WEED POPULATIONS AND CROP ROTATIONS: EXPLORING DYNAMICS OF A STRUCTURED PERIODIC SYSTEM

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1 Ecologicl pplictions, 12(4), 2002, pp y the Ecologicl Society of meric WEED POPULTIONS ND CROP ROTTIONS: EXPLORING DYNMICS OF STRUCTURED PERIODIC SYSTEM SHN K. MERTENS, 1,2,5 FRNK VN DEN OSCH, 3 ND J.. P. (HNS) HEESTEREEK 4 1 Crop nd Production Ecology Unit, Plnt Reserch Interntionl, P.O. ox 16, 6700 Wgeningen, The Netherlnds 2 Crop nd Weed Ecology Group, Wgeningen University, P.O. ox 430, 6700 K Wgeningen, The Netherlnds 3 Institute of rle Crops Reserch-Rothmsted, Hrpenden, Hertfordshire L5 2JQ, UK 4 Centre for iometry, Plnt Reserch Interntionl, P.O. ox 16, 6700 Wgeningen, The Netherlnds strct. The periodic growing of certin set of crops in prescried order, clled crop rottion, is considered to e n importnt tool for mnging weed popultions. Nevertheless, the effects of crop rottions on weed popultion dynmics re not well understood. Explntions for rottion effects on weed popultions usully invoke the diversity of environments cused y different crops tht weed popultion encounters. Using periodic mtrix model, we show tht the numer of different crops is not the sole fctor, nd tht the sequence of given set of crops cn ply n importnt role. In the model the weed popultion is structured y seed depth in the soil, nd plowing moves seeds etween lyers. For illustrtion of concepts, we use prmeter vlues thought to e chrcteristic for Polygonum persicri growing in crrots (crop ) nd spring whet (crop ) in the Netherlnds. We systemticlly exmine the popultion growth rtes for P. persicri nd their sensitivity to chnges for ll rottions of 2 6 yers sed on crops nd. We include eight scenrios tht differ in the effects of plowing nd seed survivl over winter. Differences etween rottions cn e striking. For exmple the weed popultion growth rte in the seline rottion (ssuming 100% winter survivl) is nerly 25% lower thn in rottion. The elsticity ( mesure to quntify the effect of proportionl chnges in model prmeters on popultion growth) to seedling survivl is nerly 75% higher in the yers of rottion thn in the yers of rottion. Chnging prmeter vlues chnges the reltion etween popultion dynmics nd rottion orgniztion, ut not the conclusion tht there re consequences for popultion dynmics nd mngement due to choice of rottion. While our exmple is n gronomic one, the question Does sequence mtter? nd the methods pplied should e of interest to reserchers nd mngers concerned with the periodic mngement of other ecosystems. Key words: crop rottion; elsticity nlysis; periodic mtrix model; plnt popultion mngement; Polygonum persicri; popultion dynmics, weeds; rottionl crops nd weed undnce; weed popultion mngement. INTRODUCTION Crop rottion the growing of different crops in recurring succession on the sme lnd hs long een dvocted s method to increse crop yields nd reduce inputs (Leighty 1938, Roinson 1949). Explntions for incresed yields of crops grown in rottion system rther thn s continuous monoculture crop include the periodic chnges in the environment tht prevent prticulr pests, diseses, nd weeds from dominting, nd the decresed stress on the soils y including crops tht require less cultivtion or tht increse nutrient nd orgnic-mtter inputs (Crookston 1984, Crookston et l. 1991). The development of prolemtic weed popultions is thought to e prevented y the diverse environments tht weed popultions encounter over the course of crop rottion cycle (Liemn nd Dyck 1993, Liemn nd Gllndt 1997). For Mnuscript received 20 Ferury 2001; revised 17 Octoer 2001; ccepted 19 Octoer E-mil: s.k.mertens@plnt.wg-ur.nl 25 exmple, sowing dtes, weed-control methods, nd competition from the crop will differ from seson to seson s the crop species chnge. Ech of the diverse environments is chrcteristic of given crop. crop rottion, y definition, extends over severl cropping sesons. rottion s orgniztion includes the crop species, their proportion nd order, nd the length of the rottion (numer of cropping periods). nturl question is whether the orgniztion of crop rottion, given set of crops, will ffect weed popultion growth rtes. For exmple, would rottion of lternting crrot nd whet crops result in different weed popultion growth rte thn rottion consisting of two consecutive yers of crrots followed y two consecutive yers of whet? Up to now there hve een few experimentl or theoreticl studies tht directly nd systemticlly ddress the effect of crop sequence on weed-popultion growth rtes. Most studies hve concentrted on compring continuous monoculture crop with one or more different rottions, usully in comintion with different till-

2 26 SHN K. MERTENS ET L. Ecologicl pplictions Vol. 12, No. 4 ge or weed-control tretments (Mrtin nd McMilln 1984, Schweizer et l. 1988, lckshw et l. 1994, Doucet et l. 1999, Kegode et l. 1999). Some modeling studies of weed popultion dynmics hve included crop rottions, ut did not systemticlly exmine the effect of different crop rottions sed on their set of crops (Gonzlez-ndujr nd Fernndez-Quintnill 1991, Jordn et l. 1995, Lindquist et l. 1995, Squire et l. 1997). In more generl ecologicl setting, the reliztion of the importnce of the timing of events in periodic system is not new; Drwin nd Willims (1964) found tht the seson of hunting ffected the popultion growth rte of n ge-structured popultion of rits. More recent studies hve concentrted on the frequency of nominlly periodic events, such s fire nd drought (Gotelli 1991, Gross et l. 1998, Hoffmn 1999), ut hve not exmined the effect of systemticlly different orders of events on growth rtes or on possiilities for mngement. Goluov et l. (1999) in their study of Prosopis glndulos (honey mesquite) clculted the periodic growth rtes for ll comintions of their four, yerly trnsition mtrices. However their im ws to otin confidence intervl for the periodic growth rte. The popultion dynmics of nnul weeds in crop rottion represent very simple periodic system, ut one llowing multiple venues for intervention, for exmple through chnging the rottion itself or through chnging the weed popultion s vitl rtes, through mngement, in prticulr crop. etter understnding of weed popultion dynmics in crop rottions my contriute insights pplicle to the understnding nd mngement of species in other ecosystems. Our purpose is the systemtic exmintion of the effects of different crop rottions on weed popultion dynmics nd on possiilities for mngement. Our questions pertin to