A model for the effect of photosynthate allocation and soil nitrogen on plant growth

A model for the effect of photosynthate allocation and soil nitrogen on plant growth
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  Ecological Modelling. 41 (1988) 55-65 55 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands A MODEL FOR THE EFFECT OF PHOTOSYNTHATE ALLOCATION AND SOIL NITROGEN ON PLANT GROWTH R. SIEV~,NEN 1 p. HARI 2, p.j. ORAVA 3 and P. PELKONEN 4 1 The Finnish Forest Research Institute, Department of Mathematics, Unioninkatu 40 A, SF-O0170 Helsinki 17 Finland) 2 University of Helsinki, Department of Silviculture, Unioninkatu 40 B, SF-O0170 Helsinki 17 Finland) 3 Helsinki University of Technology, Automation Technology Laboratory, Otakaari 5,4, SF-02150 Espoo 15 Finland) 4 University of Joensuu, Department of Forestry, PL 111, SF-80101 Joensuu 10 Finland) (Accepted 17 June 1987) ABSTRACT Sievanen, R., Hari, P., Orava, P.J. and Pelkonen, P., 1988. A model for the effect of photosynthate allocation and soil nitrogen on plant growth. Ecol. Modelling. 41: 55-65. A dynamical state model in which photosynthesis, the uptake of soil nitrogen, and the allocation of both photosynthates and plant nitrogen are taken into consideration in a simplified manner is developed for the growth of a plant. The dry-weight and nitrogen content of the plant are divided into above-ground and below-ground components. The dynamics of growth are presented with the aid of the plant dry weight and the nitrogen concentration of above-ground part of the plant. It is possible, using the model, to determine the effect of photosynthate allocation on the amount of growth. Predictions for photosyn- thate allocation at different soil nitrogen levels are produced using an optimality hypothesis. The results of an experiment on photosynthate allocation are described and compared with the predictions of the model. A critical discussion of the underlying assumptions, as well as of the results of the model, is also included. INTRODUCTION In a large number of plant stand-growth models the main emphasis has been placed on photosynthesis, that is, the carbon input (e.g. Promnitz and Rose, 1974; De Wit et al., 1978; Hari et al., 1982). Less attention has been focused on the allocation (or partitioning) of photosynthates, and its rela- tionship to other plant processes, e.g. nutrient and water uptake. The highly empirical approach to the partitioning of plant material applied in many plant growth models developed in ecology, agronomy and physiology is 0304-3800/88/$03.50 © 1988 Elsevier Science Publishers B.V.  56 considered a major shortcoming by Reynolds and Thornley (1982). If a plant-growth model is to be valid over a wide range of growth conditions, then adequate treatment of the partitioning of plant material and its relationship to environmental conditions is of paramount importance. One way of modelling the photosynthate allocation is based on the principle of the functional balance between various plant parts (e.g. Thorn- ley, 1972). In essence, this means that plant organs produce metabolites in amounts equivalent to those which the plant can consume. Perhaps the best-known application of this principle is the concept of root:shoot equi- librium (Davidson, 1969). It states that the root mass, multiplied by the root specific activity, is proportional to the shoot mass multiplied by the shoot specific activity. Another means of quantifying photosynthate allocation is to use hypotheses of optimality. It is assumed that a plant functions so as to optimize a certain criterion, such as the amount of photosynthates produced or the size of the seed crop (e.g. Givnish and Vermeij, 1976; Schulze et al., 1983). As far as photosynthate allocation is concerned, this means choosing an allocation pattern which optimizes the criterion over a range of environ- mental conditions. The optimality hypotheses are more general than the principle of functional balance, and they are therefore more difficult to apply to a particular case. In this paper, we present a dynamical model which incorporates photo- synthesis, nitrogen uptake, and allocation of photosynthates. The model is used for analysing the effect of a particular photosynthate allocation pattern on growth. An optimization approach is used to predict photosynthate allocation and plant biomass at different soil-nitrogen levels. This study is linked to research on intensively cultivated energy planta- tions utilizing growth models (Siev~inen, 1983). The experimental part of the present study, using Salix cv. Aquatica, was carried out in order to de- termine whether the theoretical results correspond to those obtained in the field. THE MODEL In the model the (willow) plant is considered to consist of two parts, the above-ground part (stem, branches and leaves) and the below-ground part (roots). The dry weights of these two parts are denoted by W a and Wb, respectively, and their sum is denoted by 141. The above-ground part fixes carbon and the roots take up nitrogen from the soil. The nitrogen uptake, N, is not taken into consideration in the dry weight, since it normally comprises a few per cent only. The amount of nitrogen in the plant is considered in terms of that in the above-ground part, Na, and that in the below-ground part, N b that is, N = N a + N b.  57 The main features of the model, as well as the growth conditions, are based on the following assumptions: (1) The photosynthetic rate of a plant is proportional to the dry weight of the above-ground part, W a, and depends on its nitrogen concentration, n = Na/W a. (2) The rate of nitrogen uptake of a plant is directly proportional to the soil nitrogen concentration n s, which is assumed to remain constant during the growth period, and to the dry weight of the roots, W b. (3) The photosynthetic products, as well as the nitrogen intake from the soil, are immediately allocated to the growth of above-ground and below- ground parts. (4) The coefficient for the allocation of photosynthates to the below- ground part, o~, is constant. (5) The amount of nitrogen consumed in growth in the above-ground and below-ground parts is proportional to their rate of increase in dry weight, the proportionality coefficient being higher in the above-ground part. On the basis of assumptions (1)-(4), the growth and nitrogen uptake of the plant are governed by the following equations: I/V= f(n) W a [,~Za ---- (1 -- 0~) Vv ~rb ---- O/Vgr /~ ---- fln sW b (la) (lb) (lc) where 2 means time derivative, f is a function which describes the depen- dence of the photosynthetic rate on the nitrogen concentration of the above-ground part, and fl is a constant. Following from assumption (5), the amount of nitrogen consumed in the above-ground and below-ground parts is proportional to their growth rates, respectively, but with different propor- tionality coefficients c a and c b as follows: Na = Cal/~Za, J'Qb = CbWb (ld) where c a > c b. Later on their values will be determined in terms of the model's other constants. The dynamics of growth and nitrogen uptake can actually be presented with the aid of the dry weight of the plant, W, and the nitrogen concentra- tion of the above-ground part, n. From (lb) it can be concluded that: l/~'ra (2) ~/b 1--~ If the same relation also holds for the initial values at time t = O, that is: Wa(O) (3) Wb(0) = 1 -- a  58 then also o~ Wb(t) = 1--a Wa(t)' Wa(t) = (1 - a) W(t), Wb(t)= a IV t) (4) for all t > 0. It is assumed in the sequel that (3) holds and hence (4) also holds. The values of c a and c b are determined by the requirement that nitrogen uptake and nitrogen consumption are equal, 5?= ~r + 5fb, and by the assumption that Cb/C a = k, where k < 1 (cf. assumption 5) is a constant. After some algebra they yield, together with (la-d) and (4): aflns and c b = kc b (5) Ca= f(n)(1 -- a)(l -- a + ak) On the basis of the above formulations, the differential equation describing the dynamics of n is obtained by taking derivatives of both sides of the defining equation with respect to time; thus: ft= ~--~ Na/Wa)d = /VaWa - aI'Vawa = (1 - a) f(n)(Ca-n ) (6) where the expression furthest to the right is obtained by substituting the derivatives from (la), (lb) and (ld). Note that the reasoning resulting in (6) is independent of equations (4). The dynamics of the model can now be presented with the aid of W and n by equations: I,;/= (1- a) f(n) W (7) h = fins - (1- a)n f(n) (8) (1 - + k.) where (7) is obtained by combining equations (la) and (4), and (8) follows from (6) by substitution of (5). The proper state variables of this version of the model are thus W and n and, using them, the quantities Wa, W b and N a can be computed algebraically. Function f The function f is chosen on the basis of two considerations: First, the experimental results indicate that f is a peaking-type function; i.e. there is a certain limit above which the increase in nitrogen concentration does not increase net photosynthesis, which may even be inhibited by excess nitrogen. Second, it is postulated, for mathematical reasons, that with high values of  TABLE 1 Parameter values used in the model 59 Parameter Unit Value/Range a O_<a<l k 0.25 fl gN 0.55 (g N/container) (g dw) 7 1/week 1.0 8 1.92 W(0) g dw 2.0 n, f decreases so slightly with increasing n that n f(n) -+ oo monotonically when n ~ ~. The semiempirical function: f(n) =7(lOOn)2/((lOOn) 2 8 + 8) (9) where 7 and 8 are constants, meets the above conditions. The values of 7, 8 and model's other parameters are chosen to be in approximate accordance with the results of the experiment (Table 1). STRUCTURE AND SOME PROPERTIES OF THE MODEL Equations (7) and (8) form a two-dimensional initial-value problem, the components of which are one-directionally coupled, i.e. (8) can first be solved giving n as a function of time. The solution of (7) can be readily given by the integral: W(t)= W(0)exp[(1--ot)fotf(rl(S)) ds] (10) where W(0) is the given initial value of W. The solution of (8) is most conveniently computed numerically. The properties of function f guarantee that the right-hand side of (8) has one, and only one, positive 0 (at ne). This is thus the only positive equilibrium value for n. By using, e.g., differential inequalities or Lyapunov theory, it is easily shown that n e is an asymptotically stable equilibrium with non-negative values of n as the attraction domain. The model (7)-(8) yields a type of behaviour where the nitrogen concentration tends to a steady state (Fig. 1) and the growth to strict exponential growth. The relative growth rate equals (1 - a) fine).
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