The Annual Cycle of Development of Trees and Process-Based Modelling of Growth to Scale Up From the Tree To the Stand

Abstract

Climate change affects both the annual cycle of tree development and the processes related to tree growth. The annual cycle of development manifests as observable phenological events such as leaf unfolding, flowering and leaf fall, but also includes less apparent traits, such as changes in frost hardiness and photosynthetic capacity. Seasonality in these traits can be due either to a fixed sequence of events that take place even in a constant environment, or to fluctuations in environmental factors. Thus, in a constant environment, the latter mode of development displays no seasonality. In addition, and depending on the trait considered, the internal state of development affects the tree’s capacity to respond to environmental factors. Given that the effects of climate change on the seasonality of a particular phenological trait may depend on interactions between fixed and fluctuating development traits, in order to explore these effects the entire annual cycle of development must be modelled. The processes related to tree growth include photosynthesis, respiration and allocation at the level of the individual tree; at stand level they include resource availability and biotic interactions. In this chapter we present the general theory of the annual cycle of development of trees, with examples of climate change effects on phenological traits with different mode of development for tree species in the boreal, temperate and Mediterranean zone of Europe. A process-based model on tree growth is outlined, with focus on scaling up from the tree to the stand level in time and space. Examples of climate change are presented, based on a model that couples the annual cycle of development and the growth of trees. Phenological events are characterized by responses to temperature that are under strong selective pressure. Future lines of development in this field of research include an assessment of the adaptive potential of phenological events to climate change. An example of this genetic approach is also presented.

Appendix

The annual cycle of development for boreal and temperate zone trees. See Fig. 1 for a schematic representation of the four phenological phases and processes.

Potential Rates

Phase 1: active growth

$$R_a (t) = \left\{ {\begin{array}{l} {0,T(t) < T_b } \\ {T(t) - T_b ,T(t) \ge T_b } \\\end{array}}\right.$$

(Eqn. A1)

$$R_o (t) = \left\{ {\begin{array}{l} {0,T(t) < T_{o,\min } } \\ {\frac{1}{{1 + e^{(a_o \bullet (T(t)+b_o ))} }},T(t) \ge T_{o,\min } } \\\end{array}} \right.$$

(Eqn. A2)

Phase 2: lignification

$$R_1 \left( t \right) = \left\{ \begin{array}{l} 0,T\left( t\right) < T_b\\ T\left( t \right) - T_b ,T\left( t \right) \ge T_b\\\end{array}\right.$$

(Eqn. A3)

Phase 3: rest

$$Rr(t)\left\{ \begin{array}{l} 0,T(t)T_{r,\min}\\ \frac{T(t) - T_{r,\min}}{T_{r,opt} - T_{r,\min}}, T_{r,\min} \leq T(t) \leq T_{r, opt}\\ \frac{T(t)- T_{r,\max }}{T_{r,opt}- T_{r,\max }},T_{r, opt}\langle T(t) \leq T_{r, \max}\\0, T(t)\rangle T_{r,\max }\\ \end{array} \right.$$

(Eqn. A4)

Phase 4: quiescence

$$R_o \left( t \right) = \left\{ \begin{array}{l} 0,T\left( t \right) < T_{o,\min }\\\frac{1}{{1 + e^{\left( {a_o\bullet \left( {T\left( t \right) + b_o } \right)} \right)} }},T\left( t \right) \ge T_{o,\min }\\\end{array} \right.$$

(Eqn. A5)

Frost hardiness

$$R_h \left( t \right) = \frac{1}{{\tau _h }} \bullet \left( {\hat S_h \left( t \right) - S_h \left( t \right)} \right)$$

(Eqn. A6)

$$\Delta \hat S_{hT} \left( t \right) = \left\{ \begin{array}{l}\Delta \hat S_{hT,\min } ,T\left. {\left( t \right)} \right\rangle T_{h,1}\\ \Delta \hat S_{hT,\max }\bullet \left( {1 - \frac{{T_{\min} \left( t \right) - T_{h,2} }}{{T_{h,1}- T_{h,2} }}}\right),T_{h,2}\le T\left( t \right)T_{h,1}\\ \Delta \hat S_{hT,\max} ,T\left( t \right)\langle {T_{h,2} } \\ \end{array} \right.$$

