Ecological Modelling, 65 (1993) 95-121 95 Elsevier Science Publishers B.V., Amsterdam Analysis of the cowpea agro-ecosystem in West Africa. I. A demographic model for carbon acquisition and allocation in cowpea, Vigna unguiculata (L.) Walp M. Tam6 a and J. Baumgiirtner b Biological Control Center for Africa, International Institute of Tropical Agriculture, Cotonou, Benin b Institute of Plant Sciences, Division ofPhytomedicine, ETH, Zurich, Switzerland (Received 20 November 1991; accepted 13 April 1992) ABSTRACT Tam6, M. and Baumg~irtner, J., 1993. Analysis of the cowpea agro-ecosystem in West Africa. I. A demographic model for carbon acquisition and allocation in cowpea, Vigna unguiculata (L.) Walp. Ecol. Modelling, 65: 95-121. A demographic canopy model for a photosensitive cowpea [Vigna unguiculata (L.) Walp.] variety, whose growth and development is driven by temperature, solar radiation, soil phosphate, and water is presented. The dynamics of age-structured populations of leaves, shoots, roots, peduncles, and other reproductive organs having numbers and dry matter attributes is simulated by a time-varying distributed delay model with attrition. A modified functional response model from predation theory is used to estimate the daily photosyn- thates acquisition. Dry matter allocation is simulated by a metabolic pool model. Growth and yield formation of the plant are driven by the computed ratio between carbohydrates supply and demand. Simulation results are compared with four sets of field data. Further, the model has been used to evaluate the effect of drought stress and different levels of available phosphate in the soil. The model is designed for research purposes, to be used as a tool for further studies of the cowpea agro-ecosystem. INTRODUCTION The cowpea [Vigna unguiculata (L.) Walp.] agro-ecosystem includes the plant and a large number of associated organisms, such as arthropods, Correspondence to: M. Tamb IITA-BCCA, B.P. 08-0932, Cotonou, Republic of Benin. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved 96 M. T A M O A N D J. BAUMG,~RTNER which affect yield formation (e.g., Jackai and Daoust, 1986). The interac- tions between these elements define the structure of the system and determine its dynamics. Thus, any analysis of the system leading to man- agement recommendations, including plant protection, has to consider these interactions and the way they are influenced by external factors. For this purpose, the simulation approach has often been used (Getz and Gutierrez, 1982). It can be divided into engineering and demographic approaches (Baumg~irtner and Gutierrez, 1989). In the latter case, popula- tion theory is used to formulate acquisition and allocation of dry matter as well as ageing functions which are common to all relevant elements in the system, including the plant. From a theoretical standpoint, the demo- graphic approach links classical population theory and crop physiology (Gutierrez et al., 1975, 1984b), and, in practice, facilitates the structuring of multitrophic population models (Gutierrez et al., 1984a; Baumg~irtner and Gutierrez, 1989). Ecosystem analysis requires an approach with such capa- bilities, and the demographic approach is consequently applied. Although models based on this method have already been developed and used for the analysis of several crops (e.g., Gutierrez et al., 1984b, 1987; Wer- melinger et al., 1991), the present work has features which may justify a detailed review of this model. The present analysis considers yield formation of the cowpea crop in the absence of herbivory. Though the model is incomplete, it provides the basis for studying multitrophic interactions, such as the influence of the bean flower thrips, Megalurothrips sjostedti Trybom (Thysanoptera: Thripidae) presented in subsequent work (Tamb et al., 1992). MATERIAL AND METHODS Plant growth habit The development and growth of the cowpea plant has been described in detail elsewhere (Littleton et al., 1979a, b, 1981; Summerfield et al., 1983, 1985); thus, only essential features of the plant growth habit, relevant for the understanding of the model, are considered in the present work. During the vegetative phase, the plant produces one compound leaf and one secondary shoot at each node, except for the first two nodes: they bear the cotyledons and two simple leaves, respectively. The reproductive phase is induced by the combined effect of temperature and photoperiod in most cowpea genotypes (Hadley et al., 1983). In this model, the photoperiod-sen- sitive local variety 'Kpodjigu6gu6' switches to reproductive growth after D days from sowing, each day divided into M = 10 time steps (time increment ANALYSIS OF T H E COWPEA AGRO-ECOSYSTEM 97 A t = 1 / M = 0.1 day, see below) if 1 M.D -~ Y'. (Td,-- T o ) ' ( b , +b2"/~) = 1 (1) d ' = l where the temperature T d, is obtained by forcing a sine function through the daily temperature minima and maxima, which permits the computatio n of the relevant T d, for the d ' t h time step (see Baumg~irtner et al., 1990a; Curry and Feldman, 1987). T o is the lower thermal threshold, set to 8°C after Hadley et al. (1983). ~ represents the photoperiod in hours calculated with an algorithm for 10 day averages after seedling emergence (Penning de Vries and Van Laar, 1982). The parameters b 1 and b 2 represent the intercept and the slope of the regression line between the temperature sum required for the induction of the reproductive phase and the respective photoperiod (Hadley et al., 1983). They are estimated from field data (see parameter estimation section), and their values are 0.0199 and - 0.0014195, respectively. Once the reproductive phase is initiated, new leaves, nodes, and repro- ductive structures are built through modular growth: one leaf and one inflorescence emerge at each node of both the main stem and the sec- ondary shoots. The structure and the development of the inflorescence, as well as the process of fruit production of cowpea, have been described in detail by Ojehomon (1968a,b). An inflorescence is a peduncle supporting a sequence of raceme units, each of them bearing two fertile flower buds. The term 'fruit ' refers to the developing reproductive organs irrespective of their age (flower bud, flower, and pod). The axillary buds on the secondary shoots are dormant and consequently not considered in the model. General structure o f the model Mathematical framework The model is based on the analysis of two processes, i.e. population development and population interactions, which have been reviewed re- cently by Severini et al. (1990a) and Graf et al. (1990). The demographic approach (Gutierrez et al., 1975) considers the different organisms (e.g., plants and arthropods) as populations which become the basic elements of the cowpea ecosystem. Thus, yield formation is conceived as a process controlled by interacting populations under the influence of driving