Landscape Ecology 12: 261–271, 1997. 1997 Kluwer Academic Publishers. Printed in the Netherlands. The effects of landscape composition and physiognomy on metapopulation size: the role of corridors Greg S. Anderson* and Brent J. Danielson Department of Animal Ecology, 124 Science II, Iowa State University, Ames, IA 50011-3221; *Current address: 116 Highland, New Castle, WY 82701, U.S.A. Please send all correspondence for this manuscript to B.J. Danielson, email: jessie@iastate.edu; telephone: 515-294-5248; fax: 515-294-7874 Keywords: connectivity, corridor, landscape, model, metapopulation, dispersal Abstract We develop and analyze a model that examines the effects of corridor quality, quantity, and arrangement on metapopulation sizes. These ideas were formerly investigated by Lefkovitch and Fahrig (1985) and Henein and Merriam (1990). Our simulations provide results similar to the Henein and Merriam model, indicating that the quality of corridors in a landscape and their arrangement will influence the size of a metapopulation. We then go one step further, describing how corridor arrangement alters the metapopulation, and provide a method for predicting which corridor arrangements should support larger metapopulations. In contrast to the Henein and Merriam model, we find that the number of corridor connections has no influence on the size of a metapopulation in a landscape unless there is an accompanying change in the uniformity of the distribution of corridor connections among patches. Introduction As habitat available to many organisms shifts from large, contiguous areas to small, discrete patches, it is critical to develop an understanding of how effectively populations are able to exploit new habitat arrangements. The interest in fragmentation has prompted the development of a large number of theoretical metapopulation models (e.g., Lefkovitch and Fahrig 1985; Pulliam et al. 1992; Gotelli and Kelley 1993; Gyllenberg et al. 1993; Goldwasser et al. 1994; Hanski 1994). Typically, the term metapopulation refers to a group of discrete, localized subpopulations. The majority of dynamics occur within each population, but dispersal events between populations are frequent enough to have an impact on local dynamics by either decreasing the chance of extirpation or reducing fluctuations at local sites (Taylor 1990; Kozakiewicz 1993). The amount of exchange between subpopulations within a metapopulation is dependent on a landscape’s physiognomy and composition (Dunning et al. 1992). Taylor et al. (1993) also point out that the degree of connectivity, which results from specific physiognomic arrangements of the landscape’s components, can play a significant role in regulating metapopulation dynamics. High connectivity implies that organisms can move easily between suitable habitat patches, and low connectivity indicates that movement between suitable patches is somehow difficult. Connectivity can be increased via the addition of dispersal corridors (i.e., altering composition) or by the spatial rearrangement of existing corridors (i.e., altering physiognomy) so that organisms can exploit a landscape’s resources more efficiently. The idea that connectivity influences an individual’s ability to move between habitat patches in a fragmented landscape has generated a lot of interest. Empirical studies have demonstrated that corridors can have an impact on the movement of organisms between habitat patches (Fahrig and Merriam 1985; Lorenz and Barrett 1990; Zhang and Usher 1991; Bennett et al. 1994). Despite evidence that corridors can benefit populations, there is very little known regarding the 262 effects of corridor quality, corridor arrangement, and the number of corridors connecting isolated patches. Henein and Merriam (1990) developed a model (hereafter referred to as the HM model) that described the effects of varying the quantity and quality of corridors in metapopulations of whitefooted mice (Peromyscus leucopus). Since whitefooted mice have been shown to perceive habitat quality within a landscape, it is reasonable to assume that the quality of habitat in a corridor will also affect individuals (Adler and Wilson 1987; Wegner and Merriam 1990; Barnum et al. 1992). Additional studies by Lorenz and Barrett (1990), Szacki et al. (1993), and Bennett et al. (1994) provide empirical evidence that habitat in a corridor does indeed affect corridor use. The results of the HM model demonstrate that the overall size of a metapopulation is affected by corridor quality, the arrangement of corridors, and the number of corridor connections among patches in a landscape. The questions posed by Henein and Merriam (1990) are valuable, but there appear to be inconsistencies in their model that we feel warrant further investigation. The HM model tracks groups of nestlings, juveniles, subadults, and adult mice for a 33-week breeding period followed by an overwintering period. Their model is structured so that nestlings graduate to the juvenile class after a given period. Juvenile dispersal via corridors is then calculated. A portion of surviving juveniles graduate to subadulthood, after which subadults emigrate. Finally, a portion of subadults graduate to adulthood and the adults emigrate via selected corridors. The unifying feature of all the