A Comparison of Two Approaches for Modelling Cassava (Manihot esculenta Crantz.) Crop Growth

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1 Annals of Botany 85: 77 90, 2000 Article No. anbo , available online at http: on A Comparison of Two Approaches for Modelling Cassava (Manihot esculenta Crantz.) Crop Growth V. M. GRAY Department of Molecular and Cell Biology, Uni ersity of the Witwatersrand, Pri ate Bag 3, Wits 2050, South Africa Received: 10 May 1999 Returned for revision: 23 July 1999 Accepted: 21 September 1999 Two approaches for modelling the growth and development of cassava, Manihot esculenta Crantz, are described and evaluated. The two models differ only in the hypotheses accounting for storage root growth. In model 1, assimilate allocation to storage roots is governed by the combined Chanter s (1976: Mathematical models in mushroom research and production. PhD Thesis, University of Sussex, UK) growth equation; and in model 2 the spill-o er hypothesis for assimilate allocation to storage root governs storage root growth. In both models, canopy photosynthesis generates the carbon substrate required for all growth processes. The growth rates of leaves, stems and storage roots are defined by growth equations subject to substrate saturation kinetics. A key feature of both models is that the growth demands of the stem, fibrous roots and storage roots are related to leaf demand rates. Allocation to stems and branches was modelled by means of a modified logistic growth equation which includes all the parameters and variables (number of nodes, internode lengths, stem density, stem modulus of elasticity and branch tensile strength) that define the limits of the load bearing capacity of the shoot s supportive structures. The correlation coefficients for determination of yield prediction for the models were r 0 898, P (model 1) and r 0 954, P (model 2). For a growth season of 290 d (after which leaf area index equals zero and crop growth ceases), both models simulate the sigmoidal transition from the lag to exponential phase of crop growth. Both models are equally well corroborated by observed data; however, model 1 has greater explanatory power Annals of Botany Company Key words: Allocation, simulation, model, crop growth, cassava, Manihot esculenta Crantz. INTRODUCTION Several cassava crop growth models have been developed over the past decade (Cock et al., 1979; Fukai and Hammer, 1987; Gutierrez et al., 1988; Gijzen et al., 1990; Matthews and Hunt, 1994). However, none have achieved an adequate mechanistic solution to the problem of modelling the dynamic partitioning of biomass into shoot and storage roots. The amount of biomass partitioned into storage root production during cassava crop growth can be conceptualized in terms of the mass balance equation: dr s dt A d R G l G sb G rf dc (1) dt where R s represents the storage roots, A d represents canopy photosynthetic rate, R represents the total respiratory substrate demand, G l represents the gross growth rate of leaves and petioles, G sb represents the gross growth rate of stems and branches, G rf represents the gross growth rate of fibrous roots and C represents the reserve substrate carbon pool. One interpretation of eqn (1) can be articulated as follows: the growth rate of storage roots (dr s dt) will increase if any or all of the minus terms are reduced or if A d is increased. In this understanding of eqn (1), the right-hand side represents the rate of spill-o er or rate of production of surplus substrate for storage root growth. This under- vince gecko.biol.wits.ac.za $ standing of eqn (1) has inspired the spill-o er hypothesis, which can be stated as follows: biomass allocation to storage root growth equals the spill-over or the amount of substrate carbon that remains once the growth demands of the other cassava plant components have been fulfilled (Cock et al., 1979). The two major sinks for assimilate during cassava crop growth are the woody component of the shoot and the storage roots. The balance of assimilate allocation between these two sinks determines both harvest index and yield for the cassava crop (Cock et al., 1979). In recent cassava crop growth models the assimilate demand rates for stems and branches have been based on an allometric (Gutierrez et al., 1988) or empirical relationship (Matthews and Hunt, 1994) to the total assimilate demand for leaf growth. In the growth model of Matthews and Hunt (1994) the empirical function defining the ratio of woody stem to leaf growth rate has been formulated as follows: G sb φ w (2) G l 1 φ w where φ w is the ratio of the woody component of the stem to the total above ground shoot mass [eqn (22). The righthandside of eqn (2) defines the proportion of biomass allocated to the woody fraction of the shoot relative to the amount of biomass partitioned into leaf production. In the above empirical model, as crop growth progresses, φ w increases linearly from some initial setting to a cultivar-specific 2000 Annals of Botany Company

2 78 Gray Cassa a Crop Growth Modelling maximum value, after which φ w remains constant for the remaining phase of crop growth. This hyperbolic relationship between the ratio G sb G l and φ w is not an ad hoc one, but has empirical support (Connor et al., 1981, Fig. 3). In the current approach to modelling biomass partitioning into stem and branch structures, the term φ w in eqn (2) has been redefined so that eqn (2) instead of being a simple phenomenological relationship, now becomes a function of the magnitude of the load acting on the stems and branches. This dependency of φ w on the load supported by the stems and branches provides the mechanistic or biomechanical basis for the modelling of assimilate allocation to stems and branches. Considerable progress has been made over the last three decades in developing the theoretical basis for the mechanistic modelling of biomass partitioning in plants (Thornley, 1998). However, for further advances to be made in the understanding of biomass partitioning into the load bearing structures of the plant shoot, the relationship between shoot assimilate demand and the biomechanical dynamics (or statics) of the stems and branches needs to be made theoretically explicit and transparent. This is particularly important in modelling assimilate distribution in cassava, since our current understanding of the dynamics of biomass partitioning in cassava