(Eqn. A7)

$$\Delta \hat S_{hP} (t) = \left\{ \begin{array}{l}\Delta \hat S_{hP,\min } ,NL(t)\langle {NL_{h,1} } \\ \frac{{\Delta\hat S_{hP,\max } }}{NL_{h,2}- NL_{h,1}} \bullet (NL(t) - NL_{h,1}),NL_{h,1}\leq NL(t) \leq NL_{h,2}\\\Delta \hat S_{hP,\max } ,NL(t) {NL_{h,2}} \rangle NL_{h, 2}\\0,Phase = 1 \wedge C_h= 0\end{array} \right.$$

(Eqn. A8)

Recovery of photosynthetic capacity

$$R_p= \frac{1}{{\tau _p }} \cdot \left( {\frac{1}{{1 + c_p\cdot a_p ^{ - \left( {\hat s_p- S_p } \right)} }} - \frac{1}{{1 + c_p\cdot a_p ^{\left( {\hat S_p- S_p } \right)} }}} \right)$$

(Eqn. A9)

States

Phase 1: active growth

$$S_a \left( t \right) = \int\limits_{t_{a,i} }^t {C_{aT} R_a \left( t \right)dt + C_{ap} NL\left( t \right)}$$

(Eqn. A10)

$$S_a \left( t \right) = \int\limits_{t_{a,i} }^t {C_o \left( t \right)R_o \left( t \right)dt}$$

(Eqn. A11)

end of active growth if $S_a \left. {\left( t \right)} \right\rangle S_a^*$ then:

● Phase = 2

● Sl(t) = 0

● tl,i = 0

Phase 2: lignification

$$S_l \left( t \right) = \int\limits_{t_{l,i} }^t {R_l } \left( t \right)dt$$

(Eqn. A12)

end of lignification if $S_l \left. {\left( t \right)} \right\rangle S_l^*$ then:

● Phase = 3

●

$$S_o \left( t \right) = 0$$

●

$$S_r \left( t \right) = 0$$

● t^o,i = 0

● t^o,i = 0

Phase 3: rest

$$S_r \left( t \right) = \int\limits_{t_{r,j} }^t {R_r \left( t \right)dt}$$

(Eqn. A13)

end of rest (= rest completion) if$S_r \left. {\left( t \right)} \right\rangle S_r^*$ then:

● Phase = 4

Phase 4: quiescence

$$S_o \left( t \right) = \int\limits_{t_{o,j} }^t {C_o } \left( t \right)R_o \left( t \right)a$$

(Eqn. A14)

$$C_h (t) = \left\{ {\begin{array}{l} {0,S_r (t) < S_r^* } \\ {1,S_r (t) \ge S_r^* } \\\end{array}}\right.$$

(Eqn. A15)

end of quiescence (= bud burst) if $S_o \left. {\left( t \right)} \right\rangle S_o^*$then:

● Phase = 1

●$S_a \left( t \right) = 0$

● t_a,i = 0

Frost hardiness

$$S_h \left( t \right) = \int\limits_0^t {R_h } \left( t \right)dt$$

(Eqn. A16)

$$\hat S_h \left( t \right) = \hat S_{h,\min }+ C_h \left( t \right)\left( {\Delta \hat S_{hT} \left( t \right)\Delta \hat S_{hP} \left( t \right)} \right)$$

(Eqn. A17)

$$C_h (t) = \left\{ {\begin{array}{l} {MAX\left( {0,1 - \frac{{S_o (t)}}{{S_h^* }}} \right),Phase = 1} \\ {\frac{{S_l (t)}}{{S_l^* }},Phase = 2} \\ {1,Phase = 3} \\ {MAX\left( {0,1 - \frac{{S_o (t)}}{{S_{o,h}^* }}} \right),Phase = 4} \\\end{array}} \right.$$

(Eqn. A18)

Fraction of needle area damaged by frost

$$d_f= a_f+ b_f e^{c_f .S_h \left( t \right)}$$

(Eqn. A19)

$$D\left( T \right) = \frac{1}{{1 + e^{e_f\cdot \left( {S_h \left( t \right) - T\min \left( t \right)} \right)} }}$$

(Eqn. A20)

Recovery of photosynthetic capacity

$$S_p \left( t \right) = \int\limits_o^t {R_p \left( t \right)dt}$$

(Eqn. A21)

$$\hat S_p \left( t \right) = b_p T\left( t \right)$$

(Eqn. A22)

$$K_p \left( t \right) = \frac{{S_p \left( t \right)}}{{S_p^* }}$$

(Eqn. A23)

Table A1 Variables for models of the annual cycle of trees

Table A2 Parameters for models of the annual cycle of trees