1 APPENDICES 2 APPENDIX A: A multi-species predator-prey model incorporating mycorrhizal fungi 3 We assume that the dynamics of a community composed of n plant species X=(X1, ..., Xn), 4 enemy population (Y), and mycorrhizal fungal population (M) are described by the following 5 system of differential equations: 6 7 () = () [ () + () (→ , ) − () (→ , ) − ∑ , () () − ( (, ) − ()())()] 8 () = () [∑ , (), (→ , ) − , (), (→ , ) − ()] 9 () = () [∑ ( (, ) − ()()) () − ()() − ()] =1 =1 10 11 =1 [Eqns. A 1-3] 12 13 where Xi(t), Y(t), M(t) represent population size of plant (prey) species i, enemies, and 14 mycorrhizal fungi (we do not differentiate enemy or fungal species) respectively at time t. In 15 Equation A 1, the term bi represents the per capita exponential rate of increase of Xi , ai,i is the 16 per capita intraspecific competition coefficient for plant species Xi , a i,j designates the per capita 17 interspecific competition among plant species, is the conversion rate of the per capita gain in 18 resource from fungi ( (→ ), ) to plant species i abundance, is the conversion rate of the per 19 capita loss in resource towards fungi ( ( → , )) to plant species i abundance, Ci(X,t) is the per 20 capita rate of mortality from one enemy (and XiCi(X,t) is the functional response of natural 21 enemy to plant species i), and mi corresponds to the per capita decrease in predation due to 22 mycorrhizal fungi. 1 1 In equation A 2, αM represents the conversion rate of the per capita gain in resource from 2 plant species i (, ( → , )) to fungal abundance, βM accounts for the conversion rate of the 3 per capita loss in resource towards plant species i (, (→ , )) to fungal abundance, and rM 4 represents the per capita fungi mortality rate. We assumed the per capita gain in resource (the G 5 functions) to be a Michaelis-Menten function of the fungi, and we assume the per capita loss in 6 resource (the L functions) to increase with increasing fungal population but to decrease with its 7 own population size. For the fungi, we assume G and L functions to exhibit a similar saturating 8 pattern as plant population increases. 9 10 11 12 13 g (t)M(t) g Gi (R M→Xi , ) = h i(t)+M(t) ≤ (t) (t) GM,i (→ , ) = h M,i(t)+ (t) ≤ , i M,i [Eqns. A4a-d] Li (R Xi→M , ) = li (t)M(t) ki (t)+Xi (t) LM,i (→ , ) = lM,i (t) (t) kM,i (t)+M(t) Finally, in equation A 3, di represents the conversion of functional (XiCi) to per capita 14 numerical response of enemies to plant Xi , mi the per capita protection against enemies provided 15 by the mycorrhizal associations of plant species i, e is the per capita competition coefficient for 16 the enemy population, and r the per capita exponential rate of mortality of enemies in the 17 absence of plant (i.e., starvation rate). 18 We assume that all coefficients of A 1-4 are positive, Lipchitz (meaning that each 19 coefficient is uniform continuous and does not change infinitely fast over time) and bounded 20 with respect to time, with coefficient w having upper bound wu and lower bound wl. We also 21 assume a saturating positive relationship between the per capita predation effect Ci and plant 22 species i abundance, that Ci follows uniform continuous properties such that Ci does not change 23 infinitely fast with time (Lipchitz properties). Similarly, Gi and GM,i respectively follow uniform 2 1 continuous properties such that Gi and GM,i do not change infinitely fast with time and M or Xi 2 (respectively). 