suggests that woody shoot growth occurs at the expense of storage root production (Cock et al., 1979). In the current study, the parameters and the way they influence biomass partitioning into stem and branch growth, and how this impacts on assimilate allocation to storage roots, have been defined. Thus a solution to the problem of modelling biomass partitioning into stems, branches and storage roots in cassava is proposed in this paper. With regard to the allocation of biomass to the storage roots, two cassava crop growth models are described here; one is based on the spill-over interpretation of eqn (1), and the other is based on a modified version of Chanter s equation (Chanter, 1976; Thornley and Johnson, 1990). The latter represents an alternative to modelling approaches based on the spill-over hypothesis. Both models share the key features of other cassava models, in that the growth of stems, fibrous roots, storage roots and reserves are related to substrate supply less that used for respiration and leaf growth. Recent applications of the spill-over hypothesis for modelling storage root production have not considered, in explicit fashion, the following two factors: (1) the feed-back control that the accumulating photosynthates in plant tissues exert on canopy photosynthesis; and (2) the effect storage root sink strength or photosynthetic assimilate accumulation capacity has on storage root growth rates. It will be shown that factor (1) can be applied in models based on the spill-over hypothesis; however, factor (2) cannot be incorporated into models based on the spill-over hypothesis. It will also be shown that the absence of the operation of factor (2) decreases the explanatory and predictive power of cassava crop growth models based on the spill-over hypothesis. The effects of the above two factors on assimilate fluxes into storage roots are not mutually exclusive. Decreased storage root sink capacity will result in photosynthetic assimilate accumulation in the shoot tissues which in turn will, by way of feed-back regulation, result in a reduction in the canopy photosynthetic rate. Application of Chanter s (1976) equation for modelling storage roots makes it possible to incorporate the consequences of factors (1) and (2) into the cassava crop growth model, and thus increase both the explanatory and predictive power of the model. The objectives of this study involve the following: (1) development of a mechanistic model for the dynamic partitioning of biomass into cassava shoots and fibrous roots, referred to as the core model; (2) development and comparison of two alternative models for storage root growth, one based on the Chanter s (1976) equation and the other based on the spill-over hypothesis; (3) calibration of the core model and the two storage root growth models, and then the testing of the outputs with appropriate cassava field trial data; and (4) evaluation of the two storage root models. Regarding definitions and terminology used in this study, the following should be noted. The term model 1 refers to the composite model which equals core model plus Chanter s (1976) equation for storage root growth; and the term model 2 refers to the composite model which equals core model plus spill-over hypothesis for storage root growth. COMPOSITE MODEL STRUCTURE The basic constituents of the cassava plant are: (1) the nodal units which consist of a leaf blade, petiole and internode; (2) thickened storage roots which form mainly at the base of the stem cutting that is used as planting material; and (3) the fibrous root system. The number and characteristics (length, mass density and strength) of internodes set the limits for the dimensions of stems and branches. A conceptual representation of the similarities and differences between model 1 and model 2 is given in Fig. 1. The two models differ only with respect to the relationship between the substrate pool and the storage root otherwise both models are based on the same core model (Fig. 1). Mathematically speaking, the two composite models, that is, the core model plus one of the two alternative root growth models, each consists of a set of differential equations which govern the dynamics of the 11 (model 1) or ten (model 2) primary state variables which have been used for the quantitative characterization of the basic constituents of the cassava plant (Table A1). The equations of the core model also represent or embody the hypotheses that have been formulated to explain the dynamics which govern biomass partitioning during cassava crop growth; these are the hypotheses shared by both models. The two models differ with respect to their hypotheses regarding storage root growth and assimilate allocation to the storage root. In model 1, Chanter s (1976) equation, which is based on the Logistic-Gompertz growth equation, is the hypothesis (hypothesis 1) for storage root growth and storage root assimilate allocation. In model 2, the assimilate spill-over hypothesis (hypothesis 2) governs storage root growth and storage root assimilate allocation. The description and dimensions of each model s variables are given

3 Storage root (Model 1) G rs Y rs G rf Y rf Fibrous roots Senescence Radiation Canopy photosynthesis Feed-back Control Substrate pool = Storage root (Model 2) G l Y l Leaves and Petioles Leaf area Temperature Fractional interception Stems and Branches Shoot load FIG. 1. Conceptual representation of the two cassava crop growth models. The large black arrows show the flux of carbon from canopy photosynthesis into the substrate storage pool from where it is partitioned into leaves, petioles, stems, branches, fibrous roots and storage roots. As can be seen from this conceptual scheme, models 1 and 2 differ only with respect to their relationship to the substrate pool. In model 1 the flux of assimilate into the storage root compartment depends on the value of G rs Y rs, whereas in model 2 the flux of assimilate into the storage root compartment equals the difference between the canopy photosynthesis input into and all the outputs (maintenance respiration, G l Y l, G sb Y sb, G rf Y rf ) from the substrate storage pool. The carbon flux into stems and branches is dependent on the magnitude of the load acting on the load-bearing components of the shoot. Gray Cassa a Crop Growth Modelling 79 G l, G sb, G rf and G rs (kg structural carbon m d ) are the rates for the synthesis of