further magni ed by nonlinear terms in the infection force in the models. ... small initial di erences become magni ed in nonlinear models, has been known for ...
Dynamically generated variability in plant-pathogen systems with biological control A. Kleczkowski, D. Bailey, and C.A. Gilligan
SUMMARY Using a combination of replicated microcosm experiments, simple nonlinear modelling and model tting we show that unexpected levels of variability can be detected and described in the dynamics of plant disease. Temporal development of damping{o disease of radish seedlings caused by an economically important plant pathogen, Rhizoctonia solani, is quanti ed, with and without the addition of an antagonistic fungus, Trichoderma viride. The biological control agent reduces the average amount of disease but also greatly enhances the variability amongst replicates. The results are shown to be consistent with predictions from a nonlinear model that exhibits dynamically generated variability in which small differences in the initiation of infection associated with the antagonist are later ampli ed as the pathogen spreads from plant to plant. The e ect of dynamically generated variability is mediated by the interruption of transient disease progress curves for separate replicates by an exponential decrease in susceptibility of the host over time. The decay term essentially 'freezes' the dynamics of the transient behaviour so that the solutions for di erent replicates settle on asymptotes that depend on initial conditions and parameter values. The e ect is further magni ed by nonlinear terms in the infection force in the models. A generalisation of the Lyapunov exponent is introduced to quantify the ampli cation. The observed behaviour has profound consequences for the design and interpretation of ecological experiments, and can also account for the notorious failure of many biological control strategies through the creation of 'hot spots', created by the ampli cation of plant to plant infection, where the control by the antagonist is locally unsuccessful.
Keywords: epidemiology, non-linear model, dynamically generated variability, biological control, transients, Lyapunov exponent.
Dept. of Plant Sciences, Univ. of Cambridge, Cambridge CB2 3EA, England
1
1 INTRODUCTION The importance of dynamically generated variability of population behaviour, in which small initial di erences become magni ed in nonlinear models, has been known for some time in physical systems, especially in the context of chaotic behaviour (Yao & Tong, 1994; Deissler & Farmer, 1992). Chaos has also been widely studied in biological systems (May, 1976; Grenfell et al., 1995), but chaotic behaviour requires not only a nonlinearity in the laws governing the system but also constant parameters over a comparatively long time. Such conditions rarely occur in biological populations because of uctuations and trends in environmental conditions (Grenfell et al., 1994). We show that dynamical variability may, however, occur in short-term behaviour in certain models for population growth when the approach towards a global equilibrium is interrupted. Variability in the transient behaviour, due to small di erences in initial conditions or values of critical parameters, is then re ected in the nal state (Epstein, 1995). In this paper we use a combination of replicated microcosm experimentation, simple nonlinear modelling and model tting to analyse and interpret the dynamics of biological control of an economically important plant pathogen, Rhizoctonia solani Kuhn, by a fungal antagonist, Trichoderma viride Pers ex Gray. Rhizoctonia solani is a commonly occurring soil{borne fungus that causes substantial losses due to damping{o of seedlings of many crop and other plants. It can also cause severe loss in mature plants including potatoes, rice and wheat. Biological control of damping{o diseases by species of Trichoderma has been widely advocated (Papavizas, 1985) for horticultural and other crops but commercial success has been marred by marked variability in control, even under strictly controlled conditions. We show that although the biocontrol agent reduces the average amount of disease, the variability amongst replicates is greatly enhanced in the presence of the antagonist. The results are shown to be consistent with predictions from a model in which small di erences in host susceptibility, fungal infectivity and control are ampli ed as the pathogen spreads 2
from initial inoculum in soil and later from plant to plant. The di erences are maintained because of interruption of disease progress as a result of a decline in host susceptibility.