equations in the HM model that detail the dynamics mentioned above is that none of them contain a density-dependent term. Some parameters in the model equations change according to the time of year, but there are no terms that change in relation to density. Without a density-dependent feature, all populations described by deterministic equations should behave in one of three ways, depending on initial conditions. The populations should either 1) increase indefinitely, 2) decrease to zero, or 3) remain at an unstable equilibrium from the start. The populations tracked in the HM model increase or decrease after initial seeding and then stabilize at various levels, dependent on the landscape they occupy. We feel that the HM model erroneously stabilizes at non-zero equilibria values. We have thus developed a similar model to re-examine some of the conclusions regarding corridor effects on metapopulations and to extend the analyses of corridor effects beyond that of Henein and Merriam (1990). Our model includes a density-dependent feature to allow populations to realistically stabilize at non-zero equilibria. Given the few differences between our model and the HM model, we are interested in determining if metapopulations in our simulations behave in a way that is qualitatively similar to HM metapopulations. In particular, we want to determine the effect of corridor quality, corridor quantity, and corridor arrangement on the size of a metapopulation. Methods Model description The framework developed by Henein and Merriam (1990) was innovative and allows us to effectively test each of our objectives. Therefore, we use four-patch metapopulations similar to the HM model to test for corridor effects. However, the intra-year and age-specific dynamics found in the HM model are not relevant to detailing the qualitative effects of corridors on metapopulations. Instead of adjusting model parameters for groups of juveniles, subadults, and adults and the time of year, we hold parameters constant for all animals and avoid intra-year dynamics. The populations in our model are subjected to a set of simple rules. Stated explicitly, for each patch (i): Nt+1,i = Nt,i + Bt,i + It,i – Dt,i – Et,i where B represents births, I is immigration, D is deaths, and E is emigration. The population in patch i at time t shown as Nt,i. In our model, the values of B, I, E, and D take the following forms: Bt,i = 4bNt,i 263 Fig. 1. Flow diagram of our simulation model. 3 It,i = Σ θijEj j=1 Dt,i = [1–(Nt,i + Bt,i + It,i – Et,i) + It,i – Et,i) –1/4] * (Nt,i + Bt,i Et,i = e (4bNt,i + Nt,i) The variables b and e are the constants representing the probabilities of attempting reproduction and dispersal, and θij is corridor survival from patch j to patch i. Populations in our model go through a yearly cycle where individuals are censused, breed, disperse, and are subject to overwinter mortality (Fig. 1). Following the initial seeding of animals in patches, the model consists of 3 steps. To begin with, a density-independent proportion of each patch population reproduces during each season. If an individual is selected to reproduce, it has 4 offspring which are immediately added to the population of that patch. The dispersal phase of the model follows breeding. As with reproduction, dispersal is density independent. Dispersal in our model consists of three processes. Initially, each individual in a patch is declared a disperser (with probability e) or nondisperser (with probability 1–e). This probability is independent of both density and the presence or absence of corridors of any given quality. Once the number of dispersers from each patch has been determined, each disperser independently selects an emigration route. If a patch has no corridor connections, all dispersing animals disperse into the matrix and are lost from the metapopulation. Henein and Merriam (1990) used a slightly different verbal definition for a disperser, but ultimately, dispersal in the HM model is identical to our model with respect to the effect of corridor presence or absence. When a patch is connected by more than one corridor, dispersers select randomly (without regard to corridor quality) which corridor they will use to emigrate. Following the selection of an emigration route, individuals suffer mortality with probability 1–θc while enroute to new patches. Survivorship, θc, during dispersal varies depending on the quality of the corridor selected by an animal. Differential mortality is the only feature which determines corridor quality. A higher percentage of animals die while dispersing in poor-quality corridors than those using high-quality corridors (i.e., θp < θh). Finally, surviving dispersers are added to the populations of the patches at the terminus of their respective corridor routes. During the final phase of the model, individuals are subject to overwinter mortality. As we mentioned previously, in order for populations to stabilize at non-zero equilibria they must be influenced by some sort of density-dependent factor. We have incorporated a density-dependent feature in our model in the form of overwinter mortality. Krohne et al. (1984) determined that dispersal in white-footed mice was density independent, and 264 Henein and Merriam (1990) use this observation to support their