The role of the spatial topology in trophic metacommunities: species with reduced mobility and total population size

Jou nal o Theo e ical Biology 566 (2023) 111479 A ailable online 17 Ap il 2023 0022-5193/© 2023 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by- nc-nd/4.0/). Con en s lis s a ailable a ScienceDi ec Jou nal o Theo e ical Biology jou nal homepage: www.else ie .com/loca e/yj bi The ole o he spa ial opology in ophic me acommuni ies: Species wi h educed mobili y and o al popula ion size Al onso Ruiz-He e a Depa men o Ma hema ics, Uni e si y o O iedo, Spain ARTICLE INFO Keywo ds: Addi i e in luence Popula ion abundance Managemen guidelines Spa ial opology Me acommuni ies ABSTRACT A cen al ques ion in ecology is unde s anding he in luence o he spa ial opology on he dynamics o a me acommuni y. This is no an easy ask, as mos agmen ed ecosys ems ha e ophic in e ac ions in ol ing many species and pa ches. Recen a emp s o sol e his challenge ha e in oduced ce ain simpli ying assump ions o ocused on a limi ed se o examples. These simpli ica ions make he models ma hema ically ac able bu keep away om eal-wo ld p oblems. In his pape , we p o ide a no el me hodology o desc ibe he in luence o he spa ial opology on he o al popula ion size o he species when he dispe sal a es a e small. The main conclusion is ha he in luence o he spa ial opology is he esul o he in luence o each pa h in isola ion. He e, a pa h e e s o a pai wise connec ion be ween wo pa ches. Ou amewo k can be eadily used wi h any me acommuni y, and he e o e ep esen s a uni ica ion o biological insigh s. We also discuss se e al applica ions ega ding he cons uc ion o ecological co ido s. 1. In oduc ion Habi a agmen a ion is one o he majo d i e s o e es ial biodi e si y declines and, ye , he p ac ice is p edic ed o in ensi y o e he coming yea s (Fle che e al.,2018). P omo ing he mo emen o indi iduals among isola ed egions has s ongly eme ged as a possible solu ion (Ama aseka e,2008;Leibold e al.,2004;Resasco e al.,2017). F om an ecological poin o iew, he mo emen allows a species o colonize an unoccupied egion o imp o e he sea ch o esou ces. On he o he hand, i may also inc ease he mo ali y a e due o p eda ion, s a a ion, e c. Bon e e al. (2012). This double ole o he mo emen cons an ly a ises in many eal p oblems, e.g., he op imal loca ion o a ma ine p o ec ed a ea o he cons uc ion o ecological co ido s (Haddad e al.,2014;Rassweile e al.,2012;Resasco e al., 2017). Today, unde s anding he p ecise in luence o he mo emen and he spa ial opology on he beha io o he species is o c i ical impo ance. Al hough signi ican p og ess has been made in a b oad a ie y o si ua ions, many impo an ques ions emain unsol ed (G oss e al.,2020;Guzman e al.,2019). One o he ecu en esul s in spa ial ecology is ha he popula ion dynamics obse ed a he scale o local pa ches depends la gely on he dynamics in o he habi a s h ough he mo emen o indi iduals (Abdala-Robe s e al.,2019;G oss e al.,2020;Guzman e al.,2019; Leibold e al.,2004;Zhang e al.,2021,2022). Mos agmen ed ecosys- ems con ain ophic in e ac ions wi h a high numbe o species and pa ches (G oss e al.,2020;Guzman e al.,2019;Leibold e al.,2004), making he analysis o any model a challenging ask. Due o hese E-mail add ess: [email p o ec ed]. di icul ies, many biological insigh s and managemen guidelines ha e been de i ed om he me apopula ion heo y, i.e., single species in agmen ed landscapes (F anco and Ruiz-He e a,2015;Has ings and Bo s o d,2006;Ruiz-He e a,2018;Zhang e al.,2015). Howe e , ca e mus be aken when applying hem in eal si ua ions, especially i he popula ion abundances o he in e ac ing species a e highly a iable. Ac ually, he e is g owing e idence ha communi y-le el p ocesses and spa ial a iables de e mine many ecological pa e ns (G oss e al., 2020;Guzman e al.,2019;Leibold e al.,2004;Zhang e al.,2022). Fo ins ance, Baise e al. (2013) ound ha he species so ing and pa ch dynamics models accu a ely explain many ecological p ope ies o he aqua ic ood web in he lea es o he no he n pi che plan Sa acenia pu pu ea. Ano he expe imen (Zhang e al.,2017) wi h spa ially di using labo a o y popula ions o he he e o ophic budding yeas Saccha omyces ce e esiae limi ed by an essen ial nu ien e ealed, among many o he hings, ha in a sys em o i e pa ches, he o- al popula ion abundance was ound o be highe in a homogeneous han he e ogeneous en i onmen wi h di usion (wi h he same o al esou ce le el in bo h cases). In his expe imen , Zhang e al. conside ed a pa icula spa ial opology o spa ial a angemen o he pa ches and made he ans e o indi iduals manually. The main esul s by Zhang e al. (2017) we e a he unexpec ed because he e e se conclusion had been heo e ically and expe imen ally deduced in p e ious wo ks wi h me apopula ions (DeAngelis e al.,2016;F anco and Ruiz-He e a, 2015;Zhang e al.,2015). h ps://doi.o g/10.1016/j.j bi.2023.111479 Recei ed 26 Oc obe 2022; Recei ed in e ised o m 10 Feb ua y 2023; Accep ed 24 