how the proportion of crops, their order, nd numer ffect the popultion growth rte of depth-structured seed nk nd the growth rte s sensitivity to chnges in life-cycle processes. nswering these questions with field experiments, ecuse of their long-term nture, is prcticlly difficult. We therefore find nswers through use of mthemticl model tht cptures essentil fetures of our system ut tht is simple enough to llow in-depth investigtion of the processes ffecting weed popultion dynmics in crop rottions. Our investigtion is restricted to weed with semelprous life history, growing in two crops, where its seed nk is structured y depth nd seeds cn e moved verticlly in the soil y plowing. prticulrly suitle description is periodic mtrix model. This choice lso llows ppliction of considerle ody of theory (de Kroon et l. 1986, Cswell 1989, Cswell nd Trevisn 1994, enton nd Grnt 1999, de Kroon et l. 2000). We illustrte concepts with prmeter vlues chrcteristic for Polygonum persicri L. (redshnk) growing in vriety of rottions, composed of crrots nd spring whet in the Netherlnds. METHODS Model construction The trnsition mtrices. s the ility of seed to germinte nd emerge vries with depth (Vleeshouwers 1997), nd s tillge opertions redistriute seeds in the soil (Cousens nd Moss 1990), weed seed popultion cn e considered s eing structured y the depth t which seeds re locted. We distinguish two soil lyers, where the top lyer is indexed 1 nd the ottom lyer indexed 2. The numer of seeds in soil lyers 1 nd 2 t time t cn e represented s vector, n(t): n 1(t) n(t). (1) n (t) We regrd time in discrete steps of 1 yr, where one crop is grown per yer. Ech time step the weed seed popultion in ech soil lyer cn e clculted y pplying mtrix of trnsition rules to the popultion vector resulting from the previous time step. The trnsition rules, lso clled trnsitions, descrie seed survivl, reproduction, nd movement etween soil lyers. These rules, nd therefore the mtrices, will e different depending on the prticulr crop grown ech yer. We consider two crops only, nd, nd ssume tht the trnsition rules only depend on the crop in question nd not the crop tht ws grown, sy, in the previous yer. For two-yer rottion of crops nd, the seeds will follow the trnsitions given in Fig. 1. These yerly trnsitions yield mtrices nd, for the respective crops, e.g., for crop, 2 12 (2) 21 where the element ij of mtrix is the contriution of one seed in lyer j, t time t, to the popultion of seeds in lyer i, t time t 1. The popultion over complete cycle of the rottion, is given s n(t 2) n(t) (3) i.e., first pply the trnsition mtrix nd then the mtrix (since crop is grown first, followed y crop ). In order to distinguish different yers within rottion cycle nd in keeping with the terminology of Cswell nd Trevisn (1994), we cll ech yer of rottion phse nd numer the phses with respect to some reference rottion. Defining rottion s the reference rottion, then crop occurs in phse 1, denoted (1), nd crop occurs in phse 2, denoted (2). The rottion (2) (1) is the cyclic permuttion of, nd strts with phse 2 of the reference rottion. Different cyclic permuttions of some reference rottion will hve the sme popultion growth rte, ut other properties, such s the distriution of seeds over soil lyers t the end of the rottion, will differ for ech

3 ugust 2002 WEED DYNMICS IN CROP ROTTIONS 27 FIG. 1. Trnsitions in two-yer rottion of crops nd. Weed seeds in soil re in either lyer 1 or 2, nd n the numer of weed seeds in lyer 1 or 2 t time t; ij the contriution of one seed in lyer j t time t to the popultion of seeds in lyer i t time t 1. cyclic permuttion of the reference rottion. Cyclic permuttions of given rottion will hve identicl popultion growth rtes ecuse in the long term the rottions re identicl. Properties such s the depth distriution of seeds re expected to chnge ecuse processes occurring during the previous crop ct on the depth distriution present t the strt of tht crop. More detils on these effects re given in following sections. We lso wish to distinguish rottions tht cnnot e cycliclly permuted to give identicl rottions. Rottions nd re such rottions, nd we cll them essentilly different. Essentilly different rottions re expected to produce different popultion growth rtes. In generl, for some given rottion, the dynmics re: n(t p) M n(t) (4) where M is the mtrix product of the yerly trnsitions, strting in phse h, where h will e in {1, 2, 3,..., p}, nd p is the length (period) of the rottion. We define reference rottions lexicogrphiclly, so tht the lrgest lock of consecutive crops in the rottion occurs first. Thus, for exmple, rottions consisting of consecutive locks of two crops nd two crops will hve the reference rottion, whose mtrix product is M (1) (4) (3) (2) (1). If the rottion strts with the lst crop, giving rottion, the mtrix product is M (4) (3) (2) (1) (4). For our exmple crops, nd, we systemticlly exmine ll essentilly different cropping sequences of two to six yers, s well s monocultures of oth crops. In prctice crop rottions do not usully exceed six yers. For ech rottion length, ll possile rtios of the two crops nd ll essentilly different orders re investigted. We lso exmine the effect of incresing rottion length while keeping the proportion of ech crop the sme for the series,,, nd so on. Life history underlying the trnsitions. s it is difficult to directly mesure the trnsitions ij nd ij, nd s it is not possile to directly mnipulte them ecuse they re composed of vriety of iologicl nd mngement processes, we must consider weed s life cycle nd the prmeters tht underlie trnsitions from one stge to the next. Fig. 2 illustrtes the life cycle for summer nnul weed growing in crop, where tillge occurs in the fll. y following the pths seeds tke over their life, one rrives t the trnsition vlues. For exmple, the contriution of this yer s seeds in lyer 1 to next yer s popultion in lyer 1, i.e., or, cn e roken down s follows: t time t certin frction 1 emerge from the top lyer, of which frction plnts survive, with ech surviving plnt producing seeds. The newly produced seeds re dded to the frction of seeds tht did not emerge nd tht survived, (1 1 )(1 1 ), where 1 is the frction of seeds in the top lyer tht died or were lost. The seeds FIG. 2. Weed life history used in defining the trnsition rules. See Methods: Life history underlying trnsitions nd Tle 1 for explntion of symols. The life cycle egins in the spring of one yer nd ends in the spring of the following yer.