leaf, stem, fibrous root and storage root structural dry masses, respectively, Y l, Y sb, Y rf, Y rs [kg structural mass (kg substrate carbon) are the yield conversion efficiencies for the production of leaf, stem (and branch), fibrous root and storage root structural dry masses, respectively, R m (kg substrate carbon m d ) is maintenance respiration, D l and D r (d ) are the senescence rate constants for leaves (L) and fibrous roots (R f ), respectively. The term 0 3(D l L D r R f ) represents the amount of biomass recycled back to the substrate pool following leaf and root senescence. In model 1, the substrate reserve concentration [C is the ratio of C to total plant biomass minus storage root mass. Maintenance Thus in model 1, the substrate for both shoot and storage respiration G root growth is derived from the substrate reserve pool which sb Y is localized only in the shoot component of the plant: sb C [C (4) L S t B i R f C Substrate concentration values range from 5 6 to7 5% for sugars and 6 8 to17 7% for starch in the stems of cassava (Connor et al., 1981). For model 1, a modified version of the Chanter (1976) growth equation (hypothesis 1) has been used to model storage root growth: dr s dt s0 µr [C [C K T m 10 1 R s Ω1 exp( ωt) G (5) rs where µ (d ) is the storage root growth rate constant, K T m is the substrate concentration [C that gives 0 5 µ, G rs is the rate of storage root biomass production, and Ω and ω are constants. In this equation, the storage root growth rate constant µ is modified by two factors:the first (1 R s Ω), depends linearly on the magnitude of R s as in the logistic equation, the second, exp( ω t), depends on the passage of time, and can be interpreted as the progression or state of storage organ differentiation and development, as in the Gompertz equation (Thornley and Johnson, 1990). Both the logistic and Gompertz equations are contained in eqn (5) as special cases: with ω 0, it becomes the logistic equation and the final or maximum value for R s Ω; as Ω, it reduces to the Gompertz equation. in the Appendix (Table A1). The definitions, values and units of all the parameters used in the model s equations are also given in the Appendix (Table A2). Model 1 substrate and storage root components For model 1, the rate equation for the substrate or assimilate pool is: dc dt A d G l G sb G rf G rs R Y l Y sb Y rf Y m 0 3(D l L D r R f ) rs (3) where C (kg substrate carbon m ) is the carbon substrate, A d (kg substrate carbon m d ) is canopy photosynthesis, Model 2 substrate and storage root components For model 2, a spill-over equation (hypothesis 2) for storage root growth has been derived by modifying eqn (3) as follows: dc dt A d G l G sb G rf R Y l Y sb Y m rf 0 30 D l L D r f1 R dr s dt (assuming that Y rs 1 0). A consequence of the above formulation of the spill-over hypothesis is that in model 2 no distinction has been made between the substrate pool mass and storage root mass. In model 2, given the representation of the rate of the spill-over in eqn (6), no conceptual differentiation between storage mass and (6)

4 80 Gray Cassa a Crop Growth Modelling substrate mass has been made, thus reserve substrate concentration [C is simply the ratio of storage root mass to total plant biomass: where m l, m sb, m rf and m rs (d ) are the maintenance respiration coefficients for leaves, stems, fibrous roots and storage roots, respectively. R [C s (7) L S t B i R f R s CORE MODEL STRUCTURE Canopy photosynthesis and respiration A key feature of the core model is the dependency of all growth rates on assimilate or substrate concentration and a fundamental assumption of the core model is that the growth of all components follows substrate saturation kinetics. Assimilate or substrate concentration in the core model is dependent on the rate of canopy photosynthesis. Thus canopy photosynthesis drives total crop growth and storage root growth in both of the composite models (Fig. 1). The canopy photosynthesis equation has been modified after Johnson and Thornley (1983, 1984): A d as κ 9 ln a b (8) a c: where a ξa max f A b ξακπq 2S ξακπq exp( κlai) c 2S and where A max (kg substrate carbon m s ) is the light saturated rate of photosynthesis, α (kg substrate carbon MJ ) is the quantum yield efficiency, κ is the canopy extinction coefficient and LAI is the canopy leaf area index. The day length, S, is measured in seconds and Q (MJ m d ) is the total daily photosynthetically active radiation integral. The function f A modulates A max with respect to reserve substrate concentration [C: f A K β d 0 [Cβ1 K (9) β d and K d is a constant representing the value of [C resulting in a 50% reduction in A max and the exponent β defines the shape of the inhibition response curve. The values of A max and α, derived from photosynthetic rate s. irradiance (A-I curves) measured at different temperatures (5 to 35 C) in infra-red CO gas analysis studies, show very similar response curves to temperature in a range of plants (Papadopoulos, 1992). Thus for the core model, a single dimensionless temperature dependent coefficient ξ has been used to modify the values of not only A max and α, but also the maintenance respiration coefficients with respect to temperature. The empirical function used for describing the relationship between the dimensionless coefficient ξ and temperature is given in Jones (1983, eqn 9.16, p 195). Maintenance respiration R m is defined as follows: R m ξ[m l (L P) m sb (S t B i ) m rf R f m rs R s (10) Leaf structure In the core model the rate of production of leaves and petioles as a function of substrate concentration has been defined as follows: dl dt 0 75G l D l L (11) dp dt 0 25G l D l P (12) Plant leaf growth rate is dependent on the rate of leaf formation per apex, apex number per unit area, leaf size and leaf longevity (Cock et al., 1979). On the basis of this information, a growth rate function for leaves has been defined as follows (equation modified after Matthews and Hunt, 1994): G l A No ΛL A (13) δ where A No (apices m ) is the number of growing apices per plant shoot, Λ (leaves per apex d ) is the leaf emergence rate, L A is the individual leaf area of newly produced leaves (m leaf ), δ is the specific leaf area [m (kg structural carbon). The number of apices, A No [eqn (35), is dependent on the number of primary apices, S A, initiated from the planting stake (Table A2), and on the number of branch