3 We investigate the system under the main assumption that the mycorrhizal fungal population 4 does not go to extinction nor tend towards infinity such as: 5 For every time t, ṽ ≤ () ≤ min( ) ≝ 6 Here, we demonstrate that under the main assumption and reasonable conditions on the 7 parameters and initial conditions, which we outline below, the following three statements, are 8 true: 9 10 (1) Plant and enemy populations do not tend towards infinity, therefore the solution is global in time; 11 (2) Plant and enemy populations do not go to extinction; 12 (3) When the coefficients are periodic, the system has periodic solutions. 13 Step 1: Plant and enemy populations do not tend towards infinity; rather, plant population 14 sizes are bounded above at pi , enemy population size at q. We assume a first condition (C1) 15 that the maximum sum of the per capita numerical response of the enemies in absence of 16 mycorrhizal protection (∑= ) is greater than the minimum per capita exponential 17 rate of mortality of enemies ( ), that is ∑= > . Plant population sizes are 18 bounded above at pi, and enemy population size at q as follows: + + 19 () ≤ 20 ∑ni=1 dui Ciu − ∑ni=1 dui Ciu − () ≤ ⇒ , () ≤ ≝ 21 , ⇒ , () ≤ ≝ Proof 3 , 1 If we bound the negative effects of the plant per capita interspecific competition 2 (∑=1,≠ , () ()), the per capita predation ( (, )()) and the per capita costs of 3 mycorrhizae () (→ , ) at 0, equation A 1 becomes 4 dXi () [Ineq. A 1] ≤ ()[ () − , () () + () (→ , )] 5 We can use the upper bounds of the parameters that have a positive effect on plant population 6 size (bi , αi, (→ , )) and the lower bounds of the per capita intraspecific competition 7 coefficient (aii), which has a negative effect of population size to further bound inequality A 1 as 8 follows: 9 dXi () ≤ ()[ + − , ()] [Ineq. A 2] 10 Let us assume there exists a time 0 such that (0 ) > 11 at time 0 : 12 dXi (0 ) + , , then if we write inequality A 2 <0 [Ineq. A 3] 13 In other words, is decreasing after 0 until it reaches a time 1 where (1 ) = 14 which becomes stationary or decreases. Thus, 15 For every time greater than 1 , () ≤ 16 Let us assume that the initial condition is such that 0 ≤ (0) ≤ 17 of 1 after which becomes stationary or decreases, as soon as t=0, and: 18 For every time, () ≤ + , + , , after [Ineq. A 4] , ≝ + + , , then we are in the case [Ineq. A 5] 4 1 In other words, the upper bound for the population of plant species i depends solely on its 2 maximum per capita growth rate bi, the maximum per capita increase in density of the plant 3 species i population can extract from mycorrhizal associations, and the minimum plant per capita 4 intraspecific competition (a i,i). The presence of upper bounds for plant populations that do not 5 depend on the presence of an upper bound for mycorrhizal fungal population implies that v, the 6 upper limit of mycorrhizal fungi, does exist because the fungi are limited by the amount of 7 plants. 8 By substituting the appropriate upper and lower parameter bounds into equation A 3, we can 9 bound dY() as follows: 10 dY() 11 Thus, the results presented here will hold for all saturating per capita mortality from one enemy 12 (Ci). We can further bound inequality 6 by bounding the negative effect of the per capita 13 mycorrhizal protection ( (t)) to 0: 14 15 16 dY() ≤ Y(t) [∑ni=1 [( − (t)) X (t)] − Y(t) − ] [Ineq. A 6] ≤ Y(t)[∑ni=1[( )X (t)] − Y(t) − ] [Ineq. A 7] Following the same demonstration as for , if there exists a time 0 such that (0 ) < u u ∑n i=1 di Ci − , then enemy population size is increasing after 0 until it reaches a time 1 u u ∑n i=1 di Ci − 17 where (1 ) = 18 decreases. Thus: 19 For every time greater than 1 , () ≤ 20 Assuming the initial condition is such that Y(0) ≤ 21 after which enemy population size becomes stationary or decreases as