Patch dynamics in a landscape modified by ecosystem engineers

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(and to a lesser degree, to changes in r). Given the addeddifficulty of determining analytical solutions to the morecomplex model and estimating an additional parameter,the benefits of the four-patch model seem limited. Onlyin systems where there are important differences betweenpartially recovered and fully recovered patches, e.g.between the vegetation of beaver meadows and riparianzone forest (Terwilliger and Pastor 1999, Wright et al.2002), would it be worthwhile to model the patchdynamics of the system using the four-patch model.Immigration, either from outside the boundaries ofthe system, or from patches within the system whereengineers can reproduce without having to modifyhabitat has the potential to alter the dynamics of thesystem. With even small amounts of immigration, thezero engineer steady state is no longer possible. Theeffects of immigration will be highest when engineersreside in a patch for a short time (i.e. high d), producefew successful colonizers (i.e. low n), and where degradedpatches recover rapidly (i.e. low r).Model parametersWhile the relative abundance of different patch types in alandscape can be relatively easy to determine, estimatingthe parameters of the model is somewhat more challenging.Although we were able to estimate some of themodel parameters indirectly using data on beaverpopulations and patch transitions in the central Adirondacks,we are unaware of any data set that wouldallow independent estimation of all of the model’sparameters. Future studies of the effects of ecosystemengineers on patch dynamics would benefit by structuringtheir questions in a manner that would allowinvestigators to estimate the parameters of these models.Of all the parameters, d, or the rate at which activesites are abandoned, is likely to be the easiest to estimate.Careful long-term monitoring of patch use by engineerswill yield average lifetimes of engineered patches, whichcan be converted into probabilities of patch decay. Ingeneral, organisms that exhaust patch resources graduallyrelative to the rate of resource renewal, e.g. moundbuildingdesert shrubs (Shachak et al. 1998), should tendto create patches with lower values of d than organismsthat are short-lived, e.g. leaf-tying caterpillars (Lill andMarquis 2003), or that use the resources in engineeredpatches much more rapidly than they are replenished.Estimating the number of successful colonists producedper engineered patch (n) is a bit more challenging.It can be inferred by dividing the number of newlyformed patches by the number of active patches at theprevious time step. However, independent measurementof n requires determining two variables: the number ofdispersers produced per engineered patch, and rate atwhich dispersers successfully establish new patches. Thefirst variable will be a function of birth rates and theprobability of individuals to leave their natal patch. Thesecond variable is a function of mortality rate duringdispersal and the probability that dispersing individualswill establish new colonies. In general, ecosystem engineerswith high fertility, low natal site fidelity, and lowmortality during dispersal should have high values of n.The rate at which abandoned patches recover intopotential patches (r) is probably the most difficult toestimate independently. This is because the resourcesnecessary for an engineer to recolonize a patch are likelyto accumulate steadily over time. Determining when thenecessary resources reach the critical level that separatesdegraded patches from potential patches requires athorough knowledge of the requirements of the ecosystemengineer. Furthermore, the level to which criticalresources are depleted in recently abandoned patches islikely to vary, thus the time needed for a degraded patchto recover will vary, even if recovery rates are constantacross the landscape. In general, engineers with lowresource requirements and systems with high resourcesupply rates should have high values of r.In systems where dispersing engineers are produced inpatches that have not been modified by the ecosystemengineer, one must differentiate between the proportionof successful colonists that are produced in nonengineeredpatches (i) and the proportion of coloniststhat are produced in engineered patches (n). If birth ratesare the same in the two patch types and individuals fromengineered and non-engineered patches have identicalprobabilities of successfully producing new patches, therelative importance of n and i in controlling theproportion of active patches will depend on the relativeabundance of active engineered and active nonengineeredpatches in the landscape. If however, birthrates or successful dispersal rates differ between engineeredand non-engineered patches, estimating i willrequire accurate measurements of birth rates in andsuccessful dispersal from non-engineered sites that containengineers.Estimating the rate at which partially recoveredpatches recover fully (r) (i.e. from P to F) has similarchallenges to estimating recovery rates from degradedpatches to potential patches (i.e. r). Tracking patchesover time allows estimates of the amount of timerequired from abandonment to full recovery (assumingone can set the criteria that determine full recovery).However, such an analysis cannot determine how muchof the recovery time is spent in the degraded state versusthe potential state. Independent estimation of r wouldagain require detailed understanding of the criteria usedby engineers when selecting sites, and may requiremeasurements difficult to obtain using typical methodsof surveying habitat such as the nutritional quality ofdifferent plants species (Martinsen et al. 1998). Despitethese difficulties in estimating r, we hypothesize thatOIKOS 105:2 (2004) 345