2 EXPERIMENT Using simple, repeatable and highly uniform microcosms, we quanti ed the temporal development of damping-o disease on radish seedlings, Raphanus sativus L., caused by R. solani. The disease attacks hosts in the early stages of development: mature plants rapidly become resistant to this type of parasitism (Deacon, 1980). Disease progress curves over time were obtained for ve replicate microcosms with and without addition of the antagonistic fungus, T. viride. Clear plastic boxes measuring 100 mm wide, 200 mm long and 100 mm in depth were lled with 1.0 kg of white sand (acid washed and sieved to 0.5-1.0 mm in diameter) with a moisture content of 10% by weight. Fifty radish seeds were planted in a grid to a depth of 10 mm with a spacing of 20 mm. Mycelial discs, 1.0 mm in diameter, were removed from the edge of a 5 day old colony of R. solani growing on a millipore lter over potato dextrose agar. Ten of these were placed in randomly selected positions at a depth of 5 mm. Trichoderma viride was added next to each host plant in the form of a single colonised poppy seed. There were ve replicates with and without T. viride, each treatment having the same randomisation of inoculum. All other conditions, including host density and genotype, uniformity of inoculum of R. solani and T. viride, water availability, light and temperature were strictly and identically controlled in each replicate. The boxes were sealed with clear plastic lids and incubated in a growth chamber at 23 C with a day length of 16 hours. The boxes were opened daily and the cumulative numbers of seedlings damped-o were counted. The experiment was stopped at day 19 after which the plants became stressed due to lack of nutrients and it was no longer possible to keep the conditions of the experiment constant. o
In one of the replicates with T. viride no damping{o occurred at all and the replicate is excluded from the following analyses which relate to non{zero levels of disease. For all 3
other replicates the mean values followed a sigmoidal disease progress curve with asymptotic levels of disease less than 100% ( gure 1). Trichoderma viride reduced the average amounts of disease ( gure 1a-b). Addition of the antagonist also resulted in substantially more variability amongst replicate microcosms than occurred in the control ( gures 1c-d and gure 2), F42 56 = 1:49, p = 0:08. ;
Figure 1 about here. Figure 2 about here.
3 MODEL Temporal progress of damping-o disease occurs in two phases (Brassett & Gilligan, 1988; Gilligan, 1994). There is an initial phase of primary infection arising from the introduced inoculum of R. solani, that lasts approximately eight days in the current experimental conditions. This is followed by secondary infection as the fungus spreads by mycelial growth between infected and susceptible hosts (Gilligan, 1985).
Basic model. The general form of a model for primary and secondary infection based on the simple mass action principle of contacts between susceptible plants (N ? N ) and initial i
(introduced) inoculum (P ) or infected plants (N ) is given by i
dNi dt
= (r
p
P
+ r N ) (N ? N ) s
i
;
i
(1)
(Brassett & Gilligan, 1988; Gilligan, 1994) where r and r are the rates of primary and secondary infection, respectively and N is the density of plants. The infection force p
4
s
= N ?1 N
i
dNi dt
=r
p
P
+r N s
i
is chosen for its simplicity and universality. The model, however, has a globally stable unique equilibrium representing 100% plants infected. It does not take into account the declining susceptibility of the host which may interrupt the transient approach to a global equilibrium.
Models with interrupted transients. As seedlings mature, the probability that the
remaining susceptible plants become infected decreases with time from planting. We modify equation (1) to allow for the combined e ects of declining susceptibility of hosts and infectivity of inoculum by the inclusion of a term s(t) = s0 exp(?dt), dNi dt
= (r
p
+ r N ) s(t)(N ? N )
P
s
i
i
(2)
:
We analyse the model behaviour in the next section. The basic model (2) can further be modi ed to allow for heterogeneous mixing (Liu et al., 1987) and delay in the onset of secondary infections. Each has the e ect of increasing the degree of nonlinearity in (2). Spatial correlations between infected plants are included by introducing a power law into the force of infection, , leading to dNi dt
= (r
p
P
+ r N ) s(t) (N ? N ) i
i
(3)
:
p
s
A delay is introduced in the onset of secondary infection by replacing r N in (2) with a threshold function r f (N ), where f (N ) is 0 if N N and N ? N if N > N , s
s
i
i
dNi dt
= (r
p
P
i
s
i
+ r f (N )) s(t)(N ? N ) s
i
5
i
c
:
i
i
c
(4)
The existence of such a threshold was suggested by examination of the empirical force of infection = (N ? N )?1 dN =dt calculated from the data. i
i
For the models (2) { (4) with interruption, the solution can be written in an implicit form F (N ) = expf?r P=d[1?exp(?dt)]g (see Appendix). If d = 0, then F (N ) = exp(?r P t) ! 0 as t ! 1, the models collapse into (1) and all trajectories reach the equilibrium (N ? N = 0) where all hosts are infected (model (1)). If d 6= 0 the model does not possess a single equilibrium but the solutions settle on asymptotes that depend on initial conditions and parameter values. This behaviour is caused by s(t) going to 0 as t ! 1, regardless of the value of (N ? N ). Interruption by the decay term | which can be any function decaying to 0 faster than the approach to equilibrium of an unperturbed system | essentially 'freezes' the dynamics of the transient behaviour. i