4 28 SHN K. MERTENS ET L. Ecologicl pplictions Vol. 12, No. 4 seline ( 1 1) prmeter vlues sed on Polygonum persicri growing in crrots nd spring whet under non-hericide weed-mngement regimes in the Netherlnds. TLE 1. Prmeter Description Crop Crop i seedling emergence from lyer 1 seedling emergence from lyer 2 seed mortlity lyer 1 seed mortlity lyer 2 frction of seedlings surviving weed control seed production per surviving plnt survivl over winter frction of seeds remining in lyer 1 frction of seeds moving from lyer 2 to lyer 1 frction of seeds moving from lyer 1 to lyer 2 frction of seeds remining in lyer re then moved etween lyers y plowing in the fll, with frction remining in lyer 1, i.e., moving from lyer 1 ck to lyer 1, nd then frction 1 survive over the winter to time t 1. In short, the trnsition rule for remining in lyer 1 of crop is 1 1 (1 1 )(1 1 ) 1. The other trnsitions cn e clculted in similr fshion, giving for crop : (1 )(1 ) (1 )(1 ) (1 )(1 2 ) 2 2. (5) Ech trnsition consists of two terms, the first one representing reproduction, nd the second representing survivl. The suscripts on the prmeters for seed movement ( ij ) indicte trnsfer of seeds from lyer j to lyer i. ll prmeter vlues cn depend on the crop nd cn therefore e different for crops nd. Prmeter vlues used. For the purposes of illustrtion we hve used prmeter vlues tht resemle those for Polygonum persicri, growing in crops similr to crrots (crop ) nd spring whet (crop ), under non-hericide weed-mngement regimes in the Netherlnds (Tle 1). The life cycle egins 1 pril nd continues until 31 Mrch the following yer. The depths of the soil lyers re 0 5 cm for lyer 1 nd 5 20 cm for lyer 2. Vleeshouwers (1997) hs shown tht P. persicri does not usully emerge from depths 5 cm. The prmeters for seed movement re sed on plowing mtrix experimentlly mesured y Cousens nd Moss (1990). The vlues of the prmeters for emergence nd mortlity re derived from experiments y Vleeshouwers (1997) on the emergence nd fte of P. persicri seeds in reltion to the timing of disturnce (e.g., seeded preprtion, shllow cultivtion). The timings of disturnces used y Vleeshouwers (1997) re similr to those tht would occur for seeded preprtion of crrots (crop ) nd spring whet (crop ). Experiments y Roerts nd Neilson (1980) indicte tht seed ge does not hve n importnt effect on the proility of emergence of P. persicri seedlings. Under nonhericide weed-mngement regime, the frction of seedlings surviving control () is much lower for crrots thn for spring whet, ecuse of the incresed efficcy of mechnicl nd hnd control in crrot crops. The numer of seeds produced per surviving weed () is much higher in crrots thn in whet ecuse crrot crop is less competitive thn whet crop. s there is no informtion concerning winter survivl of P. persicri seeds these prmeters ( i ) hve een set to 1. Using the prmeter vlues for crops nd in the ove setting (the seline ( 1 1) prmeter set) nd the equtions for the trnsition elements (Eqs. 5), we otin the following mtrices: (6) In the nlysis of different rottions it is useful to pprecite tht in the top lyer of crop, efore plowing, out 0.2 seeds re produced per seed ( 21 ), while in the top lyer of crop, lso efore plowing, out 9 seeds re produced per seed ( 21 ). In the ottom lyer of oth crops, 0.8 seeds per seed re produced efore plowing ( 12, 12 ). We lso investigted the effect of chnging the vlues for winter survivl in the top lyer ( 1 ) nd for seed movement ( ij ). These two prmeters were chosen ecuse there is gret uncertinty in the vlues for survivl over winter, nd ecuse in mny griculturl regions minimum-tillge cropping systems re used. In such systems plowing is crried out less frequently or plow types re used tht do not invert the soil. Not plowing is expected to decrese seed survivl over winter ecuse crop residues left on the soil surfce my, for exmple, increse microil ctivity or popultions of seed predtors. The vlues for these seven dditionl scenrios re given in Tle 2. Throughout this pper we use the terms plowing nd tillge interchngely. We lel the scenrios, for exmple, s no-till ( 1 0.5), to indicte the scenrio where plowing is not crried out t the end of the phses so tht most

5 ugust 2002 WEED DYNMICS IN CROP ROTTIONS 29 TLE 2. Prmeter vlues used for ech crop in ech scenrio. Prmeter seline No-till Scenrio 1 1 No-till No-till nd Notes: For convenience we repet the relevnt seline ( 1 1) prmeter vlues, nd for the other scenrios we show only differences with respect to the seline ( 1 1) scenrio. The nme of ech scenrio indictes in which crop tillge (plowing) is not crried out nd the level of winter survivl in the top soil lyer. See Tle 1 for description of prmeters. seeds remin in ech lyer, nd where only hlf of the seeds in the top lyer of the phses survive over the winter. We exmine the popultion growth rtes nd elsticities for ll essentilly different rottions up to six yers long. Model nlysis Growth rte. When exmining different crop rottions, question of primry importnce is how fst weed popultion grows in ech crop rottion. The growth rte over complete rottion cycle is given y the dominnt eigenvlue () of the mtrix product M (Luenerger 1979, Cswell 1989, Cswell nd Trevisn 1994). For 1 the popultion will eventully increse geometriclly, for 1 the popultion will eventully decrese geometriclly. In order to compre rottions of