Growth

growth!(; t, p, calc, biomass, WR)

Calculates the actual growth of the plant species.

potential_growth!(; calc, SLA, biomass, PAR, potgrowth_included)

Calculates the potential growth of all plant species in a specific patch.

This function is called each time step (day) for each patch. The PAR value is the photosynthetically active radiation of the day.

First, the leaf area indices of all species are calculated (see Growth.calculate_LAI). Then, the total leaf area is computed. An inverse exponential function is used to calculate the total primary production:

\[\text{totalgrowth} = PAR \cdot RUE_{max} \cdot (1 - \text{exp}(-\alpha \cdot \text{LAItot}))\]

This primary production is then multiplied with the share of the leaf area index of the individual species

calculate_LAI(; SLA, biomass, LAIs)

Calculate the leaf area index of all species of one habitat patch.

\[\begin{align} \text{LAI} &= \text{SLA} \cdot \text{biomass} \cdot \text{LAM} \\ \text{LAI}_{\text{tot}} &= \sum \text{LAI} \end{align}\]

• SLA specific leaf area [m² g⁻¹]
• LAM Proportion of laminae in green biomass [unitless], the value 0.62 is derived by [20]
• biomass [kg ha⁻¹]

There is a unit conversion from the SLA and the biomass to the unitless LAI involved.

The array LAIs is mutated inplace.

radiation_reduction(; PAR, radiation_red)

Reduction of radiation use efficiency at light intensities higher than 5 $MJ\cdot m^{-2}\cdot d^{-1}$

\[\text{Rred} = \text{min}(1, 1-\gamma_1(\text{PAR}(t) - \gamma_2))\]

The equations and the parameter values are taken from [21].

• γ₁ is the empirical parameter for a decrease in RUE for high PAR values, here set to 0.0445 [m² d MJ⁻¹]
• γ₂ is the threshold value of PAR from which starts a linear decrease in RUE, here set to 5 [MJ m⁻² d⁻¹]

comment to the equation/figure: PAR values are usually between 0 and 15 $MJ\cdot m^{-2}\cdot d^{-1}$ and therefore negative values of Rred are very unlikely

height_influence!(;
    calc, biomass, height, height_included, height_strength)

\[\text{heightinfluence} = 1 + \frac{\text{height}\cdot\text{height}_{\text{strength}}}{\text{height}_{\text{cwm}}} -\text{height}_{\text{strength}}\]

• height_strength lies between 0 (no influence) and 1 (strong influence of the plant height)
• the community weighted mean height height_cwm is calculated by Growth.community_weighted_mean_height

In these plots all three plant species have an equal biomass:

community_weighted_mean_height(; calc, biomass, height)

\[\text{height}_{\text{cwm}} = \frac{\sum \text{biomass} \cdot \text{height}}{\sum \text{biomass}}\]

temperature_reduction(; T, temperature_red)

Reduction of the potential growth if the temperature is low or too high with a step function.

\[\text{temperature_reduction}(T) = \begin{cases} 0 & \text{if } T < T_0 \\ \frac{T - T_0}{T_1 - T_0} & \text{if } T_0 < T < T_1 \\ 1 & \text{if } T_1 < T < T_2 \\ \frac{T_3 - T}{T_3 - T_2} & \text{if } T_2 < T < T_3 \\ 0 & \text{if } T > T_3 \\ \end{cases}\]

Equations are taken from [22] and theses are based on [21]. T₁ is in [22] a species specific parameter, but here it is set to 12°C for all species.

• T₀ is the lower temperature threshold for growth, here set to 3°C
• T₁ is the lower bound for the optimal temperature for growth, here set to 12°C
• T₂ is the upper bound for the optiomal temperature for growth, here set to 20°C
• T₃ is the maximum temperature for growth, here set to 35°C

water_reduction!(;
    calc,
    fun_response,
    WR,
    water_red,
    PET,
    PWP,
    WHC)

See for details: Water stress

nutrient_reduction!(;
    calc,
    fun_response,
    nutrient_red,
    nutrients)

See for details: Nutrient stress

seasonal_reduction(; ST, season_red)

Reduction of growth due to seasonal effects. The function is based on the yearly cumulative sum of the daily mean temperatures (ST).

\[\text{seasonal}(ST) = \begin{cases} SEA_{min} & \text{if } ST < 200 \\ SEAₘᵢₙ + (SEAₘₐₓ - SEAₘᵢₙ) * \frac{ST - 200}{ST₁ - 400} & \text{if } 200 < ST < ST₁ - 200 \\ SEA_{max} & \text{if } ST₁ - 200 < ST < ST₁ - 100 \\ SEAₘᵢₙ + (SEAₘᵢₙ - SEAₘₐₓ) * \frac{ST - ST₂}{ST₂ - ST₁ - 100} & \text{if } ST₁ - 100 < ST < ST₂ \\ SEA_{min} & \text{if } ST > ST₂ \\ \end{cases}\]

This empirical function was developed by [20]. In contrast to [20] SEAₘᵢₙ, SEAₘₐₓ, ST₁ and ST₂ are not species specific parameters, but are fixed for all species. The values of the parameters are based on [20] and were chosen to resemble the mean of all functional groups that were described there.

