Density Dependent Simulation of the Population

Density Dependent Simulation of the Population Dynamics of a Perennial Grassland Species, Hypochaeris radicata Hans de Kroon; Anton Plaisier; Jan van Groenendael Oikos, Vol. 50, No. 1. (Sep., 1987), pp. 3-12. Stable URL: http://links.jstor.org/sici?sici=0030-1299%28198709%2950%3A1%3C3%3ADDSOTP%3E2.0.CO%3B2-X Oikos is currently published by Nordic Society Oikos.

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/oikos.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected].

http://www.jstor.org Mon Jan 14 11:43:05 2008

OIKOS 50: 3-12. Copenhagen 1987

Density dependent simulation of the population dynamics of a perennial grassland species, Hypochaeris radicata Hans de Kroon, Anton Plaisier and Jan van Groenendael

Kroon, H. de, Plaisier, A. and van Groenendael, J. 1987.Density dependent simulation of the population dynamics of a perennial grassland species, Hypochaeris radicata. - Oikos 50: 3-12.

Hypochaeris radicata L. is a rosette-forming herb that often attains dominance after a number of years in newly established grassland like roadverges, but decreases afterwards. To evaluate the possible causes of its success and eventual decline, a transition matrix model was constructed simulating different management regimes, with parameter values estimated from preliminary field data. Mowing twice a year results in the largest initial population growth rate and the largest equilibrium density compared with the other simulated mowing frequencies (mowing once a year, uncut verge). Elasticity analysis indicated that the parameters that together form the pathway of sexual reproduction and those that represent vegetative ramification, are balanced with respect to their impact on population growth. This balance shifts with decreasing mowing frequency towards vegetative ramification. This is largely the result of variation in transitions of the pathway of sexual reproduction. Large rosette densities are maintained only when these variable transitions, such as germination and establishment of seedlings are optimal. This characterizes H. radicata as a fugitive species, rather than as a competitor, and helps to understand its replacement in a succession. H. de Kroon, A . Plaisier and J. van Groenendael, Dept of Vegetation Sci., Plant Ecology and Weed Sci., Agricultural Univ., Bornsesteeg 69, NL-6708 PD Wageningen, The Netherlands (present address of HdK: Dept of Plant Ecology, Lange Nieuwstraat 106, NL-3512 PN Utrecht, The Netherlands; present address of AP: Dept of Public Health and Social Medicine, Erasmus Univ. Rotterdam, P. 0. Box 1738, NL-3000 DR Rotterdam, The Netherlands).

Introduction

It is an accepted fact that in most types of vegetation there are continuous changes in species composition and in abundance per species (e.g. Tamm 1956, Connell and Slatyer 1977, Watt 1981a,b, Grubb et al. 1982, Willems 1983). Fluxes of species may cause problems when species are considered desirable in the habitat from a manager's or conservationist's point of view (e.g. Duffey and Watt 1971, Synge 1981). A very successful colonizer in many grassland habitats is Hypochaeris radicata L., a perennial rosette-forming species (Turkington and Aarssen 1983). Depending on the management regime, this species often outnumbers the other dicotyledons within a few years, a fact also reported by Grime (1979). Accepted 9 March 1987

0 1'

OlKOS

OIKOS 50:l (1987)

About 10 to 15 years later, however, the population size of H. radicata often decreases considerably. This study is an attempt to evaluate the key stages in the life-history of H. radicata which are responsible for the rapid increase and maintenance of a population, including different density dependent responses in various stages of the life cycle. The problem is approached by density dependent simulation using transition matrices to describe the population dynamics of the species. The model is formulated for different management regimes, and demographic transitions are calculated from preliminary field data. The responses of individuals on density and management practice are estimated from literature and unpublished results. The model is used to analyse the relative impact of these responses on

SPRING -POPULATION

SUMMER

AUTUMN-WPULATION

SPRING-POPULATION

Fig. 1. Life history diagram of Hypochaeris radicata. size-$ass subdivisions are omitted for the sake of clarity. The three main life history pathways are survival of adults (A), vegetative ramification (B) and sexual reproduction (C). The diagram shows summer and winter processes separately.

non-f lowering

population characteristics. Key stages in the life history are evaluated by means of a modified sensitivity analysis (elasticity analysis - D e Kroon et al. 1986). It was not the intention to predict the exact development of populations under different conditions (in contrast to e.g. Reader and Thomas 1977: see also Reader 1985). The results are used to tiy to explain the effects'of different management regimes on the population dynamics of H . radicata and to generate hypotheses on the possible causes of the rapid increase and eventual decline of this species in grassland habitats. The species and the study plot

Hypochaeris radicata (cat's ear) is a perennial, rosetteforming herb common in low productivity grasslands in the Netherlands. Rosettes often belong to a common taproot-system, forming a physiological unit, and sometimes individual plants are difficult to recognize. In general the genet survives much longer than the individual rosettes. As a result of a critical size for flowering, rosettes are not able to flower in their first growing season under field conditions (J. C. Vulto, unpubl.). Flowering starts in May and continues for a period of 5 to 6 wk under undisturbed conditions. The formation by the apical meristem of a flowering stalk activates axillary buds and stimulates the production of side-rosettes. The production of new leaves from the apex is inhibited, resulting generally in the death of the flowering rosette. vegetative rosettes also produce side-rosettes to a much smaller extent. U p to 20 flowering stalks, each containing one o r more flowering heads, can be found on a single rosette. Each flowering head contains between 15 and 300 seeds and this results in 300-6000 seeds per rosette (Turkington and Aarssen 1983; H. d e Kroon, pers. obs.). Germination of H . radicata takes place mainly in Sep-

