0.11 0.14 0.21 0.13 0.05 -value was 0.025 after the Dunn–Sidak correction for non-orthogonal tests. P < 0.05. < 0.01. < 0.005. Morphospecies descriptions are given in P P P Mesostigmata mitesProstigmata mites All All 0.11 0.50 Collembola All 0.64 Taxonomic groupCryptostigmata mites Morphospecies All 0.13 Table 1 – Estimated migration rates and tests for density-dependent and -independent corridor migration for the most common species and taxonomic gr The critical a ∗ ∗∗ ∗∗∗ first manipulation experiment 446 ecological modelling 202 (2007) 441–453

Table2–Model parameters Purpose of simulation Initial Parameter population

ˇ0 ˇ1 ı0 ı1 0 1 ε0 ε1 0 1 A ˝

Best estimate 42 0.167 0.002 0.042 0.001 29 0.69 0 0 0.14 0 0 0 Seasonal mortality 13 0.448 0.005 0.018 0.005 0 0 0 0 0–1 0 800 7␲/8 Variable emigration rates 42 0.167 0.002 0.042 0.001 0–29 0–0.69 0–0.3 0 0–1 0 0 0

Parameters are described in Section 2.

to the treatments. The moss was mostly H. sericeum, B. rutabu- lum and H. lacunosum var. lacunosum or I. myosuroides (Brid.) var. myosuroides, as in the neutral study and the first manipulation experiment). The first treatment consisted of a single circular patch of diameter 10 cm, which had previously been cleared of microarthropods by the method described previously. All other treatments consisted of two circular patches of diameter 10 cm, with one microarthropod-rich and one cleared patch. A corridor was either absent, or present: when present, the corridor length varied from 0 to 7 cm, and the width from 1.5 to 2.5 cm, and was either microarthropod-rich or cleared. All moss was stuck to the bare rock using glue. A prelimi- nary experiment established that the glue did not significantly reduce microarthropod immigration into the cleared moss

(F1,12 = 0.15, P > 0.05). Treatments were at least 10 cm apart and at least 10 cm from the remaining “mainland” of moss (as the positive and neutral studies). Fig. 3 – Results of the first part of the second manipulation The second and third parts also lasted 1 month (June to experiment. There was no significant difference in the July 2002 and September to October 2002, respectively). They logarithm of the abundance of all microarthropods in the consisted of just two treatments (Table 3): one isolated cleared cleared patches (open circles) by experimental treatment patch, and one cleared patch connected to a microarthropod- (F9,78 = 0.97, P > 0.05). Error bars show ±1 S.E. Table 3 gives a rich patch by a microarthropod-rich corridor of length 7 cm full description of the treatments. and width 1.5 cm (treatments 1 and 6 of the first part).

Table3–Treatments of the second manipulation experiment Treatment Corridor Corridor Corridor cleared of Notes length (cm) width (cm) microarthropods/not cleared?

1st part 1 – – – One cleared patch 2 7 0.0 – No corridor, patches separated by 7 cm 3 0 1.5 – – 4 3 1.5 Not cleared – 5 3 1.5 Cleared – 6 7 1.5 Not cleared – 7 7 1.5 Cleared – 8 7 1.5 Not cleared Left for 2 months 9 7 1.5 Not cleared Left for second month only 10 7 2.5 Not cleared – 11 7 2.5 Cleared –

2nd part 1 – – One cleared patch 2 7 1.5 Not cleared –

3rd part 1 – – One cleared patch 2 7 1.5 Not cleared –

Fig. 3 gives a pictorial representation of the treatments. ecological modelling 202 (2007) 441–453 447