vari- ables such as weather and management practices. The cowpea plant, as the central feature of the system, can be further divided into populations of leaves, shoots, peduncles, roots, and fruit, which develop through time. These subunits are likewise subjected to demographic analyses. In the 9 8 M. TAMO AND J. B A U M G A R T N E R subsequent work, the same approach will be applied to one of the major pests, the bean flower thrips M. sjostedti. Some plant components relevant to yield formation, such as reserves and photosynthetically inactive plant structures, are not subjected to demographic analyses. Each plant subunit passes through different developmental phases until it reaches the end of its physiologically active life span, provided it is not shed earlier. This developmental process is most conspicuous for fruit: they grow for most of their life span until they reach maturity and are harvested or drop. Leaves pass through a phase of growth, and then their shape remains stable until they drop. Roots, shoots and peduncles pass through a phase of growth before they become inactive and are used to store reserves. Inflorescences are directly related to the number of nodes, which continue to be produced until plant growth stops. Plant growth and the passage of subunits through phases imposes on the different populations an age structure which changes through time. The dynamics of a population with time-varying age structures is described concisely by the Von Foerster (1959) model (Wang et al., 1977; Gutierrez and Wang, 1976; Curry and Feldman, 1987) ON(a, t) ON(a, t) + - a (a )g (a , t ) for t , a > 0 (2) 0t 0a and oo U(O, t )= Jo w(a)U(a, t) da for t > 0 (3) where N(a, t) is the number of organisms of age a at time t, a(a) is the age-dependent mortality function, and to(a) is the age-dependent birth rate. Ageing and death of a cohort are expressed in equation (2), while equation (3) represents the birth of new individuals. In our case, however, a and to are not only age dependent , but influenced by complex and varying environmental conditions. Furthermore, they take into account functional and numerical responses resulting from population interactions (Baumg~irtner and Gutierrez, 1989). As a result, equations (2) and (3) must be transformed into a discrete form and evaluated numerically through simulation models (Wang et al., 1977). If the number of individuals present in the field at some point in time is assigned to age categories, the observer obtains a stage-frequency matrix (Manly, 1989). Many methods have been used for the analysis, but Severini et al. (1990a, b) argue that the very structure of the matrix implies the use of a time distributed delay model. This is primarily because the approach takes into account differences in individual developmental times, which is an important element in population theory (Bellows, 1986a, b). ANALYSIS OF T H E COWPEA AGRO-ECOSYSTEM 99 In addi t ion to the numbers of part icular organisms in general, and of the plant subunits in this work, the masses (dry matter) can also be considered as popula t ion at tr ibutes j and are s imulated by this model (Gutierrez et al., 1984a, b, 1988; Baumg~irtner et al., 1989; Wermel inger et al., 1991): mass of leaves ( j = 1), shoots ( j = 2), fruit ( j = 3), roots ( j = 4), and peduncles ( j = 5), as well as numbers of leaves ( j = 6), inflorescences ( j = 7), and fruit ( j = 8) are thus subjected to demographic analyses. The deve lopment of each at tr ibute j occurs th rough a set of k cascaded age categories (substages) as described by Vansickle 's (1977) equations: dOj , l ( t ) dt = x j ( t ) - r j , a ( t ) - txj,l( t ) " Qj,l( t ) dQi,z(t) d t = rj ' l(t) - ri'2(t) - /x i '2 ( t ) " Qi'2(t) dQj , i ( t ) dt = r L i - l( t ) -- rj'i( t ) - IzJ ' i ( t ) " QJ ' i ( t ) dQj ,k ( t ) dt rJ'k- l( t) - y j ( t ) - txj,k(t ) • Qj ,k( t ) (4) where i is the index denot ing the substage (1 < i < k ) ; j is the index denot ing the popula t ion at tr ibute specified in Table 1; k is the number of substages of popula t ion attributes. Via simulation studies, k = 25 was found to represent satisfactorily the variability in individual developmenta l times; r j , i ( t ) is the transit ion rate f rom substage i to substage i + 1; Qi,i(t) is the storage (numbers or mass) in substage i, Qj,i(t) = r j , i ( t ) " D E L j ( t ) / k , D E L j ( t ) is the expected value of the transit (developmental) t ime at t ime t in days; /zj,/(t) is the age-specific ins tantaneous attrit ion rate; x j ( t ) is the inflow rate of numbers or mass into the first substage; and yj(t) is the outflow rate of numbers or mass out of the last substage. Plants and a r th ropods are poiki lothermic organisms, whose developmen- tal t ime DELj.(t) is t empera tu re -dependen t . When the system unde r study is subjected to f luctuating tempera tures , D E L j ( t ) becomes DELj(Td,) in equat ion (5), which is def ined as the ins tantaneous value of the matura t ion t ime (Manetsch, 1976), and, consequently, a t ime-varying delay is used (Vansickle, 1977; Baumg~irtner and Severini, 1988). The same approach has been used in the crop model of Baumg~irtner et al. (1990b), but not described in detail. However, the crop models of Gut ier rez and co-workers (1984a, b, 1988), Graf et al. (1990) and Wer- 100 M. TAMO AND J. BAUMGARTNER melinger et al. (1991) are basically built on a time-invariant rather than a time-varying version. This difference is considered important enough to i-equire a detailed presentation of this model (Severini et al., 1990a, b). According to Vansickle (1977), it is convenient for simulation purposes to write (4) in terms of flow rates, reformulated as a sequence of difference equations which are the basis for the simulation model, constructed here for a time step At of 0.1 days: k - A t rj j ( Zd, + ] ) = rj,l( Zd, ) + DEL~(Td,) • {x i (Ta, ) - rj,a(Ta,)[1 + DEL~ (Ta')~_7~ ~ - - DELj(Ta,_ 1) DELj(Td,) -t-lZJ'l(rd') k k . A t rj,2( Tct, + l ) = rj,z( Ta, ) + DELj-(Td, ) DELj(Td, ) - DELj(Td,_ 1) rL](Ta, ) -r j ,2(Ta, ) • 1 + A t ' k DELj(Td') } k . A t rj,k(Ta,+ l ) = rj,k(Ta, ) + DELj(Td,) DELj(Ta' ) - DELj(Ta,_I) " r i ' k - l ( T a ' ) - Y J ( T a ' ) + A t ' k DEL'(Td') ] } 1, (5) where T a, has been explained in equation (1). In the same way, the storage in the ith element at d' becomes Qi , i (d ' ) = r j , i ( ra , ) • D E L i ( T a , ) / k (6) Model elements: ageing and net growth DELj.