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
𝑛
7
𝑑𝑋𝑖 (𝑡)
= 𝑋𝑖 (𝑡) [𝑏𝑖 (𝑡) + 𝛼𝑖 (𝑡)𝐺𝑖 (𝑅𝑀→𝑋𝑖 , 𝑡) − 𝛽𝑖 (𝑡)𝐿𝑖 (𝑅𝑋𝑖→𝑀 , 𝑡) − ∑ 𝑎𝑖,𝑗 (𝑡)𝑋𝑗 (𝑡) − (𝐶𝑖 (𝑋, 𝑡) − 𝑚𝑖 (𝑡)𝑀(𝑡))𝑌(𝑡)]
𝑑𝑡
8
𝑑𝑀(𝑡)
= 𝑀(𝑡) [∑ 𝛼𝑀,𝑖 (𝑡)𝐺𝑀,𝑖 (𝑅𝑋𝑖→𝑀 , 𝑡) − 𝛽𝑀,𝑖 (𝑡)𝐿𝑀,𝑖 (𝑅𝑀→𝑋𝑖 , 𝑡) − 𝑟𝑀 (𝑡)]
𝑑𝑡
9
𝑑𝑌(𝑡)
= 𝑌(𝑡) [∑ 𝑑𝑖 (𝐶𝑖 (𝑋, 𝑡) − 𝑚𝑖 (𝑡)𝑀(𝑡))𝑋𝑖 (𝑡) − 𝑒(𝑡)𝑌(𝑡) − 𝑟(𝑡)]
𝑑𝑡
𝑗=1
𝑛
𝑖=1
𝑛
10
11
𝑖=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, 𝛼𝑖 is the conversion rate of the per capita gain in
18
resource from fungi (𝐺𝑖 (𝑅𝑀→𝑋𝑖 ), 𝑡) to plant species i abundance, 𝛽𝑖 is the conversion rate of the per
19
capita loss in resource towards fungi (𝐿𝑖 (𝑅𝑋𝑖 →𝑀 , 𝑡)) 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 (𝐺𝑀,𝑖 (𝑅𝑋𝑖 →𝑀 , 𝑡)) to fungal abundance, βM accounts for the conversion rate of the
3
per capita loss in resource towards plant species i (𝐿𝑀,𝑖 (𝑅𝑀→𝑋𝑖 , 𝑡)) 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 , 𝑡) = h i(t)+M(t) ≤ 𝑔𝑖𝑢
(t)𝑋 (t)
𝑢
GM,i (𝑅𝑋𝑖→𝑀 , 𝑡) = h M,i(t)+𝑋𝑖 (t) ≤ 𝑔𝑀,𝑖
i
M,i
𝑖
[Eqns. A4a-d]
Li (R Xi→M , 𝑡) =
li (t)M(t)
ki (t)+Xi (t)
LM,i (𝑅𝑀→𝑋𝑖 , 𝑡) =
lM,i (t)𝑋𝑖 (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 ≤ 𝑀(𝑡) ≤ min(𝑚𝑖𝑢 ) ≝ 𝑣
𝐶𝑙
𝑖
𝑖
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 (∑𝒏𝒊=𝟏 𝒅𝒖𝒊 𝑪𝒖𝒊 𝒑𝒊 ) is greater than the minimum per capita exponential
17
rate of mortality of enemies (𝒓𝒍 ), that is ∑𝒏𝒊=𝟏 𝒅𝒖𝒊 𝑪𝒖𝒊 > 𝒓𝒍 . Plant population sizes are
18
bounded above at pi, and enemy population size at q as follows:
𝐛𝐮𝐢 + 𝛂𝐮𝐢 𝐠 𝐮𝐢
𝐛𝐮𝐢 + 𝛂𝐮𝐢 𝐠 𝐮𝐢
19
𝐗 𝐢 (𝟎) ≤
20
∑ni=1 dui Ciu 𝑝𝑖 − 𝑟 𝑙
∑ni=1 dui Ciu 𝑝𝑖 − 𝑟 𝑙
𝐘(𝟎) ≤
⇒ 𝐅𝐨𝐫 𝐞𝐯𝐞𝐫𝐲 𝐭𝐢𝐦𝐞 𝐭, 𝐘(𝐭) ≤ 𝐪 ≝
𝑒𝑙
𝑒𝑙
21
𝐚𝐥𝐢,𝐢
⇒ 𝐅𝐨𝐫 𝐞𝐯𝐞𝐫𝐲 𝐭𝐢𝐦𝐞 𝐭, 𝐗 𝐢 (𝐭) ≤ 𝐩𝐢 ≝