The moss species were the same as those in the first Prostigmata (predatory). There were 36 mite morphospecies, manipulation experiment and in the neutral study. All moss 8 Collembola morphospecies and 10 other morphospecies patches were removed from the dry-stone wall after 1 month (mostly beetles, spiders, centipedes, millipedes and pseu- (as the first manipulation experiment) for analysis, except doscorpions). for treatment 8, which was removed after 2 months. The At the end of the second manipulation experiment, per microarthropods were extracted from the moss and sorted moss patch cleared at the start, there were an average of 169 into morphospecies as before. The moss was weighed after adult Cryptostigmata mites of 7 species, 6 adult Mesostigmata being placed in the funnels to give the moss dry weight. mites of 2 species, 23 Prostigmata mites, 84 Collembola of 3 All tests were performed on the abundance of all species, and 3 other microarthropods. microarthropods, all non-predatory mites, all predatory For the tests on the first part of the experiment, none of the mites, all Collembola, and the fourteen most common hypotheses were accepted for all microarthropod individuals microarthropod species. Treatment and experimental block combined (Fig. 3). None of the species or groups of species were implemented as statistical factors and the moss dry supported Hypotheses 2, 3 nor 4, which allows us to accept weight was a covariate (to correct for differences in patch size), the modelling assumption of instantaneous migration. Addi- assuming Poisson errors corrected for overdispersion. tionally, Hypotheses 7 and 8 were never accepted. Only one The following hypotheses were tested for the first part: species (Cryptostigmata morphospecies 5: Appendix A) sup- ported the hypothesis of more individuals in the cleared patch 1. Connected patches contain more individuals than discon- after 2 months compared to 1 month (t = 3.20, P < 0.002); one nected patches due to migration across the corridor: tested 86 species (Cryptostigmata morphospecies 5 again) supported by an a priori contrast between treatments 1 and 2 and the hypothesis of more individuals in the cleared patch con- treatments 3–11. nected by the wider corridor (t = 3.01, P < 0.004); one species 2. Patches connected by a microarthropod-rich corridor con- 86 (Cryptostigmata morphospecies 8) was actually less abun- tain more individuals than patches connected by a cleared dant in the patches connected versus disconnected from corridor of the same dimensions due to individuals initially microarthropod-rich patches (t = 4.11, P < 0.0001), against in the corridor migrating across into the cleared patch: 87 expectation; and one non-predatory mite species (Cryp- treatment 6 versus 7. tostigmata morphospecies 2), one predatory mite species 3. As 2 for corridors of length 3 cm, width 1.5 cm: treatment 4 (Prostigmata morphospecies 2), all predator individuals com- versus 5. bined and one Collembola species (morphospecies 1) were 4. As 2 for corridors of length 7 cm, width 2.5 cm: treatment more abundant in the cleared patch with increasing corri- 10 versus 11. dor length (F = 9.94, P < 0.005, F = 16.21, P < 0.0005 and 5. Patches left for 2 months contain more individuals than 1,23 1,24 F = 14.59, P < 0.001, respectively), again against expectation. those left for just 1 month: treatment 8 versus 9. 1,24 There was no significant difference in abundance of any 6. The narrower the microarthropod-rich corridor the lower species or group of species between the isolated island and the abundance in the patch, due to lower migration rates the patch connected by a corridor over the three times of year, across the narrower corridor: treatment 6 versus 10. except for one predatory species (Prostigmata morphospecies 7. As 6 for cleared corridors of the same dimensions: treat- 2) where there were more individuals in the connected than ment 7 versus 11. in the isolated patch (F = 5.96, P < 0.05), and for one non- 8. The shorter the cleared corridor, the higher the abundance 1,47 predatory species (Cryptostigmata morphospecies 8) where in the patch, due to higher migration rates across the there were fewer individuals (F = 10.00, P < 0.005) (with and shorter corridor: treatment 5 versus 7. 1,47 against expectation, respectively, Fig. 4). In most cases, abun- 9. The greater the corridor length, the lower the abundance in dance varied significantly by season in agreement with Norton the cleared patch, due to lower migration rates across the (1994). longer corridors: regression of treatments 3, 4 and 6. Immigration from the surroundings into a single cleared The Dunn–Sidak correction on these nine non-orthogonal patch is high: the best estimate of the rate of immigration tests gave a critical P-value of 1 − (1 − 0.05)1/9 = 0.0057. is 280 microarthropods per month, 164 non-predatory mites, Next, the abundance of the same species and groups 22 predatory mites, 91 Collembola and 3 other microarthro- of species were compared between the two treatments: pods. The immigration rate was approximately 29 mites per (1) one cleared isolated patch (treatment 1) and (2) a month for a typical common species when the abundance cleared patch connected to a microarthropod-rich patch by was zero (for cleared patches), implying 0 − 0 and 1 =29from a microarthropod-rich corridor of length 7 cm, width 1.5 cm Eq. (2). Furthermore, assuming no immigration at the equi-