different lengths, we need men growth rte per yer. This is given y the geometric men of the cycle growth rte, ˆ (1/p) (7) where p is gin the length of the rottion cycle. Stle depth distriution nd reproductive vlue. Insight into differences in the growth rtes nd their sensitivities to chnges is gined through exmining the stle depth distriution nd reproductive vlue. The stle depth distriution is the distriution of seeds over the vrious soil lyers, pproched over the long term. The reproductive vlue is mesure of the contriution of seeds in given lyer to future popultion growth, nd is lso pproched over the long term. For n nnul weed, when emergence nd reproduction does not depend on seed ge, the reproductive vlue will depend on seed s proility of surviving until le to reproduce, nd the mount of future reproduction (cf. Cswell 1989). In periodic system the stle depth distriution nd reproductive vlues re cycliclly stle. When exmined from one projection period to the next these quntities do not chnge. They will, however, differ from one phse to the next. Mthemticlly, the stle depth distriution nd reproductive vlue correspond, respectively, to the right (w ) nd left (v ) eigenvectors ssocited with the dominnt eigenvlue () ofm. Usully the right eigenvector is normlized so tht the elements sum to 1, while elements in the left eigenvector re divided y the vlue of the first element so tht the reproductive vlues of lower lyers re reltive to tht in the top lyer. The eigenvectors re defined s T M w w v M v (8) where T indictes the trnspose nd h indictes phse in the rottion nd is in {1,2,... p} nd p is the length of the rottion. The phse index on the eigenvectors indictes the rottion mtrix with which they re ssocited. The eigenvectors, though, re chieved t the end of rottion strting with phse h. We therefore disply the eigenvectors with the finl phse with which they were chieved rther thn with the strting phse of the rottion from which they were clculted. Thus in rottion, the stle depth distriution fter phse (3) is the right eigenvector w (4), ssocited with the mtrix M (4) (3) (2) (1) (4). The left eigenvector v (4) indictes the reproductive vlue of seeds in ech lyer t the end (3) nd gives the contriution of seeds to future genertions strting with phse (4). Elsticity nlysis. The response of to perturtions in the trnsition elements nd the underlying prmeters is likely to depend on the composition of the crop rottion. These responses re usully represented s either sensitivities or elsticities. Sensitivities give the solute chnge in in response to n solute chnge in trnsition element or underlying prmeter. Elsticities give the proportionl chnge in in response to proportionl chnge in trnsition elements or underlying prmeter (de Kroon et l. 1986). We focus on elsticities, s in weed-mngement context it is more typicl to consider proportionl rther thn solute chnges in prmeters. See Cswell (2001) for further discussion of differences etween sensitivities nd elsticities. The clcultion of elsticities is sed on tht for sensitivities. n elsticity is the product of the rtio of trnsition element or prmeter to the growth rte nd of its sensitivity. Cswell nd Trevisn (1994) pro-

6 30 SHN K. MERTENS ET L. Ecologicl pplictions Vol. 12, No. 4 vide n eqution for clculting the sensitivity mtrices for the phses in periodic mtrix model: (h1) (h2) (1) (p) (p1) (h1) T SC [C C... C C C... C ] S M. (9) S C is the mtrix of sensitivities, with elements / c, of either crop or in phse h. C ij is either mtrix or, nd c ij is n element corresponding to one of these two mtrices in phse h. The first term on the right-hnd side is otined s follows: cycliclly permute the crop mtrices of the rottion so tht the hth crop mtrix (C ) occurs first, then exclude this mtrix, nd trnspose the resulting product. The second term, S M, is the sensitivity mtrix of the product mtrix M, rotted to phse h. The elements, / mij, of SM re clculted using the right (w ) nd left (v ) eigenvectors: m ij v w i j (10) m w v ij where re the elements of M, nd w v is the inner product of the right nd left eigenvectors of M (Cswell 1989). The elsticities ssocited with the trnsitions in ech phse of crop rottion cn then e clculted s c ij eij () c ij where / cij re the elements of SC, given in Eq. 9. The elsticities in ech phse sum to 1, thus the element contining the lrgest elsticity must lwys hve n elsticity The elsticities to the trnsition elements re useful guide in pointing out, in generl wy, how the growth rte will respond to chnges. However, chnges in the trnsition elements cn only e mde y djusting the prmeter vlues. Therefore it is of prcticl interest to exmine the elsticities to the prmeters, for ech phse of the rottion. The generl expression for clculting the elsticity to the underlying prmeters is otined y pplying the chin rule for differentition, giving x x gij (12) x g x for some prmeter x in mtrix G with elements g ij (Cswell 1989). In periodic system, s with the trnsition elements, one must consider the phse in which the prmeters occur, so tht the elsticity to prmeter in the hth phse is ij ij x x c ij. (13) x c x gin, / c re given y elements of ij SC (Eq. 9). Simultion nd yerly growth rtes. With the models considered here, most of the chrcteristics of ij ij weed s popultion dynmics cn e otined through direct nlysis of the mtrices. The iterted solution to