soon as t=0, and: , after which enemy population size becomes stationary or u u ∑n i=1 di Ci − [Ineq. A 8] 5 u u ∑n i=1 di Ci − , then we are in the case of 1 u u ∑n i=1 di Ci − 1 For every time , Y(t) ≤ 2 In other words, the upper bound for enemy population size depends on the lower bounds of the 3 enemy per capita exponential rate of mortality (rl), the enemy per capita competition coefficient 4 (el), and the maximum per capita numerical response of enemies to plants in absence of 5 mycorrhizal protection (∑ni=1 dui Ciu ). A first condition (C1) guarantees the existence of a strictly 6 positive upper bound q for enemy population, which is reached when the minimum per capita 7 enemy exponential rate of mortality is lower than the maximum per capita numerical response of 8 enemies to plants. Biologically, this condition means that the balance between the difference 9 between numerical response in absence of protection by mycorrhizae and mortality and 10 11 ≝q [Ineq. A 9] competition among enemies controls the enemy population. Therefore, the proof of step 1 is complete and we have demonstrated that plant (Xi) and 12 enemy (Y), population sizes are bounded above at pi , and q respectively. In particular, by the 13 principle of unique continuation (which states that two analytical functions whose difference at 14 some time becomes 0 are the same at that time), we have obtained the global existence in time of 15 the solution of the system defines by equation A 1- 4. 16 17 Step 2: Plant, and enemy populations do not go extinct; rather they remain above positive 18 ̃ respectively. We assume a second condition (C2) that is the per abundance values ̃ , 19 capita exponential rate of growth of the plant population must be greater than the sum of 20 the negative effects due to the per capita predation, the per capita interspecific competition, 21 and the per capita costs of mycorrhizal associations ( > 22 We also assume a third condition (C3) so that the per capita numerical response of enemies 23 to plants protected by mycorrhizal fungi is greater than the per capita enemy exponential 6 − ∑=,≠[, ] − ) . 1 rate of mortality (∑=( ̃ − ) > ). Plant population sizes are bounded above 2 ̃ as follows: at ̃ , and enemy population size at 3 − () ≥ ∑ − − ∑=,≠[, ] − − =,≠[, ] − ⇒ , () ≥ ̃ ≝ , , 4 5 () ≥ ̃− ) − ∑ = ( ̃≝ ⇒ , () ≥ ̃− ) − ∑ = ( 6 7 where pi and q are upper bounds for plant and enemy populations size respectively as 8 shown in Step 1. 9 10 11 Proof If we take the lower bound of the equation A 1, we get: () [ + − + − ∑=1,≠ , − , () − ( − )()] ≤ 12 Xi () [Ineq. A 10] 13 We can further bound the inequality A 10 by setting the lower bound of Xi, M, and to 0 and 14 the upper bound of Y and M to q and v: 15 () [ − − ∑=1,≠[, ] − , () − ] ≤ Xi () [Ineq. A 11] 16 If we there exists a time 0 such that − −∑ =1,≠[, ]− , > (0 ), then plant population 17 size increases until it reaches a time 1 where (1 ) = 18 plant population size is stationary or increases. Thus, − −∑ =1,≠[, ]− , after which 19 For every time greater than 1 , − −∑ =1,≠[, ]− , 7 ≤ () [Ineq. A 12] − −∑ =1,≠[, ]− 1 So if we assume that at time t=0, 2 1 after which plant population size is stationary or increases as soon as t=0, and , ≤ (0), then we are in the case of − −∑ =1,≠[, ]− 3 For every time , () ≥ 4 In other words, the lower bound for the plant population size depends on the ratio of the per 5 capita exponential growth rate when accounting for all negative effects except the per capita 6 intraspecific competition ( − 7 intraspecific competition (, ). In order to ensure that plant population size remains above 0, we 8 need to make a second