forks or apices, B ai, produced after each branching event (Table A2), where i 1, 2, 3 branching event [eqn (34). Daily leaf emergence rate Λ is defined as follows: Λ ΘΛ max f L 0 1 D D tot 1 (14) where Θ (dimensionless) is defined as the effective developmental day, D represents degree days (d C) and D tot is the total number of degree days in the growing season. Values for Θ range from 0 to 1. Maximum leaf production rate, Λ max (leaves per apex d ), is the leaf appearance rate when Θ equals one. The value Θ depends on the daily value of T D, where T D T mean T base and T base 13 C (based on data from Keating and Evenson, 1979): if 3 35 T D 22 then Θ T D 0 006T D (15A) otherwise Θ 0 (15B) thus, in the model Θ is defined only on the closed T D interval [3 35, 22. The function, f L, is a Michaelis Menten type term that modifies the value of Λ max with respect to the level of substrate supply available for leaf production and growth: f L [C (16) K L m [C

5 Gray Cassa a Crop Growth Modelling 81 Leaf area (m 2 ) Number of nodes FIG. 2. The curve describes the empirical relationship for the leaf area of new leaves s. number of nodes. The relationship is expressed in the empirical function defined in eqn (17). Following shoot emergence, the following leaf developmental events occur: (1) initially the area of each successive leaf increases until the maximum leaf size (L A,max ) is reached; and (2) thereafter the size of all subsequently produced leaves fails asymptotically to the minimum leaf area (L A,min ). where, K L m is the concentration of substrate [C for 0 5 Λ max. In the core model, leaf area (L A ) is expressed as an empirical function of the total number of nodes N sum (based on the data from Keating, Evenson and Fukai, 1982a): N L A sum d L A,max (d d ) d d (L A,max d )(d N sum ) N sum d (d d ) L A,min (17) where L A,max is the maximum leaf size, L A,min is the minimum leaf size, d and d are dimensionless constants, d and d (nodes m ) are constants, d (m ) is a constant and N sum is the total number of nodes per m. The relationship between leaf area and number of nodes as expressed in eqn (17) is depicted graphically in Fig. 2. The rate of biomass loss due to senescence of leaves (D l ) is determined by a temperature-dependent empirical function based on the data of Fukai and Hammer (1987). D D l l,max (18) T mean : where, D l,max (d ) is the maximum rate constant for senescence of cassava leaves. Cassava leaf longevity is cultivar specific and ranges from as little as 50 d (Connor and Cock, 1981) up to 160 d (Fukai and Hammer, 1987). In the core model, leaf senescence commences 50 d after shoot emergence from the planting stake. Canopy leaf area index The rate of conversion of substrate into canopy leaf area index, LAI, is expressed as a function of the leaf growth rate function, G l (kg structural carbon m d ) dlai 0 75δG dt l D l LAI (19) where δ is the specific leaf area. Stem and branch structure Assimilate partitioning into leaves and nodes in the core model is governed by eqns (13) and (20). The dry mass of internodes varies between 0 5 and 3 0 g (Cock et al., 1979). The rate of stem S t and branch B i growth as a function of the rate of leaf production [eqn (13) is defined in terms of the following logistic growth equation: ds t dt db i dt l0 ΦG 1 S t B i Ω sb 1 G (20) sb where G sb (kg structural carbon m d ) is the growth rate for the woody component of the shoot biomass, i stands for primary, secondary and tertiary branches, Φ (dimensionless) is the stem biomass partitioning factor and Ω sb is the optimal load bearing structural mass for stems and branches consisting of the specified number of nodes plus internodes produced as a consequence of leaf production at the shoot meristems after the interval t. This optimal structural shoot and branch mass which ensures the load-bearing capacity of stems and branches is fully determined in the core model by the number of nodes, internode lengths, stem density, stem modulus of elasticity, and branch tensile strength. In eqn (20) the three terms, Φ, G l and Ω sb, determine the gross growth rate of stems and branches. The stem biomass partitioning factor is defined by the following relationship: Φ φ w (21) 1 φ w where φ w is the fraction of shoot comprised of woody supportive tissues. The woody fraction of the shoot is computed as follows: φ W Ω sb (22) L P Ω sb where Ω sb S t B i (23) In the core model S t and B i are the stem and branch structural masses that make possible the shoot s loadbearing capacity. These two derived variables, S t and B i, are very simple functions of the parameters and variables that define the limits of the load-bearing capacity of the shoot s supportive structures: S t S A ρ st S L πr st (24A) B i S A B Ai ρ st B Li πr bi (24B) where S A is the number of primary stem apices, B Ai is the number of apices that arise as a result of primary, secondary, tertiary branching, S L (m) is the primary stem length, B Li (m) is primary, secondary and tertiary branch lengths; ρ st is the dry mass density of cassava woody tissues (between 300 and 400 kg DM m ) and R st and R bi (m) are stem and branch radii, respectively. Stem and branch lengths are defined as follows: S L N st N L,st (25A) B Li N bi N L,bi (25B)

6 Gray Cassa a Crop Growth Modelling where N st and N bi are the number of stem and branch nodes respectively and N L,st and N L,bi (m) are the internodal lengths for stem and branches, respectively. The mean radii of stems depend on the critical buckling downward force F st (N) that is resisted by stems. Stem radius has the following relation to the critical downward force (McNeill Alexander, 1968, pp ; Thornley and Johnson, 1990, p. 530). R st 0 F st 16S L ε st π 1 (26) where ε st is the modulus of elasticity of wet living woody tissue. The cassava stem modulus of elasticity in the model has been set at 750 MPa. The critical downward bulking force acting on the stem is defined as follows: F st g S A 9 (S t B i ) (L P) C st θ st C l θ l : (27) where g equals gravitational acceleration, C st and C l equal the proportion of carbon in stem and leaf dry biomass, respectively, and θ st and θ l equal the percentage dry mass in stems and leaves, respectively. The mean radii of branches have a functional relationship to the critical bending moment M b that is resisted by branches; and the branch radius for a given tensile strength can be defined as follows (McNeill Alexander, 1968, pp ): R bi 0 M b πt T 1 (28) where T T is the tensile strength of wet living branches and i 1, 2, 3 branches. The cassava branch tensile strength for the model has been set at 5 0 MPa. When branches, such as the primary and secondary branches, carry several different distributed loads, the bending moment at any and every cross-section is the algebraic sum of the bending moments produced by the various loads acting separately. However, the following simplifying approximation is used in the core model to estimate M bi in eqn (28) for primary, secondary and tertiary branches, respectively: M b sin Θ b 9 g B i S A B A C st θ (L P) S st A B A C l l: θ [B L B L B L (29A) M b sin Θ b 9 g 0B B ) S A B A C st θ (L P) S st A B A C l l: θ [B L B L (29B) M b sin Θ b 9 g B S A B A C st θ (L P) S st A B A C l l: θ [B L (29C) where Θ b is the branch angle (45 ). In eqn (20), the relationship between leaf production or leaf growth and the growth of the woody supporting structures is governed by the stem biomass partitioning factor Φ, and Ω sb, which represents the optimal load-bearing structural mass for stems and branches, for the number of nodes and internodes produced as a consequence of leaf production at the shoot apices. In the core model, the two factors Φ and Ω sb ensure that leaf production and woody organ growth are coupled in a mechanistic fashion. This is achieved by making the proportion of substrate carbon allocated to stems and branches dependent on the load acting on these structures; thus the leaf growth rate function G l [eqn (13) and the stem biomass partitioning factor Φ, together with the factor Ω sb, determine the overall sink demand of the load-bearing structures of the shoot in the core model. Fibrous root structure In the core model the fibrous root growth rate equation is: dr f dt G rf D r R (30) f where G rf (kg structural carbon m ) is the root growth rate and D r (d ) is the root senescence rate constant, which has been set equal to the value of D l. A modified version of the Matthews and Hunt (1994) root growth equation has been used for modelling the growth rate of fibrous roots: G rf (G l G sb )[0 05 exp( 0 005D)0 1 R f R max 1 (31) where R max (kg structural carbon m ) is the maximum root mass parameter. Stem and branch nodes In the core model the rate of production of shoot nodes as a function of the rate of leaf production has been defined by the following set of equations: dn st dt k st,n δg l L A dn b k b k b,n δg l dt L A dn b k b k b,n δg l dt L A (32A) (32B) (32C) dn b k b k b,n δg l (32D) dt L A where L A is individual leaf area, and k b, k b and k, are the b branch initiation parameters. The total number of nodes, N sum (nodes m ), is given as follows: N sum N st N b N b N (33) b The value for each of the three branch initiation parameters is equal to either 1 or 0 depending on the number of degree days, D (d C), that have accumulated since shoot emergence: k 2 bi 0ifD threshold value, and i 1 3 (34) k bi 1ifD threshold value, and i B Ai where i 1, 2, 3 branching events, i shoot apex multiplication factor and B Ai are the number of forks or apices produced at each branching event. The number of

7 shoot apices A No produced during crop development is defined as follows: A No S A ν ν ν (35) where S A is the number of primary stem apices initiated from the planting stake. The development pattern for stem nodes depends on the following conditions: if all k 2 bi 0 then k st,n 1 3 (36) if any k bi 1 then k st,n 0 When k bi 0 stems increase in length and when k bi 1 stem elongation ceases. The constants k st,n, k b,n, k b,n and k b are the node production parameters and are equal to,n either 1 or 0 depending on the number of apices (or occurrence of a branching event): 2 3 k i,n 1 if number of apices critical value k i,n 0 if number of apices critical value (37) where i stems, 1, 2 and 3 branches. In the model no more than three branching events occur during the growing season (Irikura, Cock and Kawano, 1979; Gutierrez et al., 1988). Thus the critical values for the number of apices produced for primary stems, 1,2 and 3 branches are equal to S A, S A B A, S A B A B A and S A B A B A B A, respectively. Total crop dry mass and storage root dry mass yield Total crop dry mass, W tot, expressed as tonnes dry matter (DM) per ha has been computed as follows: W tot 100 (L P) (S B i ) R f R s C C l C st C rf C rs C sub 1 (38) where C l, C st, C rf, C rs and C sub are parameters that define the proportion of carbon per unit dry biomass for leaves, stems, fibrous roots, storage roots and substrate, respectively. Storage root yield, W rs, expressed as tonnes DM per ha has been computed as follows: Gray Cassa a Crop Growth Modelling 83 W rs 10 R s C rs (39) METHODS Meteorological and en ironmental conditions The meteorological data that Keating et al. (1982a) reported for their cassava field trial site, Redland Bay, Australia (altitude 5 m, S, E), was used for the models. Meteorological data (T max, T min, Q, S) for the above site were fitted to a generalized sine function, e.g. y x x sin (πd x ) 365). Other considerations to be noted here are that the meteorological conditions applied in the models conform to those of the Southern Hemisphere coastal subtropicals that have a pronounced seasonal climate which limits the period of the effective growing season to a maximum of 9 to 10 months for plantings initiated at the end of August. Thus the run time set for the model was 290 d which is approx. 