(Ta,): the instantaneous value of the maturation delay (in days) is equal to the inverse of the instantaneous temperature-dependent develop- ANALYSIS OF THE COWPEA AGRO-ECOSYSTEM 101 T A B L E 1 Stage-specific input and a t t r i t ion ra tes for ageing and ne t growth and n u m b e r of organs expressing a d e m a n d for assimulates. Qj,i(d ') is s torage in substage i of a t t r ibu te j [see equa t ion (6)], bi, d r ep resen t s the d e m a n d ra te for growth pe r day (see metabol ic pool model) , n S is the n u m b e r of secondary shoot.s, while Cl, a and c2, d are the s u p p l y / d e m a n d rat ios calculated by the metabol ic pool model per day d j Stage: Input At t r i t ion N u m b e r of A t t r ibu te x j, i /xJ,i Substages organs affected expressing a d e m a n d 1 Leaf mass 0.01 a 0 b 2 Shoot mass 0.01 a 0 b 3 Frui t mass 0.01 a 0 b 4 Roo t mass 0.01 a (inclusive 0 b nodules) 5 Peduncles 0.01 a mass 0 b 6 Leaf n u m b e r c2, d • n s. b6, d 7 Inf lorescence 0 c n u m b e r C2,d . ns. b6,d d 8 Frui t n u m b e r Cl, d "n S" bs, d bl,d'C2,d/Ql,i(d') 1, 7 7 , Y~i=lQ6,i(d ) 0 > 7 b2"C2,d/Q2,i(d') 1, 12 Y~21Q6,i(d') 0 > 12 b3,d" Cl,d / Q3,i(d') - [ 1 - Ca, d] 1, 17 b3,d" Cl,d / Q3,i(d') 18, 25 E~ 51Q6,i(d') ba,d'C2,d/Q4,i(d') 1, 12 ~ 2 _ l Q 6 i ( d ' ) 0 > 7 b5,d'C2,d/Q5,i(d') 1, 22 0 > 22 E 22_ 1Q7,i(d') 0 n s 0 n s i -- Cl,d 1, 1 7 E25107,i(d') 0 > 1 7 a Ini t ial izat ion, b growth phase for masses, c Vegetat ive, d reproduct ive phase for the n u m b e r of inflorescences. mental rate zi(Ta,) per day: 1 D E L j ( T a , ) - zj(Ta, ) (7) In this case, z i (Td ' ) is assumed to be proportional to the tempera ture above a lower thermal threshold T o common to all j zj(Ta, ) = ~j" ( T d, - To) for T d, > To zi(Td, ) = 0 for T a, < T O (8) /3j is the derivative of the developmental rate with respect to the tempera- ture. xj(Td,): the input into the first substage x /T a , ) for each population attribute j is given in Table 1 for two different phases: an initialization phase where at the beginning of the growing season the masses of leaves, 102 M. TAM~) AND J. BAUMG,~,RTNER shoots, peduncles, nodulated roots, and fruit are arbitrarily set to 0.01 mg dry mat ter which flows via xj(Ta,) only once into the developmental process, and a growth phase, where input occurs for the numbers but not for the masses. During the vegetative phase of the plant the number of shoots is set to n s = 1. The reproductive phase is induced by the combined effect of t empera ture and photoperiod [equation (1)]. Only at this moment is the number of shoots n s set to the number of nodes n n produced so far, and calculated with the following equation, vegetative growth: ns = 1 (9a) k reproductive growth: n S = nn = Y'~ Q6,i(d*) (9b) i=1 where d * = d ' at the end of day d of the induction, and kept constant thereafter . lzj,i(Td'): the instantaneous attrition rate txj,i(Td,) represents the propor- tional growth/ loss rate for mass attributes and the losses for number attributes during the developmental process per unit of dry mat ter Q~,i(d'). It is calculated using the equations presented in Table 1. Note that growth and loss of fruit mass are mutually exclusive, i.e. the mass is ei ther increasing or decreasing during At. yj(Td'): the output rate leaving the last substage for mass attributes can ei ther represent loss (shedding of leaves and fruit), dead plant structure, or assimilates which are added to the reserves. For the leaf and fruit number we consider the output rate as the death at the end of their life span. The number of inflorescences leaving the process is not considered: the output indicates the end of the plants growing cycle. Model elements: population interactions The input rate as well as net growth are controlled by internal physiolog- ical processes, i.e. carbohydrates acquisition and allocation functions result- ing from population interactions. In this case, the model has to deal with competi t ion for resources (i.e. mass). In contrast to the above developmen- tal process in which a simulation time increment of 0.1 days was used, the budget of mass and numbers is calculated on a daily (d) basis. For the sake of simplicity, (d) is generally not expressed in the following equations. The daily s u p p l y / d e m a n d ratios c I and c2, whose index denotes alloca- tion priorities, are used to scale input and attrition rates (see Table 1). Their computat ion relies on a comprehensive model of carbon assimilation and allocation. The metabolic pool model of Gutierrez and Wang (1976) has been found particularly appropriate for this purpose (Graf et al., 1990). .ANALYSIS OF T H E COWPEA AGRO-ECOSYSTEM 103 Acquisition / / . v- r . r ; ( / t r v ~ / ~ / , , j C ~ t ~ ~ bss Output Ageing and net growth Res firation Fig. 1. The metabolic pool model is used to allocate the assimilates acquired through both photosynthesis and remobilisation of reserves to the different organs of the cowpea plant, according to their respective priorities. The model is linked with the time-varying dis- tributed delay, which represents ageing and net growth of the organs. The daily available carbohydrates (as produced by photosynthesis and drawn from reserves) flow into a pool from where they are allocated to the different developing populations following a priority scheme (Fig. 1). First priority is given to respiration, followed by the growth of fruit, which becomes second priority. The third priority is the allocation to the vegeta- tive parts (leaves, shoot, peduncles, and nodulated roots) and reserves. The maximum demand rate for the different plant attributes j is given by the following equation, in which all the variables are specific for a given day: b i = Aj -Nj" w~d" At . 'bp (10) Aj represents the genetically determined maximum demand rates estimated from field data, Nj is the number of growing organs specified in Table 1, wsd is the water supp ly /demand ratio computed with the water submodel (see below), while A~- is the sum of the effective temperatures ( T d ' - T o) for a given day with d ' = 1,M time steps (Gutierrez et al., 1988; Wer- melinger et al., 1991) 1 M A~- = ~ Y'~ (T d, - To) (11) d ' = l 104 M. TAMO AND J. BAUMGARTNER Cpp is a function of the soil phosphate available to the plant Pa, which is kept constant during the growth period as follows: e p = a ' P a b (12) where a = 0.2752 and b = 0.4307 are empirically derived constants. The total demand for all mass attributes ( j = 1..4) B is composed of two parts: (1) demand for growth proportional to tempera tures above T o , and (2) demand for respiration 4 B = Y'~ b J ( 1 - F) + b r + b M (13) j = l The conversion coefficient F for transforming assimilates into plant biomass represents the respiration for growth and is set to 30% of the net assimilation (Penning de Vries and van Laar, 1982). The demand for reserves b r is also part of the vegetative growth, but does not require conversion. For the computat ion of b r we need to consider some basic aspects of the dynamics of the reserves, which, in this model, are stored in dead shoots, peduncles, and root structures. The maximum storage capacity for reserves R~' is equal to the structures provided by dead roots, pedun- cles, and shoot material on day d = D M'D R~ = Y'~ [ y z ( T a , ) + Y4(Ta , ) + ys (T~t , ) ] ( 1 4 ) d ' = l while the real amount of reserves available on day d is M'D R a = R d _ 1 + c 2" y" {[y2(Ta,)+y4(Ta,)+ys(Ta,)] 'O.66+y1(Ta,) '0 .1} d ' = l (15) with the constraint that R d < R~ ~, while c 2 is the supp l y / dem and ratio for vegetative organs described below [see equation (24)]. The demand for reserves is simply the deficit of the storage: br = R 3 - R d (16) b M represents the demand to cover maintenance respiration and does not require conversion. It is computed according to Penning de Vries and van Laar (1982): bM= 0 . 