p
i
p
i
i
Models with varying carrying capacity. For comparative purposes we also consider
another variant of the simple model (1), which does not include the decay in susceptibility, but captures the di erences in the availability of the hosts in an alternative way. We assume that only a certain proportion of susceptible plants is available for infection, dNi dt
= (r
p
P
+ r N ) (N ? N ) s
i
i
:
(5)
Model testing. Models (1) { (2) and (4) { (5) can be solved analytically, and the solutions
are given in the Appendix. We also give the analytical solution for (3) with p = 2 (see below). The models were tested by tting to the average data with P = 10 and initial condition N = 0 and with s0 = 1. Parameters were estimated by least squares tting using FACSIMILE (Curtiss & Sweetenham, 1987). The behaviour of replicates was then examined by simulation. Parameters were allowed to vary by certain intervals about estimates obtained by tting to mean data disease progress. The minimum ranges in parameter values i
6
necessary for simulated disease progress curves to encompass the observed variability in replicate microcosms were iteratively computed. The models were deterministic except for the allowance for variation in parameter values between replicates. Simulations started at day 4 of the experiment to allow for a period of germination of the host.
4 RESULTS Average behaviour. Models (2) { (5) were successfully tted to the average data ( gures
1a-b, and table 1). Model (1) failed to predict the results of the experiment because its solutions always tend to N = N . Model (5) gives a reasonable t to the data, but there is little biological motivation for introducing the parameter , as all plants are initially susceptible. Models (1) and (5) assume that the rate of spread of infection is solely determined by the availability of susceptible tissue (the term N ? N or N ? N ). However, the ability of R. solani to infect plants is also limited by the development of plants, interrupting the progress of the infection. This is accounted for in models (2) { (4). Attention is focused below on the behaviour of the model (2) with interrupted transients. Inclusion of the nonlinear terms for heterogeneous mixing or for a delay slightly improves the goodness of t to the experimental data, see table 1. The resulting curves, however, do not di er signi cantly in the time range covered by the experiment (as shown in table 1 by the residual sum of squares). We note that tting model (3) with heterogeneous mixing to the data leads to the values of p close to 2 (p = 2:58 (2:17; 3:07) for the case without T. viride and p = 2:35 (1:54; 3:58) for the case with the control agent, the values in brackets represent approximate 95% con dence limits). The number of parameters can thus be reduced in this case by assuming p = 2 (called later a cubic model, since the right hand side of (3) is a cubic function of N ). i
i
i
i
Table 1 about here.
7
Variability among the replicates. In contrast to the well{behaved average, there is
considerable variation among the individual disease progress curves for each microcosm. This may result from di erences in initial inoculum, soil conditions and host susceptibility, as well as from the initial spatial arrangements of the inoculum. All of these may be expected to vary slightly even in a carefully regulated laboratory experiment. We model the di erences by varying the host susceptibility coecient s0 in equations (2) { (4) and the carrying capacity parameter in equation (5). The choice of s0 as the source of scatter is motivated by simplicity and lack of any biological indication otherwise. Since the rates r and r are multiplied by s0 in all models (2) { (4), the changes in s0 are equivalent to simultaneous changes in r and r . Drawing the parameters from a uniform distribution centered at appropriate mean estimates and bounded by the scatter given in table 1, we also calculated the variance for 200 replicates, and compared it with the empirical variance between replicates, gure 2. The scatter of and s0 was obtained by the least square t of the model output and observed values of the variance of the infection during the disease progress, with other parameters xed. Figures 1c and 1d show the envelopes obtained for model (2) by varying s0 around s0 = 1. Similar results can be obtained by changes in the decay parameter d. p
s
p
s
While there is little qualitative di erence in the behaviour of the variability between models, there are signi cant quantitative di erences. The observed scatter among disease progress curves for individual microcosms can be explained by much smaller variation in the parameters for models with interruption (2) | (4) than for the model with 'carrying capacity' (5) (table 1). Including nonlinear terms in the force of infection (models (3) and (4)) increases the sensitivity to small changes in parameter values compared with the models with linear , (2) and (5). The magni cation is especially apparent for the cubic model (model (3) with p = 2).