Ma ch 2023 Jou nal o Theo e ical Biology 566 (2023) 111479 2 A. Ruiz-He e a In his pape , we p opose a no el app oach o desc ibing he in luence o he spa ial opology on he popula ion abundance o a species in gene al me acommuni ies when he dispe sal a es a e small. To his ask, we will analyze a classical me acommuni y model, (see (2.1) in he nex sec ion). The e a e housand o pape s analyzing he same ques ion wi h he same ype o model and pa ame e s as we employ he e, Hayes and Ande son (2018), Zhang e al. (2017), Wang e al. (2021), Ruiz-He e a and To es (2020), Ama aseka e (2008), Sadyko and Fa nswo h (2021), Zhang e al. (2021) and Suzuki and Economo (2021). The main di icul y lies in he huge numbe o pa- ame e s and combina ions, (e.g. he e a e 2 097152 spa ial opologies o a me acommuni y made o i e nodes and wo species). Since such numbe s a e unmanageable, e en wi h compu e aid, he common p ac ice is o ocus on a educed numbe o opologies. This p ac ice could be highly p oblema ic. In pa icula , i does no allow ex apola e a gene al p ope y ega ding he in luence o he spa ial opology on he a e o a me acommuni y. The main con ibu ion o he pape is ha we deduce gene al p ope ies ha a e alid o any opology and me acommuni y. Fo example, one o he main messages is ha he in luence o he spa ial opology is he sum o he in luence o he pa hs in isola ion. In o he wo ds, we mus isualize he spa ial opologies as a collec ion o pa hs and analyze each o he pa hs in an independen manne . He e, a pa h e e s o a pai wise connec ion be ween wo pa ches. Ou esul s also iden i y ce ain pa ches and pa hs ha play a disp opo iona ely la ge ole in main aining he o al popula ion size o a a ge species (see Mouque e al.,2013 and he concep o keys one communi ies/pa ches). As s essed in he pape , he assump ion o small dispe sal a es is c ucial in ou analysis. 2. Ma e ial and me hods 2.1. Modeling amewo k and basic de ini ions We s udy he dynamical beha io o 𝑛species ha inhabi a ag- men ed ecosys em o 𝑚pa ches. A classical model o he me acommu- ni y is ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑥′ 1𝑖(𝑡) = 𝑥1𝑖(𝑡)(𝑟1𝑖−𝑟1𝑖 𝑥1𝑖(𝑡) 𝐾1𝑖 +∑𝑛 𝑗=1 𝑗≠1 𝑎1𝑗𝑖𝑥𝑗𝑖(𝑡))+∑𝑚 𝑗=1 𝑏1𝑖𝑗 𝑥1𝑗(𝑡) ⋮ 𝑥′ 𝑛𝑖(𝑡) = 𝑥𝑛𝑖(𝑡)(𝑟𝑛𝑖 −𝑟𝑛𝑖 𝑥𝑛𝑖(𝑡) 𝐾𝑛𝑖 +∑𝑛 𝑗=1 𝑗≠𝑛 𝑎𝑛𝑗𝑖𝑥𝑗𝑖(𝑡))+∑𝑚 𝑗=1 𝑏𝑛𝑖𝑗 𝑥𝑛𝑗 (𝑡) (2.1) o 𝑖= 1,…, 𝑚. In his model, 𝑥𝑙𝑖(𝑡)≥0 ep esen s he densi y o he 𝑙 h species in pa ch 𝑖and 𝑥′ 𝑙𝑖(𝑡)deno es he de i a i e wi h espec o 𝑡. The pa ame e s 𝑟𝑙𝑖 and 𝐾𝑙𝑖 a e s ic ly posi i e and deno e he maximum pe capi a a e o inc ease and ca ying capaci y o he 𝑙 h species in pa ch 𝑖, espec i ely. The di e en ypes o in e ac ion among he species in pa ch 𝑖a e de e mined by 𝑎𝑙𝑗𝑖. Fo example, i 𝑎𝑙𝑗𝑖 >0and 𝑎𝑗𝑙𝑖 <0, he e is a p eda o –p ey in e ac ion be ween he 𝑙 h species (p eda o ) and he 𝑗 h species (p ey) in pa ch 𝑖. Fo a simple o e iew, we can w i e he model o he o m 𝑋′(𝑡) = 𝑋(𝑡)𝐺(𝑋(𝑡)) + 𝐵𝑋(𝑡) wi h 𝑋(𝑡) he popula ion ec o , o ganized by pa ch, 𝐺(𝑋) he g ow h a e ma ix (a block diagonal ma ix), and 𝐵 he mo emen ma ix (block ma ix). In pa icula , he sub-ma ix 𝐵𝑙= (𝑏𝑙𝑖𝑗 )desc ibes he mo emen o he 𝑙 h species. We assume ha 𝐵𝑙=ℎ𝑙𝐶𝑙whe e ℎ𝑙≥0 is a scale pa ame e which de ines he magni ude o he mig a ion o he deg ee o mobili y o he 𝑙 h species and 𝐶𝑙= (𝑐𝑙𝑖𝑗 )is a ma ix which codi ies in he o -diagonal elemen s he spa ial opology o he spa ial a angemen o he pa ches o he 𝑙 h species. Fo 𝑐𝑙𝑖𝑗 , a s ic ly posi i e alue indica es ha he 𝑙 h species can mo e om pa ch 𝑗 o pa ch 𝑖, whe eas a ‘‘0’’ means ha he e is no such a pa h. We conside weigh ed ma ices, which include a e and equen dispe sal e en s. Fo ins ance, a la ge alue 𝑐𝑙𝑖𝑗 >0indica es ha he pa h is e y likely o be used by he indi iduals o he 𝑙 h species. The diagonal elemen s 𝑐𝑙𝑖𝑖 a e he nega i e sum o o -diagonal elemen s o columns 𝑖, e lec ing he o al amoun o emig a ion om pa ch 𝑖 o he 𝑙 h species. We obse e ha all pa ches a e isola ed o he 𝑙 h species when he ma ix 𝐶𝑙is iden ically ze o. In ou analysis, we do no impose ha all species ha e he same spa ial opology. In his manne , ou app oach ea s he spa ial opologies as a species-speci ic p ope y, ins ead o a communi y-le el ai . Sys em (2.1) is a classical model in spa ial ecology. The eade can consul (Le in,1974;Ruiz-He e a and To es,2020;Holland and Has - ings,2008;Hayes and Ande son,2018;G oss e al.,2020;Nishikawa and Mo e ,2010;Sadyko and Fa nswo h,2021;Zhang e al.,2017) and he e e ences he ein o expe imen al/ heo e ical wo ks in which (2.1) is he modeling amewo k. Fo he sake o simplici y, we ha e analyzed a me acommuni y wi h g ow h a es o he Lo ka–Vol e a ype in he main ex . Ne e heless, ou me hodology wo ks o gene al g ow h a es, (see Sec ion F in he SI). In he igu es p esen ed in his pape , we adop he ne wo k ap- p oach (A zy-Rand up and S one,2010;U ban and Kei ,2001) ha desc ibes landscapes as collec ions o habi a pa ches (nodes) linked by edges ep esen ed links be ween di e en pa ches. An a ow om node 𝑖 o node 𝑗 o he ep esen a ion o he spa ial opology o he 𝑙 