OIKO O15875 all. R. huin J. B. Diehl. and NisbetR. M. 2007. Food quality nutient limitation of seconday poduction and the stength of tophic cascades. Oikos 000: 000 000. Appendix 2. Equilibia tophic indices and stability of titophic model with dynamic stoichiomety of plants. In this second Appendix we detail a titophic model in which vaiation in nutient enichment and taits of plants and hebivoe can shape the plant s nutient content. The bitophic case has been consideed befoe by all (2004. It is woth noting that in this model thee ae no feely available nutients (i.e. R =0. Also we descibe nutients sequesteed in plants (QP which is the total amount of nutient pe unit habitat. This quantity is not epesented by its own equation but is simply the poduct of nutient content pe unit cabon (Q times amount of cabon pe unit habitat (P. equesteed nutient povides useful infomation about the bitophic and titophic systems since it can coespond to the hebivoe s nutient demands. ince most of the esults fo this model closely esemble those fom the model with static stoichiomety of the poduce (Appendix 1 we descibe esults moe biefly hee. Equilibia A hebivoe can fist invade and pesist with the plant when nutient supply ( exceeds the hebivoes ( s minimal sequesteed nutient equiement QP. The equiement is the smallest amount of nutients locked in plant (P tissues that can suppot hebivoes. Once exceeded a twospecies equilibium emeges: Q 2NL P 2NL = m ea =QP (B1a 2NL = ( QP (B1b Q 2NL = a 2NL (B1c P 2NL =QP Q 2NL (B1d whee both nutient content of plant and hebivoe biomass incease with enichment (Table B1. Plant biomass deceases with enichment until a cetain level: =QP + a q k 1+ s Q am afte which point the P equilibium switches to a cabonlimited egion whee now: P 2CL =( m+ ( ea (B3a (B2 2CL = 1 2 a + 4 q a P + 2CL a 2 (B3b
Q 2CL = 1 2 P q 2CL ap 2CL + 4k q Q + ap 2CL ap 2CL P 2CL 2 (B3c In this potion of the planthebivoe only equilibium plant biomass is fixed at the poduce s minimal cabon equiement but hebivoe biomass nutient quota and sequesteed nutient incease with (Table B1. Befoe the hebivoe in the planthebivoe only equilibium becomes cabonlimited it could also suppot a canivoe if nutient enichment exceeds: =QP + m C ( (B4 o the minimal nutient equiement of the hebivoe plus the nutients contained in the canivoes s own minimal equiement fo hebivoes m C. If satisfied a new equilibium emeges with a nutientlimited hebivoe: 3NL = m C Q 3NL = a 3NL (B5a (B5b C 3NL P 3NL = = 3NL + QP Q 3NL QP (B5c m ( QP +m ea + which equies that the atio of nutient uptake ate ( to hebivoe attack ate (a exceed the biomass of hebivoes ( 3NL o a > 3NL. At this equilibium hebivoe biomass emains fixed at the canivoe s minimal cabon equiement. Additionally nutient quota of plants emains fixed. Both canivoe and plant biomass incease with nutient enichment as does the total amount of nutient sequesteed in plants (Q 3NL P 3NL ; Eq. B10; Table B1. If the hebivoe in the twolevel system is cabonlimited the canivoe can invade if: (B5d > 3CL +Q 3CL P 2CL (B6 which says that nutient supply must exceed the nutient sequesteed in the canivoe s minimal equiement fo cabon in hebivoes ( 3CL plus the nutient sequesteed in poduces at the hebivoe s minimal cabon equiement ( P 2CL given the nutient quota of those poduces in the theelevel system ( Q 3CL. The titophic inteio equilibium that contains these tems fo cabonlimited hebivoes ( 3CL and quota ( Q 3CL becomes : 3CL = 3NL Q 3CL =Q 3NL C 3CL = ea ( Q 3CL 3CL P 2CL ea + Q 3CL (B7a (B7b (B7c
P 3CL = a ( q C 3CL ea P 2CL. (B7d ea + Q 3CL Note that both hebivoe biomass and nutient quota do not diffe fom those values in titophic case with a nutientlimited hebivoe (Eq. B5a and B5b espectively. Instead biomass of canivoes and plants continues to incease with enichment but just at a diffeent slope. Ove a nutient enichment gadient the canivoe can invade the planthebivoe system via two scenaios. In the fist the hebivoe is limited by nutients and emains so with futhe enichment. This occus when nutient content of the plant Q 3NL is elatively small: Q 3NL < m ( m+ (B8 (which is a condition that closely esembles the one fo the static case Eq. A4. The hebivoe feeding on elatively nutientpoo plants always stays nutientlimited in this titophic system because the theshold at which it would switch to cabon limitation = 3NL m + Q 3NL ( + ( ae Q 3NL +1 occus at a lowe level than equied fo the canivoe to invade (Eq. B4. The othe scenaio involves a moe nutient ich plant (i.e. one with nutient:cabon atio that exceeds the ight hand side of Eq. B8. In this