arguments for density-independent dispersal. This seems to be a common conclusion regarding small mammal dispersal (e.g., Gaines and McClenaghan 1980; but see Krebs 1992 for an exception). Accordingly, we use density-independent dispersal in our model. Terman (1993) found that reproduction in white-footed mice varied over a wide range of densities and that there was no consistent relationship between density and reproduction. Therefore, by process of elimination, we use densitydependent mortality to generate equilibrial populations. The winter survival probability in our model was 1/(Nt,i)1/4, where Nt,i is the population of patch i at time t. This value is used because it allows the patch populations to stabilize at values large enough to minimize stochastic patch-population extinctions. Note that the exact form and value of the density-dependent feature can be altered within a wide range of values without changing the qualitative results of the model (Table 1). Animals not surviving the winter are subtracted from their respective patch populations. The remaining animals are carried over to the next year when the processes of birth, dispersal, and mortality are repeated. In addition to the incorporation of a densitydependent feature, the other key difference between the HM model and ours is the stochastic nature of our model. In order to avoid the problems of dealing with fractions of animals, as is usually the case when a population (integer) is simply multiplied by proportional constants, we allow each animal to make a decision regarding reproduction, dispersal, and survival. At each decision-making event in our model, every live animal is assigned a random number. Based on a comparison between the assigned random number and the parameter in question (reproduction, dispersal, or survival), an individual either experiences an event or it does not. As an example, suppose that the expectation of surviving dispersal through a high-quality corridor is 0.8. In a deterministic model, 0.8Ni may result in a noninteger number of animals in each patch. In our stochastic model, each of the Ni individuals experience random mortality with probability 1–θc, thus Table 1. List of parameters used in model simulations. The parameters under the analysis column were the values used for the simulations presented in this paper. The columns listed alternate show two other sets of parameters that were used to check that the qualitative results of our model were parameter independent. Parameter Probability of reproducing Probability of dispersing Survival in poor corridor Survival in good corridor Overwinter survival in patch i Analysis Alternate Alternate 0.60 0.30 0.90 0.23 0.15 0.30 0.4 0.4 0.4 0.8 0.8 0.8 1/Ni1/4 1/Ni1/2 1/Ni1/4 avoiding fractions of individuals. Henein and Merriam (1990) carried fractional animals from dispersal through reproduction and then rounded the number of animals in each patch to an integer value after calculating overwinter survival at the end of each simulated year (Henein, personal communication). Our simple model only has a few parameters (Table 1). For the sake of consistency, we use Henein and Merriam’s “moderate” weekly birth rate and “moderate” weekly emigration rate for our reproductive and dispersal parameters. The survival value of θh = 0.8 for high-quality corridors was also chosen to correspond with the value found in the HM model. However, instead of using the θp = 0.73, for survival in low-quality corridors, value from the HM model, we opt to halve the survival value for high-quality corridors to θp = 0.4. Due to the stochastic nature of our simulations and the subsequent variance associated with numerous runs, a more pronounced difference between high-quality and low-quality corridors made the efforts of corridor quality easier to detect. In addition to the simulations using the parameters listed in the first column of Table 1, we ran the model for the same landscapes using the “high” and “low” birth and dispersal rates found in Henein and Merriam (1990). These simulations validate the fact that parameter changes have no effect on the qualitative results of our 265 Fig. 2. Four patch landscape arrangements used in simulations. Solid lines represent high quality corridors (θh = 0.8) and dashed lines indicate the presence of poor quality corridors (θp = 0.4). Mean metapopulations from 500 simulations are listed below each landscape. Standard deviations are given in parenthesis. model. Naturally, parameter changes affect the absolute populations in our landscapes, but all of the generalizations presented below hold true for all combinations of parameters. The parameter values in our model are roughly representative of white-footed mice and are selected to maintain similarity between our model and the HM model. Our model should not, however, be viewed as a quantitative descriptor of any particular small-mammal population. Our goal is not to simulate the dynamics of a particular real population, but to mechanistically determine the ways in which corridors influence metapopulations. In light of this, the absolute number of animals in the landscapes of our simulations is not important. Instead, the reader should focus on the number of animals in each landscape relative to the numbers in comparable