Mixture enhances productivity in a two-species forest: evidence from a modeling approach

Abstract

The effect of mixture on productivity has been widely studied for applications related to agriculture but results in forestry are scarce due to the difficulty of conducting experiments. Using a modeling approach, we analyzed the effect of mixture on the productivity of forest stands composed of sessile oak and Scots pine. To determine whether mixture had a positive effect on productivity and if there was an optimum mixing proportion, we used an aggregation technique involving a mean-field approximation to analyze a distance-dependent individual-based model. We conducted a local sensitivity analysis to identify the factors that influenced the results the most. Our model made it possible to predict the species proportion where productivity peaks. This indicates that transgressive over-yielding can occur in these stands and suggests that the two species are complementary. For the studied growth period, mixture does have a positive effect on the productivity of oak-pine stands. Depending on the plot, the optimum species proportion ranges from 38 to 74% of oak and the gain in productivity compared to the current mixture is 2.2% on average. The optimum mixing proportion mainly depends on parameters concerning intra-specific oak competition and yet, intra-specific competition higher than inter-specific competition was not sufficient to ensure over-yielding in these stands. Our work also shows how results obtained for individual tree growth may provide information on the productivity of the whole stand. This approach could help us to better understand the link between productivity, stand characteristics, and species growth parameters in mixed forests.

Appendices

Appendix

Aggregating the distance-independent individual-based model

Given a distance-independent individual-based model:

$$ \Updelta r_{i,j} = \gamma_{j} + \beta_{j} {\text{girth}}_{i,j} $$

(11)

where Δr _i,j is the radial increment of a tree i belonging to a species j between time t and time t + Δt, girth _i,j is the girth at time t for a tree i. Starting from Eq. (11), we can develop a stand model for species j using an aggregation approach. The stand can be defined with three aggregated variables for each species: the number of trees N _j , the mean radius $\bar{r}_{j}$ and the basal area G _j . The dynamic equations of these variables must be defined using Eq. (11). Since we assume that there is neither mortality nor recruitment between t and t + Δt, we have $\Updelta N_{j} = 0$. The mean radius is defined as follows:

$$ \bar{r}_{j} (t) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {r_{i,j} } $$

where $\bar{r}_{j} (t)$ is the mean radius at time t. The mean radius increment can thus be written as a function of the individual radial increments:

$$ \Updelta \bar{r}_{j} = \bar{r}_{j} (t + \Updelta t) - \bar{r}_{j} (t) = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {r_{i,j} (t + \Updelta t)} - \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {r_{i,j} (t)} = \frac{1}{{N_{j} }}\sum\limits_{i = 1}^{{N_{j} }} {\Updelta r_{i,j} } $$

It follows from Eq. (11) that:

$$ \sum\limits_{i = 1}^{{N_{j} }} {\Updelta r_{i,j} } = \gamma_{j} N_{j} + 2\pi \beta_{j} N_{j} \bar{r}_{j} $$

And the mean radius increment is given by:

$$ \Updelta \bar{r}_{j} = \gamma_{j} + 2\pi \beta_{j} \bar{r}_{j} $$

Similarly, ΔG _j can be written as a function of the individual basal area increments ($\Updelta g_{i,j}$):

$$ \Updelta G_{j} = G(t + \Updelta t) - G(t) = \sum\limits_{i = 1}^{{i = N_{j} }} {g_{i,j} (t + \Updelta t)} - \sum\limits_{i = 1}^{{i = N_{j} }} {g_{i,j} (t)} = \sum\limits_{i = 1}^{{i = N_{j} }} {\Updelta g_{i,j} } $$

where g _i,j is the basal area of a tree i and $\Updelta g_{i,j} = g_{i,j} (t + \Updelta t) - g_{i,j} (t)$. Since $g_{i,j} (t) = \pi (r_{i,j} (t))^{2}$ we can write $g_{i,j} (t + \Updelta t)$ as a function of $r_{i,j} (t)$, $r_{i,j} (t + \Updelta t)$ and $\Updelta r_{i,j}$:

$$ g_{i,j} (t + \Updelta t) = \pi (r_{i,j} (t + \Updelta t))^{2} = \pi (r_{i,j} (t) + \Updelta r_{i,j} )^{2} = \pi ((r_{i,j} (t))^{2} + 2r_{i,j} (t)\Updelta r_{i,j} + (\Updelta r_{i,j} )^{2} ) $$