systems where engineers perform qualitative transformationof the physical state of the patches they modify, e.g.beaver transforming terrestrial patches into aquaticpatches, are likely to have much lower values for r thansystems where the engineers only perform quantitativetransformations of habitats, e.g. shrub mounds increasingwater infiltration rates in desert soils (Shachak et al.1998).In systems where engineers can recolonize patchesbefore they have fully recovered, one must also determinethe degree to which engineers prefer or avoidpartially recovered patches relative to fully recoveredpatches (z). As illustrated above with data from annualbeaver surveys, if one can determine the total number ofpatches available in a landscape, this metric is notdifficult to estimate. It requires calculating the degreeto which new patches are formed in each of the twohabitat types (partially and fully recovered) relative totheir availability. In general, most systems are likely tohave values of z less than one, indicating that engineersprefer fully recovered to partially recovered patches.However, as suggested by the data from the annualbeaver surveys, it is possible for positive feedbacks tooccur whereby conditions in abandoned sites are favorablefor the establishment of species that are preferred tothose found in fully recovered sites.Implications for beaverOver the past 20 years, beaver populations on HWF haveremained relatively constant. The large jump in observedcolonies between 1985 and 1986 is due, at least in part, toan increase in the area covered during annual beaversurveys (C. Demers, pers. com.). Since 1989, the numberof active colonies on HWF has fluctuated relatively littleamong years. These data are consistent with theassumption that the beaver populations and the patchdynamics of beaver-modified habitats are at steady state.Since we were unable to independently estimate all ofthe model’s parameters, we could not directly test themodel’s predictions as to the relative abundance ofdifferent patch types in the landscape. Nevertheless,analyzing the model’s behavior using parameters estimatedfrom the annual beaver surveys reveals someinteresting features of the system. Our estimate of thenumber of successful colonists produced per engineeredpatch (n) is doubtless an overestimate. The calculation ofn ignored the role of colonists that originated from nonengineeredpatches (e.g. natural lakes). Even with theoverestimate, our estimate of n (0.39) is close to ourestimated value of the decay rate of active patches (d/0.21). In both versions of the model without animmigration term, engineers cannot persist in a systemwhere the decay rate, d, is greater than the patch creationrate, n. Thus, immigration, either from outside thesystem or from non-engineered patches, is likely to beimportant in maintaining beaver populations in thecentral Adirondacks. The habitat within the HWF isessentially identical to the surrounding area, so there isno reason to expect that HWF is receiving significantlymore dispersing beaver than it is losing through emigration.This points to the central importance of colonies oflake-dwelling beaver in maintaining the patch dynamicsof beaver-modified habitats.If the estimates of the model parameters are correct, itwould suggest that the relative abundance of differenthabitat types should be most sensitive to changes in thedecay rate of active patches (d), and the successfulcolonization rate (n). While decay rates depend primarilyon the rate at which beaver use up resources in a site andare unlikely to vary significantly over time, successfulcolonization rates could vary significantly depending onpredation rates during dispersal. Wolf reintroduction tothe Adirondacks or increasing populations of coyotescould potentially increase predation during dispersal,thereby lowering n. Based on our parameter estimates,such changes in n would lead to large decreases in theproportion of active ponds and young meadows withconcomitant increases in old meadows with welldevelopedsurrounding forests. Older meadows tend tobe much dryer than new meadows and as a result,contain a quite different community of wetland plants(Wright et al. 2003). Thus, changes in n due to increasedmortality during dispersal could have strong effects ondiversity at the landscape-scale.The decline in beaver preference for previously usedsites (z) over time is somewhat counterintuitive. If thereis variability in site quality, and beaver first selectoptimal sites (Howard and Larson 1985), one mightassume that beaver would prefer to re-use high qualitysites rather than colonize poorer quality sites, leading tohigher values of z. The relationship is in part due to theassumption that all possible sites had been colonized bythe time of the last survey. However, if we assume thatonly half of the available sites had been colonized in1999, the decline over time, while weaker, still remainssignificant (F 1,11 /25.14, r 2 /0.68, p/0.0004). It maybe that patches that are repeatedly used begin to declinein quality over time, particularly if repeated browsing bybeaver leads to dominance by non-preferred species suchas conifers or other late-successional species. There issome evidence of such an accelerated succession dynamiccaused by beaver in Algonquin Provincial Park(Fryxell 2001), thus it is certainly possible that this isoccurring in the Adirondacks as well. If so, this wouldcause beaver to increasingly avoid previously usedpatches over time, leading to the observed lower valuesof z. The model predicts that decreases in z should leadto decreased proportions of active ponds and newmeadows and increased proportions of old meadows,mirroring the effects of lower n.346 OIKOS 105:2 (2004)
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Page 6 and 7: ResultsSimple modelAs d, the decay
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Page 12 and 13: ConclusionsThe set of models develo

Patch dynamics in a landscape modified by ecosystem engineers

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