# ForGEM Probability of combination of parent trees

The probability that gametes of a mother tree Mi and father tree Fj meet can be estimated by the fraction pollen of Fj that arrive at the position of Mi, relative to the contribution to all other known and unknown father trees. The amount of pollen of any father tree arriving at the position of a mother tree depends on the the amount of pollen produced by the father tree, the distance between the target mother tree and all possible father trees, the wind direction relative to the orientation between the mother and the father, and the overlap in flowering phenology between the father and the mother trees.

The general equation for the decline of the amount of pollen with distance is [1]:

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 b slope parameter indicating the rate of decline of the amount of pollen d distance from father tree y0 maximum number of pollen produced by the father tree, at d=0

The processes affecting these 3 parameters are described in this section. It is thereby assumed that there is no incompatability between genotypes.

### fraction of pollen from father tree arriving at target mother tree

The direction of the wind affects the slope parameter, whereas the overlap in flowering phenology between the target mother tree Mi and k potential father trees determines which portion of the pollen emitted by the father tree Fj can actually pollinate a given flowering mother tree.

Thus, the fraction of Fj pollen arriving at position Mi can be described as:

$i} \, \times\, F_{j} )\; =\; \frac{\, y_{0} \left(F_{j} \right)\, \, \cdot \, e^{-b\left(M_{i} ,F_{j} \right)\; \cdot \; d\left(M_{i} ,F_{j} \right)} \, \, \cdot \, t\left(M_{i} ,F_{j} \right)}{\sum _{k}\left(\, y_{0} \left(F_{k} \right)\, \, \cdot \, \, e^{-b\left(M_{i} ,F_{k} \right)\; \cdot \; d\left(M_{i} ,F_{k} \right)} \, \cdot \, t\left(M_{i} ,F_{k} \right)\, \right)\, +\; E_{M_{i} }$

 b(Mi,Fj) slope parameter effected by of wind direction d(Mi,Fj) distance between mother tree Mi and father tree Fj y0(Fj) amount of pollen of father tree Fj at distance d= 0 t(Mi,Fj) effect of phenology, i.e. overlap in flowering phenology EMi amount of external pollen arriving at mother tree Mi

### effect of wind direction on slope parameter

$i} ,F_{j} )\; =\; b_{0} +\; m\cdot \cos \left(\alpha _{w$

 b(Mi,Fj) direction-dependent slope parameter b0 slope parameter when cos(x) = 0, i.e. at wind direction perpendicular to direction of Mi to Fj tree αw main wind direction α direction from Fj tree to Mi tree m magnitude parameter

### distance between parental trees

The Euclidean distance between father and mother trees is determined based on the following equation:

$i} ,F_{j} )\; =\; \sqrt{\left(x_{i} -x_{j} \right)^{2} +\left(y_{i} -y_{j} \right)^{2}$

### effect of phenology on flowering overlap between mother and father trees

$i},F_{j} ) = \begin{cases} |tM_{i} -tF_{j}| \le tFL} & \longrightarrow \frac{tFL- |tM_{i} -tF_{j} |} {tFL} \\ |tM_{i} -tF_{j}| > tFL} & \longrightarrow 0 \end{cases$

 t(Mi, Fj) fraction of overlap of flowering between Mi and Fj trees, relative to the flowering duration of the Mi tree tFL duration of flowering of trees tMi timing of flowering of mother tree Mi tFj timing of flowering of father tree Fj

It is assumed that the duration of flowering of trees, tFL, is the same for all genotypes.

### references

1. Degen, B., Gregorius, H.-R., & Scholz, F., (1996). "ECO-GENE, a model for simulation studies on the spatial and temporal dynamics of genetic structures of tree populations." Silvae genetica 45: 323-329.