Therefore:

$$ \Updelta g_{i,j} = 2\pi r_{i,j} (t)\Updelta r_{i,j} + \pi (\Updelta r_{i,j} )^{2} $$

It follows from Eq. 11 that:

$$ r_{i,j} (t)\Updelta r_{i,j} = \gamma_{j} r_{i,j} (t) + \beta_{j} 2\pi (r_{i,j} (t))^{2} = \gamma_{j} r_{i,j} (t) + \beta_{j} 2g_{i,j} (t) $$

and

$$ (\Updelta r_{i,j} )^{2} = \gamma_{j}^{2} + \gamma_{j} \beta_{j} 4\pi r_{i,j} (t) + \beta_{j}^{2} 4\pi^{2} (r_{i,j} (t))^{2} = \gamma_{j}^{2} + \gamma_{j} \beta_{j} 4\pi r_{i,j} (t) + \beta_{j}^{2} 4\pi g_{i,j} (t) $$

We can now express the individual basal area increment as a function of $r_{i,j} (t)$, $g_{i,j} (t)$ and the parameters of Eq. (11):

$$ \Updelta g_{i,j} = \pi \gamma_{j}^{2} + 2\pi \gamma_{j} (1 + 2\pi \beta_{j} )r_{i,j} (t) + 4\pi \beta_{j} (1 + \pi \beta_{j} )g_{i,j} (t) $$

Since $\sum\nolimits_{i = 1}^{{N_{j} }} 1 = N_{j}$, $\sum\nolimits_{i = 1}^{{N_{j} }} {r_{i,j} } = N_{j} \bar{r}_{j} (t)$ and $\sum\nolimits_{i = 1}^{{N_{j} }} {g_{i,j} } = G_{j} (t)$, we can sum the individual basal area increments to obtain the stand basal area increment:

$$ \Updelta G_{j} = \pi \gamma_{j}^{2} N_{j} + 2\pi \gamma_{j} (1 + 2\pi \beta_{j} )N_{j} \bar{r}_{j} + 4\pi \beta_{j} (1 + \pi \beta_{j} )G_{j} $$

Therefore, the system of equations for the stand model is:

$$ \left\{ \begin{gathered} \Updelta G_{j} = \pi \gamma_{j}^{2} N_{j} + 2\pi \gamma_{j} (1 + 2\pi \beta_{j} )N_{j} \bar{r}_{j} + 4\pi \beta_{j} (1 + \pi \beta_{j} )G_{j} \hfill \\ \Updelta \bar{r}_{j} = \gamma_{j} + 2\pi \beta_{j} \bar{r}_{j} \hfill \\ \Updelta N_{j} = 0 \hfill \\ \end{gathered} \right. $$

Optimum mixing proportion

Since $\Updelta G^{\prime}(x)$ is a polynomial equation of the second degree, its roots are:$x_{1} = \frac{{ - b - \sqrt {b^{2} - 4ac} }}{2a}$ and $x_{2} = \frac{{ - b + \sqrt {b^{2} - 4ac} }}{2a}$

The table below shows that for the nine plots, a is always positive so the function $\Updelta G^{\prime}(x)$ is convex. It is negative between x ₁ and x ₂ and positive for x < x ₁ et x > x ₂. x ₁ is thus a maximum for the function $\Updelta G(x)$.

Coefficients and roots of $\Updelta G^{\prime}(x)$ for the nine plots.

Plot	a	b	c	x 1	x 2
D02	1493	−64785	37871	0.593	43
D108	14688	−168693	69966	0.431	11
D20	15403	−172769	62590	0.375	11
D27	6466	−83284	37311	0.465	12
D42	5225	−74980	33143	0.457	14
D49	13152	−185208	82799	0.462	14
D534	9357	−102027	40109	0.408	10
D563	268	−31239	22837	0.736	116
D78	5457	−95660	52986	0.573	17