the model, however, ids understnding through the possiilities of grphicl representtion nd clcultion of yerly growth rtes given n initil popultion. Ech simultion strted with 10 seeds per lyer. The yerly growth rtes were clculted y dividing the popultion size t time t 1 y the popultion size t time t, once the rottion growth rte nd stle depth distriution were 99.99% of the nlyticlly clculted quntities. RESULTS Hving the sic crop mtrices nd mens of nlyzing periodic mtrix models, we cn now systemticlly exmine vriety of crop rottions. First we present results of the effects of different crop rottions on men weed popultion growth rtes per yer ( ˆ ) nd of effects on the elsticity of the growth rte to trnsition elements nd prmeters. Then, using the stle depth distriutions nd reproductive vlues, we give iologicl explntion for the differences in growth rtes nd elsticities, etween rottions nd etween the scenrios indicted in Tle 2. Effects of crop rottion on growth rte We first exmine, for ll scenrios, generl ptterns in the reltionship etween ˆ nd the proportion of ech crop. Susequently we consider the effect of crop order nd incresing rottion length. Generl ptterns in ˆ. For ech scenrio, ˆ decreses s the frction of crop increses (Fig. 3 d). The men nnul popultion growth rte of the seline ( 1 1) monoculture is out 1.79 nd tht of the seline monoculture is out The form of the decrese vries etween scenrios. For exmple, scenrios tht do not include plowing t the end of the yers result in more concve pttern in the reltionship etween ˆ nd the frction of crop compred to the seline nd no-till scenrios. In the no-till ( 1 0.5) scenrio, rottions with low frction of hve ˆ lower thn tht of monoculture of (Fig. 3). In the scenrios where plowing is not crried out in the yers, the ˆ of the monoculture is much higher thn the seline sitution. dding low proportion of crop, though, cuses lrge decrese in ˆ (Fig. 3c d). Decresing survivl over winter in the top lyer tends to decrese ˆ, ut the effect is less in rottions with high proportion of crop. Effects of crop order. For ll scenrios except where plowing is not crried out in oth crops, the ˆ for given frction of crop cn differ sustntilly depending on crop order (Tle 3). For exmple, in the seline ( 1 1) scenrio, rottion hs ˆ of 1.52, while rottion hs ˆ of For certin scenrios nd frctions of crop, crop order cn men the difference etween n incresing or decresing popultion. For exmple in the no-till ( 1 0.5) scenrio, rottion hs ˆ of while ro-

7 ugust 2002 WEED DYNMICS IN CROP ROTTIONS 31 FIG. 3. Men nnul popultion growth rtes ( ˆ ) for ll four scenrios nd ll essentilly different rottions of up to six yers of crops nd. Solid circles indicte scenrios with seed survivl over winter 1 1; open tringles indicte scenrios with () 1 0.8, nd () (d) In the no-till scenrios for prticulr crop, the vlues for the prmeters governing seed movement re such tht very little movement of seeds etween lyers occurs. See Tles 1 nd 2, nd Methods: Model construction: Prmeter vlues used for complete explntion of the prmeter vlues used in the scenrios. ttion hs ˆ of In the seline scenrios, ptterns in which ech crop is in consecutive lock hve lower ˆ thn ptterns tht hve lternting yers of crops nd. In the other scenrios, ptterns with ech crop in consecutive lock hve the highest growth rte for given rottion length nd frction of ech crop. Effects of rottion length. When rottions increse in length following the pttern,,, nd so on, the trend in ˆ tends to decrese in the seline scenrios (Fig. 4). In the no-till scenrios, the ˆ ppers to level off t 1.82, which is close to the ˆ for monoculture of in the seline scenrio (Fig. 4). In scenrios where tillge does not occur in the yers, the ˆ increses nd there is lrge effect of decresed survivl over winter (Fig. 4c d). Effects of crop rottion on elsticity of growth rte We first exmine elsticities of ˆ to trnsitions nd underlying prmeters for the seline ( 1 1) monocultures nd rottions nd. We then exmine ptterns in the highest elsticities found for ech rottion of ll eight scenrios. Elsticities to seline trnsitions per phse. In the monoculture, the highest elsticity vlue is to trnsition, which is the trnsition relted to remining TLE 3. Effect of the order of crops ( nd ) on ˆ, the men weed popultion growth rte per yer. Rottion Frction crop seline No-till No-till No-till nd

8 32 SHN K. MERTENS ET L. Ecologicl pplictions Vol. 12, No. 4 FIG. 4. Effect of incresing crop rottion length on men nnul popultion growth rte, ˆ, where equl mounts of ech crop re used nd the rottions use the simplest pttern, giving the sequence (one yer of ech crop), (two yers of ech crop),, nd so on. Drk gry rs indicte scenrios with 1 1; open rs indicte scenrios with () 1 0.8, nd () (d) In the no-till scenrios for prticulr crop, the vlues for the prmeters governing seed movement re such tht very little movement of seeds etween lyers occurs. See Tles 1 nd 2, nd Methods: Model construction: Prmeter vlues used for complete explntion of the scenrios. in the ottom lyer (Fig. 5). With vlue of 0.8, it is much higher thn the elsticities to the other trnsitions. The elsticity vlues found for the monoculture (Fig. 5) re more evenly spred, with trnsitions 12 nd 21 hving the highest vlue. Rottion (Fig. 5c) hs regulr pttern of elsticities, with the highest elsticity lternting etween the