A seasonal factor greater than one means that growth is increased by the use of already stored resources. A seasonal factor below one means that growth is reduced as the plant stores resources [20].

• ST is the yearly cumulative sum of the daily mean temperatures
• SEAₘᵢₙ is the minimum value of the seasonal effect, here set to 0.67 [-]
• SEAₘₐₓ is the maximum value of the seasonal effect, here set to 1.33 [-]
• ST₁ and ST₂ are parameters that describe the thresholds of the step function, here set to 625 and 1300 [°C d]

below_ground_competition!(;
    below,
    traitsimilarity_biomass,
    biomass, below_included,
    trait_similarity,
    below_competition_strength)

Models the below-ground competiton between plant.

Plant growth is reduced if a large biomass of plant species with similar functional traits is already present. The below_competition factor has a value between 0 and 1. For plant species $i$ with $N$ plant species present it is defined as follows:

\[\text{below_competition}_i = exp\left(-\frac{\text{below_competition_strength}}{1000} \cdot \left[\sum_{u=1}^{u=N} \text{trait_similarity}_{i,u} \cdot \text{biomass}_u\right] \right)\]

The below_competition_strength can therefore be seen as a parameter that controls the density dependence.

The trait_similarity is computed before the start of the simulation (calculation of traits similarity). and includes the traits arbuscular mycorrhizal colonisation rate (AMC) and the root surface area devided by the above ground biomass (SRSA_above).

senescence!(; sen, ST, biomass, μ)

\[\begin{align} LL &= 10 ^ { \left(log10(SLA) - 2.41\right) / -0.38} \cdot\frac{365.25}{12} \\ μ &= \frac{\text{sen_intercept}}{1000} + \frac{\text{sen_rate}}{1000} \cdot \frac{1}{LL} \\ \text{senescence} &= μ \cdot \text{SEN} \cdot \text{biomass} \end{align}\]

• LL leaf life span [$d$]
• SLA specific leaf area [$\frac{cm^2}{g}$] $\rightarrow$ this includes a unit conversion of the SLA values (in the model the unit of SLA is $\frac{m^2}{g}$)
• μ leaf senescence rate [$\frac{1}{d}$]
• SEN seasonal component of the senescence (between 1 and 3)
• sen_intercept α value of a linear equation that models the influence of the leaf senescence rate μ on the total senescence rate
• sen_rate β value of a linear equation that models the influence of the leaf senescence rate μ on the total senescence rate

The parameters $\text{sen_intercept}$ and $\text{sen_rate}$ were divided by 1000 to avoid very low numbers.

seasonal_component_senescence(;
    ST,
    Ψ₁ = 775,
    Ψ₂ = 3000,
    SENₘᵢₙ = 1,
    SENₘₐₓ = 3)

Seasonal factor for the senescence rate.

\[\begin{align*} SEN &= \begin{cases} SEN_{min} & \text{if} \;\; ST < Ψ_1 \\ SEN_{min}+(SEN_{max} - SEN_{min}) \frac{ST - Ψ_1}{Ψ_2 - Ψ_1} & \text{if}\;\; Ψ_1 < ST < Ψ_2 \\ SEN_{max} & \text{if}\;\; ST > Ψ_2 \end{cases} \\ \\ \end{align*}\]

• ST yearly accumulated degree days [$°C$]
• $Ψ₁=775$ [$°C\cdot d$]
• $Ψ₂=3000$ [$°C\cdot d$]
• $SEN_{min}=1$
• $SEN_{max}=3$

grazing_parameter(; LNCM, leafnitrogen_graz_exp)

Initialize the grazing parameter ρ.

\[\rho = \left(\frac{LNCM}{\overline{LNCM}}\right) ^ {\text{leafnitrogen_graz_exp}}\]

• LNCM leaf nitrogen per leaf mass
• leafnitrogen_graz_exp exponent of the leaf nitrogen per leaf mass in the grazing parameter
• ρ appetence of the plant species for livestock, dependent on nitrogen per leaf mass (LNCM) [dimensionless]

The function is excetued once at the start of the simulation. The grazing parameter ρ is used in the function Growth.grazing!.

grazing!(; calc, LD, biomass, relbiomass, ρ, grazing_half_factor)

\[\begin{align} μₘₐₓ &= κ \cdot \text{LD} \\ h &= \frac{1}{μₘₐₓ} \\ a &= \frac{1}{\text{grazing_half_factor}^2 \cdot h} \\ \text{totgraz} &= \frac{a \cdot (\sum \text{relbiomass}⋅\text{biomass})^2} {1 + a\cdot h\cdot (\sum \text{relbiomass}⋅\text{biomass})^2} \\ \text{share} &= \frac{ \rho \cdot \text{biomass}} {\sum \left[ \rho \cdot \text{biomass} \right]} \\ \text{graz} &= \text{share} \cdot \text{totgraz} \end{align}\]

It is thought that animals consume more in areas with greater biomass, resulting in greater trampling damage (see parameter relbiomass).