(treatment 6) for all three parts combined. librium abundance of 42 for mite B (see Fig. 1a), then 0 − 42 and 1 = 0, which gives 0 = 29 and 1 = 0.69. If immigration at 2.2.2.2. Results. For both manipulation experiments com- the equilibrium abundance is greater than zero, then 1 ≤ 0.69. bined, 38,000 individual microarthropods were counted and sorted into morphospecies (usually genus level, see 2.3. Demographic parameters Appendix A) and a further 15,000 individuals estimated. Most microarthropods belonged to the Acari (69% of indi- Demographic parameters were taken from the literature. viduals, only adults counted) and Collembola (non-predatory, Most studies of microarthropod longevity are laboratory- 31%). Within the Acari, 89% of individuals were Cryptostig- based—there are very few field data (Norton, 1994). However, mata (non-predatory), 1% Mesostigmata (predatory) and 10% the non-predatory Cryptostigmata probably live surprisingly 448 ecological modelling 202 (2007) 441–453

long in the field—from several months to 2 years (Norton, 1994). Hartenstein (1962) estimated the generation time of the predatory Mesostigmata mite Pergamasus crassipes (Mesostig- mata morphospecies 5, Appendix A) as about 2–3 months. Some Collembola species have one, and others more than one generation per year (Hopkin, 1997). Assuming Cryptostigmata mites live 12 months in the field at the equilibrium abundance (K), then ı(K) = 1/12, since the mean of the exponential distri- bution is the inverse of the rate parameter. We are forced to estimate the density-dependent mortality rates, given the lack of relevant literature. Assuming the rate at zero abundance is half that at the equilibrium abundance, then ı(0) = 1/24. Solving these simultaneous equations, and assuming an equi- librium abundance of 42 given by the average abundance of

mite B in the mainland in the Positive Study, then ı0 = 0.042 and ı1 = 0.001. By definition, the birth and death rates are equal at Fig. 4 – Results of all three parts of the second manipulation the equilibrium abundance, hence ˇ(K)=ı(K) = 1/12. If we sim- experiment. There was a significant difference in the ilarly assume that the birth rate is twice as great at zero as at logarithm of the abundance of all microarthropods in the the equilibrium abundance, then ˇ(0) = 2/12. Hence ˇ0 = 0.167 cleared patches (open circles) by time of year (F2,46 = 12.44, and ˇ1 = 0.002. P<0.00005) but not by experimental treatment (F1,46 = 0.04, P > 0.05). Error bars show ±1 S.E. Table 3 gives a full 2.4. Simulations description of the treatments. 2.4.1. Best estimate parameters One thousand simulations were performed on the popula- tion dynamics of mite B using the best estimate parameters (Table 2), in the two moss patches over 6 months. Given that

Table4–Tests for seasonality in microarthropod abundance Taxonomic groupa Mean abundance in mainland Significant difference Seasonal/not in abundances? seasonal 0 months (start) 3 months 6 months (end)

(a) Positive study *** All microarthropods 132 240 298 F2,75 = 38.2 Seasonal *** All mites 101 195 255 F2,75 = 50.8 Seasonal *** Mite A 24 33 49 F2,75 = 9.39 Seasonal *** Mite B 14 29 42 F2,75 = 19.7 Seasonal *** Mite C 3 15 12 F2,75 = 30.2 Seasonal

Mite D 8 9 14 F2,75 = 2.03 n.s. Not seasonal

All Collembola 29 37 39 F2,75 = 1.95 n.s. Not seasonal

Taxonomic group Morphospeciesb Mean abundance in mainland Significant difference Seasonal/not in abundances? seasonal 0 months (start) 6 months (end)

(b) Neutral study * Cryptostigmata mites 12 262 142 F1,43 = 4.83 Seasonal 295F1,43 = 3.12 n.s. Not seasonal 31111F1,43 = 0.02 n.s. Not seasonal

41412F1,43 = 0.13 n.s. Not seasonal 5 134 104 F1,43 = 1.97 n.s. Not seasonal 82220F1,43 = 0.02 n.s. Not seasonal