landscapes. Simulations All of the landscape patterns used for our analysis are shown in Fig. 2. The simulations used for our analysis consisted of seeding each patch in a landscape with 25 animals. The model then tracked the population of each patch in a metapopulation for 100 years. We chose 100-year simula- 266 tions because a non-stochastic version of our model (one that followed real number populations instead of whole number populations) revealed that the populations in all landscapes stabilized by 50 years. We ran the stochastic model simulations for an additional 50 years to ensure that the final population size would be close to, if not at, an equilibrial level; visual inspection of a large number of runs confirmed that this was achieved. We simulated each landscape 500 times. This large number of simulations was necessary because the large variance associated with multiple runs of each landscape tended to obscure subtle differences in the mean metapopulation sizes among various landscape arrangements. Results First, we conduct a test to determine whether or not metapopulation size is affected by the quality of corridor connections. To do this, we group landscapes that have similar numbers of corridors and similar arrangements. Since we suspect that the substitution of poor-quality connections for highquality connections would decrease overall metapopulation sizes (Henein and Merriam 1990), we regress the number of poor-quality connections against the average metapopulation size for a particular arrangement (Table 2). The lines for all seven landscape arrangements are negative and highly significant, providing clear evidence that metapopulations do decrease as the number of poor-quality corridor connections increases (given that the total number of connections remained constant). The HM model demonstrated a similar trend. Second, we test for an effect of the arrangement of corridors in a landscape on metapopulation size. For this analysis, we group landscapes that have the same number of connections and similar numbers of low-quality and high-quality connections (Fig. 3). Since we have no a priori predictions regarding the effects of different arrangements, we use one-way ANOVAs to detect differences in the mean metapopulations within each of the groups in Fig. 3. The analysis reveals that the arrangement of corridors among patches does have an effect on the overall metapopulation size in a land- Table 2. Linear regressions of landscape groups with a constant number of corridors and similar arrangements. Only the number of poor quality corridors is varied within each group. Landscape groups Total # of corridor (#’s are from Fig. 2) Connections Slope r2 p 1,2,3,4,5,6,7 8,9,10,11,12,13 14,15,16,17,18,19 20,21,22,23,24 25,26,27,28,29 30,31,32,33 34,35,36,37 0.42 0.43 0.42 0.46 0.40 0.46 0.45 0.000 0.000 0.000 0.000 0.000 0.000 0.000 6 5 5 4 4 3 3 –6.7 –8.2 –8.0 –10.4 –9.0 –13.4 –13.0 scape. These results also support the conclusions of the HM model. Third, we look for evidence that peripheral patches are less beneficial to the metapopulation than interior patches. We define a peripheral patch as one that has fewer corridor connections than the patch or patches to which it is connected. An interior patch is one that has an equal or greater number of connections than the patch or patches it is connected to (Fig. 4). Inspection of mean population values suggests that landscapes with more peripheral patches (landscape B, Fig. 4) support smaller metapopulations than landscapes with fewer peripheral patches (landscape A, Fig. 4). To test this hypothesis, we group the landscapes in Fig. 5 which contain only interior patches by our definition. For each landscape, we then compute the average metapopulation size from 500 simulations. One-way ANOVAs support our prediction that there are no differences in the metapopulation sizes between the landscapes in each group (group 1: F = 0.21, p = 0.811; group 2: F = 1.68, p = 0.187). As further evidence, we determine the mean patch populations for each patch in the landscape shown in Fig. 6. A t-test reveals that the means of the 3 peripheral patches are indeed lower than the interior patch (t = 65.658, p = 0.000). These results indicate that, where the quality of connections is constant, a factor influencing the size of a population in a particular patch is not the connections to that patch (provided it has at least one corridor connection), but the ratio of the number of connections for the patch to the number of connections of its neighbor patches. Our final question concerns the influence of the 267 Fig. 3. Landscapes used to compare the influence of corridor arrangement on metapopulation size. Landscapes are grouped so that the number of corridor connections and the number of low and high quality corridors are constant. Differences within groups were tested for using one-way ANOVA’s. F values and the probability of a greater F value are listed next to each group. number of corridor connections on metapopulation size. In light of the fact that different corridor arrangements can impact metapopulation size, caution must be exercised when testing for effects due to the number of corridors because the addition or subtraction of a corridor from a landscape inherently changes the landscape’s physiognomy. The two groups of landscapes shown in Fig. 5 contain only interior patches and have equivalent metapopulations. The fact that these groups contain landscapes with 6, 4, and 2 corridors demon- strates that metapopulation size is not influenced simply by the number of connections in a landscape, even if all the connections are of equivalent quality. This result contrasts with Henein and Merriam’s (1990) general conclusion that increasing the number of high-quality corridors increases the metapopulation and increasing the number of poor-quality corridors decreases the metapopulation. 