h species indica es he abili y o he indi iduals o his species o dispe se om pa ch 𝑖 o pa ch 𝑗. 2.2. Me hodology The mo emen associa ed wi h demog aphic p ocesses (e.g. emi- g a ion, immig a ion, coloniza ion) ope a es on compa able o smalle imescales han local oodweb dynamics (Ama aseka e,2008). In o he wo ds, pa ame e s ℎ1,…, ℎ𝑛a e smalle han he in e ac ion pa ame- e s in (2.1). This is he ocus o he pape , dispe sal phenomena, i.e, mo emen om bi h si e o dis an ep oducing si es. On he o he hand, we es ic ou sel es o me acommuni ies wi h a unique equilib- ium (𝑝11,…, 𝑝𝑛1,…, 𝑝1𝑚,…, 𝑝𝑛𝑚)wi h 𝑝𝑙𝑖 >0 o all 𝑙, 𝑖 which is a global a ac o o all non-ze o solu ions o (2.1). F om his equilib ium, we hen de ine 𝑇1(ℎ1,…, ℎ𝑛),𝑇2(ℎ1,…, ℎ𝑛), ..., 𝑇𝑛(ℎ1,…, ℎ𝑛)as he o al numbe o indi iduals o species 1, species 2, ..., species 𝑛, ac oss all nodes in he s eady s a e solu ion, espec i ely (see De ini ion 1 in SI o he p ecise exp ession o hese unc ions). No ice ha we a e excluding local ex inc ions in he me acommuni y. In ou analysis, he pa ame e s associa ed wi h he local dynamics, i.e.,𝑟𝑙𝑖,𝐾𝑙𝑖,𝑎𝑙𝑗𝑖, a e always ixed. We make explici he dependence o he deg ees o mobili y o he species on 𝑇1(ℎ1,…, ℎ𝑛),…, 𝑇𝑛(ℎ1,…, ℎ𝑛)because hey a e he key pa ame e s o unde s and he in luence o he spa ial opologies. Ou me hodology consis s o explo ing he e ec o he spa ial opologies on 𝑇1(ℎ1,…, ℎ𝑛),𝑇2(ℎ1,…, ℎ𝑛), ..., 𝑇𝑛(ℎ1,…, ℎ𝑛)by a ying he ma ices 𝐶1,…, 𝐶𝑛. Speci ically, we ollow a pe u ba ion app oach, s a ing om no dispe sal a all species and in oducing a e y small amoun o dispe sal. The e a e wo main ing edien s in ou a gumen s: (M1) 𝑇1(0,…,0), ..., 𝑇𝑛(0,…,0) a e independen o he ma ices 𝐶1, ..., 𝐶𝑛. (M2) We ha e p o ed in SI (see P oposi ion 1) ha 𝜕𝑇𝑙 𝜕ℎ𝑠 (0,…,0) de- pends on 𝐶𝑠bu no on he o he ma ices. Mo eo e , i 𝐶𝑠= 0, ha is, all pa ches a e isola ed o he 𝑠 h species, hen 𝜕𝑇𝑙 𝜕ℎ𝑠 (0,…,0) = 0. F om an applied side, (M2) says ha he in luence o he mo emen o he 𝑠 h species on he o al popula ion size o he 𝑙 h species is inde- penden o he mo emen o he o he species. Using (M1) and (M2), we can analyze 𝑇𝑙(ℎ1,…, ℎ𝑛) o small alues o ℎ1,…, ℎ𝑛. Speci ically, i 𝜕𝑇𝑙 𝜕ℎ𝑠 (0,…,0) >0( esp. <0) o a ma ix 𝐶𝑠, he mo emen o he Jou nal o Theo e ical Biology 566 (2023) 111479 3 A. Ruiz-He e a indi iduals o he 𝑠 h species in he spa ial opology associa ed wi h 𝐶𝑠 inc eases ( esp. dec eases) he o al popula ion size o he 𝑙 h species. Mo eo e , we can compa e he in luence o wo di e en opologies. I 𝐶𝑠and  𝐶𝑠a e wo di e en ma ices and 𝜕𝑇𝑙 𝜕ℎ𝑠 (0,…,0) is g ea e o 𝐶𝑠 han o  𝐶𝑠, he alue o 𝑇𝑙(ℎ1,…, ℎ𝑛)is g ea e when he 𝑠 h species mo es in he i s spa ial opology han in he second opology. Fo small alues o ℎ1,…, ℎ𝑛, we ha e ha 𝑇𝑙(ℎ1,…, ℎ𝑛) ≈ 𝑇𝑙(0,…,0) + ℎ1 𝜕𝑇𝑙 𝜕ℎ1 (0,…,0) + ⋯+ℎ𝑛 𝜕𝑇𝑙 𝜕ℎ𝑛 (0,…,0).(2.2) Thus, he bene i s/damages o he mo emen o a pa icula species wi hin a spa ial opology could be magni ied/bu e ed by he mo e- men o he o he species. Mo eo e , we maximize ( esp. minimize) 𝑇𝑙(ℎ1,…, ℎ𝑛) o small alues o ℎ1, ..., ℎ𝑛 inding he opologies ha maximize ( esp. minimize) 𝜕𝑇𝑙 𝜕ℎ1 (0,…,0), ...., 𝜕𝑇𝑙 𝜕ℎ𝑛 (0,…,0). Obse e ha we can analyze he p esence o some seden a y species wi h ou ame- wo k. I , o example, he i s 𝑙0species a e mobile and he es a e seden a y in (2.1), he o al popula ion sizes a e 𝑇1(ℎ1,…, ℎ𝑙0,0,…,0), ..., 𝑇𝑛(ℎ1,…, ℎ𝑙0,0,…,0). 3. Resul s 3.1. A gene al p ope y: The addi i e in luence o he pa hs when he species ha e educed mobili y The pa ial de i a i es 𝜕𝑇𝑙 𝜕ℎ1 (0,…,0), ..., 𝜕𝑇𝑙 𝜕ℎ𝑛 (0,…,0) ha e e y com- plex exp essions bu hey can be always exp essed as (see P oposi ion 1 in SI) 𝜕𝑇𝑙 𝜕ℎ𝑠 (0,…,0) = 𝑚 ∑ 𝑖,𝑗=1 𝑗≠𝑖 𝑐𝑠𝑖𝑗 𝑥∗ 𝑠𝑗 (𝛥𝑙𝑠𝑖 −𝛥𝑙𝑠𝑗 )(3.3) o , 𝑙, 𝑠 = 1,…, 𝑛 whe e 𝛥𝑙𝑠𝑖 and 𝛥𝑙𝑠𝑗 a e quan i ies associa ed wi h he 𝑙 h and 𝑠 h species ha depend on he local dynamics in pa ch 𝑖 and 𝑗, espec i ely; and 𝑥∗ 𝑠𝑗 is he popula ion densi y o he 𝑠 h species in pa ch 𝑗a he equilib ium in he absence o mo emen , ( ollowing he no a ion in Sec ion 2,𝑥∗ 𝑠𝑗 =𝑝𝑠𝑗 (0,…,0)). The manne o w i ing 𝜕𝑇𝑙 𝜕ℎ𝑠 (0,…,0) in (3.3) is one o he main con ibu ions o his pape because i allows us o deduce wo gene al esul s. Fi s , since (3.3) is a weigh ed sum o he o -diagonal elemen s o he ma ix 𝐶𝑠= (𝑐𝑠𝑖𝑗 ), he in luence o he spa ial opology o he 𝑠 h species on he o al popula ion size o he 𝑙 h species is he esul o he in luence o he mo emen in each pa h in isola ion. Thus, he e a e no global e ec s o he spa ial opology i sel . No ice ha he in luence o a conc e e pa h is de e mined by he biological ea u es o he depa ing and a i ing pa ches. I is wo h no ing ha common concep s in spa ial ecology such as i educibili y o modula i y (see A zy-Rand up and S one,2010) a e no c ucial ea u es o unde s anding he ole o he spa ial opologies when he species ha e educed mobili y. Second, i he mo emen o indi iduals o he 𝑠 h species om pa ch 𝑖 o pa ch 𝑗inc eases ( esp. dec eases) he o al popula ion size o he 𝑙 h species, hen he mo emen om pa ch 𝑗 o pa ch 𝑖dec eases ( esp. inc eases) i . To jus i y his