• LD daily livestock density [livestock units ha⁻¹]
• κ daily consumption of one livestock unit [kg d⁻¹], follows [23]
• ρ appetence of the plant species for livestock, dependent on nitrogen per leaf mass (LNCM) [dimensionless], initiliazed by the function Growth.grazing_parameter
• relbiomass: relative biomass of the patch in relation to the mean biomass of the whole grassland, is calculated by Growth.calculate_relbiomass! [-]
• grazing_half_factor is the half-saturation constant [kg ha⁻¹]
• equation partly based on [22]

Influence of grazing (livestock density = 2), all plant species have an equal amount of biomass (total biomass / 3) and a leaf nitrogen content of 15, 30 and 40 mg/g:

• leafnitrogen_graz_exp = 1.5

• leafnitrogen_graz_exp = 5

Influence of grazing_half_factor:

mowing!(;
    calc,
    mowing_height,
    days_since_last_mowing,
    height,
    biomass,
    mowing_mid_days)

\[\begin{align} \lambda &= \frac{\text{mown_height}}{\text{height}}\\ \text{mow_factor} &= \frac{1}{1+exp(-0.1*(\text{days_since_last_mowing} - \text{mowing_mid_days})}\\ \text{mow} &= \lambda \cdot \text{biomass} \end{align}\]

The mow_factor has been included to account for the fact that less biomass is mown when the last mowing event was not long ago. Influence of mowing for plant species with different heights ($height$):

Visualisation of the mow_factor:

trampling_parameter(; trampling_factor)

Intialize the trampling factor.

To avoid a very small trampling factor for paramter inference, the trampling factor is divided by 10000 and the unit [ha m⁻¹] is added.

The trampling_factor is used in the function Growth.trampling!.

trampling!(; calc, LD, biomass, relbiomass, height, trampling_factor)

\[\begin{align} \text{trampled_proportion} &= \text{height} \cdot \text{LD} \cdot \text{trampling_factor} \\ \text{trampled_biomass} &= \min(\text{relbiomass} ⋅ \text{biomass} ⋅ \text{trampled_proportion}, \text{biomass}) \\ \end{align}\]

It is assumed that tall plants (trait: height) are stronger affected by trampling. A linear function is used to model the influence of trampling.

It is thought that animals consume more in areas with greater biomass, resulting in greater trampling damage (see parameter relbiomass).

Maximal the whole biomass of a plant species is removed by trampling.

• biomass [$\frac{kg}{ha}$]
• relbiomass: relative biomass of the patch in relation to the mean biomass of the whole grassland, is calculated by Growth.calculate_relbiomass! [-]
• LD daily livestock density [$\frac{\text{livestock units}}{ha}$]
• trampling_factor [$ha$], this parameter is initiliazed with the function Growth.trampling_parameter
• height canopy height [$m$]

calculate_relbiomass!(; calc, p)

Relative biomass of the patches in relation to the mean biomass of the overall grassland.

\[\text{relbiomass} = \frac{\text{patch_biomass}}{\text{mpatch_biomass}}\]

• relbiomass relative biomass of each patch [-]
• patch_biomass sum of the biomass of all species in one patch [kg ha⁻¹]
• mpatch_biomass mean of the sum of the biomass of all species in all patches [kg ha⁻¹]

clonalgrowth!(; p, calc)

Yearly clonal growth.

\[\begin{align} \text{growth_factor} &= \frac{0.05}{\text{nneighbours}} \\ \text{crowded_factor} &= \min(\frac{\text{msurrounded_biomass}}{\text{biomass_target}}, 2.0) \\ \text{clonalgrowth} &= \text{growth_factor} \cdot \text{crowded_factor} \cdot \text{biomass} \\ \end{align}\]

The biomass is transferred from the home patch to the neighbour (target) patches. This is done for all patches once per year.

• clonalgrowth: biomass that is transferred from the home to the target patch [kg ha⁻¹]
• nneighbours: number of neighbour patches of the home patch. For a grid this value lies between 2 (edge) and 4 (middle).
• msurrounded_biomass: mean biomass of the home and the (upto 4) neighbour patches [kg ha⁻¹]
• biomass_target: biomass of the target patch [kg ha⁻¹]
• growth_factor: proportion of biomass that is transferred from the home patch to one neighbour patch. This factor is modified by the crowded_factor [-]
• crowded_factor: factor to adapth clonal growth based on the biomass distribution of the patches in the direct surroundings. The value lies between 0 (no clonal growth due to high surrounded biomass) and 2 (high clonal growth due to high own biomass).