Paper Trail
The Correlated Random Walk and the Rise of Movement Ecology
William F. Fagan
Department of Biology, University of Maryland, College Park, Maryland 20742 Justin M. Calabrese
Smithsonian Conservation Biology Institute, National Zoological Park, 1500 Remount Rd., Front Royal, Virginia 22630
Understanding why, how, and when animals move is essential to many areas of ecology and related
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dynamics, ecosystem engineering, and conservation biology all hinge upon knowing what critters are PRYLQJ IURP ZKHUH WR ZKHUH LQ D ODQGVFDSH LQFOXGLQJ LQIRUPDWLRQ RQ KRZ TXLFNO\ KRZ UHJXODUO\
DQGE\ZKDWURXWHWKH\WUDYHO7KHFRPSOH[LWLHVLQYROYHGLQVXFKSURFHVVHVKDYHVSDZQHGWUHPHQGRXV HIIRUWVLQERWK¿HOGUHVHDUFKZKHUHJRDOVLQFOXGHPHDVXULQJDQGFKDUDFWHUL]LQJVXFKPRYHPHQWVDQG WKHRUHWLFDOUHVHDUFKZKHUHJRDOVLQFOXGHH[SORULQJWKHQDWXUHDQGSRWHQWLDOFRQVHTXHQFHVRIPRYHPHQW Ecologists today routinely receive some training in both empirical and theoretical research, and in the UROHRIVWDWLVWLFDODQDO\VHVDQGPRGHO¿WWLQJDVDZD\RIOLQNLQJWKHWZRSHUVSHFWLYHV+RZHYHUWKDW KDVQRWDOZD\VEHHQWKHFDVH6SDWLDOTXHVWLRQVLQHFRORJ\ZHUHORQJDQDUHDZKHUHWKHJXOIEHWZHHQ theory and reality was particularly wide. In part, this was due to the additional mathematical challenges of spatial models, but it also due to the perhaps greater technological challenges of measuring and FRQWH[WXDOL]LQJDQLPDOPRYHPHQWV
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was built when the article “Analyzing insect movement as a correlated random walk” was published in Oecologia. This paper, which represented a collaboration between ecologist Peter Kareiva and mathematician Nanako Shigesada, is a milestone along the Paper Trail because it marks a critical link between the abstract world of ecological theory and the hands-on way in which ecologists actually collect data on individual animals. Even now, this paper, which has been cited almost 500 times, continues to DWWUDFWLQWHUHVWDVDNH\QH[XVOLQNLQJWKHUHDOPVRIWKHUXPSOHGVKLUWVDQGWKHPXGG\ERRWV.DUHLYDDQG 6KLJHVDGD¶VSDSHUKHOSHGWUDQVIRUPWKHTXDQWLWDWLYHVWXG\RIDQLPDOPRYHPHQWIURPDSXUHO\WKHRUHWLFDO venture into an integrative science, where theory and data are merged to generate new understanding.
&RPELQLQJ FOHDU SURVH DQG LQVWUXFWLYH HTXDWLRQV .DUHLYD DQG 6KLJHVDGD ZDV RQH RI WKH
¿UVWSDSHUVLQHFRORJ\WRSURYLGHDFRQFUHWHWUDFWDEOHOLQNDJHEHWZHHQVSDWLDOHFRORJLFDOPRGHOVDQG IHDWXUHV WKDW FRXOG EH UHDGLO\ REVHUYHG²DQG TXDQWL¿HG²E\ ¿HOG ELRORJLVWV .DUHLYD DQG 6KLJHVDGD (1983) introduced a generalized two-dimensional correlated random walk (CRW) model to ecology, and demonstrated how it could be parameterized by decomposing an individual animal’s movement path into a series of movement steps and turning angles. The CRW was a clear advance in spatial ecology because it dealt with an obvious discrepancy between previously used, simple (uncorrelated) random walks and HPSLULFDOUHDOLW\²QDPHO\WKDWPRYLQJDQLPDOVYHU\IUHTXHQWO\H[KLELWGLUHFWLRQDOSHUVLVWHQFH
204 Bulletin of the Ecological Society of America, 95(3)
Paper Trail
The key to their approach was to write the model in terms of the moments of the step length and turn DQJOHGLVWULEXWLRQVZKLFKLVLPSRUWDQWIRUWZRUHDVRQV)LUVWWKHVHPRPHQWVGRQRWUHTXLUHFRPSOH[
statistical methods to estimate, and can instead be calculated from directly from movement path data via simple paper-and-pencil formulas. Second, by focusing on the statistics of the step length and turn angle distributions instead of making particular distributional assumptions, Kareiva and Shigesada ensured that their model would apply to a wide range of ecological scenarios. The mean step length, PHDQVTXDUHGVWHSOHQJWKDQGPHDQFRVLQHRIWXUQDQJOHVDUHQRZVWDQGDUGVWDWLFVXVHGWRVXPPDUL]H movement paths and parameterize movement models.