claim, we obse e ha he con ibu ion o he mo emen o he 𝑠 h species om pa ch 𝑗 o pa ch 𝑖in (3.3) is 𝑐𝑠𝑖𝑗 𝑥∗ 𝑠𝑗 (𝛥𝑠𝑙𝑖 −𝛥𝑠𝑙𝑗 ). Pa icula ly, since 𝑐𝑠𝑖𝑗 𝑥∗ 𝑠𝑗 ≥0, i 𝛥𝑙𝑠𝑖 −𝛥𝑙𝑠𝑗 <0, ( esp. >0) he mo emen o he 𝑠 h species in his pa h educes ( esp. inc eases) he o al popula ion size o he 𝑙 h species independen ly o 𝑐𝑠𝑖𝑗 . The analysis o he pa h om pa ch 𝑖 o pa ch 𝑗in ol es he sign o 𝛥𝑙𝑠𝑗 −𝛥𝑙𝑠𝑖 which is he opposi e o 𝛥𝑙𝑠𝑖 −𝛥𝑙𝑠𝑗 . To acili a e unde s anding, we s udy a me acommuni y made o h ee compe i o s wi h educed mobili y in a landscape o 10 nodes, (see Fig. 1, i s ow). Wi h he pa ame e s gi en in Fig. 2, we collec he di e en alues o 𝛥𝑙𝑠𝑖 in Table 1. The in luence o he spa ial opology o species 1 on i s o al popula ion size is he sum o he in luence o he 10 pa hs, namely he pa h om pa ch 1 o 2, om 2 o 3, om 3 o 4,..., and om 10 o 1. In his pa icula case, he pa h Table 1 Values o 𝛥𝑙𝑠𝑖. These pa ame e s de e mine he in luence o he di e en pa hs in he spa ial opologies o Fig. 1. See SI Sec ion B1. 𝑗 𝛥11𝑗𝛥12𝑗=𝛥13𝑗 𝑗= 1 1.211 −0.110 𝑗= 2 1.325 −0.172 𝑗= 3 1.448 −0.241 𝑗= 4 1.578 −0.315 𝑗= 5 1.719 −0.396 𝑗= 6 1.869 −0.484 𝑗= 7 2.032 −0.580 𝑗= 8 2.208 −0.685 𝑗= 9 2.4 −0.8 𝑗= 10 2.838 −1.064 om pa ch 10 o pa ch 1 dec eases i s o al popula ion size, and he es o he pa hs inc eases i . The e o e, i we emo e hese las pa hs in he spa ial opology (see Fig. 1 second ow), he o al popula ion size o species 1 dec eases (see Fig. 2 ed cu e). Gene ally speaking, he mo emen o indi iduals o a species owa ds pa ches o lowe biological quan i ies gene ally inc eases i s o al popula ion size. In he me acommuni y discussed in Fig. 1/Table 1, he pa h om pa ch 10 o pa ch 1 educes o al abundance because he indi iduals goes o a pa ch wi h lowe compe i ion. The analysis o he in luence o he spa ial opology o species 2 on he o al popula ion size o species 1 is analogous. Speci ically, we ha e o analyze he con ibu ion o he 10 pa hs in isola ion. In his case, he pa h om pa ch 10 o pa ch 1 has a nega i e con ibu ion on he o al popula ion size o he i s species, and he es o he pa hs ha e a posi i e con ibu ion. In an analogous manne , we can analyze he in luence o he hi d species on he o al popula ion size o he i s species. In summa y, he o al popula ion size o a species in a me acom- muni y depends on wo ac s: he pa ame e s associa ed wi h he local dynamics in all pa ches, and he spa ial opologies. When he species ha e a educed mobili y, he in luence o he spa ial opologies is he esul o he in luence o each pa h in isola ion. In u n, he analysis o he in luence o a conc e e pa h depends on he pa ame e s associa ed wi h he local dynamics o he a i ing and depa ing pa ches, ia he quan i ies 𝛥𝑠𝑖𝑗 o (3.3). 3.2. Examples o applying he model o di e en me acommuni ies The gene ali y o he modeling amewo k in (2.1) masks many phenomena ha depend on he ype o me acommuni y. To a oid his p oblem, we will apply ou esul s o simple me acommuni ies made o wo species in landscapes wi h a gene al numbe o pa ches, say 𝑚. The educed numbe o species in he me acommuni ies allows us o e-w i e model (2.1) and o mula (3.3) in a mo e iendly manne . 3.2.1. P eda o –p ey me acommuni ies Conside he equa ions ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥′ 𝑖(𝑡) = 𝑥𝑖(𝑡)(𝑟𝑖−𝑟𝑖 𝑥𝑖(𝑡) 𝐾𝑖 −𝛼𝑖𝑦𝑖(𝑡)) + ∑𝑚 𝑗=1 𝑐𝑖𝑗 ℎ1𝑥𝑗(𝑡) 𝑦′ 𝑖(𝑡) = 𝑦𝑖(𝑡)(𝑠𝑖−𝑠𝑖 𝑦𝑖(𝑡) 𝑄𝑖 +𝛽𝑖𝑥𝑖(𝑡)) + ∑𝑚 𝑗=1 𝑐𝑖𝑗 ℎ2𝑦𝑗(𝑡) (3.4) o 𝑖= 1,…, 𝑚 wi h 𝑥𝑖(𝑡)and 𝑦𝑖(𝑡) he popula ion densi ies o a p ey and a gene alis p eda o in pa ch 𝑖, espec i ely. In (3.4),𝑟𝑖is he maximum pe capi a a e o inc ease, 𝐾𝑖is he ca ying capaci y and 𝛼𝑖is he p eda ion a e o he p ey in pa ch 𝑖. The spa ial opology o he p ey is de e mined by he ma ix 𝐶= (𝑐𝑖𝑗 )and ℎ1≥0s ands o i s deg ee o mobili y. In an analogous manne o he p eda o , we de ine 𝑠𝑖,𝑄𝑖,𝛽𝑖, 𝐶and ℎ2. Model (3.4) has linea unc ional ela ionship and logis ic g ow h o he p eda o wi hou p ey. No ice ha he model i s in he modeling amewo k gi en in (2.1) bu i is no he mos common s uc u e o p eda o –p ey models. Al hough ou esul s a e applicable Jou nal o Theo e ical Biology 566 (2023) 111479 4 A. Ruiz-He e a Fig. 1. Spa ial opologies employed in Fig. 2. Fig. 2. Rep esen a ion o he o al popula ion size o he i s compe i o . Fixed pa ame e s 𝑟𝑙𝑖 =𝐾𝑙𝑖 = 1 and compe i ion a es 𝑎𝑙𝑗1= −0.1,𝑎𝑙𝑗2= −0.15,𝑎𝑙𝑗3= −0.2, 𝑎𝑙𝑗4= −0.25,𝑎𝑙𝑗5= −0.3,𝑎𝑙𝑗6= −0.35,𝑎𝑙𝑗7= −0.4,𝑎𝑙𝑗8= −0.45,𝑎𝑙𝑗9= −0.5and 𝑎𝑙𝑗10 = −0.6 o all 𝑖= 1,…,10;𝑙, 𝑗 = 1,2,3wi h 𝑙≠𝑗. Fo simplici y, we assume ha all compe i o s ha e he same deg ee o mobili y, i.e., ℎ1=ℎ2=ℎ3=ℎ. The en ies o he ma ices 𝐶𝑙in bo h me acommuni ies a e 𝑐𝑙𝑖𝑗 = 0.2i he ou e om pa ch 𝑗 o pa ch 𝑖exis s o he 𝑙 h species. The blue and ed cu es ep esen 𝑇1(ℎ, ℎ, ℎ)in he opologies o he i s and second ows in Fig. 1, espec i ely. See Table 1 o he alues o 𝛥1𝑗𝑖. o gene al models (see Sec ion F in SI), we always impose he p esence o a globally s able equilib ium. The dynamics in pa ch 𝑖in he absence o mo emen , i.e.,ℎ1=ℎ2= 0, is gi en by ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥′ 𝑖(𝑡) = 𝑥𝑖(𝑡)(𝑟𝑖−𝑟𝑖 𝑥𝑖(𝑡) 𝐾𝑖 −𝛼𝑖𝑦𝑖(𝑡)) 𝑦′ 𝑖(𝑡) = 𝑦𝑖(𝑡)(𝑠𝑖−𝑠𝑖 𝑦𝑖(𝑡) 𝑄𝑖 +𝛽𝑖𝑥𝑖(𝑡)). (3.5) In his case, he (local) coexis ence s a e is 𝑥∗ 𝑖=𝐾𝑖𝑠𝑖(𝑟𝑖−𝛼𝑖𝑄𝑖) 𝑟𝑖𝑠𝑖+𝛼𝑖𝛽𝑖𝐾𝑖𝑄𝑖 and 𝑦∗ 𝑖=(𝑠𝑖+𝛽𝑖𝐾𝑖)𝑄𝑖𝑟𝑖 𝑟𝑖𝑠𝑖+𝛼𝑖𝛽𝑖𝐾𝑖𝑄𝑖 . The p ey is excluded in (3.5) when 𝑟𝑖−𝛼𝑖𝑄𝑖≤0. Thus, o p e en he p esence o local ex inc ions o he p ey in he absence o mo emen , we suppose ha 𝑟𝑖−𝛼𝑖𝑄𝑖>0 o 𝑖= 1,…, 𝑚. We deno e by 𝑇1(ℎ1, ℎ2)and 𝑇2(ℎ1, ℎ2) he o al popula ion sizes o he p ey and he p eda o in (3.4), espec i ely. In he SI (see P oposi ion 2), we ha e ob ained ha 𝜕𝑇1 𝜕ℎ1 (0,0) = 𝑚 ∑ 𝑖,𝑗=1 𝑗≠𝑖 𝑐𝑖𝑗 𝑥∗ 𝑗(1 𝑟𝑖−𝛼𝑖𝑄𝑖 −1 𝑟𝑗−𝛼𝑗𝑄𝑗),(3.6) 𝜕𝑇1 𝜕ℎ2 (0,0) = 𝑚 ∑ 𝑖,𝑗=1 𝑖≠𝑗 𝑐𝑖𝑗 𝑦∗ 𝑗(−𝛼𝑖𝐾𝑖 𝑟𝑖(𝑠𝑖+𝛽𝑖𝐾𝑖)+𝛼𝑗𝐾𝑗 𝑟𝑗(𝑠𝑗+𝛽𝑗𝐾𝑗)),(3.7) Jou nal o Theo e ical Biology 566 (2023) 111479 5 A. Ruiz-He e a Fig. 3. Spa ial opologies o he p ey and p eda o ( i s columns) and ep esen a ion o he o al popula ion size o he p ey. We analyze model (3.4) wi h pa ame e s 𝑟𝑖=𝑠𝑖=𝑄𝑖=𝐾𝑖= 1,𝛼1= 0.1,𝛼2= 0.25,𝛼3= 0.9,𝛼4= 0.95,𝛽𝑖= 0.1 o 𝑖= 1,2,3,4. We assume ha bo h species ha e he same deg ee o mobili y, i.e., ℎ1=ℎ2=ℎ. The blue cu e ep esen s 𝑇1(ℎ, ℎ)in he ep esen ed opologies. No ice ha wi h hese pa ame e s, he ep esen ed opologies maximize he o e all popula ion size o he p ey. The ed cu e ep esen s 𝑇1(ℎ, ℎ)in he opologies ha minimize he o al popula ion size o he p ey. Speci ically, we change he sense o all he pa hs in he ep esen ed opologies. The en ies o he ma ices a e 𝑐𝑖𝑗 = 0.2o 𝑐𝑖𝑗 = 0.2i he ou e om he pa ch 𝑗 o he pa ch 𝑖exis s. 𝜕𝑇2 𝜕ℎ1 (0,0) = 𝑚 ∑ 𝑖,𝑗=1 𝑗≠𝑖 𝑐𝑖𝑗 𝑥∗ 𝑗(𝛽𝑖𝑄𝑖 𝑠𝑖(𝑟𝑖−𝛼𝑖𝑄𝑖)−𝛽𝑗𝑄𝑗 𝑠𝑗(𝑟𝑗−𝛼𝑗𝑄𝑗)),(3.8) and 𝜕𝑇2 𝜕ℎ2 (0,0) = 𝑚 ∑ 𝑖,𝑗=1 𝑖≠𝑗 𝑐𝑖𝑗 𝑦∗ 𝑗(1 𝑠𝑖+𝛽𝑖𝐾𝑖 −1 𝑠𝑗+𝛽𝑗𝐾𝑗),(3.9) wi h 𝑥∗ 𝑗, 𝑦∗ 𝑗 he densi y o popula ion o he p ey and he p eda o in pa ch 𝑗a he equilib ium in he absence o mo emen , espec i ely. When bo h species ha e educed mobili y, he mo emen o he p ey om pa ch 𝑗 o pa ch 𝑖inc eases i s own o al popula ion size (see Exp ession (3.6)), p o ided 𝑟𝑗−𝛼𝑗𝑄𝑗> 𝑟𝑖−𝛼𝑖𝑄𝑖,(3.10) (see Rema k 1 in SI). I he e e se inequali y is sa is ied, he mo e- men dec eases i . No e ha he p ey is close o being excluded in pa ch 𝑖when 𝑟𝑖−𝛼𝑖𝑄𝑖≈ 0. In ligh o (3.10), p omo ing he di usion o hose pa ches, i.e., sou ces close o becoming sinks, is highly ecommended o enhance he o al popula ion size o he p ey. Gene ally speaking, o mula (3.6) sugges s ha he mo emen o he p ey o ‘‘less quali y’’ pa ches inc eases i s own popula ion densi y. The manne o measu e he quali y o he pa ch 𝑖is ia he quan i y 𝑟𝑖−𝛼𝑖𝑄𝑖, ha depends on he in insic biological ea u es o he pa ch and p eda ion p essu e. Following he same logic wi h exp ession (3.7), he mo emen o he p eda o om pa ch 𝑗 o pa ch 𝑖con ibu es posi i ely o he o al popula ion size o he p ey, p o ided 𝛼𝑖𝐾𝑖 𝑟𝑖(𝑠𝑖+𝛽𝑖𝐾𝑖)<𝛼𝑗𝐾𝑗 𝑟𝑗(𝑠𝑗+𝛽𝑗𝐾𝑗). In ui i ely, 𝛼𝑖𝐾𝑖 𝑟𝑖(𝑠𝑖+𝛽𝑖𝐾𝑖)is a quan i y ha measu es he p eda o s’ damages on he p ey in pa ch 𝑖. The e o e, he ecommenda ion o enhance he o al popula ion size o he p ey is o p omo e he mo emen o he p eda o owa ds pa ches whe e hey p o oke less damages o he p ey. The analysis o he in luence o he spa ial opologies on he o al popula ion size o he p eda o is analogous using exp essions (3.8) and (3.9). Wi h he p e ious discussion, we ha e he p ecise desc ip ion o he ole o any pa h in he me acommuni y when he popula ions ha e a educed mobili y. The in luence o he spa ial opologies will be he sum o he in luences men ioned abo e. In pa icula , i we wan o desc ibe he opologies ha maximize he o al popula ion size o he p ey when hey ha e educed mobili y, we ha e o cons uc he opologies wi h only pa hs wi h posi i e con ibu ion o he o al popula ion size o he p ey. They a e always di ec ed g aphs (see Fig. 3) because i a pa h has a posi i e ( esp. nega i e) con ibu ion, he pa h in he opposi e sense, i.e., exchanging he depa ing and a i ing pa ches, has a nega i e ( esp. posi i e) con ibu ion. Thus, di ec ed mo emen s a e hose wi h he highes in luence o he o al popula ion size o he species in me acommuni ies. 3.2.2. A me acommuni y wi h a mobile compe i o and a seden a y com- pe i o Conside he sys em ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥′ 𝑖(𝑡) = 𝑥𝑖(𝑡)(𝑟𝑖−𝑟𝑖 𝑥𝑖(𝑡) 𝐾𝑖 −𝛼𝑖𝑦𝑖(𝑡)) + ∑𝑚 𝑗=1 𝑐𝑖𝑗 ℎ1𝑥𝑗(𝑡) 𝑦′ 𝑖(𝑡) = 𝑦𝑖(𝑡)(𝑠𝑖−𝑠𝑖 𝑦𝑖(𝑡) 𝑄𝑖 −𝛾𝑖𝑥𝑖(𝑡)) (3.11) o all 𝑖= 1,…, 𝑚. In his model, 𝑟𝑖, 𝑠𝑖and 𝐾𝑖, 𝑄𝑖a e he maximum pe capi a a es o inc ease and ca ying capaci ies o he compe i- o s in pa ch 𝑖, espec i ely. The pa ame e s 𝛼𝑖, 𝛾𝑖≥0 ep esen he compe i ion a es in pa ch 𝑖. In (3.11), we assume ha he second compe i o is seden a y, i.e.,ℎ2= 0. Analogously o he p eda o –p ey me acommuni y discussed abo e, we assume ha 𝑟𝑖−𝛼𝑖𝑄𝑖>0 𝑠𝑖−𝛾𝑖𝐾𝑖>0 o all 𝑖= 1,…, 𝑚 o exclude he p esence o local ex inc ions in he absence o mo emen . No ice ha he (local) coexis ence s a e is 𝑥∗ 𝑖=𝐾𝑖𝑠𝑖(𝑟𝑖−𝛼𝑖𝑄𝑖) 𝑟𝑖𝑠𝑖+𝛼𝑖𝛾𝑖𝐾𝑖𝑄𝑖 and 𝑦∗ 𝑖=(𝑠𝑖−𝛾𝑖𝐾𝑖)𝑄𝑖𝑟𝑖 𝑟𝑖𝑠𝑖+𝛼𝑖𝛾𝑖𝐾𝑖𝑄𝑖 . We deno e he o al popula ion sizes o he compe i o s by 𝑇1(ℎ1,0) and 𝑇2(ℎ1,0). In he SI (see Sec ion D), we ha e ob ained ha 𝜕𝑇1 𝜕ℎ1 (0,0) = 𝑚 ∑ 𝑖,𝑗=1 𝑗≠𝑖 𝑐𝑖𝑗 𝑥∗ 𝑗(1 𝑟𝑖−𝛼𝑖𝑄𝑖 −1 𝑟𝑗−𝛼𝑗𝑄𝑗),(3.12) 𝜕𝑇2 𝜕ℎ1 (0,0) = 𝑚 ∑ 𝑖,𝑗=1 𝑗≠𝑖 𝑐𝑖𝑗 𝑥∗ 𝑗(−𝛾𝑖𝑄𝑖 𝑠𝑖(𝑟𝑖−𝛼𝑖𝑄𝑖)+𝛾𝑗𝑄𝑗 𝑠𝑗(𝑟𝑗−𝛼𝑗𝑄𝑗)).