Characterizing the variability. In the theory of chaotic dynamics, the magni cation
of di erences in initial conditions is usually characterised by the Lyapunov exponent as a 8
measure of trajectory departure and variance magni cation (Yao & Tong, 1994; Deissler & Farmer, 1992). Standard de nitions are not applicable to the system considered here, as the solutions always settle on a nite asymptote. Nevertheless, we can discuss the dependence of equilibria on the parameters and on initial conditions (for example on the value of P ). De ne in terms of a partial derivative of the nal state of the infection with respect to a given parameter j
j
j
=
@ (Ni1 ) @j
Ni1 j
!?
1
(6)
so that if is changed into (1 + ), N 1 becomes N 1 (1 + ) (with being a small number). is the j -th component of a vector of parameters, ~, e.g. ~ = (r ; r ; s0 ; P ) for (2). Then j
j
i
i
j
p
j
2
! var (N 1 ) mean (N 1) ?2 = var ( ) mean ( ) i
i
j
j
s
(7)
;
where mean ( ) and var ( ) represent the mean and variance of over di erent realisations of . The analogue of the Lyapunov exponent can then be de ned as j
j
j
j
j
= ln( ) j
for any of the parameters. Values of and with = s0 (models (2) { (4)) and = (model (5)) are listed in table 1. j
j
If there is no interruption in the transient behaviour (d = 0), N ! N regardless of the parameters and ! ?1 for any parameter. For the model with carrying capacity () varying, = 1 and = 0 for = whereas = ?1 for other parameters. The magni cation factor was estimated by simulating the models (2) { (5) and comparing the nal variance in N 1 with the variance in the parameters, equation (7). i
j
i
9
In contrast to the dynamical Lyapunov exponent, there is no sharp threshold around = 0, but high and positive values of suggest a strong dependence of the outcome of the experiment on the particular parameter (table 1).
5 DISCUSSION We have shown that large variability in the outcome of the epidemics in microcosms can be explained by relatively small variations in the rates of disease transmission. The variability in environmental conditions, susceptibility of plants and initial spatial arrangements a ects the rates of growth and hence changes the transient behaviour. Interruption of the latter results in the large variation in the nal state. For the case with T. viride, additional variability in r and r amongst replicates can be generated by spatial variability in the success of T. viride in controlling disease within microcosms. Fast growth due to secondary infection can occur locally within a microcosm, where T. viride is locally unsuccessful. p
s
The initial scatter is ampli ed in the systems where the infection progress is interrupted by change in susceptibility, so that time is also a limiting factor as well as the availability of susceptible tissue. Similar biological situations were described by Kiyosawa (1972) for rice blast caused by Pyricularia oryzae. Note that the infection process can also be interrupted by other factors such as harvests and changes in environmental conditions (temperature, humidity). A similar mechanism by which a progress of a viral infection in human population is periodically interrupted by a schooling system leads to a long{time chaotic behaviour in measles time series (Grenfell et al., 1995). Nonlinear terms, while changing the average dynamics only slightly, increase the magni cation factor signi cantly. Models with varying rates give a much better prediction for variability, both in magnitude and in behaviour than a conventional estimate of binomial variance associated with departures of a quantal variable from the average of a model with xed parameters, cf. gure 2. The latter is more appropriate for the system where each time point comes from an independent replicate rather than from sequential observations of the same replicate as in this and 10