9 5 months. T ABLE 1. Model calibration results Total dry mass (kg DM m ) Storage root yield (kg DM m ) Model 1 r 0 96 r 0 99 Model 2 r 0 96 r 0 98 Regression results for simulated s. observed results for total and storage root dry mass for five intervals (83, 134, 174, 212, 252 d after planting); observed data from Fukai and Hammer (1987). Numerical analysis Numerical solutions for the models were obtained using Dynamodl (a Delphi 3 based modelling shell which incorporates a fourth-order Runge-Kutta integration algorithm). For the numerical analysis a step-increment ( t) of0 1 d was used. Dynamodl has been developed by A. Carter and myself, and is available from me on request. The models focus on the simulation of biomass partitioning dynamics in a cassava crop grown under optimal conditions; therefore the major modelling assumption was that crop growth was not constrained by any limiting factors such as nutrients or water supply. Data for an irrigated cassava crop, as reported by Fukai and Hammer (1987) were used for the calibration and validation of the model. Model parameters were classified into two sets (Table A2): (1) the set of independent parameters (P l ) include all parameters whose values are derived from empirical data and were not adjusted as a consequence of model calibration; and (2) the set of dependent parameters (P c ) include all parameters whose values were dependent on model calibration. Model calibration and model corroboration Numerical simulation models cannot be validated, verified or confirmed (Oreskes et al., 1994). However, numerical simulation models can be used to evaluate hypotheses, that is, to either falsify or corroborate the hypotheses upon which the model is based. The Redland Bay experimental results were used as the baseline data for model calibration and model corroboration (Keating et al., 1982a, b, c; Fukai and Hammer, 1987). The long growing season data of Fukai and Hammer (1987, Fig. 5), with the planting initiated in September, was used for model calibration. Calibration of the model was achieved by adjusting the values of the calibration or dependent parameters until the r for simulation output s. observed values fell between 0 96 and The r for simulation outputs of the two models s. observed results are presented in Table 1. Parameters selected for model calibration were subjected to sensitivity analysis (data not shown). Sensitivity was defined as the ratio of the fractional change in output variables (total mass and storage root yield) to the fractional change in the selected parameters. Sensitivity tests were conducted by adjusting the values of the calibration parameters by 5%. With regard to the magnitude of the model s sensitivity to parameter adjustment, the calibration parameters for model 1 were ranked in the following order, ω µ K T m K d B Ai and S A N Lst and N L,bi K L m β D l,max ; and for

8 84 Gray Cassa a Crop Growth Modelling Planting month TABLE 2. Model test results Observed (kg DM m ) Model 1 (kg DM m ) Model 2 (kg DM m ) Sept Oct Nov Dec Jan Simulated s. observed r r P P Correlation coefficients for predicted results s. observed values for total biomass yields produced by crops initiated at different planting dates; observed data from Keating et al. (1982b). model 2 the parameters were ranked as follows, K d N Lst and N L,bi K L m B Ai and S A β D l,max. The outputs of model were most sensitive to the values of parameters that defined storage root sink strength and storage root growth dynamics. In model 2 the outputs were most sensitive to the values of parameters that defined photosynthetic rates and shoot assimilate demand. Except for parameters K d and K L m, the values for the remaining calibration parameters shared by both models were the same; and with regard to the independent parameters, the values were the same for both models (Table A2). In order to calibrate model 2, the values for the parameters K d and K L m had to be increased from 0 05 to and 0 04 to 0 325, respectively (Table A2). This difference in the values of K d and KL m for the two models arises because the two models do not have similar conceptions regarding the nature of the substrate pool. For example, the substrate pool sizes and concentrations are very different for the two models; in model 1 the substrate pool excludes the storage root biomass, whereas in model 2 the substrate pool equals the storage root biomass [as depicted in eqn (6). This is the reason why the parameter values for K d and K L m have different magnitudes in the respective models. As a consequence of the differences in the natures and magnitudes of the substrate pools for the two models, calibration of model 2 relative to model 1 required a decrease in the level of substrate feedback control on the rate of canopy photosynthesis and a decrease in the substrate affinity constant for leaf production. In addition, calibration of model 2 was impossible: (1) if the core model term f A [eqn (9) was excluded from the canopy photosynthesis equation [eqn (8); and (2) if the canopy feedback response parameter β 8 0. Thus, β 8 0 is the default setting for both models. This, in turn, results in eqn (9) working in both models as a continuous switching-function with regard to substrate feedback regulation of the rate of canopy photosynthesis. The two models were corroborated by comparing the predicted biomass yields for different planting times with the observed biomass yields derived from data not used for model calibration (Keating et al., 1982 b). There was no significant difference with regard to the total biomass yield predictions of model 1 and model 2 (Table 2). Hence on statistical grounds it can be argued that both models are equally well corroborated with respect to the observed data. This is an important issue and will be addressed more fully below. RESULTS AND DISCUSSION Biomass partitioning Simulated outputs for biomass partitioning are represented in Fig. 3. Figure 4 and Table 3 show the linear relationships for the distribution of dry matter between storage root and total plant mass. Cassava crop growth progresses through two phases, an initial lag phase lasting 40 to 100 DAP (days after planting), and an exponential phase during which root bulking occurs (Fig. 5). In cassava, it has been demonstrated that the relationship between storage root mass and total plant mass is a linear one (Boerboom, 1978; Connor et al., 1981). This linearity shows that storage root bulking rate kept pace with the rate of crop growth. To describe the distribution of dry matter over storage roots of cassava, a simple model has been developed by Boerboom (1978). It was based on the assumption that the relation between the weight of storage roots y and total weight x can be Dry mass (kg m 2 ) Days Rf Lp Rs Sb Cr FIG. 3. The dynamics of biomass allocation between storage roots (R s ), stems and branches (S b ), leaves and petioles (L p ), fibrous roots (R f ), and storage reserves (C r ), as simulated by model 1. The simulated growth curve for the storage roots, relative to the patterns displayed by the other components, shows the two phases of storage root growth and development characteristic of cassava growth dynamics: initial sink limiting lag phase; the second phase characterized by exponential growth and competition with other organs.

