The Gaussian likelihood function used to fit the plant—N and plant—plant interaction models is as follows:
L ( P ,
 x
,
 y
,
 xy
)
 j
T 

1


 2

1
1
  xy exp


 2 ( 1

1
 xy
)
2



( x j

 x
2 x
ˆ j
)
2

2
 xy
( x j
 x
ˆ j
)( y j
 x
 y
 y
ˆ j
)

( y j

 y
2 y
ˆ j
)
2











.
Here, the likelihood for each set of parameters (P) is based on the variance of variable x (e.g. available N), the variance of y (e.g. plant population density) and their covariance (ρ xy
).
The structured Gaussian likelihood function that includes an extra parameter (α) to relate the modelled values of mean July air temperature to the chironomid assemblage data that they are based on is as follows:
L ( P ,
 x
,
 y
,
 xy
,

)
 j
T 

1


 2

1
1
  xy exp

 2

1

1
 xy

2



 x j
 

2 x j

2

2
 xy
 x j
 
 j x

 y j
 j
  y j
 j

2 y

2 y











.
This likelihood function was used to fit the plant—temperature models.
In the plant—N models with N-dependent population growth, x represents N availability and y represents plant population density. The models describe the change in x and y over time given an instantaneous population growth rate (r) and a function that describes the uptake of N that is used to increase population growth; this can either a linear or saturating (i.e. Michealis-Menton) function. Additionally, these models were applied with or without a logistic density-dependent limit on population growth. The loss rate of N from the system (γ x
) and the mortality rate of the plant population (γ y
) are described as a linear function of the current density.

Jeffers, Bonsall, Froyd, Brooks & Willis
N-Dependent Population Growth Models
Linear uptake
Linear uptake with density-dependent population growth
Saturating uptake
Saturating uptake with density dependent population growth dx dt
   axy
  x x dy dt
 raxy
  y y dx dt
   axy
  x x dy dt
 raxy ( 1
 y
K
)
  y y dx dt
   axy
1
 bx
  x x dy dt
 r axy
1
 bx
  y y dx dt
   axy
1
 bx
  x x dy dt
 r axy
1
 bx
( 1
 y
K
)
  y y
Here, λ is a constant, background supply of N to the ecosystem and γ x
, reflects a linear, densitydependent loss of nitrogen out of the system (i.e. the loss of N cannot exceed that which is available at any point of time). The conversion of available nitrogen into increased plant population growth is reflected by a (although in the saturating uptake model, this is explained by a and b). The density-dependent plant population mortality rate is represented by γ y
. In models with density-dependent population growth, the parameter K refers to the carrying capacity of the population.
The linear N uptake model without density-dependent controls on population growth provided the best description of the relationship between birch and pine population dynamics and available N. The maximum likelihood estimated (MLE) parameters and their standard errors for the pine—N and birch—N models are listed below.
Linear uptake of nitrogen model: dx
   axy
  x x dy dt
 raxy
  y y dt
Parameters Pine
λ 0.489 (0.1)
Birch
0.062 (0.003)
γ
N r
0.0001 (0.0001)
2.66 (0.336)
3.275 (1.947)
0.009 (0.007) a
γ x
4.812 (0.764)
16.986 (5.322)
45.531 (1.028)
0.00025 (0.00027)
λ has dimensions of L -2 T --1 , γ
N has dimensions of T --1 ,r has dimensions of L -2 T --1 , a has dimensions of 1 and
γ x has dimensions of T --1 .
The N-dependent population growth with a positive feedback model assumes that increasing plant mortality (i.e. as decaying plant biomass) leads to greater available N via an additional model parameter (α).
Supplementary Information p.
2

Jeffers, Bonsall, Froyd, Brooks & Willis
N-Dependent Population Growth Models with Positive Feedback on Available N
Linear uptake with feedback dx dt
   axy
  x x
  y y dy dt
 raxy
  y y
Linear uptake with feedback and densitydependent population growth dx dt
   axy
  x x
  y y dy dt
 raxy ( 1
 y
K
)
  y y
Saturating uptake with feedback dx dt
   axy
1
 bx
  x x
  y y dy dt
 axy r
1
 bx
  y y
Saturating uptake with feedback and density dependent population growth dx dt
   axy
1
 bx
  x x
  y y dy dt
 axy r
1
 bx
( 1
 y
K
)
  y y
There were no populations for which N-dependent population growth with a positive feedback on available N was the AIC-inferred best model.
The N independent population growth model assumes that the relationship between plant population density and N availability is based solely on the positive feedback of lost plant biomass is converted into available N (as described above). In these models, there is no effect of available N on population growth; therefore, plant population growth is limited only by density dependence. Two forms of density dependence were considered here: logistic and exponentially decaying effects.
N independent population growth with positive feedback on available N
Logistic density dependence dx dt
    x x
  y y dy dt
 ry ( 1
 y
K
)
  y y
Exponentially decaying density dependence dx dt
    x x
  y y dy dt
 ry exp(
 ay )
  y y
The N independent population growth models with exponentially increasing density-dependent controls on population abundance provided the best description of the relationship between oak and alder population dynamics and available N. The MLE parameters and their standard errors for the oak—N and alder—N models are listed below.
Supplementary Information p.
3

Jeffers, Bonsall, Froyd, Brooks & Willis
Nitrogen independent population growth: dx
    x x
 