condition (C2) that is the per capita exponential growth rate must be 9 greater than the sum of the negative per capita effects due to predation, mycorrhizal associations , ≝ ̃ [Ineq.A 13] − ∑=1,≠[, ] − ) and the maximum per capita 10 costs, and interspecific plant competition ( > 11 this condition means that plant growth rate must be high enough to compensate all negative 12 effects except intraspecific competition. − ∑=1,≠[, ] − ). Biologically, 13 14 By substituting this upper bound () of mycorrhizal fungal population size (M) and the 15 appropriate upper and lower parameter bounds into equation A 3, we can bound 16 Y(t)[∑ni=1( − )X (t) − Y(t) − ] ≤ 17 If there exists a time 0 such that (0 ) < 18 increases until it reaches a time 1 where (1 ) = 19 population size is stationary or increases. Thus, dY() as follows: [Ineq. A 14] ∑ ̃− =1( )− 8 dY() , then enemy population size ∑ ̃− =1( )− , after which enemy ∑ ̃− =1( )− 1 For every time greater than 1 , 2 If we assume that the initial condition that 3 case of 1 (1 ) = 4 For every time t, () ≥ 5 In other words, the lower bound for enemy population size depends on the ratio of the maximum 6 per capita enemy exponential rate of mortality ( ) and the difference between the lower value of 7 the per capita numerical response of enemies to plants in the presence of mycorrhizal protection 8 (∑=1( ̃ − )) and the maximum per capita competition effect among enemies 9 ( ). In order to ensure that the lower bound of enemy population is greater than 0, we need to ∑ ̃− =1( )− ≤ () ∑ ̃− =1( )− [Ineq. A 15] ≤ (0), then we are in the as soon as t=0, and: ∑ ̃− =1( )− ≝ ̃ [Ineq. A 16] 10 create a third condition (C3) so that the per capita numerical response of enemies to plants 11 protected by mycorrhizal fungi is greater than the per capita exponential rate of mortality of 12 enemies (∑=1( ̃ − ) > ). Biologically, this condition means that the per capita 13 numerical response of enemies to plants protected by mycorrhizal associations needs to remain 14 greater than the per capita exponential rate of mortality of enemies. If this condition were not 15 met, the enemy population would go to extinction. If this condition is met, then the first 16 condition, that is the maximum per capita numerical response of enemies to plants must be 17 greater than the minimum per capita enemy exponential rate of mortality, is also met. 18 Furthermore, when this condition is met, the upper bound of the abundance of the mycorrhizal 19 fungal population is also met (() ≤ min( )) in the case of one species but not in the case of 20 n plant species. In other words, the domain of coexistence will be additionally constrained by the 21 condition on the mycorrhizal fungal population. 9 1 Therefore, assuming the mycorrhizal fungal population has an upper bound, only two conditions 2 are required to achieve plant species coexistence: (1) the per capita exponential rate of growth 3 of the plant population must be greater than the sum of the negative per capita effects due 4 to predation, interspecific competition, and the costs of mycorrhizal associations; and (2) 5 the per capita numerical response of enemies to plants protected by mycorrhizal fungi is 6 greater than the per capita enemy exponential rate of mortality (∑=( ̃ − 7 ) > ). 8 9 Step 3. If the coefficients of A 1-4 are periodic and mycorrhizal fungal population remains 10 between an upper and a lower bound, plant, enemy, and mycorrhizal fungal populations 11 exhibit periodic dynamics. In other words, there exists a periodic solution to the system 12 described by A 1-4.We assume that the coefficients are periodic with the same period T>0. 