0 1 - R + Y'~ Y'~e j , i (d ' ) 'a j " 2 " e (T'' 25,/,0 (17) i=l j = l ANALYSIS OF THE COWPEA AGRO-ECOSYSTEM 105 where 6j represents the coefficients at 25°C equal to 0.03 ( j = 1), 0.15 ( j -- 2), 0.01 ( j = 3), 0.01 ( j = 4), and 0.01 ( j = 5), respectively (Penning de Vries and van Laar, 1982). The supply is calculated as follows: the functional response model derived from Frazer and Gilbert (1976) by Gutierrez et al. (1981, 1988), Baumg~irtner et al. (1989) and Wermel inger et al. (1991) was used to estimate the rate of daily photosynthesis, P, per day P = B" {1 - e E s(LAI)'Cp(I, O)/B]} (18) The potentially producible carbon, Cp is the result of the conversion of solar radiation into dry mat ter equivalents (Loomis and Williams, 1963): I Co - 3.875.0 (19) where I represents the incoming radiation (in c a l m 2 d a y - l ) and 0 indicates planting density. The light capture rate, s, is a function of the leaf area index LAI and the light extinction coefficient, 3' which was set to 0.62 (Littleton et al., 1979b) s = 1 - e [-z''LAI] (20) k LAX = 0 " u " ~ Q l , i ( d ' ) ' h f (21) i = 1 The LAI is computed by multiplying the daily leaf area, given as the product of the leaf mass and the slope u of the regression (with intercept 0) between observed leaf area and leaf mass, with a planting density of 0. On cultivars producing large seeds, such as the variety under study, early leaf senescence and abscission are observed at the reproductive nodes where pods are developing (Pate et al., 1983). The mechanism driving this process is believed to depend on phytohormones (Nooden and Murray, 1982; Neumann and Nooden, 1984). This is accounted for by reducing the photosynthetically active leaf area when the ratio between the daily com- puted values of the number of inflorescences and the cube of the number of growing pods r n < 15. The scaling factor hf is subsequently calculated as follows: k E Q7,i(d') / = 1 ( 2 2 ) h f = 0.21 + 0.05 • k 16 3 106 M. TAMO AND J. BALIMGARTNER The s u p p l y / d e m a n d ratios c~ and c 2 are computed as follows: b 3 + b 5 c 1 = 1 for P ' > 1 - F P ' " (1 - F) b 3 + b 5 c 1 = b3 + b5 for P ' < 1 ~ e 2 ~ - - - 0 for c I < 1 P ' - [ C l " ( b 3 + b 5 ) ] = for c~ = 1 ¢2 ( l - F ) - l ' ( b l -q- b 2 + b 4 + bs) + b r where (23) (24) P ' = P + R - b g (25) P is the gross photosynthesis [equation (18)], while P ' represents the amount of daily available assimilates in the metabolic pool allocated via c~ to fruit and peduncles, and via c 2 to leaves, shoot, peduncles, roots, and reserves (Fig. 1). Water submodel The influence of drought stress on growth and yield formation of cowpea has been documented by Turk et al. (1980) and Turk and Hall (1980a,b,c). During the second planting season in Benin, which begins early September , the rains may stop before the end of October. By that time, some cowpea crops may not have reached the flowering stage and the effect of drought stress is substantial. Hence, it is necessary to link the plant model to a submodel for soil water balance and water stress. The details of the water submodel used in this work are described by Gutierrez et al. (1988). Briefly, it uses the computed water s u p p l y / d e m a n d ratio Wsa to adjust the demand rates for plant growth by assuming a water holding capacity of 600 1 in a root envelope of 1 m 3 [see equation (10)]. Water supply is calculated with the same functional response model described in (18), while water demand is est imated with the Ritchie (1972) model. Parameter est imation The model required detailed data on the dynamics of masses and numbers of plants grown without pest stress. Hence, the data used for the estimation of the model parameters were collected in cowpea fields kept pest free with Cymbush Super E D ® (66% Cypermethr in + 33% Dimethoate , applied with the Electrodyn ® method). For this purpose, ANALYSIS OF THE COWPEA AGRO-ECOSYSTEM 107 three fields were planted at the I ITA Biological Control Center for Africa in Abomey-Calavi on May 17th, 1988 (0.25 ha), on September 13th, 1988 (0.13 ha), and on May 7th, 1990 (0.24 ha), respectively. The sowing was performed manually with three cowpea grains per hole at a spacing of 0.25 × 0.75 m. The seedlings were thinned to one plant per hole 2 weeks after emergence. The variety under study was 'Kpodjigu6gu6', a local cultivar which is slightly photosensitive, has an indeterminate and semi-erect growth habit, and is harvested after 70-75 days. The length of leaves and peduncles on different nodes, the length of the different internodes, as well as the length of the growing pods, were recorded on 20 plants in the field. These measurements were repeated on the same plants every second day during the growing cycle under study. At the same time, individual leaves, stem cuttings between two nodes, pedun- cles, and pods from 20 plants of the same field were first measured, then dried in paper bags at 104°C for 24 h, and subsequently weighed. This was needed to determine the specific weights for each organ. The dry matter at each sampling date was estimated by multiplying the specific weights by the respective lengths measured in the field. To obtain the growth rates, the dry masses calculated in this way were plotted against physiological time, which could have been expressed by simulation time increments [equation (11)]. In this case, however, the physiological time horizon in units of T A B L E 2 Parameters for the time distributed delay model: j represents the index for the different a t t r i b u t e s , ~[~j is the change in the daily developmental rate with respect to temperature. The potential demand rate for mass and for numbers Aj is given for the three different phenological phases of the plant: germination (germ), vegetative phase until anthesis of the reproductive structures (veg), and reproductive phase (repr) afterwards j Attr ibute /3j Aj germ veg repr 1 Leaf mass 0.00159 1.634 4.521 4.521 2 Shoot mass 0.00151 0.771 1.719 1.719 3 Fruit mass 0.00138 0 0 f(i) a 4 Root mass 0.00151 1.1565 2.578 1.719 5 Peduncles mass 0.00238 0 0 f(i) b 6 Leaf number 0.00222 0.008 0.04 0.04. ns 7 Intl. number 0.00222 0 0 0.04. ns 8 Fruit number 0.00145 0 0 0.024 a f ( i ) = 0.00708.e[0.0112.(i-0.5).29], where i denotes the index of the substages i = 1-25 in which the reproductive structures are growing. hf(i)=O.OO65"e[°°21°6(i-°5)17], where i denotes the index of the substages i = 1-22 in which the peduncles are growing. 