trn- (1),(3) (2),(4) sitions 12 nd 21, governing movement etween lyers. (see Methods: Model construction: Elsticity nlysis for explntion of symols.) The pttern of elsticities for rottion is more complicted (Fig. 5d). Overll the highest elsticity is for trnsition (1), tht of remining in the ottom lyer during the first crop. In the other phses, the trnsitions governing movement of seeds etween lyers nd of remining in the ottom lyer tend to hve the highest elsticity vlues. Elsticities of underlying prmeters. s with the elsticities for the trnsitions, the crop rottion influences the impct of chnges in prticulr prmeter on the growth rte (Tle 4). We gin focus on the seline ( 1 1) rottions nd, nd lso compre them with the monocultures of nd. In the next section we will investigte the resons for differences in elsticities. The highest elsticity vlues tend to e to the prmeters for survivl over winter ( i ), in either or oth of the soil lyers. In monocultures of nd, winter survivl in the ottom lyer is most importnt, ut in crop it hs much lrger elsticity thn in crop. In phse of rottion, survivl over winter in the top lyer is most importnt, while in phse winter survivl in the ottom lyer is most importnt, nd hs much lrger elsticity thn monoculture. For rottion, winter survivl in the ottom lyer hs higher elsticity in the first nd the lst phses thn in the middle two phses. The vlue of the prmeter for seedling survivl () is likely to depend hevily on weed-control methods nd is therefore the process over which frmer usully hs the most influence. It lso hs the sme elsticity s the prmeter for seed production () ecuse in this model is density independent nd multipliction with yields single prmeter seeds produced per emerged seedling. The elsticity of differs etween rottions nd etween phses in rottion. In monoculture of the elsticity of is low compred to the elsticities of the other prmeters, while in monoculture of, hs n elsticity similr to mny of the other prmeters. In rottion the elsticity of is lower in the phses compred to n monoculture. In the phses the elsticity of is lrger reltive to

9 ugust 2002 WEED DYNMICS IN CROP ROTTIONS 33 FIG. 5. Elsticities of trnsition elements in the seline ( 1 1) scenrio. The trnsitions for crop or of phse h re indicted s follows: c, lck; c, drk gry; c, light gry; c, open rs. c ij monoculture. In rottion the elsticities of during the phses re lso very low, while during the phse they re somewht lrger thn in monoculture ut not nerly s lrge s during the phse of rottion. The elsticities for do not necessrily remin the sme for the sme crop in different phses within rottion. For exmple in the seline ( 1 1) rottion, the elsticity of is much greter during phse (3) thn during ny other phse (Fig. 6). Elsticity ptterns in ll scenrios. n overview of the effect of chnging prmeter vlues cn e seen in grphs of the highest elsticity to trnsition per rottion ginst the frction of crop (Fig. 7 d). s it is not possile to include the rottion or phse in the grphs, we give these results in the ppendix. In the TLE 4. Elsticity of weed growth rte to seline ( 1 1) prmeter vlues. Prmeter Monoculture (1), (3) (2), (4) (1) (2) (3) (4) Notes: For rottion, only vlues for yers 1 nd 2 re given, s yers 3 nd 4 re the sme s yers 1 nd 2, respectively. See Tle 1 for explntion of prmeters.

10 34 SHN K. MERTENS ET L. Ecologicl pplictions Vol. 12, No. 4 FIG. 6. Elsticities of the prmeter for seedling survivl () in rottion (seline [ 1 1] scenrio). seline scenrios the vlues of the highest elsticities pper to increse with the frction of crop (Fig. 7). Crop order, however, cn cuse differences in the vlues s well s the trnsition nd phse in which they occur. For the seline rottions where consecutive crops occur, then the highest elsticities re to. When nd crops lternte, then the highest elsticities re to trnsitions 12 nd 21. In the no-till scenrios, the highest elsticity vlues re to trnsition for ll rottions (Fig. 7). s the frction of crop increses, then the elsticity vlues pproch 1. In the no-till scenrios the reltion is more complicted (Fig. 7c). There is some pttern: Rottions with mjority of crop tend to hve the highest elsticity to trnsition, while those with mjority of crop hve the highest elsticity to trnsition. In etween, the dominnt elsticity is to trnsition, ut other trnsitions my lso crry the highest elsticity. In the no-till in oth the nd scenrios, the pttern of trnsitions crrying the highest elsticity is similr to tht for the no-till scenrios ut the vlues tend to e close to 1 (Fig. 7d). Cuses of differences in growth rtes nd elsticities Using the seline ( 1 1) monocultures nd rottions nd, we elucidte the cuses of some of the differences descried in the preceding susections. ecuse we hve exmined smll set of the infinite numer of theoreticlly possile rottions, some points my pper s outliers in the figures of FIG. 7. Highest vlues of elsticities to trnsition elements for ll scenrios nd ll essentilly different rottions up to six yers long. For pnel () the weed seed survivl over winter 1 0.8, nd for () (d) In the no-till scenrios for prticulr crop, the vlues for the prmeters governing seed movement re such tht very little movement of seeds etween lyers occurs. See Tles 1 nd 2, nd Methods: Model construction: Prmeter vlues used for complete explntion of the scenrios. This figure is sed on the dt in the ppendix.