tember and October. Because of the lack of a dormancy mechanism (Turkington and Aarssen 1983) germination in spring is much less important and there is no permanent seedbank. A dry summer clearly increases mortality of rosettes from all phenological classes. For reviews on H. radicata see Turkington and Aarssen (1983) and Aarssen (1981). The population studied is localized in a road verge near Barneveld in the centre of the Netherlands. The verge consists of a sandy subsoil covered with a layer of humus. In constructing the road verge in 1972 a seed mixture of the following composition was used: Festuca rubra L. (several cv's) 75%, Festuca ovina L. 20% and Agrostis capillaris L. 5%. No dicotyledon species were introduced. As a rule this type of verge is mown twice a year but other mowing frequencies are applied. The model In describing population dynamics with matrix models the choice of appropriate state variables and time steps is very important (Werner and Caswell 1977, Law 1983). In comparing 'age' and 'stage' as a criterion Werner and Caswell (1977) demonstrated the superiority of 'stage' in describing the population dynamics of a plastic monocarpic perennial (Dipsacus sylvestris Huds.). Similarly a division into stages was chosen to describe the individuals of the H . radicata population. In the model four developmental stages are recognized that are further split into four size categories. The stages are seeds (S), juveniles from seeds (R,), side rosettes (R,) and mature rosettes (R,). The latter two categories are subdivided into: vegetative (V) and generative (G). The size is estimated by the number of leaves times the length of the longest leaf and the total size spectrum is subdivided into four classes. To obtain more uniformity in the transitions between the different stages a distinction has been made between

Fig. 2. Simplified diagram of the life cycle of H. radicata, with the three main pathways drawn separately. The rosette subdivisions into vegetative - generative categories and into size classes are omitted for the sake of clarity. Elasticities are presented for a colonizing population under mowing frequency 0. See text for further explanation.

YEAR (i)

SPRING SURVIVAL ( A )

YEAR (i.1)

mature rosettes

mature rosettes

; !

(C)

I;\

172- 112--+$

+

, ,

P--,2Q

I '

I I RAMIFICATION (8)

I

SPRING

-.190-

-D

$!!_307-.

; r,-,ll6-,+, a 8 I i e

--

I 41-

I

I i

juvenile rosettes

rosettes

.-

-

!L. -

-- - - - - -

i

i

-- - ----

i i

I 079-

- - ,112- - ,116-

-.-

i

-+seeds

+ramification

rosettes

summer- and winter-processes. A simplified flow-diagram of the transitions is given in Fig. 1. Each rosette has the potential to survive andlor to flower and/or to ramify at the onset of the season. In spring 5 stages are present (ignoring size differences), converging into 3 stages in autumn. This is caused by the fact that in spring the distinction between V and G rosettes is made. A t the end of the summer nearly all flowering rosettes die and winter survival is only affected by the size of the remaining rosettes. As a consequence of the absence of a seedbank it is assumed that seed dispersal occurs in autumn after which a seed either germinates or dies. In contrast to side rosettes, juvenile rosettes, though being of the same age, d o not flower o r ramify (Fig. 1). After one growing season both side rosettes and juvenile rosettes become mature. The transition probabilities are arranged into a summer (A,) and winter (A,) matrix. The initial stage vector contains the spring population (N,, (t)). The intrinsic properties of the model were calculated using an iteration procedure, with steps of one year:

N,, ( t + l ) = A

* N,, (t)

in which A is a year-transition matrix derived from:

In this form the model is linear and time-invariant. It is relatively simple to replace fixed matrix elements by variables that are a function of the number of individuals. This is achieved by using a multiplication factor (S) that reduces in a density dependent fashion elements in the transition matrix. The shape of this function can be described in several ways. Most authors use a more or less exponential decrease (e.g. Smouse and Weiss 1975, Guckenheimer et al. 1977, Travis et al. 1980). These functions have the disadvantage that density dependent effects are most pronounced at small population sizes. For an establishing population in a relatively open habitat, as is the case in a newly constructed roadside verge, this is unrealistic. The 'reflected sigmoid increase function' of Usher (1972), which is steepest at intermediate densities, seems more appropriate. Usher's function is modified into the following equation:

with A and B constant and S,,, as the minimum value of the multiplication factor, corresponding to a maximum population size possible in the field (100-120 rosettes m-'). The values of the parameters A and B shape the curve S, and are fixed empirically at 16.0 and 3.75 respectively for all cases. Different strengths of reduction for various phases in the life cycle are achieved solely by varying S,,,.

A = A, * A, (cf. Sarukhan and Gadgil 1974). The iteration was continued until the intrinsic properties of the model had reached stability. These properties are the maximum eigenvalue, equivalent to the population growth rate, and the left and right eigenvector giving the stable distribution of reproductive values (v) and the stable stage distribution (w) respectively.