1 0 8 M. T A M ( ) A N D J. B A U M G A R T N E R 1000 - 800 / Eli:3 I :~ ~0o , . / o <> o ._o) 400 - 0 C/ , e/ , . . o:=#:F "-u . . , 0 200 400 600 800 t ime [daydegrees ] Fig. 2. An example of the parameterization of the model: estimation of the maximum growth ra tes for the masses of one leaf (<>) and one shoot in te rnode ( , ) with l inear growth ra tes in the initial growing phase. The growth of one fruit ( [ ] ) is exponent ial . These ra tes are given in Table 2. daydegrees was calculated by forcing a sine wave through daily tempera- ture extremes and integrating it above the developmental threshold T o (Frazer and Gilbert, 1976; Gilbert et al., 1976) set to 8°C (Hadley et al., 1983). This allowed measurement of physiological events such as growth or shedding of particular organs in daydegrees, in order to determine the number of substages i where organs were expressing a demand or were affected by attrition in the delay process (Table 1) on the one hand, and to calculate /3j (Table 2) as the reciprocal of the duration of development of the respective attribute on the other. In addition, the number of dayde- grees observed until the induction of the reproductive phase was used for the computation of the parameters b I and b 2 in equation (1). Plant growth parameters were estimated for three distinct phenological phases: a germination phase, from the sowing to the fall of the cotyledons, followed by a vegetative phase until the induction of the reproductive phase [equation (1)]. The increase of attributes is either constant (leaves, shoots, roots) or exponential (peduncles and fruit) at the beginning of the growing phase and decreases thereafter, presumably due to restrictions imposed by natural physiological processes (Fig. 2). The initial growing phase is therefore considered to reflect the potential growth of the plant and used to approximate the demand rates h t (Table 2). Model validation and evaluation The validation of the model is based on the visual comparison of the curves generated by the simulation model with four sets of data on growth ANALYSIS OF T H E C O W P E A A G R O - E C O S Y S T E M 109 and development of the 'Kpodjigu6gu6' variety. Three fields were planted at the Center in Abomey-Calavi on September 23rd, 1987 (second season 1987), on March 22nd, 1988 (early first season 1988), and on May 12th, 1989 (first season 1989). One field was planted in Sohedji, northern Zou Province, on June 4th, 1989 (first season 1989). In all the fields under study, the planting densitity was 5.33 plants per m 2. The dry weight of leaves, shoots, peduncles (second season 1987 only) and reproductive structures was measured bi-weekly at the Center, and weekly in Sohedji, using the same methodology as described in the parame- ter estimation procedures. The daily weather data used to drive the model (including the water submodel) were recorded at the Center, and consisted of maximum and minimum temperature (°C), solar radiation (cal m 2 day-l) , average wind speed (km h-~), average relative humidity (%), and rainfall (mm day-l) . For the validation of the field data collected in Sohedji, the weather data from the meteorological station in Savalou, 7 km away, were used. The values of the available phosphate and water in the soil were estimated through simulation studies. To evaluate the effect of drought stress and soil phosphate level on plant growth and yield formation, changes in the plant growth pattern were simulated with different levels of water and soil phosphate available to the plant at the beginning of the growing season using the weather data and the planting date of the first season 1989 at the station. The evaluation of drought stress was performed by reducing the measured rainfall by 50, 70 and 75%. The influence of soil phosphate was tested by reducing the values of available phosphate by 20, 40 and 60%. RESULTS AND DISCUSSION Validation The driving variables, as well as their influence on the supp ly /demand ratios and on the growth patterns, are depicted in Figs. 3-6 for four different cropping seasons. Stress occurs whenever the supply does not meet the demand. The number of days after planting is abbreviated as DAP. Second season 1987 (Fig. 3) Rainfall was concentrated during the vegetative phase of plant growth (Fig. 3B). Nevertheless, the soil water available allowed the formation of consistent grain yield, which was limited by the low available phosphate 1 1 0 M. TAMO AND J. BAUMG,~RTNER .. _ t ~ A 3O E 20 .~ - - "~ 100 .. t i l l ' ' L.J,. I ® 3000 i[" | ~ - 1ooo ~ 1"olD ~ E m o.o ..Q t~. 8 ~ 0.2 E lOOOO o beginning of the reproductive phase E ~ . m ~ ~ 20000 = ~ . . . . x./~-°"-~'~'.... ~P/O ~ I I I I I I 0 14 28 42 56 70 (20/0W87) (07/10/87) (21110/87) (04/11/87) (18/11/87) (02/12187) time [DAP] Fig. 3. Model validation: driving variables (A, daily temperature extremes; B, radiation and rainfall per day) and growth pattern of the crop planted on September 23 rd, 1987, at the Center in Abomey-Calavi [C, carbohydrates available per day, (upper area) gross photosyn- thesis, (lower area) reserves; D, supply/demand ratio for vegetative ( ) and for reproductive ( . . . . . . ) organs; E, observed (symbols) and simulated (lines) weights of plants components]. ANALYSIS O F T H E C O W P E A A G R O - E C O S Y S T E M 111 content in the soil (Pa = 3.5 ppm). The initial value for soil water was set to 510 1 m -3. During the early reproductive phase, the incoming solar radia- tion was relatively constant, allowing a regular increase of the simulated gross photosynthesis until the development of the first pods (Fig. 3C). Thereafter, the photosynthetically active leaf area was presumably reduced by senescence hormones, and the reserves stored mainly in shoots, pedun- cles, and roots were remobilized to meet the increasing demand of the growing pods. This is reflected in the patterns of the supp ly /demand ratios for both vegetative and reproductive organs (Fig. 3D). At the end of the germination phase, the cotyledons fell off and, at the same time, the nodulation process induced by Rhizobia increased the root demand, lead- ing to a stress situation for the seedling. This was the cause of the first drop in the supp ly /demand ratio of the vegetative organs, whereas subsequent reductions during the vegetative phase were mainly caused by fluctuations in solar radiation. The major decline during the reproductive phase was probably due to both the effect of the senescence hormones and the increased demand of the growing pods. The combined effect of these two factors on the supp ly /demand ratio for reproductive organs induced the shedding of flower buds, flowers and young pods, so that the available assimilates could be allocated entirely to the surviving growing pods. Simulation of the dry matter acquisition for leaves, shoots, peduncles, and fruit compared satisfactorily with field data, as shown in Fig. 3E. Early first