13 We consider the application φ that maps an initial state of the system to the solution of 14 system at generation T, that is: 15 φ [(X1(0),··· ,Xn(0),Y (0))] = (X1(T),··· ,Xn(T),Y (T)) 16 From steps 1 and 2, we know that there exists a compact set Dn+1 that is an invariant set 17 under the flow of the differential system. So φ maps the compact set Dn+1 to itself. 18 Therefore, according to the Brouwer ﬁxed-point theorem, we know that the continuous 19 application φ has a ﬁxed point. This fixed point is an initial data for which the solution is T- 20 periodic. 21 Conjecture. The study of the stability of the general periodic system is beyond the scope of this 22 study but preliminary analysis of the discrete model and simulations (Table 3 in Appendix C) 23 suggested stability. The challenge will consist in finding a suitable Liapounov-type function. 10 1 APPENDIX B: Analyses of the effects of bi-directional mutualism on a prey-predator system with a 2 type III functional response, and of the effects of predation on a bi-directional mutualism system. 3 4 1. What is the effect of bi-directional mutualism on a prey-predator system with a type III functional 5 response? 6 We assume that the dynamics of a plant species (X), interacting with an enemy population (Y), 7 and a fixed mycorrhizal fungal population (M) are described by the following system of 8 differential equations: 9 () = ()[ + (→ , ) − (→ , ) − [(, ) − ]() − ()] 10 dY() = Y(t)[(C(, ) − mM)(t) − eY(t) − r] 11 Where each parameter is similar to the one described in the main article. 12 We replace the functions C, G, and L with their respective expression, and we obtain the 13 following system of two differential equations [Eqn. B 3-4]: 14 () = () [ + h+M − k+X(t) − (1+()2 − ) () − ()] 15 dY() = Y(t) [ ( 16 We set up each differential equation to 0 to investigate the null isoclines describing where the 17 plant and natural enemy populations are stable (Yx, Yy). This approach results in two trivial 18 isoclines (X=0 and Y=0), as well as two more complicated isoclines: gM () 1+()2 lM () [Eqn. B 3-4] − mM) (t) − eY(t) − r] gM lM − − h+M k+X − 1+2 + 19 = 20 Which exists if (1+ 2 ) ≠ , and: 21 [Eqn. B 1-2] [Eqn. B 5] 2 = (1+ 2 ) − mM r [Eqn. B 6] − 22 11 1 These two non-trivial isoclines lead to a complex isocline plane that can be analyzed graphically 2 (Fig. 5). Specifically, we are interesting in understanding how the presence of a fixed population 3 of mycorrhizal fungi (M) influences the isocline plane. 4 Increasing the abundance of mycorrhizal fungi rotates the isocline associated with the 5 population of natural enemies (Yy) towards the right (Fig. 5). In contrast, the isocline associated 6 with the population of plant shrinks in the two dimensions (Y and X, Fig. 5). Such change in the 7 isoclines can lead to dramatic change in the equilibrium reached by the systems. However, in 8 general the increase in mycorrhizal fungi abundance dampens the effects of natural enemies, 9 which results in an increase in abundance and a decrease in the natural enemy abundance at the 10 equilibrium. This effect can stabilize or destabilize the system (Fig. 6). A too abundant 11 population of fungi can push both natural enemies and plant species to extinction as the costs of 12 hosting mycorrhizal fungal associations increase (Fig. 6). 