(3.13) The analysis o he in luence o he spa ial opology on he o al popula ion size o he compe i o s when he mobili y deg ee is educed is simila o ha in model (3.6). We unde line he ecommenda ion o p omo ing he di usion o he mobile compe i o owa ds egions whe e i is close o being excluded o enhance i s own o al popula ion size. 4. P ac ical implica ions The p e ious esul s p o ide new guidelines o manage s who wish o maximize he o al popula ion size o a a ge species in me acommuni ies. We ake ad an age o he ollowing insigh s: (R1) The in luence o he mo emen o a species wi hin a spa ial opology on he o al popula ion size o a a ge species is he esul o he con ibu ion o each pa h in isola ion. (R2) The mo emen o indi idual o a species owa ds pa ches o lowe biological quali ies gene ally inc eases i s o al popula ion size. Jou nal o Theo e ical Biology 566 (2023) 111479 6 A. Ruiz-He e a Fig. 4. Spa ial opologies (le and cen e columns) and ep esen a ion o he o al popula ion size o he i s compe i o . We analyze a me acommuni y made o h ee compe i o s in a landscape o en pa ches wi h he same pa ame e s as hose in Fig. 1. The ed and black cu es ep esen 𝑇1(ℎ, ℎ, ℎ)in he opologies o he i s and second ows, espec i ely. In o he wo ds, we add he pa h om pa ch 1 o pa ch 10 in he spa ial opology o he i s compe i o . No ice ha he bene i o his pa h is almos he same as he bene i s o he pa hs om pa ches 1 o 2, 2 o 3 and so on, (see Fig. 2 blue cu e). 4.1. Which is he addi ional pa h ha maximizes he o al popula ion size o a a ge species? The answe o his ques ion depends on he abili y o he es o he species o use he addi ional pa h. I only he a ge species can use he co ido , manage s should cons uc a pa h om he ‘‘s onges sou ce’’ o he ‘‘weakes sou ce’’. As emphasized in Fig. 4, he in oduc ion o a single pa h in a spa ial opology could p oduce a no iceable inc emen o he o al popula ion size. When o he species can use he addi ional pa h as well, he esul o he in luence o he mo emen o he di e en species is ha d o p edic . Howe e , i he a i ing pa ch is close o becoming a sink o he a ge species, he bene i s o he mo emen o ha species no mally p e ail. 4.2. Consequences o he loss o dispe sal ou es Gi en ha some pa hs ha e nega i e in luence on he o al popu- la ion size o some species, he loss o pa hs in he spa ial opologies pe se does no necessa ily h ea en he biodi e si y. On he o he hand, he in luence o a pa h on he o al popula ion size o a species only de- pends on he biological ea u es o he a i ing and depa ing pa ches. Pa icula ly, he in luence o he loss o pa hs on he o al popula ion size o he species does no depend on he numbe o isola ed clus e o he esul ing landscapes. We illus a e his phenomenon in Fig. 5 wi h a p eda o –p ey me acommuni y o eigh pa ches in which pa ches 3 and 5 ha e he same biological ea u es. In his me acommuni y, he bidi ec ional pa h joining pa ches 3 and 8 and he one joining pa ches 5 and 8 ha e he same in luence because hey in ol e pa ches wi h iden ical biological ea u es. 5. Discussion This pape o e s a cohesi e amewo k o s udy he in luence o he spa ial opology on he o al popula ion size o a species in ophic me acommuni ies. This is a long-ou s anding ques ion wi h deep epe - cussions in conse a ion and managemen , (see he open ques ion (c) in G oss e al.,2020;Guzman e al.,2019;Zhang e al.,2021). We ha e analyzed a gene al model, mo ing beyond simple spa ial opologies o a educed numbe o pa ches. As s essed in Guzman e al. (2019), ophic me acommuni ies a e no well unde s ood and ou knowledge is s ill unde de elopmen . I is wo h no ing ha he bene i s/damages o he spa ial opologies o di usi e mo emen s can in ol e a conside able a ia ion o he o al popula ion size o a species, magni ying o dilu ing any managemen s a egy. Ou main con ibu ion was o p o ide use ul exp essions o he pa ial de i a i es o he o al popula ion size o a species wi h espec o hei deg ees o mobili y. The key di icul y o his ask comes om he high numbe o pa ame e s in ol ed in (2.1), obs uc ing e en he use o any symbolic ma hema ical algo i hm. 5.1. A uni ied pe spec i e o he ole o he spa ial opologies The mo emen o many species is associa ed wi h demog aphic p ocesses, Ama aseka e (2008,2003), Guzman e al. (2019) and S auss e al. (2019). When all species exhibi his ype o mo emen , ha is, low dispe sal a es, he in luence o he spa ial opologies on he o al popula ion size o a species is he sum o he in luence o each pa h in isola ion. In o he wo ds, we ha e o isualize he spa ial opologies as a collec ion o pa hs and analyze he con ibu ion o each o hem independen ly. The con ibu ion o a speci ic pa h on he o al popula- ion size o a species depends on he g ow h a es, ca ying capaci ies, and he in e ac ion among he species o he a i ing and depa ing pa ches. Gene ally speaking, he mo emen o indi iduals om high- quali y pa ches (i.e. low compe i ion) o low-quali y (high