Elasticity analysis A n important tool in analysing a model is the use of sensitivity analysis. A general formula for the sensitivity in matrix projection models reads:

Tab. 1. Effects of mowing. Relative values of parameter groups are given. *denotes a third flowering the magnitude of which depends on the size of a generative rosette. See text for further explanation. Mowing frequency

Seed production Germination and establishment of seedlings Survival of iuvenile rosettes ~amificationby vegetative rosettes generative rosettes

0

1

2

1

312

312*

1 1

1 918

918

1

1 314

27132

1

2 918

s,, = GAiGa,, = v,w,i (Caswell 1978)

where s,, is the sensitivity of population growth rate A. for small perturbations in matrix element a,,; v and w are the left and right eigenvector respectively. The interpretation of sensitivities gives problems when parameters with highly different numerical values are to be compared (De Kroon et al. 1986). Therefore, based on the original derivative, a relative measure of sensitivity was developed, defined as e,, = blnAl6ln a,, = (a,,lh)(bA/Ga,,). This index is termed elasticity, and can be interpreted as a measure that quantifies the relative contribution of a,, to the population growth rate (De Kroon et al. 1986). It is important to note that the elasticity is an intrinsic property of a transition matrix, calculated from the values of its elements and its eigenvectors. As with Caswell's (1978) original measure, elasticities are additive between matrix elements. A n interesting property of elasticities is that they give a single value for a single life-history pathway. This is illustrated in Fig. 2. where elasticity values are given for the three main pathways in the life cycle of H . radicata: survival, sexual reproduction and ramification. Note that the life cycle is drawn over two subsequent years since seeds and side rosettes need this period to reach maturity. At any point in the life cycle, the elasticities of all transitions belonging to a certain pathway sum to identical values. For instance in the pathway of sexual reproduction, the contribution of seed production to the growth rate of the population is essentially the same as the contribution of survival of juvenile rosettes. Being part of the same pathway. both transitions are completely dependent upon one another in their separate effect on A. Parameter estimations Transition matrices were constructed for Hypochaeris radicata populations under various mowing regimes. Es-

timates of the parameter values of the transition matrices are mainly based on one year of demographic field work (1983), a quantification of seed production for two years and several sowing experiments to establish the emergence and survival of seedlings (Van Ast et al. 1987). With the available data and general knowledge of the species an extrapolation has been made to a situation of a colonizing population when there is no mowing. Three situations are modelled: 1) no mowing, 2) mowing once in June, and 3) mowing twice, in June and in September. owing has a number of consequences. Mowing in June removes the flowering stalks of approximately half of the generative rosettes (the other half produces reproductive organs later in the season). This induces a second flowering with approximately twice the number of flowering stalks compared with the first flowering (H. de Kroon, unpubl.). This results in an increase of the average seed production over one season (Tab. 1). Mowing a second time in September induces a third flowering period. The number of extra seeds produced in September is size dependent, because during the summer the different rosette sizes have different survival probabilities. In total, mowing twice maximizes seed production. On the other hand, the formation of additional flowering stalks reduces the number of axillary buds available for vegetative propagation. Therefore. in the model ramification of generative rosettes is reduced by mowing. Mowing results in a more open vegetation, even in a colonizing phase shortly after the construction of the roadside verge, and, as for many herbs, this results in increased survival probabilities of vulnerable stages such as seedlings, juvenile rosettes and side rosettes (see e.g. Goldberg and Werner 1983). In the model the survival of seedlings and side rosettes is affected only by mowing in September, since they emerge late in the season. Note that ramification by generative rosettes when mowing twice is affected simultaneously by a decrease (factor 314) as a result of a reduced number of buds available, and by an increase (factor 918) as a result of a more open vegetation (Tab. 1). Mature rosettes are strong competitors (Turkington and Aarssen 1983) with large horizontally oriented leaves that are hardly shaded by the rest of the vegetation. Their survival is assumed to be unaffected by mowing. While the population is expanding after establishment a dense turf is developing at the same time. The effects of increasing interspecific competition and increasing population density are not analysed separately but integrated into one reduction function which is related to the density of the population. Several phases in the life cycle are made density dependent. The survival of juvenile rosettes is reduced because of increased shading with increasing density. Young side rosettes are affected as well. but less so because the newly formed ramets are connected to the taproot system and to some extent protected from shad-

Tab. 2. Minimum values (S,,,) of the parameter groups that are made density dependent. Smi,values are given as proportions of the transitions under colonizing conditions. Mowing frequency

Germination and establishment of seedlings Survival of juvenile rosettes Ramification

118 112 213

118 9/16 213

114 9/16 314

ing by grasses by other rosettes of the same taproot. Germination and establishment of seedlings are two processes that are also affected by turf formation. Both are incorporated in a single model parameter, and are highly dependent on the number of gaps in the vegetation. The number of gaps decreases considerably with increasing density. Seed production itself is assumed to be density independent because of the competitive ability of established rosettes. The strength of the density dependent reduction is given by the S,,,-values. Tab. 2 lists values as proportions of the maximum values under colonizing conditions for three mowing regimes. Mowing is assumed to reduce the effect of density because the number of gaps increases and shading decreases. Consequently, the S,,,-values for mowing frequencies 1and 2 are increased by the same proportions as used in Tab. 1.

rosettes many mature plants have established as well in comparison with the other mowing frequencies. The mature rosettes raise the seed production of the population and permit a rapid future increase. The model indicates a small colonizing ability for H. radicata under reduced mowing (frequencies 1 and 0). There are also differences in the stage structure of the populations at stable densities (Fig. 4 right). The number of juvenile rosettes increases with increasing mowing frequency but the relative differences are smaller compared with Fig. 4 left. The number of rosettes belonging to other categories is relatively constant over the mowing frequencies. With increasing mowing frequencies, stable densities seem to be maintained by means of seeds more than by ramification o r survival of adult rosettes. Aspects of the life history

To put these simulation results in a life history perspective it is necessary to analyse what parts of the life cycle are most important for growth and maintenance of the population under different mowing regimes. This is done by means of elasticity analysis. The elasticities of three main life history pathways