9 74 103 F1,43 = 1.26 n.s. Not seasonal

All Collembola All 328 240 F1,43 = 2.63 n.s. Not seasonal

Abundance of the most common mite species and all Collembola combined in the mainland moss in the (a) positive study at the start, intermediate point and end of the experiment and in the (b) neutral study at the start and end of the experiment only. a Species descriptions are not available. b See Appendix A for morphospecies descriptions. ∗ P < 0.05. ∗∗∗ P < 0.0005. ecological modelling 202 (2007) 441–453 449

there is no independent measure of the emigration rate, ε0 and 3. Results ε1 were initially set to zero. For simplicity, both patches were assumed to be at the equilibrium abundance of 42 at the start of the experiment. 3.1. Best estimate parameters

Using the parameter best estimates, the abundance of mite 2.4.2. Seasonal mortality B never fell to zero in either patch for any of the simula- Microarthropod populations have often been observed to fluc- tions (Fig. 5a), implying that the rescue effect was not relevant. tuate seasonally (Luxton, 1981). There was strong evidence of Nevertheless, there was a weak trend of increasing final abun- seasonal variation in mite abundance in the positive but not dance with increasing density-independent migration rate in the neutral study, and no evidence for seasonal variation in (Fig. 5b). Furthermore, the magnitude of the corridor effect did abundance of Collembola in either study (Table 4). not vary greatly for a reasonable range of density-dependent Given the seasonality of mite abundance in the positive birth and death rates (parameters for which there is neither study, seasonal mortality rates were assumed (subject to a literature nor empirical data). However, there was very little minimum of zero): difference in the simulated final abundance between the con-    nected and disconnected corridor treatments (Fig. 5b) based N t ı i (t) = ı 1 + A sin + ˝ + ı Ni(t). (6) on the best estimate range of the corridor migration rates. It 0 6 0 was impossible to recreate the strong corridor effect of the Positive Study by adjusting either the birth and death rates (to The intrinsic rate of increase was then:- simulate the shorter lifespan of the Mesostigmata and Collem-    bola), or the density-dependent migration ( ). t 1 r(t) = ˇ − ı 1 + A sin + ˝ (7) 0 0 6 at time t months, with amplitude A and phase ˝ in both patches. Mortality rates in the two patches are assumed to be in phase. Then parameter values were chosen (Table 2) in an attempt to recreate the corridor effect of mite B (Fig. 1a). The emigra- tion rate was set to zero as before. Initially, the immigration rate was also set to zero. The instantaneous expected lifespan (calculated as the inverse of the mortality rate for a Poisson process) is long for much of the year (in keeping with the lit- erature), but much shorter for a brief period. One thousand simulations were performed for each migration rate between zero and one (covering the range measured in the first manip- ulation experiment), and the average final population of the two patches recorded.

2.4.3. Variable emigration rates Microarthropods search for suitable microhabitats as dictated by the demands of various stages of their life history (Norton, 1994). A single moss patch of diameter 10 cm may not encom- pass a sufficiently large area. Suppose an individual leaves a patch at any instant with a probability that is equal for both the complete- and broken-corridor treatments. For the complete- corridor treatment, suppose further that of this probability, there is a further chance that an individual will migrate across the corridor to the adjacent patch. Then the emigration rate will be lower in the complete- compared to the broken- corridor treatment. Fig. 5 – (a) A single typical simulation of the population Mite B was chosen to attempt to recreate the corridor dynamics of mite B in two connected patches (shown in effect based on emigration rates depending on the patch con- black and grey) over 6 months based on best estimate nectivity. The birth and death (now not seasonal) rates were parameters (Table 2). (b) A simulated weak increase in best estimates, and the migration rate was varied within abundance at the end of the experiment (6 months) with a reasonable range. One thousand model simulations were increasing migration. The migration rate for the then performed with the density-independent emigration broken-corridor treatment is assumed to be zero and for rate varying from zero to 0.3, at three reasonable levels of the complete-corridor treatment between 0.05 and 0.21 immigration (Table 2). For simplicity, the density-dependent (indicated by the area between the dotted lines). Error bars emigration rate was set to zero. represent 95% confidence intervals. 450 ecological modelling 202 (2007) 441–453