case the canivoe invades when the hebivoe in the twodimensional system is cabonlimited. That hebivoe emains cabonlimited with futhe enichment until a esouce limitation theshold fo hebivoes in titophic chains (Eq. B9 is cossed. Afte this point the hebivoe actually becomes nutientlimited again. As with the static case thee is no scenaio in which a hebivoe is nutient limited when the canivoe invades and then becomes cabon limited. As with the static stoichiomety model it seems counteintuitive at fist glance that poduction of hebivoes could should switch fom cabon limitation to nutient limitation with enichment in the thee species chain. To undestand this esult one must appeciate that the minimal nutient demands of hebivoe in the pesence of pedatos ( QP 3 inceases fom the minimal nutient equiement without pedatos QP to: (B9 in the twospecies chain (Eq. B1a QP 3 =Q 3NL P 3NL 3NL = +ea +m (B10 This equiement inceases because of motality imposed by the canivoe: the hebivoe must eat moe sequesteed nutient to compensate fo this highe motality ate. (A paallel phenomenon occus fo plants and thei nutient equiements when they ae gazed; Gove 1997. Although nutients sequesteed in plants do incease with enichment when the hebivoe is cabonlimited they do not incease steeply enough. Thus eventually poduction of hebivoes in the titophic system becomes nutient limited. Tophic indices One can calculate esponse of the tophic indices fo plants and hebivoes following the same methods used fo the static stoichiomety case. That is many of the esponses of the indices to change in paametes can be calculated fom the individual esponse of the numeato (biomass in the theelevel system and the esponse of the denominato (biomass in the two level system to a change in a paamete and the quotient ule fom calculus (i.e. Eq. A12. The esults (Table 2 A1 closely match those fo the static stoichiomety case. As with that fist model esponse of plant biomass in ti
tophic chains with nutientlimited hebivoes (P 3NL to inceases in nutient:cabon atio of hebivoes ( depends upon enichment (. Below a cetain level of enichment a = C 3NL QP +2 m m q a C C (B11 plant biomass deceases with ; above this theshold it inceases (Table B1. As fo the static stoichiomety model this switch in sign of P 3NL affects esponse of the plant tophic index to when diffeent esouces limit hebivoes in bitophic and titophic chains (Table 2. tability Elsewhee we aleady shown that bitophic system is always stable egadless of the esouce limiting hebivoe gowth (all 2004. Fo titophic systems with cabonlimited hebivoes the Jacobian matix fo a cabonlimited hebivoe (J CL is: J CL = a ap ( Q 2 Q ( 2 ae 0 0 C 0 whee we have dopped subscipts denoting tophic level and esouce limitation status. Meanwhile the Jacobian matix fo a nutientlimited hebivoe (J NL is: J NL = ' ( a ap ( Q 2 Q ( 2 eaq 0 ea C + 0 C 0 + tability analysis of the theedimensional models depends upon the thee Routhuwitz citeia A 1 > 0 A 3 > 0 and A 1 A 2 A 3 > 0 whee A 1 A 2 and A 3 ae the coefficients of the Jacobian matix chaacteistic polynomial. In the Jacobian matix fo the cabonlimited hebivoe (J CL the fist citeion (A 1 > 0 is always met. The second citeion (A 3 > 0 is (B12 (B13 always met (because Q >ea. The thid citeion is met as long as: P 3CL > a Q 3CL a C C 3CL Q 3CL (B15 which is met as long as the theelevel is feasible. When hebivoes ae nutientlimited one can show that the model is stable based on the signs of the associated Jacobian matix (J NL. The fist Routhuwitz citeion is met since J 11 < 0 implying that A 1 = (J 11 + J 22 > 0. The second citeion is met since A 3 = J 11 J 23 J 32 > 0 always. The thid citeion (A 1 A 2 A 3 > 0 is met since J 11 J 22 (J 11 + J 22 + J 22 J 23 J 32 > 0 always. ence the theelevel food chain model with a nutient o cabon limited hebivoe is stable.
Table B1. igns of patial deivatives of equilibial biomass of plants (P j and hebivoes ( j deived fom the model with dynamic stoichiometic of the poduce whee + indicates incease in the equilibium with an incease in the paamete indicates a decease in the equilibium and 0 indicates no change. Responses of nutient quota (Q ae pesented in the main text (Table 2. Resouce a Quantity b Nutient supply Maximum Min. plant Gaze nut:c efficiency e nut:c Cabon P 2 0 0 0 2 P 3 3 + + 0 + 0 0 0 0 0 Nutient P 2 + 2 P 3 3 + + 0 + + c 0 0 0 0 a Cabon means that cabon limits hebivoes; nutient indicates nutient limitation. b ubscipt efes to numbe of tophic levels ( 2 fo bilevel 3 fo tilevel. c This sign changes as nutient enichment cosses the paticula theshold (Eq. B11.