season 1988 (Fig. 4) The crop was planted before the onset of the rainy season, so that irrigation (4 m m m - 2 day -1) was necessary until the first important rain. Thus, the initial value of the available soil water was set to 476 1 m-3. With the exception of the early vegetative growth, the plant did not suffer from drought stress, because precipitation was regularly distributed over the whole growing cycle (Fig. 4B). Cloudy skies during this period were responsible for wide fluctuations in solar radiation, which had a major impact on both the production of assimilates (Fig. 4C) and on the supp ly /demand ratios (Fig. 4D). Stress induced by nodulation, as well as the decline caused by the senescing leaves, were accentuated by the relatively small photosynthate production caused by the low incoming radiation. Compared with the second season 1987, the higher soil phosphate content (P~--6 ppm) gave a considerable increase in dry matter produc- tion. Comparison of model predictions with field data showed that the com- bined dry matter of shoot and peduncles was slightly overestimated before pod elongation, and underest imated thereafter (Fig. 4E). A plausible 112 M. TAMO AND J. BAUMGARTNER 40 2O _ o ~ E ~ E ~ .~ - ~ lOO ==8"~ ~'~ 50 c 4000 2000 - D 1.0 0.6 0.2 E 30000 10000, ~, beginning of the reproductive phase r= , , . . . . . ~ • ~ ~ hs OOt +pe'd uncles " . . . I I I I I 14 28 42 56 70 (22/03/88) (05/04188) (19/04/88) (03/0~88) (17/05188) (31/05/88) t i m e [ D A P ] Fig. 4. Model validation: driving variables (A, daily temperature extremes; B, radiation and rainfall per day) and growth pattern of the crop planted on March 22 nd, 1988, at the Center in Abomey-Calavi [C, carbohydrates available per day, (upper area) gross photosynthesis, (lower area) reserves; D, supply/demand ratio for vegetative ( ) and for reproductive ( . . . . . . ) organs; E, observed (symbols) and simulated (lines) weights of plant components]. A N A L Y S I S O F T H E C O W P E A A G R O - E C O S Y S T E M 113 40 .= "~ 30 E 2o (5 ~ 200 ,<~ ,--'o E 150 o ~. E ~:; loo ,~ 50 8000 o c ¢o >, "o 4000 ,Iz A t max 1 B ......... radiatio. I . . ~ _ h d ,- . i. C ~ I"°ID ~ -- 0.6 L, Q . 8 ~ 0.2 E 40000 E • ~ 20000 o+ | I o (12J0~/89) beginning of the reproductive phase . . . . . . . . . . . . . . . . . . . . ! / - , , :i / leaves .m- - - ~ , - / , ~ ; " ~ I I I I I 14 28 42 56 70 (26A05/~) (09~o/89) (23J06/89) (07107/1~) (21~07/89} time [DAP] Fig. 5. Model validation: driving variables (A, daily temperature extremes; B, radiation and rainfall per day) and growth pattern of the crop planted on May 12 tn, 1989, at the Center in Abomey-Calavi [C, carbohydrates available per day, (upper area) gross photosynthesis, (lower area) reserves; D, supply/demand ratio for vegetative ( ) and for reproductive ( . . . . . . ) organs; E, observed (symbols) and simulated (lines) weights of plant components]. 114 M. TAM() AND J. BAUMG,~RTNER explanation for this discordance can be found in E1-Sharkawy et al. (1984), who demonstrated that, for field beans (Phaseolus vulgaris L.), stomatal conductance was mainly influenced by air humidity. Hence, irrigation of the crop could not prevent stomatal closure under low air humidity conditions, leading to a decrease in photosynthetic activity. This aspect was not included in the present model. First season 1989 (Fig. 5) The amount of soil water available at the beginning of this season was set to 525 1 m -3. The vegetative phase until 40 DAP was characterized by low rainfall (Fig. 5B), which reduced the water supp ly /demand ratio causing slight stress around 20 DAP. Superphosphate was applied at a rate of 80 kg per ha before planting, in order to approach optimal growth conditions (Pa -- 13.5 ppm). Peak values of over 8 g assimilates per day were predicted by the model (Fig. 5C). The shortage of photosynthate production due to daily fluctuations in solar radiation, combined with the water stress described above, had already induced a decline of the supp ly /demand ratio for vegetative organs during the vegetative phase, as depicted in Fig. 5D. The simulated dry matter allocated to leaves, shoots plus peduncles, and fruit closely corresponded to the field data (Fig. 5E). First season 1989 (northern Zou) (Fig. 6) As in the other first season plantings, rainfall was equally distributed over the whole cropping cycle, and the daily changes in solar radiation (Fig. 6B) accounted for much of the fluctuations of the photosynthetic activity (Fig 6C). The estimated value for available water in the root envelope at the beginning of the growing period was 510 1 m -3. In contrast to the fertilized field planted at the Station 2 weeks earlier, this field, planted in Sohedji (northern Zou department), provided the validation base for poor soil conditions (Pa = 2.5 ppm). During most of the vegetative phase, the demand of vegetative organs, scaled by the low phosphate level, was covered by the supply, indicating that apparently the plant was not stressed (Fig. 6D). Thereafter, the increasing demand of the growing pods com- bined with a reduction in photosynthesis due to senescing leaves led to an early depletion of the reserves (Fig. 6C). The trend of the dry matter curves depicted in Fig. 6E reflect the poor fertility of the soil. The simulation curve of fruit dry matter reached a plateau at 16 g five days after the last sampling date. Evaluation of drought stress (first season 1989) The effect of the simulated proportional reduction of rainfall on the soil water level and on the water supp ly /demand ratio is presented in Fig. 7A A N A L Y S I S O F T H E C O W P E A A G R O - E C O S Y S T E M 115 ~3 40 = "~ 30 E 2 0 , ; i .o 150 _o~,_ E -- ~ 10o m 5 0 t m a x t rain I I r 'r"' I , . I . , I , , , . , I_l . = . . . . . . . I , . I J . l . , C i m ~, 2000 - m looo- r- D ~ 1.0 m E = ..D I~ i ~ 0.2 7 ~, beginn ing of the reproduct ive phase ~ i ---~ !, 15000 fru, O) E 5000 / " ' ; ~ .~o~o/1//o "-. I I I I I 0 14 28 42 56 70 (04/06/80) (18/06/80) (02/07/89) (16,'07/80 ) (30/07/89) (13/08/89) t i m e [ D A P ] Fig. 6. Model validation: driving variables (A, daily temperature extremes; B, radiation and rainfall per day) and growth pattern of the crop planted on June 4 th, 1989, in Sohedji, northern Zou Province [C, carbohydrates available per day, (upper area) gross photosynthe- sis, (lower area) reserves; D, supply/demand ratio for vegetative ( ) and for reproduc- tive ( . . . . . . ) organs; E, observed (symbols0 and simulated (lines) weights of plant compo- nents]. 116 M, TAM() AND J. BAUMGARTNER 600 550 500 0 450 ¢0 1.0 ~ 0.8 ~ o " ~ 0.6 - 0 . 4 0.2 50000 40000 - ,f-, m • o 30000 - " 20000 - ._~ 10000- o I o A B - , . ~ i i J I I I I I 14 28 42 56 7O t i m e [ D A P ] Fig. 7. Model evaluation; simulation of drought stress by proportional reduction of the rainfall during the first season 1989 at the Center in Abomey-Calavi. Evaluation of the effect of 100% ( ), 50% ( . . . . . . ), 30% ( ), and 25% rainfall ( . . . . . . ) A, on the available soil water; B, on the water supp ly-demand ratio; C, on the dry weight acquisition for leaves (1)and fruits (f). A N A L Y S I S O F T H E C O W P E A A G R O - E C O S Y S T E M 117 50000 40000- ////"-" -" w- .