9 Gray Cassa a Crop Growth Modelling Storage root DM (kg) Dry mass (tonnes/ha) Model 1 Total Model 1 Yield Model 2 Total Model 2 Yield Total Yield Total plant DM (kg) FIG. 4. Using the data in Fig. 3, the graph of storage root dry mass s. the sum of all dry mass components, i.e. total plant dry mass, approximates a linear relationship, which can be interpreted as follows. For the linear relationship between total plant dry mass x and storage root dry mass y, the two parameters of the linear regression y a bx, can be used to represent the following: (1) b efficiency of storage root production (ESRP); and (2) the intercept of the axis defined by the ratio a b, equals the initial plant dry mass at which stage root production starts and thus marking the end of the lag phase of storage root growth. With leaf senescence carbon is recycled into the substrate pool and becomes available for storage root growth. Thus at the end of the growing season while no net increase in total biomass or shoot mass occurs, recycled carbon is available for storage root growth. Biomass expressed as kg dry mass m. represented by the linear regression y a bx. The magnitude of the regression coefficient b represents the portion of substrate carbon diverted to storage roots. The regression coefficient has been defined as the efficiency of storage root production (ESRP). The harvest index HI approaches ESRP as x. The initial plant weight (ISS) at which storage root production starts is represented by the intercept with the x-axis (c) and can be calculated according to c a b. It should be noted that there is a degree of autocorrelation in the Boerboom (1978) linear plot. However, the purpose of the analysis in Table 3 is to demonstrate that the simulation outputs of both models conform to the observed patterns reported by Boerboom (1978). The degree of similarity that exists between observed and simulated results for the two parameters, ESRP and ISS, gives a measure of the validity of the models (Table 3) Days FIG. 5. Model 1 and 2 simulation outputs for total crop dry mass production and storage root yield. In model 1 assimilate allocation to storage roots is governed by Chanter s (1976) equation; and in model 2 the spill-o er hypothesis for assimilate allocation to storage root governs storage root growth. The Chanter growth equation generates a smoother sigmoid curve for storage root growth compared to the spill-over function. The spill-over function of model 2 results in: (1) earlier initiation of storage root bulking, as no distinction exists in model 2 between the assimilate reserve pool and the storage root compartment; and (2) perturbations or on-off switching behaviour with regard to assimilate allocation to storage roots during branching events. With regard to total crop biomass production both models produce characteristic sigmoid growth curves. At the end of the growing season, carbon is recycled from the senescent leaves, back into the substrate pool and becomes available for storage root growth. Thus at the end of the growing season while total mass falls due to leaf senescence, some storage root growth occurs. Storage root growth dynamics In the models, cassava storage root growth and development displays only the first two phases of storage organ dominance (Fig. 5) which are characteristic of other crops [sweet potato (Ipomoea batatas), potato (Solanum tuberosum), onion (Allium sp); the initial phase when storage organ growth is sink-limited; the second phase when there is competition between storage and other organs; and the third phase when storage organs are such strong sinks that all assimilates are allocated to them (Boerboom, 1978; Kooman and Rabbinge, 1996). This difference TABLE 3. Obser ed and simulated alues for the efficiency of storage root production (ESPR) and initial plant mass (ISS) at which storage root production starts ESPR ( b) [kg storage root DM (kg) total DM) s.e. b [kg storage root DM (kg total DM) a (10 kg DM m ) s.e. a (10 kg DM m ) ISS ( c) (10 kg DM m ) r Observed Model Model s.e. standard error of the coefficients a and b. The regression coefficient b equals ESPR and ISS equals the ratio a b; where: y a bx, y is the storage root mass and x represents total plant mass. Observed data from Fukai and Hammer (1987).