.DUHLYDDQG6KLJHVDGDHVWDEOLVKHGWKHPHDQVTXDUHGGLVSODFHPHQW06'DOVRFDOOHGWKHQHW VTXDUHGGLVSODFHPHQWDVDVWDQGDUG\DUGVWLFNIRUMXGJLQJWKHDSSURSULDWHQHVVRIWKH&5:IRUSDUWLFXODU GDWDVHWV7KHFHQWHUSLHFHRIWKHLUSDSHUZDVDFORVHGIRUPH[SUHVVLRQIRUWKHH[SHFWHGYDOXHRIWKH MSD under the CRW model in terms of the moments of the step length and turn angle distributions.
&RPSDULQJWKHREVHUYHG06'WRWKDWSUHGLFWHGE\WKH¿WWHG&5:PRGHODOORZVXVHUVWRJDXJHKRZZHOO the CRW describes their data. The combination of biological plausibility, mathematical tractability, and a clear connection between model and data established the CRW as a simple, yet nontrivial null model against which real animal movements could be compared. The authors were clear that, while the CRW LVPRUHUHDOLVWLFWKDQDVLPSOHUDQGRPZDONLWLVVWLOODUDGLFDOVLPSOL¿FDWLRQRIUHDODQLPDOPRYHPHQW ,PSRUWDQWO\WKH\QRWHGWKHVSHFL¿FZD\VLQZKLFKUHDODQLPDOPRYHPHQWVGHYLDWHIURPWKH&5:PD\
UHYHDOLPSRUWDQWELRORJLFDOLQVLJKWVLQWRWKHXQGHUO\LQJPRYHPHQWSURFHVV)RUH[DPSOHDQREVHUYHG MSD that increases consistently faster than model predictions suggests more directed movement than FDQEHFDSWXUHGZLWKD&5::KHQFRXSOHGZLWKERRWVWUDSFRQ¿GHQFHLQWHUYDOVUHÀHFWLQJSDUDPHWHU uncertainty (Turchin 1998), Kareiva and Shigesada’s (1983) approach provides an unambiguous gauge RIWKHGHJUHHWRZKLFKDPRUHFRPSOLFDWHGDQGELRORJLFDOO\UHDOLVWLFPRYHPHQWPRGHOLVMXVWL¿HG,QRXU opinion, this is one of the most important and lasting contributions of their work.
%HFDXVH GLIIXVLRQ PRGHOV FDQ EH GHULYHG IURP WKH &5: YLD WKH GLIIXVLRQ DSSUR[LPDWLRQ WKH DSSURDFKLQLWLDWHGE\.DUHLYDDQG6KLJHVDGDKHOSHGWRSURYLGHDSDWKZD\EHWZHHQWKHTXDQWLWLHV
¿HOGHFRORJLVWV¶FDQREVHUYHPRYHPHQWSDWKVRILQGLYLGXDOVDQGWKHSRSXODWLRQOHYHOGLIIXVLRQUDWHV HPSOR\HGE\WKHRUHWLFLDQV7XUFKLQ)RUH[DPSOH7XUFKLQVKRZHGKRZKDELWDWVSHFL¿F individual movement behaviors could be estimated via CRW methods and then translated into the VSDWLDOGLVWULEXWLRQRIWKHSRSXODWLRQYLDDGLIIXVLRQDSSUR[LPDWLRQ2WKHUDUHDVRIVSDWLDOHFRORJ\KDYH VXEVHTXHQWO\HPXODWHGWKLVLQGLYLGXDOOHYHOWRSRSXODWLRQOHYHOXSVFDOLQJDSSURDFKLQFOXGLQJPRPHQW HTXDWLRQV IRU VSDWLDO SRSXODWLRQ G\QDPLFV VSDWLDOO\ H[SOLFLW PHWDSRSXODWLRQ PRGHOV DQG LQGLYLGXDO based spatial population models. Collectively, these frameworks have fundamentally changed the character of spatial ecology by forging links between observable, individual-level phenomenon and WKHLUSRSXODWLRQRUFRPPXQLW\OHYHOFRQVHTXHQFHV
The CRW approach has steadily grown and developed over the years to become the workhorse of PRGHUQPRYHPHQWHFRORJ\0DQ\H[FLWLQJDGYDQFHVLQVSDWLDODQDO\VHVRIDQLPDOPRYHPHQWFDQWUDFH DQLQWHOOHFWXDODQFHVWU\WR.DUHLYDDQG6KLJHVDGD)RUH[DPSOHFRPSRVLWHUDQGRPZDONPRGHOV DOORZPRYHPHQWEHKDYLRUWRYDU\LQVSDFHDQGWLPHDQGFDQKHOSFRQWH[WXDOL]HPRYHPHQWE\OLQNLQJ behavioral changes to environmental covariates (Benhamou 2014). Behavioral change point analyses (Gurarie et al. 2009) take a similar approach, but are based on a continuous-space analog of the discrete