~ 30000- " 2 0 0 0 0 - 10000 - 0 rlllllllllII~JIIllffilllllII~lli111dlJlllilfllllilllll llllIJt,llllli~lll* 7 14 21 28 35 42 49 56 63 70 time [DAP] Fig. 8. Model evaluation: the influence of different levels of available soil phosphate on the dry weight acquisition of leaves (1) and fruits (f). The simulation curves were generated for the first season 1989 at the Center in Abomey-Calavi using 100% ( ), 80% ( ...... ), 60% ( ), and 40% ( ...... ) of the available phosphate present in the soil. and B, respectively. The soil water curve indicates that 25% of the observed rainfall drives the system to the permanent wilting point (476 1 m -3) for the first time at 38 DAP (see Fig. 5B). Despi te precipitations occurring after 40 DAP, which allowed the unreduced soil water curve to reach field capacity (600 1 m-3) , the 25% curve dropped again to the permanent wilting point during the pod filling phase. A reduction of 50% of the observed precipitation did not affect the growth of the plant. Even with a reduction of 70%, grain yield was reduced by only 21%. However, a further 5% decrease caused considerable yield loss (42%) (Fig. 7C). The resulting yield loss was caused mainly by the shedding of flower buds and young pods. This tendency compares favorably to the results of Turk et al. (1980). Evaluation of the influence of different levels of soil phosphate (first season 1989) The proport ional reduction of the phosphate level in the soil caused a general decrease in plant growth (Fig. 8). Here , lower leaf growth resulted in lower production of assimilates, and consequently lower grain yield. The results of this evaluation are very similar to the simulation curves validated in Figs. 3E, 4E, and 6E. 1 1 8 M. TAM0 AND J. BAUMGARTNER CONCLUSIONS The work demonstrates the suitability of the demographic approach for the analysis of cowpea growth and development. Since the approach has been successfully applied to a variety of different crops (see Graf et al., 1990), this investigation confirms its general applicability to ecosystem analysis. The value of the cowpea model stems primarily from two qualities. First, its solid theoretical foundation based on demographic principles (Gutierrez and Wang, 1976; Baumg~irtner et al., 1990a; Severini et al., 1990a, b). Second, the model is capable of satisfactorily representing crop growth under a variety of different growing conditions, such as different weather patterns, soil properties, and management practices. The model also permits the assessment of yield limiting factors, which is of great value as herbivory will be added to the model in the future. In the absence of herbivory the soil qualities, particularly the availability of phosphate, appear to be a major hindrance for growth. Where the soil meets the demands of the plant, and irrigation is disregarded, the interplay between radiation and precipitation appears to control yield formation at large. High crop growth rates occurred under a regular supply of water, provided that the number of cloudy days was not limiting for photosynthe- sis. ACKNOWLEDGEMENTS This study was funded by the Swiss Development Cooperation and carried out as a special project of the Biological Control Program (BCP) of IITA. We wish to thank Dr. H.R. Herren, Director of BCP, and Prof. V. Delucchi, former Head of the Division of Phytomedicine, ETH Zurich, for the logistic support, and, together with all the colleagues of both institutes, for the good working ambience. Great help in collecting the data was given by Mr. B. Hettin, Mr. M. Azokpota, Mr. C. Assou, Ms. T. Sossavi, Mr. B. Dato, and Mr. M. Fassi. We are grateful to Ms. I. Olaleye, who illustrated the graphics, and to colleagues at IITA for reviewing the manuscript. REFERENCES Baumg~irtner, J. and Gutierrez, A.P., 1989. Simulation techniques applied to crops and pest models. In: R. Cavalloro and V. Delucchi (Editors), PARASITIS 88. Proc. Sei. Congr., Barcelona, 25-28 October 1988. Bolefin de Sanidad Vegetal, Fuera de Serie 17: 175-214. Baumg~irtner, J. and Severini, M., 1988. Microclimate and arthropod phenologies: the leaf miner Phyllonorycter blancardella F. (Lep.) as an example. In: F. Prodi, F. Rossi and G. Cristoferi (Editors), Agrometeorology. Editrice Compositori, Bologna, pp. 225-253. A N A L Y S I S O F T H E C O W P E A A G R O - E C O S Y S T E M ], 19 Baumg~irtner, J., Bieri, M., Klay, A., Genini M. and Zahner, Ph., 1989. Fungicide side effects on the dynamics of an acarine predator -prey system in apple orchards: an explorative study with simulation models. In: C. Gessler, D.J. Butt and B. Koller (Editors), Integrated Control of Pome Fruit Diseases, Vol. II. WPRS Bull., 12(6): 317-335. Baumg~irtner, J., Severini, M. and TamS, M., 1990a. Modelli demografici per la fenologia e l ' interazione fra specie nella gestione dei sistemi agricoli. Societh Italiana di Fitoiatria. Convegno Nazionale 'Modelli euristici e operativi per la difesa integrata in agricoltura'. Caserta, September 27-29th (in press). Baumg~irtner, J., Wermelinger, B., Hugentobler, U., Delucchi, V., Baronio, P., De Berardi- nis, E., Oertli J.J. and Gessler, C,, 1990b. Use of a dynamic model on dry matter production and allocation in apple orchard ecosystem research. Acta Hortic., 276: 123-139. Bellows, T.S., Jr., 1986a. Impact of developmental variance on behavior of models for insect populations. I. Models for populations with unrestricted growth. Res. Popul. Ecol., 28: 53-62. Bellows, T.S., Jr., 1986b. Impact of developmental variance on behavior of models for insect populations. II. Models for populations with density dependent restrictions on growth. Res. Popul. Ecol., 28: 63-67. Curry, G. and Feldman, R.M., 1987. Mathematical foundations of population dynamics. Texas A&M University Press, College Station, 246 pp. EI-Sharkawy, M.A., Cock, J.H. and Held, A.A., 1984. Water use efficiency in cassava. II. Differing sensitivity of stomata to air humidity in cassava and other warm-climate species. Crop Sci., 24: 503-507. Frazer, B.D. and Gilbert, N., 1976. Coccinellids and aphids. A quantitative study of thc impact of adult ladybirds (Coleoptera, Coccinellidae) preying on field populations of pea aphids (Homoptera, Aphididae). J. Entomol. Soc. B.C., 73: 33-56. Getz, W.M. and Gutierrez, A.P., 1982. A perspective on systems analysis in crop production and insect pest management. Annu. Rev. Entomol., 27: 447-466. Gilbert, N., Gutierrez, A.P., Frazer, B.D. and Jones, R.E., 1976. Ecological Relationships. Freeman, Reading, 157 pp. Graf, B., Baumgiirtner, J. and Gutierrez, A.P., 1990. Modeling agroecosystem dynamics with the metabolic pool