Proof
3
𝐚𝐥𝐢,𝐢
1
If we bound the negative effects of the plant per capita interspecific competition
2
(∑𝑛𝑗=1,𝑗≠𝑖 𝑎𝑖,𝑗 (𝑘)𝑋𝑗 (𝑘)), the per capita predation (𝐶𝑖 (𝑋, 𝑡)𝑌(𝑡)) and the per capita costs of
3
mycorrhizae 𝛽𝑖 (𝑡)𝐿𝑖 (𝑅𝑋𝑖→𝑀 , 𝑡) at 0, equation A 1 becomes
4
dXi (𝑡)
𝑑𝑡
[Ineq. A 1]
≤ 𝑋𝑖 (𝑡)[𝑏𝑖 (𝑡) − 𝑎𝑖,𝑖 (𝑡)𝑋𝑖 (𝑡) + 𝛼𝑖 (𝑡)𝐺𝑖 (𝑅𝑀→𝑋𝑖 , 𝑡)]
5
We can use the upper bounds of the parameters that have a positive effect on plant population
6
size (bi , αi, 𝐺𝑖 (𝑅𝑀→𝑋𝑖 , 𝑡)) 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 (𝑡)
𝑑𝑡
𝑙
≤ 𝑋𝑖 (𝑡)[𝑏𝑖𝑢 + 𝛼𝑢𝑖 𝑔𝑢𝑖 − 𝑎𝑖,𝑖
𝑋𝑖 (𝑡)]
[Ineq. A 2]
10
Let us assume there exists a time 𝑡0 such that 𝑋𝑖 (𝑡0 ) >
11
at time 𝑡0 :
12
dXi (𝑡0 )
𝑑𝑡
𝑏𝑖𝑢 +𝛼𝑖𝑢 𝑔𝑖𝑢
𝑙
𝑎𝑖,𝑖
, then if we write inequality A 2
<0
[Ineq. A 3]
13
In other words, 𝑋𝑖 is decreasing after 𝑡0 until it reaches a time 𝑡1 where 𝑋𝑖 (𝑡1 ) =
14
which 𝑋𝑖 becomes stationary or decreases. Thus,
15
For every time 𝑡 greater than 𝑡1 , 𝑋𝑖 (𝑡) ≤
16
Let us assume that the initial condition is such that 0 ≤ 𝑋𝑖 (0) ≤
17
of 𝑡1 after which 𝑋𝑖 becomes stationary or decreases, as soon as t=0, and:
18
For every time, 𝑋𝑖 (𝑡) ≤
𝑏𝑖𝑢 +𝛼𝑖𝑢 𝑔𝑖𝑢
𝑙
𝑎𝑖,𝑖
𝑏𝑖𝑢 +𝛼𝑖𝑢 𝑔𝑖𝑢
𝑙
𝑎𝑖,𝑖
, after
[Ineq. A 4]
𝑙
𝑎𝑖,𝑖
≝ 𝑝𝑖
𝑏𝑖𝑢 +𝛼𝑖𝑢 𝑔𝑖𝑢
𝑏𝑖𝑢 +𝛼𝑖𝑢 𝑔𝑖𝑢
𝑙
𝑎𝑖,𝑖
, 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(𝑡)
𝑑𝑡
as follows:
10
dY(𝑡)
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 (𝑚𝑖𝑙 𝑀(t)) to 0:
14
15
16
𝑑𝑡
dY(𝑡)
𝑑𝑡
≤ Y(t) [∑ni=1 [(𝑑𝑖𝑢 𝐶𝑖𝑢 − 𝑚𝑖𝑙 𝑀(t)) X𝑖 (t)] − 𝑒 𝑙 Y(t) − 𝑟 𝑙 ]
[Ineq. A 6]
≤ Y(t)[∑ni=1[(𝑑𝑖𝑢 𝐶𝑖𝑢 )X𝑖 (t)] − 𝑒 𝑙 Y(t) − 𝑟 𝑙 ]
[Ineq. A 7]
Following the same demonstration as for 𝑋𝑖 , if there exists a time 𝑡0 such that 𝑌(𝑡0 ) <