Fig. 7 – Simulated impact of the migration rate (0)onthe average abundance of mite B at the end of the 6 months experiment based on seasonal mortality rates and no immigration. Note the significant increase in abundance with increasing migration. The final abundance in the broken-corridor treatment (assuming no migration) is 10 in the positive study, and 10.5 by simulation. The final abundance in the complete-corridor treatment (assuming a migration rate in the range 0.05–0.21 from the first manipulation experiment, indicated by the dotted lines) is 27 in the positive study, and 11.1 by simulation. Error bars represent the 95% confidence intervals.

of decline (Fig. 8), and the lower the average final abundance (Fig. 9). Furthermore, assuming no immigration for simplicity, Fig. 6 – (a) The intrinsic rate of population increase (r) the model can explain the corridor effect for mite B assum- varies over time due to seasonal mortality rates. (b) When ing emigration rates of 0.10 for the complete- and 0.30 for the migration rate ( 0) is low or zero (shown here), the the broken-corridor treatments (Fig. 9). The corridor effect is rescue effect rarely (or never) occurs (typical simulations for very insensitive to the patch-to-patch migration rate, but is mite B displayed), whereas (c) at higher migration rate (here weakened with increasing immigration (Fig. 9). The impor- 0 = 0.1), the rescue effect occurs occasionally. Populations tant message here is that we can find parameter values that in the two patches are shown as separate lines. accurately recreate the corridor effect if we assume that the Immigration and emigration are set to zero (Table 2). emigration rate depends on patch connectivity.

3.2. Seasonal mortality

When the intrinsic capacity of population increase (r)isnega- tive, the abundance generally decreases, and vice versa (Fig. 6). The average final abundance (Fig. 7) and the chance of the occurrence of the rescue effect (Fig. 6) both increased with the migration rate. With some immigration, the rescue effect can occur even at low or zero patch-to-patch migration, so weak- ening the corridor effect. Although the corridor effect has been recreated, the magnitude of the effect has not (Fig. 7). Further simulations (not shown) demonstrate that the corridor effect is stronger when the mortality rates are out of phase in the Fig. 8 – Typical simulations of the change in two patches, since the rescue effect is more likely to occur. microarthropod abundance over time, according to the

density-independent emigration rate (ε0). ε0 is zero 3.3. Variable emigration rates (representing the mainland patch) (top line), an intermediate rate of 0.15 (representing the Abundance generally falls over time for positive emigration complete-corridor patch) (middle line) and a high rate of rates. The greater the emigration rate, the greater the rate 0.30 (representing the broken-corridor patch) (lower line). ecological modelling 202 (2007) 441–453 451

