Patch dynamics in a landscape modified by ecosystem engineers

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variables, and this effect is largest at low values of r (Fig.6B). Adding immigration also affects the relationshipbetween n and the steady state values of the statevariables at low values of the parameter, but themagnitude of the effect is much larger (Fig. 6C). Thedifferences between the steady state values of the statevariables in the model with and without immigrationbegin to decrease at n/d. For all values of all threeparameters, the effect of adding immigration (at i/0.1)is to reduce the proportion of P* while increasing theproportion of A* and D*.Partial recovery modelAdding a fourth patch type to the model causes severalimportant changes to the behavior of the model.Increasing d causes A* to decrease steadily and F* toincrease steadily while D* and P* reach a maximum atintermediate values of d (Fig. 7A). Not surprisingly,increasing r causes D* to decrease (Fig. 7B). Interestingly,varying r only results in a slight increase in P*except at low values of r presumably because patches inthe P state are quickly transformed into A accountingfor the increase in A* as r increases. Increasing r has anegative effect on F*, particularly at low values of r. Atvalues of n/d, F* and P* steadily decrease as nincreases while A* and D* increase (Fig. 7C). Increasingr has a negligible effect on A* and D*, and servesprimarily to increase F* while decreasing P* (Fig. 7D).Varying z, the degree of discrimination againstpartially recovered patches relative to fully recoveredpatches, causes the most interesting changes in thesteady state values of the state variables. Increasing z,or causing the engineer to prefer sites in the partiallyrecovered state to fully recovered patches, causes bothA* and D* to increase (Fig. 7E), presumably since itessentially increases the number of patches that areavailable to colonization. Interestingly, increasing zcauses F* to decrease. This is because, as the willingnessof engineers to colonize partially recovered patchesincreases, partially recovered patches tend to be colonizedand converted to active patches before they canfully recover. P* shows a unimodal relationship with z,with a maximum at intermediate values of z. Thisrelationship represents a balance between the directincrease in the rate at which P is converted into A andthe indirect effect of increasing A on the production of P(via an increase in D) as z increases. At both highervalues of d and lower values of n, the peak in P*, occursat lower values of z.Although this model is significantly less analyticallytractable than the simpler models, under certain conditions,the system behaves essentially like the simplesystem discussed above. Specifically, as r approaches 1,particularly at values of z close to (or greater than) 1, theFig. 7. Dynamics of steady state values for state variables of thecomplex (4-patch) model in response to changes in the modelparameters. In simulations where they were held constant, d/0.21 (A), r/0.25 (B), n/0.39 (C), r/0.01 (D), and z/1.21(E).OIKOS 105:2 (2004) 343
Fig. 8. Effect of varying the rate of recovery from partiallyrecovered patches (P) to fully recovered patches (F) and level ofpreferences for previously used versus fully recovered habitat (z)on the difference between the simple (3-patch) model andcomplex (4-patch) model in the steady state proportion of activepatches (A). Values of z greater than one indicate a preferencefor previously used habitat. In all simulations, d/0.21, r/0.25and n/0.39.values of the state variable at steady state approach thoseof the simple model with the same parameters (Fig. 8).DiscussionModel predictionsThe three versions of the model presented here producequantitatively different predictions about the proportionof engineered landscapes in different habitat types as theparameters are varied. However, all three models agreeon several important qualitative predictions aboutengineered landscapes (Table 2). Landscapes will tendto have large proportions of active patches (A) whenecosystem engineers are efficient in their resource use,producing many new colonists while only graduallydegrading the resources of the patch, and when resourcerenewal occurs rapidly after engineers abandon a site.Engineers that create landscapes dominated by abandoned(D) patches would produce large numbers ofcolonizers by rapidly depleting the resource levels of apatch, and leaving abandoned patches that recover veryslowly. In most respects, the proportion of potential sitesin the simple model (P) reacts to changes in theparameters in a manner similar to the proportion offully recovered sites (F) in the more complex model.Ecosystem engineers that create patches that producefew new colonizers and are abandoned quickly, yetrecover rapidly should create landscapes dominated bythese fully recovered patch types. In landscapes bestdescribed by the partial recovery model, partiallyrecovered patches (P) will be most abundant whenabandoned patches rapidly recover to a state sufficientto allow recolonization, but only slowly regenerate to thefully recovered state.In many cases, sites currently used by ecosystemengineers, and those recently abandoned are easilydistinguished from patches that have not been modifiedby engineers. For example, pocket gophers form distinctmounds of loose soil in many prairie ecosystems (Huntlyand Inouye 1988), grizzly bears create extensive patchesof tilled soil in alpine meadows while foraging for lilybulbs (Tardiff and Stanford 1998), tilefish and grouperexcavate marine sediments (Coleman and Williams2002), and leaves occupied by shelter-building Gelechiidcaterpillars are strikingly tied together, Lill and Marquis2003. Because these states are readily identifiable, itshould be feasible to compare the relative abundance ofdifferent patch types in landscapes where the sameengineer operates, but where values of the parametersare likely to be different (e.g. predation risk is higherthus lowering the number of successful colonizers, orproductivity is higher, thereby speeding up recovery fromabandoned sites). Such an analysis would serve as acritical test of whether these models successfully capturethe relationship between the population dynamics of anecosystem engineer and the dynamics of the patches itcreates.Differences between modelsAdding a fourth patch type to represent habitat that ispartially recovered, yet still capable of being engineereddoes not alter the fundamental patch dynamics of themodel. In both the original and partial recovery versionsof the model, the proportion of the landscape that willbe in the active and degraded states at steady state reactssimilarly to changes in the parameters d, r, and n.Furthermore, the steady state proportion of potentialpatches (P) in the simple model behaves similarly to thesteady state proportion of fully recovered patches (F) inthe complex model with respect to changes in d and nTable 2. Summary of the parameter combinations that lead to high relative abundance of each of the patch types in the threemodels.Patch type 3-Patch model without immigration 3-Patch model with immigration 4-Patch modelA / r, n;¡/ d / r, n;¡/ d / r, n;¡/ dD / n; ¡/ r; intermediate d / d, n;¡/ r / n; ¡/ r; intermediate dP / d; ¡/ n / d, r; ¡/ n, i / r; ¡/ n, r; intermediate d, zF N.A. N.A. / d, r;¡/ r, n,z344 OIKOS 105:2 (2004)
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Patch dynamics in a landscape modified by ecosystem engineers

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