u u
𝑙
∑n
i=1 di Ci 𝑝𝑖 −𝑟
𝑒𝑙
, then enemy population size is increasing after 𝑡0 until it reaches a time 𝑡1
u u
𝑙
∑n
i=1 di Ci 𝑝𝑖 −𝑟
17
where 𝑌(𝑡1 ) =
18
decreases. Thus:
19
For every time 𝑡 greater than 𝑡1 , 𝑌(𝑡) ≤
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:
𝑒𝑙
, after which enemy population size becomes stationary or
u u
𝑙
∑n
i=1 di Ci 𝑝𝑖 −𝑟
[Ineq. A 8]
𝑒𝑙
5
u u
𝑙
∑n
i=1 di Ci 𝑝𝑖 −𝑟
𝑒𝑙
, then we are in the case of 𝑡1
u u
𝑙
∑n
i=1 di Ci 𝑝𝑖 −𝑟
1
For every time 𝑡, 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 𝑝𝑖 ). 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
𝑒𝑙
≝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 𝒑̃𝒊 , 𝐚𝐧𝐝 𝒒
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 (𝒃𝒍𝒊 > 𝜷𝒖𝒊
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
𝒍𝒖
𝒊𝒗
𝒌𝒍𝒊
− ∑𝒏𝒋=𝟏,𝒋≠𝒊[𝒂𝒖𝒊,𝒋 𝒑𝒋 ] − 𝑪𝒖𝒊 𝒒 ) .
1
rate of mortality (∑𝒏𝒊=𝟏(𝑑𝑖𝑙 𝐶𝑖𝑙 𝑝̃𝑖 −𝑑𝑖𝑢 𝑚𝑖𝑢 𝑣𝑝𝑖 ) > 𝒓𝒖 ). Plant population sizes are bounded above
2
̃ as follows:
at 𝒑̃𝒊 , and enemy population size at 𝒒
3
𝒃𝒍𝒊 − 𝜷𝒖𝒊
𝐗(𝟎) ≥
𝒍𝒖𝒊 𝒗
𝒍𝒖 𝒗
𝒖
𝒖
∑𝒏
𝒃𝒍𝒊 − 𝜷𝒖𝒊 𝒊 𝒍 − ∑𝒏𝒋=𝟏,𝒋≠𝒊[𝒂𝒖𝒊,𝒋 𝒑𝒋 ] − 𝑪𝒖𝒊 𝒒
𝒍 − 𝒋=𝟏,𝒋≠𝒊[𝒂𝒊,𝒋 𝒑𝒋 ] − 𝑪𝒊 𝒒
𝒌𝒊
𝒌𝒊
⇒ 𝐟𝐨𝐫 𝐞𝐯𝐞𝐫𝐲 𝐭𝐢𝐦𝐞 𝐭, 𝐗(𝐭) ≥ 𝒑̃𝒊 ≝
𝒂𝒖𝒊,𝒊
𝒂𝒖𝒊,𝒊
4
𝑙 𝑙
5
𝐘(𝟎) ≥
𝑢
𝑢
̃−𝑑𝑖 𝑚𝑖 𝑣𝑝 ) − 𝒓𝒖
∑𝒏
𝒊=𝟏 (𝑑𝑖 𝐶𝑖 𝑝
𝑖
𝑖
𝒆𝒖
𝑙 𝑙
̃≝
⇒ 𝐟𝐨𝐫 𝐞𝐯𝐞𝐫𝐲 𝐭𝐢𝐦𝐞 𝐭, 𝐘(𝐭) ≥ 𝒒
𝑢
𝑢
̃−𝑑𝑖 𝑚𝑖 𝑣𝑝 ) − 𝒓𝒖
∑𝒏
𝒊=𝟏 (𝑑𝑖 𝐶𝑖 𝑝
𝑖
𝑖
𝒆𝒖
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:
𝑋𝑖 (𝑡) [𝑏𝑖𝑙 + 𝛼𝑖𝑙 𝐺𝑖𝑙 − 𝛽𝑖𝑢
𝑙𝑖𝑢 𝑀𝑢
𝑘𝑖𝑙 +𝑋𝑖𝑙
𝑢
𝑢
− ∑𝑛𝑗=1,𝑗≠𝑖 𝑎𝑖,𝑗
𝑝𝑗 − 𝑎𝑖,𝑖
𝑋𝑖 (𝑡) − (𝐶𝑖𝑢 − 𝑚𝑖𝑙 𝑀𝑙 )𝑌(𝑡)] ≤
12
𝑑Xi (𝑡)
𝑑𝑡
[Ineq. A 10]
13
We can further bound the inequality A 10 by setting the lower bound of Xi, M, and 𝐺𝑖𝑙 to 0 and
14
the upper bound of Y and M to q and v:
15
𝑋𝑖 (𝑡) [𝑏𝑖𝑙 − 𝛽𝑖𝑢
𝑙𝑖𝑢 𝑣
𝑘𝑖𝑙
𝑢
𝑢
− ∑𝑛𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗
𝑝𝑗 ] − 𝑎𝑖,𝑖
𝑋𝑖 (𝑡) − 𝐶𝑖𝑢 𝑞] ≤
𝑑Xi (𝑡)
[Ineq. A 11]
𝑑𝑡
𝑢
16
If we there exists a time 𝑡0 such that
𝑙 𝑣
𝑢
𝑢
𝑏𝑖𝑙 −𝛽𝑖𝑢 𝑖 𝑙 −∑𝑛
𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗 𝑝𝑗 ]−𝐶𝑖 𝑞
𝑘
𝑖
𝑢
𝑎𝑖,𝑖
> 𝑋𝑖 (𝑡0 ), then plant population
𝑢
17
size increases until it reaches a time 𝑡1 where 𝑋𝑖 (𝑡1 ) =
18
plant population size is stationary or increases. Thus,
𝑙 𝑣
𝑢
𝑢
𝑏𝑖𝑙 −𝛽𝑖𝑢 𝑖 𝑙 −∑𝑛
𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗 𝑝𝑗 ]−𝐶𝑖 𝑞
𝑘
𝑖
𝑢
𝑎𝑖,𝑖
after which
𝑢
19
For every time 𝑡 greater than 𝑡1 ,
𝑙 𝑣
𝑢
𝑢
𝑏𝑖𝑙 −𝛽𝑖𝑢 𝑖 𝑙 −∑𝑛
𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗 𝑝𝑗 ]−𝐶𝑖 𝑞
𝑘
𝑖
𝑢
𝑎𝑖,𝑖
7
≤ 𝑋𝑖 (𝑡)
[Ineq. A 12]
𝑢
𝑙 𝑣
𝑢
𝑢
𝑏𝑖𝑙 −𝛽𝑖𝑢 𝑖 𝑙 −∑𝑛
𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗 𝑝𝑗 ]−𝐶𝑖 𝑞
𝑘𝑖
1
So if we assume that at time t=0,
2
𝑡1 after which plant population size is stationary or increases as soon as t=0, and
𝑢
𝑎𝑖,𝑖
≤ 𝑋𝑖 (0), then we are in the case of
𝑢
𝑔 𝑣
𝑢
𝑢
𝑏𝑖𝑙 −𝛽𝑖𝑢 𝑖 𝑙 −∑𝑛
𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗 𝑝𝑗 ]−𝐶𝑖 𝑞
𝑘
3
For every time 𝑡, 𝑋𝑖 (𝑡) ≥
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 (𝑏𝑖𝑙 − 𝛽𝑖𝑢
7
𝑢
intraspecific competition (𝑎𝑖,𝑖
). 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
𝑖
𝑢
𝑎𝑖,𝑖
𝑙𝑖𝑢 𝑣
𝑘𝑖𝑙
≝ 𝑝̃𝑖
[Ineq.A 13]
𝑢
− ∑𝑛𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗
𝑝𝑗 ] − 𝐶𝑖𝑢 𝑞) and the maximum per capita
𝑙𝑖𝑢 𝑣
10
costs, and interspecific plant competition (𝑏𝑖𝑙 > 𝛽𝑖𝑢
11
this condition means that plant growth rate must be high enough to compensate all negative
12
effects except intraspecific competition.