* (
 y y ) dy
 ry * exp(
 ay )
  y y dt dt
Parameters
λ
γ
N
α
Oak
2.775 (0.137)
4.6 (0.024)
0.00011 (0.00003)
Alder
0.00002188 (0.000002)
0.014 (0.001)
0.00000299 (0.00000001) r a
0.072 (0.008)
0.000448 (0.00001)
0.094 (0.01)
0.00043331 (0.000014)
γ x
0.000617 (0.0002) 0.00002156 (0.0000009)
λ has dimensions of L -2 T --1 , γ
N has dimensions of T --1 , α has dimensions of 1, r has dimensions of L -2 T --1 , a has dimensions of 1 and γ x has dimensions of T --1 .
In the plant—temperature models, increasing mean July air temperature is hypothesized to have a positive effect on tree population growth via one of two functional forms: saturating or exponential decline. The saturating effect describes exponentially increasing population growth given increased temperature at low temperatures followed by an asymptotic response at a maximum temperature beyond which population growth no longer responds to higher temperatures. Alternatively, the exponential decline model assumes that each increment of temperature increase has an exponentially declining effect on population growth rate. In these models, x describes the dynamics of mean July air temperature and y describes the dynamics of plant population density. Temperature dynamics are independent of plant population density and are described as a function of temperatures at previous points in time. Plant population density dynamics involve an instantaneous rate of increase (r) that is modified by either of the alternative functions for describing the effect of temperature on population growth and by density-dependent limitations on growth (i.e. logistic density-dependence). There is also a linear, density-dependent mortality rate (γ).
Plant—Temperature Models
Saturating effect
Exponentially decaying effect dx dt
=x t-1
+
1 a x t=2 𝑑𝑦 𝑑𝑡
= 𝑟 ( 𝑏𝑥𝑦 𝑎+𝑐𝑥
) (1 − 𝑦
𝐾
) − 𝛾 𝑦 𝑦 𝑑𝑥 𝑑𝑡
= 𝑥 𝑡−1
+
1 𝑎 𝑥 𝑡=2 𝑑𝑦 𝑑𝑡
= 𝑟 𝑦 𝑒𝑥𝑝(−𝑏𝑥) (1 − 𝑦
𝐾
) − 𝛾 𝑦 𝑦
Supplementary Information p.
4

Jeffers, Bonsall, Froyd, Brooks & Willis
In the temperature equation, a determines the fractional impact of previous temperatures on current temperatures. In the plant population equation, the parameters a, b and c together describe the temperature-dependent plant population growth rate.
The exponentially decreasing effect of temperature on population growth provided the best fit between climate dynamics and the population dynamics of all four tree species. The MLE parameters and their standard errors for the plant—temperature models are listed below.
Exponentially declining effect of temperature on population growth: 𝑑𝑥 𝑑𝑡
= 𝑥 𝑡−1
+
1 𝑎 𝑥 𝑡=2 𝑑𝑦 𝑑𝑡
= 𝑟𝑦𝑒𝑥𝑝(−𝑏𝑥) (1 − 𝑦
𝐾
) − 𝛾 𝑦 𝑦
Parameters Pine Oak Birch Alder a r b
K
0.759 (0.089)
0.002 (0.00012)
396.803 (0.383)
24283 (798.963)
163.072 (2.311)
36.923 (4.253)
172.937 (6.499)
151263 (16248)
39.649 (2.942)
11.636 (1.635)
6.733 (0.409)
17828.(8792.847)
25.256 (0.716)
1.046 (0.128)
15.727 (0.495)
57962 (6103)
γ 1.42 (0.113) 19.781 (1.839) 4.053 (1.481) 6.615 (0.564)
γ has dimensions of T --1 , r has dimensions of L -2 T --1 , a has dimensions of 1, b has dimensions of 1 and K has dimensions of L -2 .
Plant-plant interactions between pairs of tree species were represented by the Lotka-Volterra competition model, dx dt
 r x x (
K x
 x
K x
  y
) dy
 r y y (
K y dt
 y
  x
) .
K y
In this model, x refers to the density of one plant population and y refers to the density of another. It includes an instantaneous rate of increase (r), logistic density-dependent growth
(determined by the carrying capacity, K) and a competition coefficients (α and β) that describes the per-capita effect of one species on the population growth rate of the other. This coefficient can be positive or negative, thus this model can demonstrate four forms of plant—plant interactions: competition (-,-), facilitation (+,-) or mutualism (+,+). The MLE parameters and their standard errors for the plant—plant interaction model for each species pair are listed below.
Supplementary Information p.
5

Jeffers, Bonsall, Froyd, Brooks & Willis r x
α
K x r y
Lotka-Volterra competition model
Parameters Pine (x) - Oak (y)
β
K y
Parameters r x
0.165 (0.005)
-0.035 (0.046)
9098.203 (224.34)
0.00003 (0.064)
-0.344 (0.442)
0.754 (8269.022)
Oak (x) – Birch (y)
10.022 (1.523)
Pine (x) –Birch (y)
6.7 (0.132)
0.584 (0.038)
12062.93 (1038.63)
9.101 (0.117)
0.001 (0.0002)
13463.336 (10.381)
Oak (x) – Alder (y)
0.067 (0.006)
Pine (x) –Alder (y)
2.606 (1.801)
1.086 (0.398)
3392.593 (592.308)
2.253 (0.769)
4.018 (0.059)
10671.742 (1978.88)
Birch (x) – Alder (y)
8.439 (0.01)
α
K x r y
1.366 (0.349)
2666.26 (1250.43)
9.581 (1.361)
-0.456 (0.442)
485.011 (2857.045)
2.428 (0.267)
-0.249 (1.234)
14991.11 (2841.56)
11.637 (5.714)
β
K y
2.033 (1.303)
14716.88 (4870.8)
-1.817 (0.49)
98.247 (420.434)
12.077 (0.715)
65647.281 (28380.164)
γ has dimensions of T --1 , r x
and r y have dimensions of L -2 T --1 , α and β have dimensions of 1, and K x and K y
have dimensions of L -2 .
Supplementary Information p.
6