268 Fig. 4. Two landscapes that contain peripheral and interior patches. Darkened patches represent peripheral patches. They have fewer connections than the patch(es) to which they are connected. Landscape A has 2 peripheral patches and 2 interior patches. Landscape B has 3 peripheral patches and 1 interior patch. Note that the metapopulation size of A is significantly greater than B (t-test). Fig. 5. Groups of landscapes containing all interior patches. One-way ANOVA’s demonstrate that the mean metapopulation size is the same within each group despite differences in the number of connections. Discussion Our model clearly demonstrates that the quality of corridors connecting patches in a landscape does influence the size of a metapopulation. The results of the HM model are similar. Given the structure and assumptions of our model, this result is not surprising. Two of the assumptions of our model are that dispersal and reproduction are densityindependent. This means that a constant proportion of the population leaves a patch each year. This proportion leaves regardless of quantity or quality of corridors connecting a given patch to other patches in a system. Thus, if a patch has no corridor connections, the B, D, and E terms in equation 1 remain unchanged, but I = 0 because our model assumes that all animals in the matrix are lost to mortality. This results in a low metapopulation for a landscape with no corridor connections (landscape 50, Fig. 2). A patch connected by high-quality corridors will have few animals lost enroute, and I will approach the loss due to E. Poor-quality connections result in an I that is close to 1/2 of E. It is clear that the addition of each poor-quality corridor in a system formerly containing only high-quality connections results in a decrease in I for the two patches linked by the poor corridor. It is this reduction in immigration that results in the overall decrease in metapopulation size. It may be unrealistic to assume that all animals entering the matrix are lost to the population. Fig. 6. A landscape composed of 1 interior and 3 peripheral patches. The population of the interior patch (2) is significantly larger than the average populations of the 3 peripheral patches (1, 3, and 4) (t-test). However, even if the rules of our model allowed animals to reach new patches by dispersing through the matrix, the model results would be the same. A corridor, by definition, allows a better chance of survival during dispersal than movement through an unfavorable matrix (Forman and Godron 1981). If we allowed animals to disperse through the matrix, it would be analogous to adding a third type of corridor (i.e., very-poor quality corridors) to our model, and we would still see the same trend of landscapes with higher-quality connections supporting larger metapopulations. Where we differ with Henein and Merriam (1990) is with regard to their conclusion that increasing the number of high-quality connections increases metapopulation size but increasing the number of low-quality connections decreases the 269 size of a metapopulation. Certainly, where isolated patches can be connected by high-quality corridors to other clusters of connected patches, the addition of the corridor will result in an increase in the metapopulation. The Lefkovitch and Fahrig (1985) model, the HM model, and our model quite clearly show this. However, given that all patches are connected, adding, rearranging, or even deleting (compare landscape 30 with 48 and 33 with 49, Fig. 2) corridors can result in increasing the equilibrial metapopulation only if the alterations result in more equitable distributions of connections among patches within the landscape. If the number of corridors is increased in such a fashion that the ratio of interior to peripheral patches remains the same, the additional corridors will have no effect on the equilibrial metapopulation. While this phenomena may, in fact, occur in the other models, it was not shown or discussed. Given the assumption that the number of emigrants and survival during emigration represent a constant proportion of each patch population, we feel our results are more easily explained. Here again, it is important to keep in mind that the only element of the models altered by changing corridor arrangement, quality, and quantity is I for each patch. If the quality of corridors remains constant, then I should not change for any patch regardless of the number of connections. To illustrate the point, consider landscapes 20 and 48 in Fig. 2. Landscape 20 has 4 high-quality connections whereas landscape 48 has only 2. For reference, the top most patch in each landscape we refer to as patch 1, the middle