most ecological experiments. This raises a more general question regarding the applicability of standard methods of averaging results over many replicates. In a nonlinear system the mean value may not be a good representation of the state (Epstein, 1995), as even if the experiment is carried out under apparently identical conditions, the outcomes can be very di erent. Prediction of control strategies based on the average curve will greatly underestimate the e ects of such 'hot spots' due to local ampli cation by secondary infection in the eld. This may account for the notorious failure of many biological control strategies for plant disease to be transferred from the laboratory to the eld (Papavizas, 1985). One of the most important issues in theoretical ecology is the question of patchiness, its origin and in uence on persistence of metapopulations (Anderson & May, 1986; Hassell et al., 1991; Gilpin & Hanski, 1991; Hanski, 1994; Levin, 1994). The above example presents one possible mechanism whereby interaction between natural variability and nonlinear, timedependent dynamics can produce patches of di erent loads of infection. Here, a microcosm is a bounded system of favourable ('hot spots') and unfavourable sites for infection, equivalent to an uncoupled patch in a metapopulation. Di erences in the parameter values for the rates amongst microcosms re ect spatial heterogeneity within patches as well as di erences between patches. Hence, although the models considered here are not spatially explicit, they take implicit account of hierarchical spatial heterogeneity between sites within patches and between patches. The model and experimental system can easily be extended to allow couplings amongst patches. Our model is essentially deterministic; stochastic variation enters only in the rates amongst di erent replicates but thereafter these rates remain constant within replicates. Further work is needed on the interaction of stochastic variation and dynamically generated variability. We suggest that dynamically generated variability and the related analysis of the evolution of associated probability distributions (Horsthemke & Lefever, 1984), has profound consequences for the design and interpretation of ecological experiments as well as for consideration of appropriate statistical distributions in model tting (Gilligan, 1990; Kareiva, 1995). More studies on both the experimental and theoretical aspects are necessary in order to predict the shape of the probability distribution and its development in 11
time and to qualify and quantify the role of nonlinearities in the dynamics of the mean and variance of epidemiological variables. This work was supported by grants from the Biotechnology and Biological Sciences Research Council which we gratefully acknowledge.
APPENDIX The Appendix lists analytical solutions for the equations presented in the paper.
Model (1) and (5). Model (1) yields the solution Ni =
with = r by
p
P
p
P
;
+ r N . The solution for the model (5) is similar, but with N modi ed s
Ni =
and 0 = r
rp P (1 ? exp(? t)) N rpP + rs N exp(? t)
rp P (1 ? exp(? 0t)) N rpP + rsN exp(? 0t)
+ r N . s
For the models (2) { (4) we present the solutions in an implicit form F (N ) = expf?r P=d[1 ? exp(?dt)]g. i
p
Model (2). For the model with the linear force of infection, the function F (N ) is given i
by
r P + r N r P ? rpPrp Prs N F (N ) = N ? N N p
s
i
+
p
i
i
The implicit equation can be further solved for N to obtain i
12
:
Ni =
rp
1 ? exp ? 1 ? e? =d P + r N exp (? (1 ? e? ) =d)
rp P
dt
N
dt
s
:
In particular, if t ! 1 Ni1 =
rpP (1 ? exp (? =d)) N rp P + rs N exp (? =d)
:
Model (3). This equation can be easily solved for p = 2, but not in the general case. For the cubic model (p = 2),
,
rpP + rs Ni2 (N ? Ni)2
F (N ) = i
0q r Pr N exp @ p
s
2
+ r N2
rpP
s
rp P N2
!?
2(
rp P rp P +rs N 2 )
s r !1 arctan r P N A ; s
i
p
Model (4). For the model with the delay, 8 < F (N ) = : i
N
? i N
N
N
? c N
N
p
r P +r
s (Ni ?N ?Ni
N
. c )
p
r P N
13
? c N
? rpP
rp P +rs (N?Nc )
, if N , if N
i
< Nc ;
i
> Nc .