𝑘𝑖𝑙
𝑢
− ∑𝑛𝑗=1,𝑗≠𝑖[𝑎𝑖,𝑗
𝑝𝑗 ] − 𝐶𝑖𝑢 𝑞 ). Biologically,
13
14
By substituting this upper bound (𝑣) 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(𝑑𝑖𝑙 𝐶𝑖𝑙 −𝑑𝑖𝑢 𝑚𝑖𝑢 𝑣)X𝑖 (t) − 𝑒 𝑢 Y(t) − 𝑟 𝑢 ] ≤
17
If there exists a time 𝑡0 such that 𝑌(𝑡0 ) <
18
increases until it reaches a time 𝑡1 where 𝑌(𝑡1 ) =
19
population size is stationary or increases. Thus,
dY(𝑡)
𝑑𝑡
as follows:
[Ineq. A 14]
𝑑𝑡
𝑙 𝑙
𝑢 𝑢
𝑢
∑𝑛
̃−𝑑
𝑖
𝑖=1(𝑑𝑖 𝐶𝑖 𝑝
𝑖 𝑚𝑖 𝑣𝑝𝑖 )−𝑟
𝑒𝑢
8
dY(𝑡)
, then enemy population size
𝑙 𝑙
𝑢 𝑢
𝑢
∑𝑛
̃−𝑑
𝑖
𝑖=1(𝑑𝑖 𝐶𝑖 𝑝
𝑖 𝑚𝑖 𝑣𝑝𝑖 )−𝑟
𝑒𝑢
, after which enemy
𝑙 𝑙
𝑢 𝑢
𝑢
∑𝑛
̃−𝑑
𝑖
𝑖=1(𝑑𝑖 𝐶𝑖 𝑝
𝑖 𝑚𝑖 𝑣𝑝𝑖 )−𝑟
1
For every time 𝑡 greater than 𝑡1 ,
2
If we assume that the initial condition that
3
case of 𝑡1 𝑌(𝑡1 ) =
4
For every time t, 𝑌(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 (𝑟 𝑢 ) 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
(∑𝑛𝑖=1(𝑑𝑖𝑙 𝐶𝑖𝑙 𝑝̃𝑖 − 𝑑𝑖𝑢 𝑚𝑖𝑢 𝑣𝑝𝑖 )) and the maximum per capita competition effect among enemies
9
(𝑒 𝑢 ). In order to ensure that the lower bound of enemy population is greater than 0, we need to
𝑒𝑢
𝑙 𝑙
𝑢 𝑢
𝑢
∑𝑛
̃−𝑑
𝑖
𝑖=1(𝑑𝑖 𝐶𝑖 𝑝
𝑖 𝑚𝑖 𝑣𝑝𝑖 )−𝑟
𝑒𝑢
≤ 𝑌(𝑡)
𝑙 𝑙
𝑢 𝑢
𝑢
∑𝑛
̃−𝑑
𝑖
𝑖=1(𝑑𝑖 𝐶𝑖 𝑝
𝑖 𝑚𝑖 𝑣𝑝𝑖 )−𝑟
𝑒𝑢
[Ineq. A 15]
≤ 𝑌(0), then we are in the
as soon as t=0, and:
𝑙 𝑙
𝑢 𝑢
𝑢
∑𝑛
̃−𝑑
𝑖
𝑖=1(𝑑𝑖 𝐶𝑖 𝑝
𝑖 𝑚𝑖 𝑣𝑝𝑖 )−𝑟
𝑒𝑢
≝ 𝑞̃
[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 (∑𝑛𝑖=1(𝑑𝑖𝑙 𝐶𝑖𝑙 𝑝̃𝑖 − 𝑑𝑖𝑢 𝑚𝑖𝑢 𝑣𝑝𝑖 ) > 𝑟 𝑢 ). 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 (𝑀(𝑡) ≤ min(𝑚𝑖𝑢 )) in the case of one species but not in the case of
𝐶𝑙
𝑖
𝑖