Results Changes in population size and structure under density dependence

The three management regimes are simulated using density dependent reduction functions (Tab. 2). All simulations were initiated with 100 seeds m-'. With increasing mowing frequency the initial growth rate increases and this ultimately results in larger population sizes (Fig. 3). At a certain density population growth ceases and the number of rosettes starts oscillating as a result of density dependent feedback. These oscillations are very small for mowing frequency 0 and 1 so that the population growth closely resembles a logistic curve. The simulations are continued until the point of approximate stability (about 30 yr in this case). Beyond this point the density functions no longer validly describe the further development of the population. The asymptotic behaviour of the model is explored separately in the Appendix. Large differences in colonizing ability are especially apparent. Five years after initiation the population with mowing frequency 2 has attained a considerable size (Fig. 4 left). Although there is a large group of juvenile

time (years)

Fig. 3. Simulated population growth under density dependent influences for three mowing frequencies. The initial population growth rates are 1.270, 1.431and 1.946 for mowing frequencies 0, 1 and 2 respectively.

0x

1x

Ox

2x mowing frequency

1x

2x mowing frequency

Fig. 4. Simulated stage structures for three mowing frequencies. (left) populations 5 yr after initiation, (right) populations at stable density. R, = juvenile rosettes, R, = side rosettes, R,,, = mature rosettes. Dark areas indicate the proportion of flowering rosettes.

(survival, vegetative ramification and sexual reproduction - cf. Fig. 2) were calculated for growing populations in the colonizing phase and for populations with a stable density (Fig. 5). In general, the survival of mature rosettes is of little importance. Much greater is the impact of ramification and of sexual reproduction. Both are of the same magnitude and their ratio is dependent upon specific conditions. With increasing mowing frequency the importance of seed production increases relative to the importance of ramification for colonizing as well as stable populations. However, seed production is less important in stable populations. These results of the elasticity analysis on the life history of H. radicata do not give precise information with respect to single transitions within a single pathway, because all transitions in the same pathway in the life cycle have essentially the same elasticity (see Fig. 2). Moreover, elasticities are by definition relative so that underlying numerical changes in transitions are obscured. For example, the increase in the elasticity of the pathway of sexual reproduction versus the pathway of ramification with increasing density (see Fig. 5 ) can be caused by an increase of the number of seeds produced as well as by a decrease in the number of side rosettes formed by an average rosette, or both.

For the pathways of ramification and sexual reproduction the relative shifts in parameter values between colonizing and stable conditions are given in Tab. 3 for different mowing frequencies. The pathway of survival needs no further consideration because its impact is negligible compared with the others (Fig. 5). The reduction is most pronounced for the germinationlestablishment parameters, and is the main cause for the increase in importance of ramification versus sexual reproduction with increasing density as given in Fig. 5. This result suggests that the balance between sexual reproduction and ramification is one of the major characteristics in the life history of H. radicata. This balance is mainly influenced by variation in sexual reproduction, whereas the possibilities for ramification seem more constant.

Discussion Verification of the model

In general H. radicata is a plant of various kinds of manmade habitats like frequently cut or heavily grazed

Elasticity ,400 1

mowing frequency a ,;.:

1

....;... .. .:.:

..:,.. .. .... .... .. .. .... ..: ..

... ..

;;;;: A

B

:!:' A

C

COLONIZING POPULATION

C

... :....

..:.

. .. . ... ....:... ..... .... .... .. j':'....

... ... ..

..:.: . .......... .;.: ..

0

;;:;

B

B

1

..:.:

::..........

A

The life history of Hypochaeris radicata

2x

1x

0x

A

C

B

C

STABLE POPULATION Fig. 5. Elasticities for three main life history pathways for populations under different simulated management regimes. A = survival of adult rosettes, B = vegetative ramification, C = sexual reproduction.

grasslands (Turkington and Aarssen 1983). Bakker et al. (1983) reported a large increase of H. radicata in a former meadow, now continuously grazed, in comparison with another part of the same meadow, now in use for hay making. In roadverge grasslands in the Netherlands its abundance is determined to a large extent by the mowing regime. In stands mown twice a year H. radicata often is the most abundant dicotyledon, whereas in uncut verges the population consists of a few scattered individuals or is completely absent. In a few years' time this situation resulted after mowing was stopped in a roadside verge previously mown twice a ye& (Tab. 4). Struik (1967) showed that H . radicata reached its highest production in continuously grazed Pastures and in lawns. Since the mean dry weight per plant was smallest in the latter treatment, the number of rosettes was very large. The relative dry weight of floral organs increased with decreasing stand defoliation, but was smaller in the uncut lawn. Grazing induced an optima1 sexual reproduction. The i n p u t df the model consisted of the responses of plant individuals on various mowing frequencies. The simulation indicated that plant density increased with increasing stand defoliation, which is in agreement with the field data mentioned above.