suggests that mortality may have been seasonal only in the positive study, thereby explaining why the corridor effect was found only in the positive study. Factors influencing the differ- ence in mortality between the studies may have included the following: higher overall densities in the neutral study; differ- ent moss species; and different times of the year. Furthermore, the winter was mild during the neutral study, but very cold with snow during the positive study (A. Gonzalez, personal communication). Mortality may then have been temporarily very high in the positive study, thereby allowing the rescue effect to operate, and hence the corridor effect. Alternatively, mortality may have been seasonal in both cases, but out of phase only in the positive study. Fig. 9 – Effect of the density-independent emigration rate Two further conditions were necessary for the hypothesis (ε ) on abundance of mite B after 6 months for three levels 0 to hold. Firstly, immigration from the surrounding environ- of immigration. High (circles), intermediate (triangles) and ment must be low so that populations driven to extinction no (squares) immigration represent density-independent at times of high mortality can be rescued only by migration rates ( ) of 29, 15 and 0, and density-dependent rates ( )of 0 1 from the adjacent patch. Immigration was high at the site 0.690, 0.357 and 0, respectively. Assuming no immigration, of the neutral study (second manipulation experiment). Thus an emigration rate of 0.10 for the complete-corridor although mortality may have been seasonal in the neutral treatment simulates the final abundance of 27 in the study, any corridor effect was probably negated by high immi- positive study, and an emigration rate of 0.30 for the gration. Secondly, the corridor effect assumes reasonably high broken-corridor treatment simulates the final abundance of migration between patches, higher than measured in the first 10 in the positive study. Points represent 1000 simulations. manipulation experiment. The migration rate in the neutral The 95% confidence error bars are too small to be visible on study may not have been high enough to produce the corridor this plot. effect. It would be interesting to measure the immigration and migration rates at the site of the positive study to see whether 4. Discussion they fall within the range required by the hypothesis. The model also assumes that the birth rate increases The most important result of this paper is that even when with reducing abundance. Alternatively, if there is an Allee the lifespan of a species is long compared to the time-scale in effect, then the rescue effect may become even more impor- question, we have two mechanisms that predict that corridors tant at times of high mortality. Populations that fall below may act to reduce the rate of species extinctions. Both can the Allee threshold may be rescued from extinction by explain why the species richness of microarthropods inhab- migrants from the adjacent patch, if this is above the thresh- iting patches of moss has been found to be greater in the old (Amarasekare, 1998). The has indeed been presence of a connecting corridor in the Positive Study (Gilbert recorded for mites in the laboratory (Stamou and Asikidis, et al., 1998; Gonzalez et al., 1998), but not in the Neutral Study 1989). (Hoyle and Gilbert, 2004). First, incorporating a seasonal pat- Now I turn to the second mechanism: emigration rates. No tern to mortality caused some simulated populations to die empirical studies have measured whether emigration rates to out, which were then occasionally rescued by migrants from the surrounding matrix are lower for patches connected by a the adjacent patch. The second mechanism assumes that the corridor (Hudgens and Haddad, 2003). Such emigration would rate of emigration into the surrounding matrix is lower for require that an individual has a reasonable chance of locat- patches connected by a corridor. Soil microarthropods live for ing the corridor, and prefers to move through the corridor several months to a couple of years (see Methods), a substan- rather than the surrounding matrix. Indeed, corridors have tial proportion of the duration (6 months) of the two studies. been observed to channel movement (Berggren et al., 2002). With such a low turnover in population, and without either of When leaving a patch, microarthropods may search in a way the two mechanisms, it was possible to recreate only a very that maximizes their chance of finding suitable habitat, per- weak corridor effect over a reasonable range of movement haps using cues such as pheromone trails (Verhoef et al., 1977). parameter values. The main message is that it is important Indeed, microarthropods are able to find conspecifics, since to identify the mechanisms by which corridors may function. they generally aggregate (Usher, 1975). Alternatively, isolated A weak corridor effect was reproduced by invoking sea- patches may be drier than connected patches, perhaps due to sonal mortality, and hence seasonal longevity. A strong effect an exchange of moisture across the connecting corridor. Emi- was simulated only by assuming mortality rates that were out gration and death rates may then be higher and birth rates of phase in the two patches. This could occur at the scale lower in the isolated patch, leading to lower microarthropod of a nature reserve only when the reserves are tens or hun- abundance and the corridor effect. However there is no evi- dreds of kilometers apart, or separated by several hundreds of dence that the moisture content of moss patches depends on metres in altitude. Mortality rates were probably in phase in patch connectivity (Hoyle and Gilbert, 2004). In any case, the the moss-microarthropod microecosystem since the patches corridor effect can be simulated using lower emigration rates were physically close together. The abundance of most species for patches connected by a corridor. Immigration need not be was seasonal in the positive but not in the neutral study. This very low (contrary to the previous hypothesis), but the corridor 452 ecological modelling 202 (2007) 441–453