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 (∑𝒏𝒊=𝟏(𝑑𝑖𝑙 𝐶𝑖𝑙 𝑝̃𝑖 −
7
𝑑𝑖𝑢 𝑚𝑖𝑢 𝑣𝑝𝑖 ) > 𝒓𝒖 ).
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
𝑑𝑋(𝑡)
𝑑𝑡
= 𝑋(𝑡)[𝑏 + 𝛼𝐺(𝑅𝑀→𝑋 , 𝑡) − 𝛽𝐿(𝑅𝑋→𝑀 , 𝑡) − [𝐶(𝑋, 𝑡) − 𝑚𝑀]𝑌(𝑡) − 𝑎𝑋(𝑡)]
10
dY(𝑡)
𝑑𝑡
= Y(t)[𝑑(C(𝑋, 𝑡) − mM)𝑋(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
𝑑𝑋(𝑡)
𝑑𝑡
= 𝑋(𝑡) [𝑏 + 𝛼 h+M − 𝛽 k+X(t) − (1+𝑤𝑐𝑋(𝑡)2 − 𝑚𝑀) 𝑌(𝑡) − 𝑎𝑋(𝑡)]
15
dY(𝑡)
𝑑𝑡
= Y(t) [𝑑 (
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
𝑐𝑋(𝑡)
1+𝑤𝑐𝑋(𝑡)2
lM
𝑐𝑋(𝑡)
[Eqn. B 3-4]
− mM) 𝑋(t) − eY(t) − r]
gM
lM
−𝛽
−𝑎𝑋
h+M
k+X
𝑐𝑋
−𝑚𝑀
1+𝑤𝑐𝑋2
𝑏+𝛼
19
𝑌𝑋 =
20
Which exists if 𝑚(1+𝑤𝑐𝑋 2 ) ≠ 𝑀, and:
21
[Eqn. B 1-2]
[Eqn. B 5]
𝑐𝑋
𝑑𝑐𝑋 2
𝑌𝑦 = 𝑒(1+𝑤𝑐𝑋 2 ) −
𝑑mM𝑋
𝑒
r
[Eqn. B 6]
−𝑒
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:
𝜕𝑋
𝜕𝑋
15
|𝜕𝑋
𝜕𝑌
𝜕𝑌
|
𝜕𝑌
𝜕𝑋
𝐴=
=|
𝜕𝑌
𝐶=
2𝑑𝑌𝑐𝑋
1+𝑤𝑐𝑋 2
(1 −
𝜕𝑋
𝜕𝑋
𝑐𝑋 2
1+𝑤𝑐𝑋 2
𝐵 = 𝑋(𝑚𝑀 −
) − 𝑚𝑑𝑌𝑀
𝐷 = 𝑑𝑋 (
𝑐𝑋
1+𝑤𝑐𝑋 2
𝑐𝑋
1+𝑤𝑐𝑋 2
)
− 𝑚𝑀) − 2𝑒𝑌 − 𝑟
|
𝜕𝑋
16
The expression of A ( 𝜕𝑋) is more complicated than the other partial derivatives but can be
17
simplified as follows:
18
𝐴=𝑏+
19
We can assess how M qualitatively influences the structure of the Jacobian Matrix. In particular,
20
we should note that as M increases
𝛼𝑀
ℎ+𝑀
−𝛽
𝑙𝑀
𝑋
2𝑐𝑌𝑋
𝑐𝑤𝑋 2
(1 −
)−
(1 −
)−
2
𝑘+𝑋
𝑘+𝑋
1+𝑤𝑐𝑋
1+𝑤𝑐𝑋 2
21
1) B increases
22
2) C decreases
12
2𝑎𝑋 + 𝑚𝑌𝑀
[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
𝜕𝐴
𝜕𝑀
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 𝜕𝑀 < 0 , which could stabilize the system
9
(Eqn. B1-2). Otherwise, mycorrhizal fungi might destabilize the prey-predator system (Fig. 6).