In interpreting the results of the simulation and the elasticity analysis one must take into account the organization of rosettes in a taproot system. Vegetative ramification guarantees the survival of the genet even when the mother rosette dies. It is probably relatively little affected by environmental conditions and crowding. The large elasticities for ramification suggest that the survival of taproots is an important characteristic in the life history of H. radicata. Two functions are apparent. Firstly, it guarantees the maintenance of the population in years when sexual reproduction is not successful. As such it can be regarded as a dormancy mechanism (cf. Harper and White 1974). Secondly, many side rosettes are capable of flowering in the year after their formation and their seed production can contribute considerably to the total sexual reproduction (elasticities up to 43% of those of the total seed production). From the results it appears that an increase in initial growth rate under increasing mowing frequency (Fig. 3) is the result of positive effects of mowing on seed production (by inducing an extra flowering period) and on the establishment of seedlings (by creating a more open vegetation). The role of sexual reproduction is reduced relative to that of vegetative propagation with increasing density mainly as a result of difficulties in the germination/establishment phase. This is in accordance with observations in psammophyte communities in Poland (Symonides 1979). Nevertheless, results of the simulations indicate that differences in seed production between the mowing regimes play an important role in the maintenance of the different population sizes. This is a remarkable result. In general, more o r less stable populations with high densities primarily rely on survival and vegetative spread for their maintenance (Sarukhan 1974, Abrahamson 1980, but see Barkham and Hance 1982). For H. radicata reproduction seems to be more important than for other perennial rosette forming dicotyledons like Plantago lanceolata (Van Groenendael and Slim 1987), Ranunculus repens (Sarukhan and Gadgil 1974), Hieracium floribundum (Thomas and Dale 1975) and Hieracium pilosella (Bishop et

Tab. 3. Percentual reductions in parameter values for three mowing regimes in two life history pathways of H. radicata (B ramification and C - sexual re~roduction).w hen ~ o p u l a t i o n s develop towards stable conditions.

.

Mowing frequency

~ ~ ~ i (B) f i ~ ~ t i ~ ~ Seed production (C) Ciermination and establishment of seedlings (C) Survival of juvenile rosettes (C)

0

1

2

14 0

19 0

25 0

38 22

49 25

71 42

Tab. 4. Percentage cover of Hypochaeris radicata in permanent quadrats of 1x4 m, placed in variously managed sections of 50 m of a roadverge near Barneveld, opposite the locality of the population under study, normally mown twice a year. Means and standard deviations of two replicates are presented over a four year period (after H. Heemsbergen, unpubl. res.). -

-

-

Year Mown twice a year (June and October) Mown once a year (June) Mown once every two years (June) No mowing

1982

1983

1984

1985

17.5f 3.5 3.5? 0.7 16.5f 2.1 25.0f 14.1

20.0f 7.1 6.5f4.9 2 . 5 f 0.7 6.5k2.1

15.0f7.1 2.5f0.7 1 . 5 f 0.7 0

9 . 0 f 1.4 3 . 0 f 1.4 0 0

al. 1978). These differences may be related to the population fluxes of the species. For H. radicata the turnover of genets is high with an estimated half-life of about 8 months (J. van Groenendael, unpubl.). In comparison, the estimated half-life of genets of Plantago lanceolata is at least 20 months (Van Groenendael 1985b). This can be related to the meristematic organization of H . radicata in which the flowering rosette does not survive, in contrast with Ranunculus spp. (Sarukhan and Harper 1973) and Plantago lanceolata (Van Groenendael 1985a). After flowering the rosettes of both Hieracium spp. die as well, but here other factors, assumedly not applicable to H. radicata, are responsible for the small role of sexual reproduction for maintaining the population. For Hieracium floribundum, for instance, the percentage flowering is density dependent. This increases the longevity of mature rosettes in crowded situations (Thomas and Dale 1975, Reader and Thomas 1977). Together with vegetative propagation this is responsible for the maintenance of a dense population. Another illustrative experiment is reported by Davy and Bishop (1984) with regard to Hieraciumpilosella. Increased nutrition stimulated flowering and,on account of the terminal inflorescences, therefore rosette mortality. Stolon formation increased simultaneously, but the establishment of daughter rosettes was reduced in the denser vegetation. The net effect was a decline of the population, especially in ungrazed swards. Hypochaeris radicata lacks high rosette survival probabilities under crowded conditions, and vegetative reproduction mechanisms that can compensate for the continuous loss of rosettes. Together with the small probability of an individual seed to grow to adult size a very large seed production is necessary to guarantee sufficient recruitment. Its dependence on sexual reproduction makes H . radicata less suited to maintain itself in a succession, in contrast to many other perennials (Grime 1979). On the other hand, its large fecundity together with an adequate dispersal mechanism makes the species a rapid colonizer. Therefore, H. radicata may be characterized as a fugitive species, rather than as a competitor, despite its perennial life cycle. To allow this species to persist in a grassland, it is necessary to maintain the grassland in an early successional stage by regular mowing. -

-

APPENDIX: Stability in a non-linear population model Although the asymptotic behaviour of the model presented here falls beyond its biological validity, an analysis of this behaviour is useful from a theoretical point of view. A non-linear model can develop towards a stable equilibrium, a stable limit cycle or a regime of irregular oscillations (chaos) (e.g. May 1976, Guckenheimer et al. 1977). For one-dimensional models the mathematical boundary conditions for these possibilities can be described to a large extent (e.g. May and Oster 1976. Witten 1980, Pacala and Silander 1985). Especially the chaotic region has been analysed intensively in recent years including applications for biological systems (see e.g. Schaffer 1985, Schaffer and Kot 1986 and references therein). However, non-linear models with two or more dimensions (matrix projection models with several state variables) have received little attention until now (Guckenheimer et al. 1977, Witten and D e la Torre 1984). Their behaviour produces great mathematical complexities which are as yet largely unsolved. The limit behaviour of the three simulation runs of H . radicata falls into two different categories. For mowing frequency 0 and 2 the oscillations induced by the density reduction functions are damped, resulting in stable equilibria (N(t) = 55.8 and 105.5 respectively). For mowing frequency 1 the oscillations increase with time until a stable two-point cycle is reached (with N(t) values of 50.0 and 114.3 respectively). The observed differences can only be analysed in crude mathematical terms. From current knowledge it appears that a smaller initial growth of the population (e.g. Guckenheimer et al. 1977, Hassell 1980, Kot and Schaffer 1984) and a flatter reduction function (Usher 1972, May and Oster 1976, Pacala and Silander 1985) promote the stability of the system. As a result, a coefficient of density dependence (C,,), proportional to the chance for a more complicated behaviour. can be formulated as:

where A,,, is the initial growth rate and C,, is a 'coefficient of non-linearity', expressing the strength of the reduction functions applied in the model. Note that h,,, and C,, are given the same weight, for convenience, as any information in the literature on their relative importance is lacking. The extent to which each of the re-

duction functions used in the model of H. radicata, contributes to the non-linearity of the system (C,,) depends on: 1. the steepness of the reduction function itself, and 2. the importance of the matrix elements on which the reduction function works. This importance corresponds with the extent to which total population size N(t) is affected by this set of matrix-elements, and can be expressed by their elasticities, that quantify the impact of each element on the growth rate of the population. The justification for this is that the importance of a matrixelement for changing N(t) is proportional to its impact on A (A. Otten, pers. comm.). The steepness of the reduction function is given by its first order derivative. The coefficient of non-linearity can now be formulated as : C,, = Z(f,'*e,) With f,' as the absolute value of the first order derivative of reduction function f influencing parameterset i, and e, as the added elasticities of this group of parameters, both calculated for an approximately stable population density. Calculating the coefficients of non-linearity for the three modelled populations of H. radicata and inserting them into ( 1 ) produces the largest coefficient of density dependence for mowing frequency 1: mowing frequency C,, ( * l o o ) : Cdll( * l o o ) :

0 0.675 0.857

1

2

0.727 1.040

0.133 0.258

It appears that stable equilibria are possible both with a small initial population growth rate and steep reduction functions (mowing frequency 0) and vice versa (mowing frequency 2 ) . If the initial growth is intermediate and the reduction functions relatively steep, stable limit cycles develop. Simulations with other hypothetical populations revealed that the values of the C,, proposed usually match the limit behaviour of the model (but not always). This suggests that the weighting of A,,, relative to C,, in (1) needs further attention. Acknowledgements - We are much indebted to S. ter Borg, T. de Jong. P. Klinkhamer, M. Usher and M. Williamson for valuable comments on the manuscript. A. Otten gave mathematical support during the study, L. de Nijs carried out the demographic field work and H. Klees prepared the figures.

References Aarssen. L, W. 1981.The biology of Canadian Weeds Hypochrieris radicata L. - Can. J . Plant Sci. 61: 365-381. Abrahamson. W. G. 1980. Demography and vegetative reproduction, - In: Solbrig, O, T, (ed,), Demography and evelution in plant populations, Blackwell, Oxford,pp, 89-106. Ast. A. van. ter Borg, S. J. and van Groenendael, J. hl. 1987. De oorzaak van de achteruitgang van Biggekruid in onze bermen. - De Levende Natuur 88: 88-93. Bakker3 J , P,, de Bie, S,. Dallinga, J. H , , Tjaden3P, and de Vries. Y , 1983, Sheep grazing as management tool for OIKOS 50.1 (1987)

heathland conservation and regeneration in the Netherlands. - J. appl. Ecol. 20: 541-560. Barkham, J. P, and Hance, C. E. 1982. Population dynamics of the wild daffodil (Narcissus pseudonarcissus). 111. Implications of a computer model of 1000 years of population change. - J. Ecol. 70: 323-344. Bishop, G. F., Davy, A. J. and Jefferies, R. L. 1978. Demography of Hieracium pilosella in a breck grassland. - J . Ecol. 66: 615429. Caswell, H. 1978. A general formula for the sensitivity of population growth rate to changes in life history parameters. - Theor. Popul. Biol. 14: 215-230. Connell, J. H. and Slatyer, R . 0 . 1977. Mechanisms of succession in natural communities and their role in community stability and organization. - Am. Nat. 110: 1119-1144. Davy, A. J. and Bishop. G . F. 1984. Responses of Hieracilkm pilosella in breckland grass-heath to inorganic nutrients. J. Ecol. 72: 319-330. Duffey, E. and Watt, A. S. 1971. The scientific management of animal and plant communities for conscrvation. - Blackwell, Oxford. Goldberg, D. E . and Werner. P. A. 1983. The effects of size of opening in vegetation and litter cover on seedling establishment of goldenrods (Solidago spp.). - Oecologia (Berl.) 60: 149-155. Grime, J. P. 1979. Plant strategies and vegetation proccsses. Wiley, Chichester. Groenendael, J. M. van 1985a. Teratology and metameric plant construction. - New Phytol. 99: 171-178. - 1985b. Differences in life histories between two ecotypes of

Plantago lanceolata L. - In: White. J . (ed.), Studies on

plant demography: a festschrift for John L. Harper. Aca-

demic Press, London, pp. 51-67.

- and Slim, P. 1987. The contrasting dynamics of two popula-

tions of Plantago lanceolara classified by age and size. - I.

Ecol. (in press).