10 86 Gray Cassa a Crop Growth Modelling between cassava and the other crops may be related to the fact that cassava is a perennial crop that continues to grow for 2 or more years. After the first phase, which ends when total plant mass equals ISS, the second phase, which is characterized by competition for assimilates between storage roots and the rest of the plant, commences. Both models generate the appropriate sigmoid curves for storage root growth; however, model 1 generates an accurate simulation of the duration of the lag phase and of the timing of the transition from lag to exponential growth. Application of the spill-over hypothesis [eqn (6) in the composite model 2 results in the commencement of storage root bulk at a much earlier stage of crop development relative to model 1 (Table 3, Fig. 5). Biomass partitioning into stems and branches The two major sinks for assimilate during cassava crop growth are the woody component of the shoot and the storage roots. The balance of assimilate allocation between these two sinks determines both harvest index and yield for the cassava crop (Cock et al., 1979). Assimilate allocation to stems and branches in models 1 and 2 are based on mechanistic assumptions. Total stem biomass produced per growing season varies with cultivars and was correlated with the total number of leaves produced: M Mex 59 produced 12 tonnes stem dry mass (DM) per hectare 300 DAP, and the total number of leaves produced per plant equalled 1050; M Col 22 produced 4 6 tonnes stem DM per hectare 300 DAP, and the total number of leaves produced per plant equalled 649 (Connor et al., 1981; Connor and Cock, 1981). The models described in this paper are based on the cultivar M Aus 10 which produces between 1100 and 1200 leaves m and has a stem biomass of between 8 and 9 tonnes DM ha under irrigation (Keating et al., 1982a, b; Fukai and Hammer, 1987). Model 1 predicts a stem biomass of 8 2 tonnes DM ha (1111 leaves produced) and model 2 predicts a stem biomass of 8 5 tonnes DM ha (1360 leaves produced). These results demonstrate that the stem biomass partitioning hypothesis of eqn (20) is consistent with the data. Total crop mass and storage roots yields Simulation outputs for total crop production and storage root yield are represented in Fig. 5. Experimental results for total dry mass production and storage root yield for an irrigated cassava crop obtained after 252 d in the growth season at Redland Bay were 26 6 and 16 2 tonnes DM ha, respectively (Fukai and Hammer, 1987, Fig. 5). The output of the two models for total dry mass (W tot ) and yield (W rs ) after 252 d were 26 W tot 27 tonnes DM ha and 16 W rs 17 tonnes DM ha, respectively. Other field trial data for the humid tropical sites give an average yield of 10 8 tonnes DM ha at 300 DAP for crops grown at CIAT (Veltkamp, 1986). For humid tropical sites in Africa, such as the International Institute of Tropical Research (IITA), cassava yields of 16 tonnes DM ha at 300 DAP have been reported (Gutierrez et al., 1988). Matthews and Hunt (1994) cited values of 28 5 tonnes ha for total crop dry mass production and 16 4 tonnes ha for storage root dry mass production after 300 d as default production values for cassava crops grown under CIAT conditions. Under favourable experimental conditions, cassava can yield as much as 25 to 30 tonnes DM ha (Best and Hargrove, 1993). Other reports suggest that the storage root yield potential for cassava can be as high as 30 tonnes DM per ha per year (Cock, 1976). CONCLUSIONS To understand the growth of cassava it is necessary to know if (1) the strength of stem assimilate demand affects the growth and development of the storage root, (2) the root sink capacity limits root growth; and (3) the strength of storage root demand affects growth and development of the shoot. On a practical level these questions are best dealt with by means of grafting experiments. But how well are these questions dealt with by the alternative cassava crop growth modelling approaches? Arbitration between the two models can proceed along the lines of first deciding which model is the most likely, and secondly, which has the greatest explanatory power. With regard to likelihood, given the data and a mathematical description of the two models, the question can be posed: How likely are the data, given the model? The statistical data (Tables 1 3) show that both models were very similar with regard to predictive power. Also, the Wilcoxon s matched pairs rank test for model 1 s. model 2 showed that there was no significant difference between the predictions of the two models (P ). Therefore, on statistical grounds both models are very similar. In order to evaluate the merits of the two models further, the models need to be tested by confrontation with other data. For example, it can be argued that the grafting data of Cours (1951), cited by Hunt, Wholey and Cock (1977) is best accounted for by model 1, because: (1) model 1 is fully modular, the growth of all components are defined by independent growth functions and different parameters; and (2) hypothesis 1, as expressed by eqn (5) contains parameters which determine both inherent storage root growth dynamics and inherent sink strength of the storage root. This means that in model 1, with the exclusion of the effect of the variable [C on storage root growth, the inherent storage root growth dynamics are unaffected by what goes on in the rest of the plant. These conditions do not hold for model 2; for example, hypothesis 2 gives a description of storage root growth that is dependent on assumptions and considerations regarding assimilate production by the shoot, thus eqn (6) for storage root growth in model 2 is not functionally equivalent to eqn (5) for storage root growth in model 1. Therefore, while model 1 can be used to predict or explain the results of grafting experiments, model 2 fails in this respect. In the grafting experiments, it was reported that if a scion of a clone (H 33) which had a low root starch fraction (38%) was grafted onto a stock (H 31) which had a high root starch fraction (54%), the resulting plant had a high root starch fraction similar to that of H 31 and not of H 33 (Cours, 1951; cited

11 by Hunt et al., 1977). Similar results were also obtained when a Manihot glazio ii scion was grafted onto a cassava stock and for grafts made between cassava and M. flabellifolia (Mogilner et al., 1967; cited by Hunt et al., 1977). With regard to explanatory power, which of the two hypotheses (or mathematical descriptions) gives the best account for the above three questions? Model 1 can account for all three questions, whereas model 2 can account for only the first question. Therefore model 1 is superior in terms of likelihood and explanatory power. From a Popperian perspective the following principles regarding the standing of the two models can be roughly stated as follows. Model 1 is better than model 2 if, and only if, (1) model 1 has greater empirical content than model 2; (2) model 1 can account for the successes of model 2; and (2) model 1 is not yet falsified but model 2 is falsified by grafting experiments, if failure to explain a result is taken to be logically equivalent to falsification of a hypothesis. ACKNOWLEDGEMENTS Stephanie Dyer is thanked for the information on the strength of woody materials. Andrew Carter is thanked for his considerable contribution to the development of Dynamodl. The work was supported by the Mellon Foundation Grant and African Products Pty Ltd. Thanks to J.H.M. Thornley and an anonymous referee for many helpful suggestions and comments. LITERATURE CITED Best R, Hargrove TR Cassa a: The latest facts about an ancient crop. Cassava Program: International Center for Tropical Agriculture. Boerboom BWJ A model of dry matter distribution in cassava (Manihot esculenta Crantz). 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