13 We can also get similar results by investigating the general Jacobian Matrix associated 14 with Equations B3-4: 15 | | = =| = 2 1+ 2 (1 − 2 1+ 2 = ( − ) − = ( 1+ 2 1+ 2 ) − ) − 2 − | 16 The expression of A ( ) is more complicated than the other partial derivatives but can be 17 simplified as follows: 18 =+ 19 We can assess how M qualitatively influences the structure of the Jacobian Matrix. In particular, 20 we should note that as M increases ℎ+ − 2 2 (1 − )− (1 − )− 2 + + 1+ 1+ 2 21 1) B increases 22 2) C decreases 12 2 + [Eqn. B 7] 1 3) D decreases 2 From the Routh-Hurwitz criteria, we know that a continuous system of 2 differential equations is 3 stable if the trace (A+D) is negative while the determinant (AD-BC) is positive. Therefore, the 4 impact of M on the system will depend on its effect on A. We can evaluate this effect, by taking 5 the partial derivative of A with respect to M: 6 7 From this equation, we can conclude that if the protection against natural enemy conveyed by 8 mycorrhizal fungi to the plant is small enough, then < 0 , which could stabilize the system 9 (Eqn. B1-2). Otherwise, mycorrhizal fungi might destabilize the prey-predator system (Fig. 6). = − (ℎ+)2 − (1 − )+ + + [Eqn. B 8] 10 11 2. What is the effect of predation on a bi-directional mutualism system? 12 We assume that the dynamics of a plant species (X), interacting with a fixed enemy population 13 (Y), and a mycorrhizal fungal population (M) are described by the following system of 14 differential equations: 15 () = ()[ + (→ , ) − (→ , ) − [(, ) − ()]() − ()] [Eqn. B 8-9] 16 () 17 We replace the functions C, G, and L with their respective expression, and we obtain the 18 following system of two differential equations [Eqn. B 10-11]: 19 () 20 () g X(t) l X(t) = () [ − − ] h + X(t) k + M(t) = ()[∑=1 , (), (→ , ) − , (), (→ , ) − ] gM(t) lM(t) () = () [ + h+M(t) − k+X(t) − [1+()2 − ()] − ()] 21 13 [Eqn. B 10-11] 1 In order to study the behavior of the system, we investigated how natural enemy population 2 alters the expression of the general Jacobian matrix: 3 4 | = | = | | 2 = − − 2 (ℎ + ) + − 2 (ℎ + ) + || 2 = − (1 − ) − ℎ + + + 2 = − As the natural enemy population increases, we note that: 5 1) A decreases 6 2) B2 increases 7 3) C2 remains negative 8 Therefore, if A decreases enough to become negative and B2 increases and becomes positive, the 9 stability will occur when D2 is negative according to Routh-Hurwitz criteria. Negative D2 means 10 that the costs of the mycorrhizal associations to the fungi must exceed a fraction of the difference 11 between the benefits of the association to the fungi and the fungal mortality rate (Fig. 7). In other 12 words, there needs to be non-negligible costs of the AM fungal associations to the fungi. 13 However, as for the effect of mycorrhizal fungi on the plant-enemy system, the abundance of 14 natural enemies must be high enough to stabilize the system, but not too high or plant population 15 might be drawn to extinction (Fig. 7). 14 1 APPENDIX C. 2 3 Table 1. Parameter values used for the simulations depicted in Figures 2 and 3. Parameter Figure 2 A Figure 2 B Figure 3 A Figure 3 B b 0.42 0.42 0.56 0.53 a 0.09 0.53 0.89 0.96 α 0.81 0.23 0.84 0.84 0.19 1*X/(1+1*0.15X12 ) 0.19 0.19 0.19 1*X/(1+1*0.13X2) 1*X/(1+1*0.15X2) 1*X/(1+1*0.60X2) m 0.23 0.12 0.23 0.05 G 0.46M/(0.11+M) 0.19M/(0.18+M) 0.75M/(0.11+M) 0.75M/(0.11+M) L 0.63M/(0.32+ X) 0.63M/(0.32+ X1) 0.63M/(0.32+ X) 0.53M/(0.32+ X) 0.25 0.68 0.25 0.68 0.42 0.28 0.42 0.75 0.56X/(0.19+X) 0.56X/(0.19+X) 0.56X/(0.19+X) 0.56X/(0.19+X) LM 0.18X/(0.33+M) 0.75X/(0.33+M) 0.18X/(0.33+M) 0.28X/(0.33+M) rM 0.05 0.12 0.05 0.12 0.07 0.02 0.07 0.05 1.00 0.93 1.00 0.96 0.07 0.02 0.07 0.07 0.7 0.7 0.7 0.7 1.05 1.05 1.05 1.05 2.6 2.6 2.6 2.6 β C Type Plant parameters αM βM GM e d Mycorrhizal fungi parameters Enemy parameters r X(0) Y(0) 4 M(0) Initial population density 5 6 7 8 9 10 11 12 13 14 15 16 17 18 15 1 Table 2. Parameter values used