effect weakens with increasing immigration. The migration anticipated only from those experiments that encompass a rate estimated from the first manipulation experiment again period of high mortality and hence low mainland abundance. allows for only a weak corridor effect based on the emigra- The phase lag in mortality between adjacent patches could tion hypothesis. The moss species of the neutral study may in principal be estimated by recording microarthropod abun- have contained more microhabitats than the species of the dance in a small area of moss sampled along a transect of positive study and hence the emigration rates of the neutral mainland moss at frequent time intervals throughout the year. study may have been lower than in the Positive Study. This All models make important assumptions, so it is important would imply that the emigration rate hypothesis and hence to capture the most relevant processes of the system under the corridor effect was relevant only to the positive study. study. The model should demand only data that are practi- Each patch was connected to two other patches in the positive cally available from experiments. The principle of parsimony study, but only one patch in the neutral study. Emigration may should apply to processes for which data are not available. then have been much lower in the connected compared to the In this case, mortality was assumed constant over the life of isolated patches in the positive study, since the microarthro- an individual, to avoid the need to track each microarthro- pods would have a greater chance of encountering a corridor pod, and emigration and migration were initially assumed when dispersing. Other factors could have led to a difference to be density-independent, even though microarthropods are in emigration between the studies include: different times of known to aggregate (Usher, 1975). The following refinements the year; seasonal movement (second manipulation experi- could be added to the model, but I suggest that they would ment; Norton, 1994); and as mentioned above, higher overall introduce spurious accuracy: allow for mortality in the cor- densities in the neutral study. Thus the strong corridor effect ridor; assume a finite corridor migration time; assume a of the positive study could well have been due to a combina- non-reproductive juvenile stage; and extend the model to tion of both seasonal mortality and connectivity-dependent many species at more than one , since species per- emigration rates. sistence depends on interaction with other species (Holyoak, Unfortunately, it is not possible to observe the microarthro- 2000). pods move, since they are usually buried deep in the moss. It was therefore necessary to use the manipulation experiments Acknowledgements as an indirect method to estimate the degree of immigration and migration. The first manipulation experiment was con- I thank Roy Norton, Pablo Martinez, Hans Klompen, Maciej ducted not in the field, but under controlled conditions. The Skorupski and Warren Welbourn for mite identification, Brian estimated migration rates should then be viewed as indica- Cave for collembola identification, Anthony Smith for moss tions rather than exact values. Although the results of the identification, Mike Bonsall for modelling advice, and Francis second manipulation experiment are consistent with those Gilbert, Doug Levey and an anonymous referee for comments of the Neutral Study, it is surprising that it provided no evi- on an earlier version of the paper. I was funded by NERC. dence of migration across the corridor (contrary to the first experiment), and no evidence that cleared patches left for 2 months contained more individuals than those left for 1 Appendix A. Supplementary data month. A pilot study found no evidence that immigration is lower in cleared patches stuck to the rock with glue compared Supplementary data associated with this article can be found, to patches not stuck down. Nevertheless, the in the online version, at doi:10.1016/j.ecolmodel.2006.11.008. may be lower for glued patches compared to patches growing naturally on rock. If so, all the cleared patches in the sec- references ond manipulation experiment may have reached the carrying capacity (40% of that for patches not glued) after 1 month. This experiment would then be unable to detect any migration that Amarasekare, P., 1998. Interactions between local dynamics and did occur. To test this idea, the experiment could be repeated dispersal: insights from single species models. Theor. Popul. with the moss sampled after a shorter period, perhaps one Biol. 53, 44–59. week. The second manipulation experiment certainly demon- Beier, P., Noss, R.F., 1998. Do habitat corridors provide strated that immigration at the site of the neutral study is high. connectivity? Conserv. Biol. 12, 1241–1252. The seasonal mortality hypothesis (but not the emigration Berggren, A., Birath, B., Kindvall, O., 2002. Effect of corridors and hypothesis) requires low immigration. It would be instructive habitat edges on dispersal behavior, movement rates, and movement angles in Roesel’s bush-cricket (Metrioptera roeseli). to measure immigration at the site of the positive study to see Conserv. Biol. 16, 1562–1569. whether this was consistent with this hypothesis. To test the Berthet, P.L., 1964. Field study of the mobility of oribatei (Acari) emigration hypothesis, it may be possible to trap emigrants using radioactive tagging. J. Anim. Ecol. 33, 443–449. by surrounding moss patches with sticky tape. The hypothe- Brown, J.H., Kodric-Brown, A., 1977. Turnover rates in insular sis then predicts fewer emigrants from patches connected by : effect of immigration on extinction. 58, a corridor. The seasonal mortality hypothesis could be tested 445–449. in the following way. The abundance of microarthropods in Burkey, T.V., 1997. Metapopulation extinction in fragmented landscapes: using bacteria and communities as the mainland moss should be monitored at regular inter- model . Am. Nat. 150, 568–591. vals throughout the year. The test of corridor effectiveness Collinge, S.K., 2000. Effects of grassland fragmentation on insect as implemented in the positive and neutral studies should species loss, colonization, and movement patterns. Ecology be repeated on several occasions. A corridor effect should be 81, 2211–2226. ecological modelling 202 (2007) 441–453 453

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