approach. Mitt. Schweiz. Entomol. Ges., 63: 465-476. Gutierrez, A.P. and Wang, Y., 1976. Applied population ecology: models for crop produc- tion and pest management. In: G.A. Norton and C.S. Holling (Editors), Pest Manage- ment. I IASA Proceedings Series 4, Pergamon Press, Oxford, pp. 255-280. Gutierrez, A.P., Falcon, L.A., Loew, W., Leipzig, P.A. and van den Bosch, R., 1975. An analysis of cotton production in California: a model for Acala cotton and the effects of defoliatiors on its yield. Environ. Entomol., 4: 125-136. Gutierrez, A.P., Baumgiirtner, J.U. and Hagen, K.S., 1981. A conceptual model for growth, development and reproduction in the ladybird beetle, Hippodamia conuergens (Cole- optera: Coccinellidae). Can. Entomol., 113: 21-33. Gutierrez, A.P., Baumg~irtner, J.U. and Summers, C.G., 1984a. Multitrophic models of predator -prey energetics. Can. Entomol., 116: 923-963. Gutierrez, A.P., Pizzamiglio, M.A., Dos Santos, W.J., Tennyson, R. and Villacorta, A.M., 1984b. A general distributed delay time varying life table plant population model: cotton (Gossypium hirsutum L.) growth and development as an example. Ecol. Modelling, 26: 231-249. Gutierrez, A.P., Schulthess, F., Wilson, L.T., Villacorta, A.M., Ellis, C.K. and Baumg~irtner, J.U., 1987. Energy acquisition and allocation in plant and insects: a hypothesis for the possible role of hormones in insect feeding patterns. Can. Entomol., 119: 109-129. 120 M. TAMO AND J. BAUMG,g, RTNER Gutierrez, A.P., Wermelinger, B., Schulthess, F., Baumg~irtner, J.U., Herren, H.R., Ellis, C.K. and Yaninek, J.S., 1988. Analysis of biological control of cassava pests in Africa. I. Simulation of carbon, nitrogen, and water dynamics in cassava. J. Appl. Ecol., 25: 901-920. Hadley, P., Roberts, E.H., Summerfield, R.J. and Minchin, F.R., 1983. A quantitative model of reproductive development in cowpea (Vigna unguiculata (L.) Walp.) in relation to photoperiod and temperature, and implications for screening germplasm. Ann. Bot., 51: 531-543. Jackai, L.E.N. and Daoust, R.A., 1986. Insect pests of cowpeas. Annu. Rev. Entomol., 31: 95-119. Littleton, E.J., Dennett, M.D., Elston, J. and Monteith, J.L., 1979a. The growth and development of cowpeas (Vigna unguiculata) under tropical field conditions. 1. Leaf area. J. Agric. Sci. Camb., 93: 291-307. Littleton, E.J., Dennett, M.D., Monteith, J.L. and EIston, J., 1979b. The growth and development of cowpeas (Vigna unguiculata) under tropical field conditions. 2. Accumu- lation and partition of dry weight. J. Agric. Sci. Camb., 93: 309-320. Littleton, E.J., Dennett, M.D., Elston, J. and Monteith, J.L., 1981. The growth and development of cowpeas (Vigna unguiculata) under tropical field conditions. 3. Photosyn- thesis of leaves and pods. J. Agric. Sci. Camb., 97: 539-550. Loomis, R.S. and Williams, W.A., 1963. Maximum crop productivity: an estimate. Crop Sci., 3: 67-72. Manetsch, T.J., 1976. Time-varying distributed delay models and their use in aggregative models of large systems. IEEE Trans. Syst., Man Cybern., 6: 647-553. Manly, B.F.J., 1989. A review of methods for the analysis of stage-frequency data. In: L. McDonald, B.F. Manly and J. Logan (Editors), Estimation and Analysis of Insect Populations. Lecture Notes in Statistics 55, Springer, Berlin, pp. 3-69. Neumann, P.M. and Nooden, L.D., 1984. Pathway and regulation of phosphate transloca- tion to the pods of soybean plants. Physiol. Plant., 60: 166-170. Nooden, L.D. and Murray, B.J., 1982. Transmission of the monocarpic senescence signal via the xylem in soybean. Plant Physiol., 69: 754-756. Ojehomon, O.O., 1968a. The development of the inflorescence and extra-floral nectaries of Vigna unguiculata (L.) Walp. J. West Aft. Sci. Assoc., 13: 93-111. Ojehomon, O.O., 1968b. Flowering, fruit production and abscission in cowpea, Vigna unguiculata (L.) Walp. J. West Afr. Sci. Assoc., 13: 227-234. Pate, J.S., Peoples, M.B. and Atkins, C.A., 1983. Post anthesis economy of carbon in a cultivar of cowpea. J. Exp. Bot., 34: 544-562. Penning de Vries, F. and Van Laar, H. (Editors), 1982. Simulation of Plant Growth and Crop Production. Center for Agricultural Publishing and Documentation, Wageningen, 308 pp. Ritchie, J.T., 1972. Model for predicting evaporation for a row crop with incomplete cover. Water Resour. Res., 8: 1204-1213. Severini, M., Baumg~irtner, J. and Ricci, M., 1990a. Theory and practice of parameter estimation of distributed delay models for insect and plant phenologies. In: R. Guzzi, A. Navarra and J. Shukla (Editors), Meteorology and Environmental Sciences. World Scientific and International Publisher, Singapore, pp. 674-719. Severini, M., Baumg~irtner, J., Seifert, M. and Ricci, M., 1990b. The analysis of poikilother- mic population development by means of time distributed delay models. Computer Science and Mathematical Methods in Plant Protection, Int. Workshop, Parma, Novem- ber 7-9, 1990 (in press). ANALYSIS OF THE COWPEA AGRO-ECOSYSTEM 121 Summerfield, R.J., Minchin, F.R., Roberts, E.H. and Hadley, P., 1983. Cowpeas. In: W.H. Smith and S. Yoshida (Editors), Potential Productivity of Field Crops Under Different Environments. IRRI, Los Bafios, Philippines, pp. 249-280. Summerfield, R.J., Pate, J.S., Roberts, E.H. and Wien, H.C., 1985. The physiology of cowpeas. In: S.R. Singh and K.O. Rachie (Editors), Cowpea Research, Production and Utilization. John Wiley, Chichester, pp. 65-101. Tam6, M., Baumg~irtner, J. and Gutierres, A.P., 1992. Analysis of the cowpea agro-ecosys- tem in West Africa. II. Modelling the interactions between the cowpea and the bean flower thrips Megalurothrips sjostedti (Trybom) (Thysanoptera, Thripidae). Ecol. Mod- elling (in press). Turk, K.J. and Hall, A.E., 1980a. Drought adaptation of cowpea. 2. Influence of drought on plant water status and relations with seed yield. Agron. J., 72: 421-427. Turk, K.J. and Hall, A.E., 1980b. Drought adaptation of cowpea. 3. Influence of drought on plant growth and relations with seed yield. Agron. J., 72: 428-433. Turk, K.J. and Hall, A.E., 1980c. Drought adaptation of cowpea. 4. Influence of drought on water use, and relations with growth and seed yield. Agron. J., 72: 434-439. Turk, K.J., Hall, A.E. and Asbell, C.W., 1980. Drought adaptation of cowpea. 1. Influence of drought on seed yield. Agron. J., 72: 413-420. Vansickle, J., 1977. Attrition in distributed delay models. IEEE Trans. Syst. Man Cybern., 7: 635 -638. Von Foerster, H., 1959. Some remarks on changing populations. In: F. Stohlman, Jr. (Editor), The Kinetiks of Cellular Proliferation. Grune & Stratton, New York, pp. 382-407. Wang, Y., Gutierrez, A.P., Oster, G. and Daxl, R., 1977. A population model for plant growth and development: coupling cotton-herbivore interactions. Can. Entomol., 109: 1359-1374. Wermelinger, B., Baumg~irtner, J. and Gutierrez, A.P., 1991. A demographic model of assimilation and allocation of carbon and nitrogen in grapevines. Ecol. Modelling, 53: 1-26.