Grubb, P. J . , Kelly, D . and Mitchley, J. 1982. The control of relative abundance in communities of herbaceous plants. In: Newman, E. I. (ed.), The plant community as a working mechanism. Blackwell, Oxford, pp. 79-97, Guckenheimer, J., Oster, G. and Ipaktchi, A. 1977. The dynamics of density dependent population models. - J . Math. Biol. 4: 101-147. Harper, J. L. and U7hite,J . 1974. The demography of plants. Ann. Rev. Ecol. Syst. 5: 419463. Hassell, M. P. 1980. Some consequences of habitat heterogeneity for population dynamics. - Oikos 35: 150-160. Kot. M. and Schaffer. W. M. 1984. The effects of seasonalitv on discrete models of population growth. - Theor. Popui Biol. 26: 340-360. Kroon. H. dc, Plaisier, A , , van Groenendael, J. and Caswell. H. 1986. Elasticity: the relative contribution of demographic parameters to populat~ongrowth rate. - Ecology 67: 1427-1431. Law, R. 1983. A model for the dynamics of a plant population containing individuals classified by age and size. - Ecology 63: 224-230. May, R. M. 1976. Simple mathematical models with very complicated dynamics. - Nature. Lond. 261: 459367. - and Oster. G. F. 1976. Bifurcations and dynamic complexity in simple ecological models. - Am. Nat. 110: 573-599. Pacala. S. W. and Silander. J . A. 1985. Neiehborhood models of plant population dynamics. I . Single-species models of annuals. - Am. Nat. 125: 385411. R, 1985, Temporal in recruitment and mortality for the pasture weed Hieraciunzfloribundum: imof population plicationsfor a - J. a ~ ~ l . Ecol. 22: 175-185. - and Thomas, A , G , 1977, Stochastic of patch

formation by Hieracium floributldum (Compositae) in

ab"doned pasture. - Can. J . Bat. 55: 3075-3079.

Sarukhan. J. 1974. Studies on plant demography: Ranunculus ~

~

11

repens L., R. bulbosus L. and R. acris L. 11. Reproductive strategies and seed population dynamics. - J. Ecol. 62: 151-177. - and Gadgil, M. 1974. Studies on plant demography: Ranunculus repens L., R. bulbosus L. and R. acris L. 111. A mathematical model incorporating multiple modes of reproduction. - J . Ecol. 62: 921-936. - and Harper, J. L. 1973. Studies on plant demography: Ranunculus repens L.. R. bulbosus L. and R. acris L. I. Population flux and survivorship. - J. Ecol. 61: 675-716. Schaffer, W. M. 1985. Order and chaos in ecological systems. Ecology 66: 9S106. - and Kot, M. 1986. Chaos in ecological systems: the coals that Newcastle forgot. - Trends Ecol. Evol. 1: 58-63. Smouse, P. E . and Weiss, K. M. 1975. Discrete demographic models with density-dependent vital rates. - Oecologia (Berl.) 21: 205-218. Struik, G . J. 1967. Growth habitats of dandelion, daisy, catsear, and hawkbit in some New Zealand grasslands. - N. Z. J. Agr. Res. 10: 331-344. Symonides, E. 1979. The structure and population dynamics of psammophytes on inland dunes. IV. Population phenomena as a phytocenose - forming factor (a summing-up discussion). - Ekol. Pol. 27: 259-281. Synge, H. 1981. The biological aspects of rare plant conservation. - Wiley, Chichester. Tamm, C. 0. 1956. Further observations of the survival and flowering of some perennial herbs, I. - Oikos 7: 273-292. Thomas, A. G. and Dale, H. M. 1975. The role of seed production in the dynamics of established populations of Hier-

acium floribundum and a comparison with that of vegetative reproduction. - Can. J. Bot. 53: 3022-3031. Travis. C. C., Post, W. M., DeAngelis, D. L. and Perkowski, J. 1980. Analysis of compensatory Leslie matrix models for competing species. - Theor. Popul. Biol. 18: 1 6 3 0 . Turkington, R. and Aarssen, L. W. 1983. Biological Flora of the British Isles. 156. Hypochoeris radicata L. - J . Ecol. 71: 999-1022. Usher. M. B. 1972. Developments in the Leslie matrix model. - In: Jeffers, J. N . R. (ed.), Mathematical models in ecology. Blackwell, Oxford, pp. 29-60. Watt, A. S. 1981a. A comparison of grazed and ungrazcd Grassland A in East Anglian Breckland. - J. Ecol. 69: 499508. - 1981b. Further observations on the effects of excluding rabbits from Grassland A in East Anglian Breckland: the pattern of change and factors affecting it (1936-73). - J. Ecol. 69: 509-536. Werner, P. A. and Caswell, H. 1977. Population growth rate and age versus stage-distribution models for teasel (Dipsacus sylvestris Huds.). - Ecology 58: 1103-11 11. Willems. J. H . 1983. Species composition and above ground phytomass in chalk grassland with different management. Vegetatio 52: 171-180. Witten, M. 1980. Some generalized conjugacy theorems and the concept of fitness and survival in logistic growth models. - Bull. Math. Biol. 42: 507-528. - and de la Torre, D. L. 1984. Biological populations obeying difference equations: the effects of stochastic perturbations. - J. Theor. Biol. 111: 493-507.

http://www.jstor.org

LINKED CITATIONS - Page 1 of 4 -

You have printed the following article: Density Dependent Simulation of the Population Dynamics of a Perennial Grassland Species, Hypochaeris radicata Hans de Kroon; Anton Plaisier; Jan van Groenendael Oikos, Vol. 50, No. 1. (Sep., 1987), pp. 3-12. Stable URL: http://links.jstor.org/sici?sici=0030-1299%28198709%2950%3A1%3C3%3ADDSOTP%3E2.0.CO%3B2-X

This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR.