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CRW. Mechanistic home range models (Moorcroft and Lewis 2013) are based on CRWs and allow home ranges of individuals (or groups) to arise naturally from realistic movement behavior and interactions EHWZHHQ WKH LQGLYLGXDO DQG LWV HQYLURQPHQW DQG FRQVSHFL¿FV7KHVH DSSURDFKHV H[WHQG .DUHLYD DQG 6KLJHVDGD¶VLGHDVLQLPSRUWDQWGLUHFWLRQVE\DOORZLQJPRYHPHQWEHKDYLRUWRGHSHQGRQFRQWH[W
While great strides have been made in building biological realism into CRW-based models, it is now FOHDUWKDWWKLVIUDPHZRUNLVQHDULQJLWVOLPLWV)RULQVWDQFHPDQ\DQLPDOVH[KLELWPRYHPHQWEHKDYLRUV that repeat at regular intervals (e.g., daily, seasonally). Animals may also use memory to navigate, or PD\ DYRLG UHFHQWO\ H[SORLWHG DUHDV ZKHQ IRUDJLQJ$OO RI WKHVH ELRORJLFDO UHDOLWLHV DQG PDQ\ RWKHUV violate the Markovian assumption under which Kareiva and Shigesada (1983) derived their results.
0HPRU\DYRLGDQFHUHSHWLWLRQDQGRWKHUELRORJLFDOFRPSOH[LWLHVLQWURGXFHORQJWHUPDXWRFRUUHODWLRQV into the movement paths of individuals, which the CRW and other Markovian movement models cannot DFFRPPRGDWH RU XWLOL]H$ QHZ IURQWLHU RI PRYHPHQW HFRORJ\ LV WR UHOD[ WKH ¿UVWRUGHU 0DUNRYLDQ assumption such that movement models can use the information contained in long-term autocorrelations to identify critical behaviors (Fleming et al. 2014a, b). Doing so will allow ecology to go beyond purely random movement, and to begin incorporating real biological mechanisms into movement models. It is remarkable to note that Kareiva and Shigesada saw this frontier on the horizon over 30 years ago. That it remains an open challenge for movement ecology is testament both to the enduring contributions of WKHLUSDSHUDQGWRWKHLQKHUHQWGLI¿FXOW\LQWDNLQJWKHQH[WPDMRUVWHSEH\RQGWKHLUSLRQHHULQJZRUN Literature cited
Benhamou, S. 2014. Of scales and stationarity in animal movements. Ecology Letters 17:261–272.
Fleming, C. H., J. M. Calabrese, T. Mueller, K. A. Olson, P. Leimgruber, and W. F. Fagan. 2014a.
)URP¿QHVFDOHIRUDJLQJWRKRPHUDQJHV$VHPLYDULDQFHDSSURDFKWRLGHQWLI\LQJPRYHPHQWPRGHV across spatiotemporal scales. American Naturalist. In press.
Fleming, C. H., J. M. Calabrese, T. Mueller, K. A. Olson, P. Leimgruber, and W. F. Fagan. 2014b.
1RQ0DUNRYLDQPD[LPXPOLNHOLKRRGHVWLPDWLRQRIDXWRFRUUHODWHGPRYHPHQWSURFHVVHV0HWKRGVLQ Ecology and Evolution. In press.
Gurarie, E., R. D. Andrews, and K. L. Laidre. 2009. A novel method for identifying behavioural chang-
es in animal movement data. Ecology Letters 12:395–408.
Kareiva, P. M., and N. Shigesada. 1983. Analyzing insect movement as a correlated random walk.
Oecologia 56:234–238.
Moorcroft, P. R., and M. A. Lewis. 2013.Mechanistic home range analysis.(MPB-43). Princeton Uni-
versity Press, Princeton, New Jersey, USA.
Turchin, P. 1991. Translating foraging movements in heterogeneous environments into the spatial dis-
tribution of foragers. Ecology 72:1253–1266.
Turchin, P. 1998. Quantitative analysis of movement: measuring and modeling population redistribu-
tion in animals and plants. Sinauer Associates, Sunderland, Massachusetts, USA.
206 Bulletin of the Ecological Society of America, 95(3)