for simulations depicted in Figure 4. Parameter Rare species Common species b 0.7 0.32 a 0.49 0.28 α 0.11 0.54 β 0.11 0.35 0.54X1/(1+0.54*0.42X12+0.37*0.13X22) 0.37X2/(1+0.37*0.13X22+0.54*0.42X12) m 0.07 0.14 G 0.65M/(0.14+M) 0.77M/(0.65+M) L 0.54M/(0.19+X1) 0.14M/(0.79+X2) C Type Plant parameters a12 0.18 αM βM GM LM Mycorrhizal fungi parameters 0.7 0.23 0.09 0.32 0.61X1/(0.68+X1) 0.35X2/(0.37+X2) 0.19X1/(0.35+M) 0.21X2/(0.42+M) rM e d 0.07 Enemy parameters 0.07 1 0.26 r X(0) Y(0) 2 M(0) 0.07 Initial population density 5 6.45 3.5 1.2 3 4 5 6 7 8 9 10 11 12 13 16 1 2 3 Table 3. Influence of the parameters to the solution presented in Figure 4. N means that coexistence was not feasible; F means that species coexisted and their abundances were fixed; NP means that the solution was not periodic but coexistence occurred; and P means that the solution was periodic. 4 Parameter Type bi aii αi βi pi pii Plant mi gi hi li Ki aij αM βM gM hM Mycorr hizal fungi lM kM rM e d Enemy r X(0) Y(0) M(0) Initial populati on density Effect of decreasing F, P (0.35-0.7) N, P (0.25) NP, P (0.40-0.49) P P NP sometimes P N P sometimes NP NP sometimes P P P P sometimes NP P P sometimes NP P NP sometimes P P F (0), P N, P (0.14-0.54) P C P sometimes NP N, P (0.10-0.18) F, P (0,38-0.7) P P P sometimes NP F, P (0.33-0.61) P P sometimes NP P sometimes NP P P sometimes NP P sometimes F P sometimes F or NP N Value 0.7 (rare) 0.32 (common) 0.49(rare) 0.28(common) 0.11(rare) 0.54(common) 0.11(rare) 0.35(common) 0.54(rare) 0.37(common) 0.42(rare) 0.13(common) 0.07(rare) 0.14(common) 0.65(rare) 0.77(common) 0.14(rare) 0.65(common) 0.54(rare) 0.14(common) 0.19(rare) 0.79(common) 0.18 0.7(rare) 0.23(common) 0.09(rare) 0.32(common) 0.61(rare) 0.35(common) 0.68(rare) 0.37(common) 0.19(rare) 0.21(common) 0.35(rare) 0.42(common) 0.07 Effect of increasing P (0.7-0.95), N, P (1.23-1.24), N P (0.32-0.79), F (0.80-1.12), P P NP and sometimes P P (0.11-0.29),NP P P with a sometimes NP P sometimes NP P C P sometimes NP NP sometimes P NP sometimes P P (0.14-0.46),NP NP sometimes P P P (0.14-0.68),NP P sometimes NP P sometimes NP P (0.14-0.51) sometimes NP, NP P, NP(0.21-0.33), P P P (0.18-0.68), N P sometimes NP P (0.23-0.29), N P (0.11-0.33), F P (0.32-0.84), F P sometimes NP P (0.35-0.36), N P P P (0.19-0.68) sometimes NP, F P (0.21-0.55), F P NP sometimes P P (0.07-0.19), F P sometimes NP P sometimes NP P sometimes NP P sometimes NP P sometimes NP P sometimes NP P sometimes NP 0.07 1 (rare) 0.26(common) 0.07 5(rare) 6.45(common) 3.5 P sometimes NP P sometimes NP 1.2 P sometimes NP and N 5 17 P sometimes NP P sometimes NP P sometimes NP P sometimes NP P sometimes NP 1 2 3 Table 3. Parameter values used for simulations depicted in Figure 6 and 7. Parameter Figure 6 A Figure 6 B Figure 7 A Figure 7 B Figure 7 C b 0.98 1.02 0.32 0.25 0.98 a 0.07 0.09 0.74 0.19 0.19 α 0.56 0.09 0.25 0.25 0.25 β 0.75 0.89X/(1+0.26*0. 89X2) 0.75 0.79X/(1+0.19*0.7 9X2) 0.86 0.96 0.96 m 0.40 0.12 0.72 0.33 0.35 G 0.16M/(0.35+M) 0.09M/(0.42+M) 0.33M/(0.33+M) 0.14M/(0.26+M) 0.14M/(0.26+M) L 0.58M/(0.07+ X) 0.67M/(0.56+ X) 0.19M/(0.40+ X) 0.19M/(1.00 + X) 0.19M/(1.00 + X) 1.00 0.18 0.18 0.51 0.09 0.09 C Plant parameters αM βM GM Mycorrhizal fungi parameters 0.46 X/(0.11+X1) 0.23 X/(0.81+X1) 0.23 X/(0.81+X1) LM 0.16 X/(0.04+M) 0.07 X/(0.07+M) 0.07 X/(0.82+M) rM 0.88 0.30 0.26 0.14 0.02 0.02 1 1 1 1 1 1 e d Enemy parameters r X(0) Y(0) M(0) Initial population density 0.04 0.04 0.07 0.07 0.04 0.04 1 1 1 1 18 19

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# APPENDICES APPENDIX A: A multi-species predator

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