References Population Dynamics of the Wild Daffodil (Narcissus Pseudonarcissus): III. Implications of a Computer Model of 1000 Years of Population Change J. P. Barkham; C. E. Hance The Journal of Ecology, Vol. 70, No. 1. (Mar., 1982), pp. 323-344. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28198203%2970%3A1%3C323%3APDOTWD%3E2.0.CO%3B2-S

Demography of Hieracium Pilosella in a Breck Grassland G. F. Bishop; A. J. Davy; R. L. Jefferies The Journal of Ecology, Vol. 66, No. 2. (Jul., 1978), pp. 615-629. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28197807%2966%3A2%3C615%3ADOHPIA%3E2.0.CO%3B2-E

Mechanisms of Succession in Natural Communities and Their Role in Community Stability and Organization Joseph H. Connell; Ralph O. Slatyer The American Naturalist, Vol. 111, No. 982. (Nov. - Dec., 1977), pp. 1119-1144. Stable URL: http://links.jstor.org/sici?sici=0003-0147%28197711%2F12%29111%3A982%3C1119%3AMOSINC%3E2.0.CO%3B2-H



Response of Hieracium Pilosella in Breckland Grass-Heath to Inorganic Nutrients A. J. Davy; G. F. Bishop The Journal of Ecology, Vol. 72, No. 1. (Mar., 1984), pp. 319-330. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28198403%2972%3A1%3C319%3AROHPIB%3E2.0.CO%3B2-L

The Demography of Plants J. L. Harper; J. White Annual Review of Ecology and Systematics, Vol. 5. (1974), pp. 419-463. Stable URL: http://links.jstor.org/sici?sici=0066-4162%281974%295%3C419%3ATDOP%3E2.0.CO%3B2-J

A Model for the Dynamics of a Plant Population Containing Individuals Classified by Age and Size Richard Law Ecology, Vol. 64, No. 2. (Apr., 1983), pp. 224-230. Stable URL: http://links.jstor.org/sici?sici=0012-9658%28198304%2964%3A2%3C224%3AAMFTDO%3E2.0.CO%3B2-H

Bifurcations and Dynamic Complexity in Simple Ecological Models Robert M. May; George F. Oster The American Naturalist, Vol. 110, No. 974. (Jul. - Aug., 1976), pp. 573-599. Stable URL: http://links.jstor.org/sici?sici=0003-0147%28197607%2F08%29110%3A974%3C573%3ABADCIS%3E2.0.CO%3B2-W

Neighborhood Models of Plant Population Dynamics. I. Single-Species Models of Annuals Stephen W. Pacala; J. A. Silander, Jr. The American Naturalist, Vol. 125, No. 3. (Mar., 1985), pp. 385-411. Stable URL: http://links.jstor.org/sici?sici=0003-0147%28198503%29125%3A3%3C385%3ANMOPPD%3E2.0.CO%3B2-R

Studies on Plant Demography: Ranunculus Repens L., R. Bulbosus L. and R. Acris L.: II. Reproductive Strategies and Seed Population Dynamics Jose Sarukhan The Journal of Ecology, Vol. 62, No. 1. (Mar., 1974), pp. 151-177. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28197403%2962%3A1%3C151%3ASOPDRR%3E2.0.CO%3B2-U



Studies on Plant Demography: Ranunculus Repens L., R. Bulbosus L. and R. Acris L.: III. A Mathematical Model Incorporating Multiple Modes of Reproduction Jose Sarukhan; Madhav Gadgil The Journal of Ecology, Vol. 62, No. 3. (Nov., 1974), pp. 921-936. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28197411%2962%3A3%3C921%3ASOPDRR%3E2.0.CO%3B2-Y

Studies on Plant Demography: Ranunculus Repens L., R. Bulbosus L. and R. Acris L.: I. Population Flux and Survivorship Jose Sarukhan; John L. Harper The Journal of Ecology, Vol. 61, No. 3. (Nov., 1973), pp. 675-716. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28197311%2961%3A3%3C675%3ASOPDRR%3E2.0.CO%3B2-M

Order and Chaos in Ecological Systems William M. Schaffer Ecology, Vol. 66, No. 1. (Feb., 1985), pp. 93-106. Stable URL: http://links.jstor.org/sici?sici=0012-9658%28198502%2966%3A1%3C93%3AOACIES%3E2.0.CO%3B2-I

Hypochoeris Radicata L. (Achyrophorus Radicatus (L.) Scop.) Roy Turkington; Lonnie W. Aarssen The Journal of Ecology, Vol. 71, No. 3. (Nov., 1983), pp. 999-1022. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28198311%2971%3A3%3C999%3AHRL%28R%28%3E2.0.CO%3B2-Q

A Comparison of Grazed and Ungrazed Grassland A in East Anglian Breckland A. S. Watt The Journal of Ecology, Vol. 69, No. 2. (Jul., 1981), pp. 499-508. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28198107%2969%3A2%3C499%3AACOGAU%3E2.0.CO%3B2-C

Further Observations on the Effects of Excluding Rabbits from Grassland A in East Anglian Breckland: The Pattern of Change and Factors Affecting it (1936-73) A. S. Watt The Journal of Ecology, Vol. 69, No. 2. (Jul., 1981), pp. 509-536. Stable URL: http://links.jstor.org/sici?sici=0022-0477%28198107%2969%3A2%3C509%3AFOOTEO%3E2.0.CO%3B2-9



Population Growth Rates and Age Versus Stage-Distribution Models for Teasel (Dipsacus Sylvestris Huds.) Patricia A. Werner; Hal Caswell Ecology, Vol. 58, No. 5. (Sep., 1977), pp. 1103-1111. Stable URL: http://links.jstor.org/sici?sici=0012-9658%28197709%2958%3A5%3C1103%3APGRAAV%3E2.0.CO%3B2-3