𝛼𝑀
= − (ℎ+𝑀)2 −
𝛽𝑙
𝑋
(1 −
)+
𝑘+𝑋
𝑘+𝑋
[Eqn. B 8]
𝑚𝑌
𝜕𝐴
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
𝑑𝑋(𝑡)
𝑑𝑡
= 𝑋(𝑡)[𝑏 + 𝛼𝐺(𝑅𝑀→𝑋 , 𝑡) − 𝛽𝐿(𝑅𝑋→𝑀 , 𝑡) − [𝐶(𝑋, 𝑡) − 𝑚𝑀(𝑡)]𝑌(𝑡) − 𝑎𝑋(𝑡)]
[Eqn. B 8-9]
16
𝑑𝑀(𝑡)
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
𝑑𝑋(𝑡)
𝑑𝑡
20
𝑑𝑀(𝑡)
g 𝑀 X(t)
l𝑀 X(t)
= 𝑀(𝑡) [𝛼𝑀
− 𝛽𝑀
− 𝑟𝑀 ]
𝑑𝑡
h𝑀 + X(t)
k 𝑀 + M(t)
𝑑𝑡
= 𝑀(𝑡)[∑𝑛𝑖=1 𝛼𝑀,𝑖 (𝑡)𝐺𝑀,𝑖 (𝑅𝑋𝑖→𝑀 , 𝑡) − 𝛽𝑀,𝑖 (𝑡)𝐿𝑀,𝑖 (𝑅𝑀→𝑋𝑖 , 𝑡) − 𝑟𝑀 ]
gM(t)
lM(t)
𝑐𝑋(𝑡)
= 𝑋(𝑡) [𝑏 + 𝛼 h+M(t) − 𝛽 k+X(t) − [1+𝑤𝑐𝑋(𝑡)2 − 𝑚𝑀(𝑡)] 𝑌 − 𝑎𝑋(𝑡)]
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
𝜕𝑋
| 𝜕𝑋
𝜕𝑀
𝜕𝑋
𝜕𝑋
𝜕𝑋
𝐴=
𝜕𝑋
𝜕𝑀| = |
|
𝑀𝑔𝑀 𝑋
𝑀𝑙𝑀 𝑋
𝜕𝑀
𝐶2 = −𝛼𝑀
− 𝛽𝑀
2
(ℎ𝑀 + 𝑋)
𝑘𝑀 + 𝑀
𝜕𝑀
𝑔𝑀
𝑙
−𝛽
2
(ℎ + 𝑀)
𝑘+𝑋
||
𝑔𝑀 𝑋
𝑙𝑀 𝑋
𝑀
𝐷2 = 𝛼𝑀
− 𝛽𝑀
(1 −
) − 𝑟𝑀
ℎ𝑀 + 𝑋
𝑘𝑀 + 𝑀
𝑘𝑀 + 𝑀
𝐵2 = 𝑌𝑚 − 𝛼𝑋
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