compe i ion) ones leads o he inc ease o i s own popula ion size. This mo emen is a manne o eleasing compe i i e p essu e so ha indi iduals ha would ha e los o compe i ion in he sou ce can s ill su i e by emig a ing o low-quali y pa ches (see c owding e ec s men ioned in Debinski and Hol ,2000). No ice ha compe i ion s eng h is composed o bo h he compe i ion a e and he abundance o he species. Thus, i is possible o ha e a case whe e mo ing o a pa ch wi h ewe indi iduals does elease indi iduals om compe i ion, e en i he compe i ion s eng h is highe . The mo emen o indi iduals om high quali y pa ches o low quali y ones also echoes he long-s anding explo a ions o escue e ec and sou ce–sink dynamics. Ou esul s indica e ha p omo ing escue e ec s o a species, i.e., he mo emen o indi iduals om Jou nal o Theo e ical Biology 566 (2023) 111479 7 A. Ruiz-He e a Fig. 5. Spa ial opologies and ep esen a ion o he o al popula ion size o he p ey in bo h opologies. We analyze a p eda o –p ey me acommuni y (model (3.4)) in a landscape o eigh pa ches. Fixed pa ame e s 𝑟𝑖=𝐾𝑖=𝑠𝑖=𝑄𝑖= 1 and 𝛽𝑖= 0.1.𝛼1=𝛼7= 0.15,𝛼2=𝛼6= 0.3,𝛼3=𝛼5= 0.9,𝛼8=𝛼4= 0.95. Fo simplici y, we assume ha bo h species ha e he same deg ee o mobili y, i.e., ℎ1=ℎ2=ℎ. The en ies o he ma ices 𝐶𝑙in bo h me acommuni ies a e 𝑐𝑙𝑖𝑗 = 0.2i he ou e om pa ch 𝑗 o pa ch 𝑖exis s, (we a e assuming he same spa ial opology o bo h species). Unde hese condi ions, 𝑇1(ℎ, ℎ, ℎ)coincides in bo h opologies. Since he biological ea u es o pa ches 3 and 5 a e he same, he in luence o he bidi ec ional pa h joining pa ches 3 and 8 is he same as ha o he bidi ec ional pa h joining pa ches 5 and 8. Bo h cu es coincide by his eason. No ice ha an indi idual has access o all pa ches in opology 1 bu no in opology 2. su ounding habi a s o a oid local ex inc ions, leads o a conside able inc emen o i s o e all popula ion size in me acommuni ies wi h low dispe sal a es. The analysis o he o al popula ion o a species has a long a- di ion in ecology, s a ing wi h he wo k by F eedman and Wal man (1977) o simple me apopula ions o wo pa ches, (see Ruiz-He e a and To es,2018;Zhang e al.,2020,2017 o ecen esul s in his di ec ion). A common conclusion o hese pape s is ha he o al pop- ula ion size o a species can exceed he sum o he ca ying capaci ies o he isola ed pa ches. Pu di e en ly, he mo emen o he species in a me acommuni y can ha e a posi i e in luence on a species. In compa ison wi h hose wo ks, ou main con ibu ion is o ex end he analysis o landscapes wi h complex spa ial opologies and any numbe o species. Recen ly, Zhang e al. (2017) analyzed he o al popula ion abundance o a mobile consume in a consume - esou ce sys em wi h i e pa ches. They hypo hesized ha a consume popula ion di using in a landscape wi h a he e ogeneously dis ibu ed inpu o exploi able enewed limi ing esou ce can each a g ea e o al biomass han a popula ion di using in a space wi h he same o al inpu o esou ces dis ibu ed homogeneously, (see Hypo hesis 3). They ejec ed heo e i- cally and expe imen ally his hypo hesis. Ne e heless, his conclusion should be aken wi h cau ion because he spa ial opology o he consume did no en e in he game. Acco ding o ou esul s, he spa ial opologies gene ally play a negligible in luence on he o al popula ion size o a species in homogeneous landscapes. Howe e , he spa ial opologies no mally play a ema kable ole in he e ogeneous landscapes. Unde he biological condi ions in Zhang e al. (2017), i seems ha hey conside ed a spa ial opology wi h a nega i e in luence on he consume . Howe e , his is no always ue. In ac , we ha e p o ided a spa ial opology in SI (see Fig. E1 in SI) wi h a posi i e in luence on he consume and o which he hypo hesis s a ed by Zhang e al. holds. In ag eemen wi h p e ious wo ks Haddad e al. (2017), a gene al insigh o his pape is ha he o al biomasses o he species chie ly depend on he spa ial opologies. Neglec ing hese a iables in ol e he lack o a key ac o in any me acommuni y. Mos pape s in spa- ial ecology ocus on a educed numbe o examples o he spa ial opologies. Some popula op ions a e he ully connec ed opology, whe e di ec dispe sal om one pa ch o ano he is possible, o he so- called E dös–Rényi andom g aphs (A zy-Rand up and S one,2010; G oss e al.,2020). We s ess ha hese op ions ne e con ain he mos bene icial pa h s uc u es o popula ion abundances, because hey in oduce symme ic (bidi ec ional) dispe sal pa hs. As men ioned in Sec ion 3, i he mo emen o a species in a pa h om pa ch 𝑖 o pa ch 𝑗inc eases ( esp. dec eases) i s own o al popula ion size o he o al popula ion size o o he species, he mo emen in he pa h om pa ch 𝑗 o pa ch 𝑖dec eases ( esp. inc eases) i . This ema k implies ha symme ic spa ial opologies no mally ha e less in luence on he o al popula ion sizes o he species han, o ins ance, he dend i ic o di ec ed g aphs p esen ed in i e ine me acommuni ies. This esul echoes ecen heo e ical and expe imen al indings, see Heino e al. (2015) and he e e ences he ein. F om a conse