11 ugust 2002 WEED DYNMICS IN CROP ROTTIONS 35 seline ( 1 1) depth distriution of weed seeds (frction in ech lyer), seed reproductive vlues (reltive to vlue in lyer 1) t the end of the indicted phse, nd growth rtes per phse in oth lyers nd for the totl popultion. TLE 5. Seed depth distriution Crop rottion Phse Lyer 1 Lyer 2 Monoculture (1), (3) (2), (4) (1) (2) (3) (4) Reproductive vlue, lyer nnul popultion growth rte Lyer 1 Lyer 2 Totl the popultion growth rtes nd elsticities (Figs. 3 nd 7). Such points re not outliers in sttisticl sense ecuse vlues re due to the sme deterministic processes t work for ny other point. Therefore the resoning pplied elow to differences etween rottions nd cn e used to explin differences etween other rottions, s well s to differences etween scenrios. Depth distriutions nd seline ( 1 1) growth rtes. In oth monocultures, the ottom lyer hs higher proportion of seeds thn the top lyer (Tle 5), reflecting how plowing moves lmost ll seeds from the top lyer to the ottom lyer nd moves only out third of the seeds in the ottom lyer to the top lyer. The frction of seeds in the top lyer of is, however, out twice tht found in the top lyer of crop. In crop, due to high seedling survivl nd reproduction y seeds in the top lyer, greter proportion of the totl popultion ends up in the ottom lyer compred to crop. In the modeled crop rottions, the depth distriution of seeds chieved fter ech phse only pproches tht of the corresponding monoculture crop. This occurs ecuse the distriution resulting fter ech phse depends on the preceding distriution s well s on the trnsition vlues of the current phse (Tle 5). For exmple, when seline crop precedes seline crop, there will e greter proportion of seeds in the top lyer efore the phse compred to monoculture (Tle 5). Therefore more seeds will e produced during the phse compred with monoculture nd they will e on the top lyer efore plowing. fter plowing the proportion of seeds on the ottom lyer is greter thn in monoculture. This is ecuse efore plowing there ws greter proportion nd numer of seeds in the top lyer thn in monoculture. Plowing moved more seeds to the ottom lyer nd fewer seeds to the top lyer, leding to decline in the numer of seeds in the top lyer nd very lrge increse in the ottom lyer. The overll growth rte for phse following n phse is thus lrger thn the yerly growth rte in monoculture (Tle 5). Similr resoning cn e used to show tht, fter n phse, if the initil distriution of seeds is higher in the ottom lyer compred with n monoculture, then the proportion nd solute numer of seeds in the top lyer will increse, s will the totl numer of seeds over oth lyers. Such sitution occurs when the preceding phse is crop. The stle depth distriution for given phse in rottion cn e thought of s trnsient distriution when compred to monoculture sitution. For the seline rottions nd, the different vlues of ˆ cn e explined s follows. Compred with the phses of rottion, the phses of result in higher proportion of seeds in the top lyer nd thus lso greter popultion in the top lyer (Tle 5). These seeds cn then produce mny more seeds during the following phse. Similrly, the phses of result in higher proportion nd numer of seeds in the ottom lyer thn the phses of. Therefore, efore plowing in the following yer, fewer seeds re lost from the ottom lyer thn from the top lyer. Consequently during ech phse of, the popultion will grow fster thn in the sme crop in rottion, leding to higher growth rte over the entire rottion cycle. The lternting pttern of crops thus increses the popultion of the top lyer y the end of n yer, nd sets the popultion up for nother round of high reproduction in the next yer (Fig. 8). In contrst, t the end of phse (1) in rottion, while the proportion of seeds in the top lyer is similr to tht of the phses of, during the second phse very few seeds will e produced ecuse it is n phse. In fct, the popultion declines more thn in n monoculture (Tle 5). Phse (2) cts s rke, slowing the yerly growth rte in ech of the following phses compred to the corresponding crop growth rtes in rottion (Fig. 8). Reproductive vlues, depth distriutions, nd seline elsticities. Unlike the growth rtes, the elsticity of the growth rte to chnges in trnsitions depends on wht will hppen to seed in the future s well s the proility of which lyer it will end up in t the end of the previous crop. Elsticity vlues therefore depend

12 36 SHN K. MERTENS ET L. Ecologicl pplictions Vol. 12, No. 4 FIG. 8. Simulted popultion dynmics of seline ( 1 1) scenrio. roken lines nd open circles indicte the popultion in the top lyer, nd thick lines nd solid circles indicte the popultion in the ottom lyer. oth on the reproductive vlue nd the depth distriution of seeds. While resons for differences in the stle depth distriutions depend on previous distriutions nd trnsitions, differences in the reproductive vlues re due to differences in future environments. In n monoculture the reproductive vlue of the lower lyer is more thn 3.5 times tht of n individul in the top lyer, i.e., most contriutions to future genertions come from seeds in the ottom lyer (Tle 5). This is ecuse prior to plowing in crop, the seed popultion decreses less in the ottom lyer. On the other hnd, in monoculture, ecuse of high seed survivl nd reproduction in the top lyer efore plowing, the reproductive vlue of seed in the top lyer is 5 times tht of seed in the ottom lyer. In rottion, the different sequences of following crops will lter the reproductive vlues of ech phse. For exmple, fter phse of rottion, the reproductive vlue of the top lyer is lmost 15 times tht of the ottom lyer, which contrsts shrply with tht found in monoculture of (Tle 5). In the rottion, the seeds in the top lyer t the end of n phse cn produce mny new offspring during the following yer. Furthermore the newly produced seeds will e moved to the ottom lyer prior to the next phse where their chnces of survivl re higher thn in the top lyer. Such resoning cn e used to explin the different reproductive vlues in the phse nd in other rottions. Understnding the cuses of differences in reproductive vlues, we cn now explin the differences in the elsticities. The high elsticity found in trnsition 12 of rottion, for exmple, cn e ccounted for y the high reproductive vlue of seeds in the top lyer nd ecuse most of the seeds tht re in the top lyer cme from the ottom lyer. Likewise, the trnsition 21 hs high elsticity ecuse most of the seeds in the ottom lyer were in the top lyer efore plowing, nd once in the ottom lyer they hve high reproductive vlue. Therefore mngement prctices tht decrese the proportion of seeds in the top lyer t the end of n phse or the mount of reproduction during the phse re likely to hve the lrgest impct on the growth rte. This result concurs with tht of the elsticities of the prmeters, where, for exmple, winter survivl in the top lyer of nd in the ottom lyer of oth hve high elsticities. Through considering the stle