Appendix A. Stability of the logistic semi-discrete model.
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1 Ecological Archiv E89-7-A Elizava Pachpky, Rogr M. Nib, and William W. Murdoch. 8. Bwn dicr and coninuou: conumr-rourc dynamic wih ynchronizd rproducion. Ecology 89:8-88. Appndix A. Sabiliy of h logiic mi-dicr modl. In hi appndix w driv h local abiliy condiion and h priod on h conumrrourc boundary for h cald mi-dicr modl wih logiic pry and a yp funcional rpon. Th wihin-yar dynamic of h linar modl ar dcribd by df ( f f fc d dc c d db f d (A. wih iniial condiion f ( v, c( w, b (, (A. and bwn-yar dynamic ar dcribd by v + w ( b( + c( + f(. (A.3 Dicr map A bfor, w can wri c( w. (A.4 W can rwri h quaion for f( a df f f w f. (A.5 d
2 L y. Thi man ha f dy df, (A.6 d f d and o dy y( w, (A.7 d which ha a oluion Thrfor, y ( d +. (A.8 w / τ w / τ+ w / τ v f ( v Thi man ha w ( / τ τ w ( /. (A.9 + b( v w ( / d. v + τ τ w ( / (A. W can hu wri down h map rlaing { v+, w+ } o {, } w ( / + τ τ w ( / v + v w a v v h( v, w, (A. w v d w k v w ( / + + ( τ τ w ( / v +, w. (A. Equilibria
3 Thr ar hr poibl yp of quilibrium a in hi ym: (a boh conumr and rourc abn (, ; (b rourc a carrying capaciy and conumr abn (, ; and (c boh conumr and rourc prn (v, w. Our primary inr i in h hird of h which w rfr o a a poiiv quilibrium. Equaion (A. can b olvd o yild an xprion for v in rm of w: v w( / τ τ w( /. (A.3 Th quilirbrium w can b obaind by ubiuing hi xprion ino (A. and rarranging rm o g w( / v d+. (A.4 τ τ w( / v + Plugging in h xprion for v and wih a bi of algbra, w obain w( / w( / d (A.5 τ ( / w( / τ w τ τ w( / + τ τ w( / Thi xprion can b implifid o yild ( τ τ w( / w( / + τ τ w( / w( / d (A.6 Thi, in urn, can b xprd a ( log τ τ w( / + τ τ w( / w( /. (A.7 3
4 Thi implifi o ( aw( / log[ ], (A.8 and o w obain h quaion for w ( w ( which dcrib h dniy of h conumr a h bginning of h yar. (A.9 For fuur rfrnc, no ha hi impli ha w can wri w( / ( / (A. and w( / ( ( ( / (. (A. Alo, if w dfin ( ( g ( ( / (, (A. w can wri v a v ( / g(. (A.3 d Linar abiliy analyi To linariz h ym, l v% v v b h dviaion from h rourc quilibrium and w% w w b h dviaion from h conumr quilibrium. W can wri h h v h( v+ v%, w+ w% h( v, w + v% + + w%, (A.4 v w k k w k( v+ v%, w+ w% k( v, w + v% + + w% (A.5 v w whr all parial driva ar valuad a h quilibrium (v, w. 4
5 Th acobian of hi ym i h h dv dw. (A.6 k k dv dw Sabiliy of h zro quilibrium (, Th lmn of h acobian for h (, quilibrium ar,,, and. Th characriic quaion for hi ym i hu: λ ( + λ (A.7 wih oluion λ and λ. implying ha h (, quilibrium i unabl if >. Sabiliy of h rourc only quilibrium (, Th lmn of h acobian for hi quilibrium ar, (A.8 ( + (, (A.9, and (A.3 ( + Th characriic quaion i. (A.3 ( λ( ( + λ, (A.3 which ha oluion λ - and λ - (+. 5
6 Sinc - i alway <, h quilibrium i unabl if - (+ >. In h main x, w rfr o h xprion - (+ a λ, h maximum gomric growh facor of h conumr (quaion (8 in h main x. Sabiliy of h non-zro qulibirium (v, w For hi quilibrium, ( /. (A.33 Th xprion for can b wrin a v g ( ( ( δ g ( v d ( v ( d g( ( v d+ ( / + +. (A.34 L ( / σ. Noic (from quaion (A.3 ha a quilibrium g( ( / +. (A.35 v d Subiuing ino h prviou quaion, and implifying w obain g( ( σ v d v. (A.36 σ Subiuing h xprion for v from quaion (A.3, w obain σ σ ( σ ( ( ( g g d d g ( d. (A.37 Th xprion for can b wrin a g( g( τ g( w dv d. v + ( v + (A.38 6
7 Uing quaion (A.4, w can wri or g( g( τ w v d v ( v + ( g( g( τ w w v v ( v + d. (A.39 (A.4 Th xprion for can b wrin a g( g( T v d+ + v d ( g τ g( v d w τ τ w, (A.4 + v + or r( / ( g ( τ τ ( d τ ( / ( ( ( d g τ w d ( / ( / ( ( g ( τ ( g τ + +. (A.4 Subiuing xprion for w and for v from quaion (A.9 and (A.3, w obain ( / ( ( / vw (, (A.43 or r( / vw (. (A.44 Inring hi ino h xprion for w obain 7
8 ( ( d (. ( v + g( τ τ g( ( / v + ( (A.45 Th condiion for abiliy can b xprd in rm of quanii B and B, whr and B ( + (A.46 B (A.47 (Gurny and Nib 998. Th boundary whr h abl quilibrium wich o h ovrcompnaing cycl of priod i dfind by B+ B. (A.48 Th boundary whr h abl quilibrium wich o conumr-rourc cycl occur whn B. Th priod on h conumr-rourc boundary i dfind by whr π Priod co B ω ( ω (A.49. (A.5 Plugging in h xprion for B w obain g( τ τ g( ( ( ( / ( / v + ω + ( d ( ( v + (A.5 W ud Mahmaica o find h ovrcompnaion and conumr-rourc abiliy boundari. Th cod accompani hi papr a an Ecological Archiv upplmn. 8
9 LITERATURE CITED Gurny, W. S. C., and R. M. Nib Ecological Dynamic. Oxford Univriy Pr, Nw York, Nw York, USA. 9
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