patch as patch 2, the lower left patch as patch 3, and the lower right patch as patch 4. If we think about the dynamics in patch 1 in each landscape, it becomes clear why the number of connections has no effect on the size of a metapopulation. During dispersal, patch 1 in landscape 20 loses a proportion (eN) of its population. Half of eN goes down each corridor, arriving in patches 2 and 4 respectively. In return, patch 1 receives eN/2 from both patches 2 and 4. The net result is that patch 1 loses eN animals and gets back eN minus a certain proportion [1–θc)eN] that died during dispersal. Patch 1 in landscape 48 has only 1 corridor, but the dispersal loss is still eN. All eN animals emigrate to patch 2. Since patch 2 also has only 1 connection, patch 1 receives [(1–θc)eV] back. In both these landscapes, all of the patches lose eN animals to dispersal and get back (1–θc)eN animals. This situation will hold true as long as all the patches in the landscape have the same number of corridor connections. When a landscape is composed of patches with a variable number of connections (i.e., interior and peripheral patches), the situation becomes somewhat more complicated. The results of our model, as well as the HM model, demonstrate that corridor arrangement will affect metapopulation size. Our model shows a consistent trend that can be explained by the structure of I in equation 1 and demonstrates that landscapes with the greatest number of interior patches will support the largest metapopulations. Recall that the only portion of equation 1 modified by altering the characteristics of the corridor connections in a landscape is I. This quantity is dependent on two things. First, it is affected by θc which is corridor survival. Naturally, higher values of θc result in a higher number of emigrants reaching their destinations and, thus, larger metapopulations. Second, the value of I is influenced by eN which is the total number of emigrants leaving each patch. Recall that all patches have the same, constant proportion of animals emigrating regardless of corridor connections. When a given patch has no connection, eN animals leave and are lost in the matrix and none immigrate from other patches. This results in substantially lower populations in unconnected patches. If a patch has one connection, eN animals disperse through that corridor. If the corridor leads to another patch that has no other additional connections, then the patch receives θceN animals back through immigration. This situation changes if the first patch is connected to a patch that has additional corridors. If the second patch has one additional corridor, patch 1 is returned θceN2/2 and if the second patch has two additional connections, patch 1 receives θceN2/3 animals. The best illustration of this point is landscape 34 in Fig. 2. In this case, patch 2 has 3 corridor connections, while patches 1, 3, and 4 have one connection apiece. During the simulations, patches 1, 3, and 4 lose eNi animals to dispersal. Each of these patches only receives θceN2/3 animals from immigration. All three of these patches are losing substantially more animals from emigration than are being 270 replaced by immigration. Meanwhile, patch 2 loses eN2 animals to dispersal and gains θceN1 + θceN3 + θceN4 through immigration. If patches 1, 3, and 4 support less than 1/3 the animals that patch 2 supports, then the return to patch 2 is less than it loses to dispersal. Our simulations show that the populations of the peripheral patches are more than 1/3 of the interior patch (Fig. 6), so patch 2 actually has more animals immigrating than emigrating. The overall result of the dynamics described above is a reduction in the metapopulation size for any landscape containing peripheral patches. The greater the ratio of peripheral patches to internal patches in a landscape, the lower the metapopulation will be. Our model provides clear evidence that corridor quality and arrangement will influence metapopulation dynamics in a landscape of interconnected patches. These results support the conclusions found by Henein and Merriam (1990) using a similar model. The HM model did not, however, provide an explanation of why the corridor arrangement in a landscape should alter the size of a metapopulation. We believe that landscapes with a greater ratio of peripheral to interior patches will support smaller metapopulations. In contrast to the HM model, our model shows that the number of corridor connections in a landscape does not influence metapopulation size unless it also alters the peripheral:interior patch ratio. We feel that what Henein and Merriam (1990) labeled as an effect of the number of corridor connections was actually a change due to a subtle shift in the relative numbers of peripheral and internal patches. This emphasizes the need to carefully consider how changes in a landscape’s composition also alters its physiognomic properties. Overall, both models provide evidence that the size of a metapopulation will be affected by qualitative characteristics of corridors within a landscape. Acknowledgements Versions of this manuscripts were critically read by Kirk Moloney, Bill Clark, and Kringen Henein. We are extremely grateful for their efforts. 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