References Anderson, R.M., & May, R.M. 1986. The invasion, persistence and spread of infectious diseases within animal and plant communities. Phil.Trans.R.Soc.Lond.Biol., 314, 533{ 570. Brassett, P.R., & Gilligan, C.A. 1988. A discrete probability model for policyclic infection by soil-borne plant parasites. New Phytol., 109, 183{191. Curtiss, A.R., & Sweetenham, W.P. 1987. FACSIMILE User Guide. Oxfordshire: UK Atomic Energy Authority, Harwell Laboratory. Deacon, J.W. 1980. Introduction to modern mycology. Basic Microbiology, vol. 7. Oxford: Blackwell Scienti c Publications. Deissler, R.J., & Farmer, J.D. 1992. Deterministic noise ampli ers. Physica D, 55(1-2), 155{165. Epstein, I.R. 1995. The consequences of imperfect mixing in autocatalytic chemical and biological systems. Nature, 374, 321{327. Gilligan, C.A. 1985. Construction of temporal models: III. Disease progress of soil borne pathogens. Pages 67{105 of: Gilligan, C.A. (ed), Mathematical Modelling of Crop Disease. Advances in Plant Pathology, vol. 3. London: Academic Press. Gilligan, C.A. 1990. Antagonistic interactions involving plant-pathogens - tting and analysis of models to nonmonotonic curves for population and disease dynamics. New Phytologist, 115(4), 649{665. Gilligan, C.A. 1994. Temporal aspects of the development of root disease epidemics. In: Campbell, C.L., & Benson, D.M. (eds), Epidemiology and Management of Root Diseases. Berlin, Heidelberg, New York: Springer-Verlag. 14
Gilpin, M., & Hanski, I. (eds). 1991. Metapopulation Dynamics: Empirical and Theoretical Investigations. London: Academic Press. Grenfell, B.T., Kleczkowski, A., Ellner, S.P., & Bolker, B.M. 1994. Measles as a case-study in nonlinear forecasting and chaos. Philosophical Transactions of the Royal Society of London Series A-Physical Sciences and Engineering, 348(1688), 515{530. Grenfell, B.T., Kleczkowski, A., Gilligan, C.A., & Bolker, B.M. 1995. Spatial heterogeneity, nonlinear dynamics and chaos in infectious diseases. Statistical Methods in Medical Research, 4, 160{183. Hanski, I. 1994. Spatial scale, patchiness and population-dynamics on land. Philosophical Transactions of the Royal Society of London Series B-Biological Sciences, 343(1303), 19{25. Hassell, M.P., May, R.M., Pacala, S.W., & Chesson, P.L. 1991. The persistence of hostparasitoid associations in patchy environments .1. A general criterion. American Naturalist, 138(3), 568{583. Horsthemke, W., & Lefever, R. 1984. Noise-induced transitions. Berlin, Heidelberg, New York: Springer Verlag. Kareiva, P. 1995. Ecology - predicting and producing chaos. Nature, 375, 189{190. Kiyosawa, S. 1972. Mathematical studies on the curve of disease increase - a technique for forecasting epidemic development. Ann. Phytopath. Soc. Japan, 38, 30{40. Levin, S.A. 1994. Patchiness in marine and terrestrial systems - from individuals to populations. Philosophical Transactions of the Royal Society of London Series B-Biological Sciences, 343(1303), 99{103. Liu, W.M., Hethcote, H.W., & Levin, S.A. 1987. Dynamical behavior of epidemiological models with nonlinear incidence rates. Journal of Mathematical Biology, 25(4), 359{ 380. May, R.M. 1976. Simple mathematical models with vary complicated dynamics. Nature, 261, 459{467. 15
Papavizas, G.C. 1985. Trichoderma and Gliocladium - biology, ecology, and potential for biocontrol. Ann. Rev. Phytopathol., 23, 23{54. Yao, Q.W., & Tong, H. 1994. Quantifying the in uence of initial values on nonlinear prediction. Journal of the Royal Statistical Society Series B- Methodological, 56(4), 701{725.