depth distriution, nd reproductive nd prmeter vlues, the trnsition nd prmeter elsticities for other rottions cn similrly e explined. Explining differences etween scenrios. The different ptterns found etween scenrios in the growth rtes nd elsticities cn rodly e understood y considering the nlysis of the seline monocultures nd simple rottions presented in the lst section. s exmples we consider why rottions with lternting yers of nd hve lower growth rtes in the notill scenrios thn in the seline scenrios, why not plowing in oth crops leds to little effect of crop order nd why growth rtes of certin rottions in the no-till ( 1 0.5) scenrios cn e lower thn tht of the crop monoculture. Finlly we exmine resons for the effects of incresing rottion length on ˆ. Differences in elsticity ptterns etween nd within scenrios cn e explined y considering how seeds re distriuted over the soil lyers nd their cpcity for future reproduction. Such n exercise follows the resoning used erlier for the seline ( 1 1) nd rottions nd therefore we do not devote spce to it here. In rottions in which tillge is not crried out for either crops or, the ˆ re lower for those rottions in which crops nd lternte. In the no-till scenrios, this is ecuse, for n lternting pttern of crops nd, seeds produced in crop remin in the top lyer nd then re mostly removed during the following phse. In the no-till scenrios, seeds tht were uried t the end of the yer re not rought ck to the surfce efore the next yer nd so cnnot reproduce. The effect of consecutive locks of crops in the no-till scenrios is to either store seeds until they cn e rought ck to the surfce fter the first phse (no-till scenrios) or to cuse lrge uild up in the popultion during the yers tht is not offset

13 ugust 2002 WEED DYNMICS IN CROP ROTTIONS 37 y decreses during the phses (no-till scenrios). Considering the elsticity ptterns found in the nlysis of the seline ( 1 1) rottion (Fig. 5), this is not surprising result. When plowing is not crried out in oth crops, the lck of difference etween ˆ for rottions tht differ only in crop sequence is ecuse not plowing effectively decouples the dynmics of the two lyers. Thus two, nerly seprte popultions re creted nd therefore order of multipliction of mtrices nd crop order does not hve lrge effect. Structure, however, still does ply role s ˆ for ech rottion is not simple (geometric) verge of the monoculture growth rtes. The decoupling of the lyers lso cuses the highest elsticity of ech phse to e close to 1 ecuse the nnul growth rtes re dominted y single trnsition. Rottions with ˆ lower thn tht of the lowest monoculture my occur when introduction of nother crop with higher monoculture ˆ moves seeds to lyer where they will e removed fster thn they will e replced y the increse cused y the crop with higher monoculture ˆ. For the no-till, ( 1 0.5) scenrios, the mjority of seeds in monoculture of crop will e in the ottom lyer. The inclusion of single crop in rottions of 3 to 5 crops, permits plowing so tht mny seeds re moved to the top lyer where they re removed from the popultion. The low survivl over winter compenstes for the slight increse in popultion during the yer. The ptterns oserved in Fig. 4, for the effects of incresing rottion length, cn lso e explined y the sme resons for the differences in crop order. When rottions re extended to common rottion length, the shorter rottions just hve more repeting units. Conversely, longer rottions hve longer spns of ech crop in consecutive lock. In the seline scenrios, rottions with ech crop in lock hd the lowest ˆ for given numer nd frction of ech crop. For the other scenrios the opposite ws the cse. It is not cler, however, whether the ˆ will rech limit if rottion length were further extended. In the no-till scenrios, ˆ ppers to level off, ut it my lso e incresing very slowly. DISCUSSION Our gol hs een to show how different types of crop rottions ffect weed popultion dynmics. We hve used periodic mtrix model nd exmined vriety of crop rottions, nd the effects of crop order, rottion length, nd proportion of ech crop. Of prime importnce is the conclusion tht the order of crops will ffect the popultion growth rte. Our conclusion rests on the form of the model, i.e., tht the popultion is structured, nd tht life-cycle prmeters chnge with the crop eing grown. Only when life-history prmeter vlues led to non-structured popultion, does the order hve little effect on the popultion growth rte. The sequence of crops in rottion lso ffects the sensitivity of the growth rte to chnges, t oth the levels of the trnsition elements nd the underlying prmeters. The differences in growth rtes nd elsticities etween rottions, nd etween phses within rottions, re in generl due to how mixtures of crops, in comprison to monocultures of ech crop, lter oth the distriution of seeds over soil lyers nd the contriution of seed in prticulr lyer to future genertions. elow we consider extensions of our pproch, nd the implictions for weed mngement nd understnding of crop rottions. Extension of the method The pproch we hve tken in this pper cn e extended to include spects such s density dependence, effects of environmentl vrition on vitl rtes, nd incresing the numer of soil lyers. Doing so is not likely to chnge our qulittive conclusion nd comes t the expense of nlyticl trctility, therefore complicting the interprettion of results. For exmple, periodiclly fluctuting popultions due to density-dependent vitl rtes my oscure the effects of the crop rottion cycle. Our interest is not in forecsting wht the popultion will e, rther it is in projecting wht the popultion would e should the given conditions (prmeter vlues) remin constnt (Cswell 1989). Understnding popultion projections provides sound sis for developing more complex models, while in the empiricl ren it results in more precise hypotheses nd thus in etter experimentl designs. Division of the soil column into more lyers is likely to lower weed popultion growth rtes over rottion cycle ecuse seeds will reside much longer in lyers from where they cnnot emerge nd thus hve greter chnce of losing viility. In situtions where ccurte quntittive prediction of the future popultion is of interest, division of the soil into more lyers or use of n integrl projection model would most likely e necessry (Esterling et l. 2000). ecuse weed popultions cn e structured in mny other wys, such s y seed ge, size of rhizomes, or seed position reltive to ridge, our pproch cn e pplied to other situtions esides those where the seed popultion is structured y depth in the soil nd the seeds re moved y plowing. Implictions for mngement The outcomes of decisions concerning crop rottion sequences hve implictions for weed mngement, in oth the long nd short term. s different rottions cn hve different growth rtes, the mount of time to rech trget weed popultion will e different for ech rottion. If other, non-weed-mngement spects re equl, then the rottion with the lowest weed popul-