a ion pe spec i e, his pape has p o ided a s a egy o inc ease he o al popula ion size o Jou nal o Theo e ical Biology 566 (2023) 111479 8 A. Ruiz-He e a a a ge species a ying he spa ial opology. Howe e , in la ge, mul i- species communi ies, s uc u es ha con e highe o e all popula ion sizes o some species could cause educ ion in o he s. Thus, he al- e a ion o spa ial opologies should be employed o species speci ic objec i es. 5.2. An al e na i e in e p e a ion o some classical concep s in spa ial ecology The numbe o dispe sal connec ions, o connec i i y, and he e en- ness o hei dis ibu ion among pa ches a e usual measu es in spa ial ecology ha o e look c ucial in o ma ion o he spa ial opologies. The analysis o hese a iables alone can lead o appa en ly con adic o y esul s. Fo example, he e a e heo e ical and expe imen al esul s in which he e ogeneous dis ibu ions o pa hs in he spa ial opologies nega i ely/posi i ely impac he abundance o a popula ion (see G oss e al.,2020;Ba e and G oss,2016 and he e e ences he ein). Ou esul s sugges he e a e pa hs wi h posi i e in luence on he abundance o a species and pa hs wi h nega i e in luence. Mo eo e , he in luence o a pa h is de e mined by he biological ea u es o he a i ing and depa ing pa ches. Pa icula ly, inc easing he numbe o dispe sal connec ions damages a a ge species when mos in oduced pa hs ha e a nega i e in luence. On he o he hand, di e en species o a me a- communi y no mally ha e spa ial opologies based on hei dispe sal ai s (e.g. wind dispe sed s. bi d dispe sed seeds, walking s. lying species). The common assump ion o sha ing he same spa ial opology is o e simpli ying, specially when he e exis e uges o compe i ions wi hin he landscape o g ea di e ences among he mo emen abil- i ies o he popula ions. Fo example, he di ec ional wa e low and i s in luence on he mo emen o a popula ion de e mine i s spa ial opology in i e me acommuni ies. Assuming he same opology o all species would in ol e he in oduc ion o ic i ious pa hs o some species and he emo al o eal pa hs o o he ones. We s ess ha he opologies a e no mally e y di e en among he species in he mos bene icial si ua ion o a a ge species (see Fig. 3). Gene ally speaking, imposing he same opology o all species is a condi ion ha dilu es he in luence o he spa ial a iables on he o al popula ion size o he species. O iginally, a keys one species was de ined as a species wi h a c ucial ole in communi y s uc u e and/o ecosys em unc ioning, (see he classical wo k by Paine in he six ies on ocky in e idal communi- ies Paine,1966). The e a e many me acommuni ies in na u e in which some habi a s play a disp opo ional in luence o species ec ui men and species di e si y. Wi h hese p o o ypical examples, Mouque e al. (2013) ex end he concep o keys one species o communi ies and ecosys ems. Ou esul s sugges ha he concep o keys one commu- ni y can be in e p e ed a he le el o connec ions, no only habi a s. In e es ingly, hese keys one elemen s a e ound a e y dispe sal a es, i.e., when he communi ies a e a he isola ed. Fo example, in Fig. 3, he pa hs o he opology in ol es an inc emen o one hi d o i s o al abundance. 5.3. Limi a ions and u u e esea ch di ec ions This pape has o e ed a numbe o biological insigh s o gene al me acommuni ies. Ne e heless, ca e mus be aken when applying hem in eal si ua ions. We ha e imposed wo c ucial assump ions: he absence o local ex inc ions and small dispe sal a es. Local ex inc ions in me acommuni ies a e a he common in na u e (Ama aseka e,2008; F anco and Ruiz-He e a,2015;Leibold e al.,2004), bu hey we e neglec ed in his pape . The i s na u al ques ion will be o ex end ou analysis o me acommuni ies ha allow local ex inc ions. The analysis o highly mobile species equi es a di e en app oach and new phenomena eme ge. No e ha he op imal spa ial opology sugges ed in Sec ion 3is ne e ecommended o popula ions wi h a high deg ee o mobili y. I he lux o indi iduals is ela i ely high on a di ec ed g aph Fig. 6. Rep esen a ion o wo opologies in which he addi i e in luence o he pa hs is no alid o highly mobile species. like ha in Fig. 3, he whole popula ion ends o occupy a unique pa ch. Thus, he o e all popula ion size in ha opology will be smalle han in he opology made o isola ed nodes. We men ion ha he addi i e in luence o he pa hs men ioned in Sec ion 3is no alid o highly mobile species ei he . Fo example, he in oduc ion o a pa h om pa ches 1 and 3 in Fig. 6 does no ha e in luence on a highly mobile species. Biologically, he bene i o his new pa h is educed because he access o pa ch 1 o pa ch 3 passing h ough pa ch 2 is a he simple o a highly mobile indi idual. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing inan- cial in e es s o pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape . Acknowledgmen s I would like o hank he edi o and he anonymous e iewe s o hei eedback and commen s, which imp o ed he quali y o he manusc ip . 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