16
List of Figures 1
2
Comparison of the simulations (solid lines) and experimental values with (circles) and without (squares) Trichoderma viride. Simulation parameters for the model (2) were obtained by a nonlinear least-square t and are listed in table 1. (a) and (b): The average values for the case without Trichoderma viride, (a), and with Trichoderma viride, (b). (c) and (d): Individual replicates compared with simulated curves based on the parameters estimated for the mean data (central curve) and the envelopes (other curves) obtained by varying s0 around s0 = 1 by 31% for the case without Trichoderma viride, (c), and by 65% for the case with Trichoderma viride, (d) : : : : : : : : :
18
Variance for the experimental data: (a) without and (b) with T. viride ( lled squares and circles, respectively), and of the equation (2) with s0 varied as in table 1 (uniform distribution, 200 replicates, dashed line), compared with the binomial estimate based on the formula N Y (1 ? Y ) where Y is an average proportion of infected plants in all microcosms at a given time Y = N =N (based on the simulations, thick line) : : : : : : : : : : : : : : : : : : : : : :
19
i
17
Number of diseased plants
(a)
(b)
30
30
20
20
10
10
0
0 4
9
14
19
24
4
9
Number of diseased plants
(c)
14
19
24
19
24
(d)
50
50
40
40
30
30
20
20
10
10
0
0 4
9
14
19
24
4
Time (days)
9
14
Time (days)
Figure 1: Comparison of the simulations (solid lines) and experimental values with (circles) and without (squares) Trichoderma viride. Simulation parameters for the model (2) were obtained by a nonlinear least-square t and are listed in table 1. (a) and (b): The average values for the case without Trichoderma viride, (a), and with Trichoderma viride, (b). (c) and (d): Individual replicates compared with simulated curves based on the parameters estimated for the mean data (central curve) and the envelopes (other curves) obtained by varying s0 around s0 = 1 by 31% for the case without Trichoderma viride, (c), and by 65% for the case with Trichoderma viride, (d) 18
(a) 150
Variance
100
50
0 4
9
14
19
24
19
24
Time (Days)
(b) 150
Variance
100
50
0 4
9
14 Time (Days)
Figure 2: Variance for the experimental data: (a) without and (b) with T. viride ( lled squares and circles, respectively), and of the equation (2) with s0 varied as in table 1 (uniform distribution, 200 replicates, dashed line), compared with the binomial estimate based on the formula N Y (1 ? Y ) where Y is an average proportion of infected plants in all microcosms at a given time Y = N =N (based on the simulations, thick line) i
19
List of Tables 1
Parameters, residual sum of squares for the t of N to the data and multiplication factors for the models (2) { (5). The values in brackets correspond to approximate 95% con dence intervals. The number of degrees of freedom for the residual sum of squares is also given in brackets next to the value of RSS. Scatter measures the relative variation of the parameters around the values obtained by the least square t to the average data, s0 around 1 for models (2) { (4) and around the values listed in the table for model (5) : i
20
20
rp rs d Nc
Linear model (2)
Cubic Threshold model (3) model (4) Without T. viride
Carrying capacity model (5)
0.0018 (0.0012 { 0.0027) 0.014 (0.009 { 0.021) 0.182 (0.144 { 0.229) | |
0.0036 0.0045 (0.0032 { 0.0040) (0.0035 { 0.0058) 0.0033 0.038 (0.0024 { 0.0046) (0.020 { 0.075) 0.335 0.267 (0.298 { 0.377) (0.209 { 0.341) | | | 3.8 (2.5 { 5.6) Residual sum of squares 10.7 (11) 15.0 (10) Multiplication factors 10% 19% 6.1 2.86 1.8 1.05
0.0040 (0.0039 { 0.0047) 0.013 (0.011 { 0.016) |
RSS (df) 20.6 (11) Scatter
31% 1.65 0.5
0.51 (0.49 { 0.53) | 12.3 (11) 45% 1 0
Table 1: Parameters, residual sum of squares for the t of N to the data and multiplication factors for the models (2) { (5). The values in brackets correspond to approximate 95% con dence intervals. The number of degrees of freedom for the residual sum of squares is also given in brackets next to the value of RSS. Scatter measures the relative variation of the parameters around the values obtained by the least square t to the average data, s0 around 1 for models (2) { (4) and around the values listed in the table for model (5) i
21
Table 1 | ctd. With T. viride rp rs d Nc
0.0014 (0.0011 { 0.0019) 0.007 (0.004 { 0.011) 0.101 (0.065 { 0.157) | |
RSS (df) 7.5 (11) Scatter
65% 1.31 0.27
0.0021 0.0027 (0.0019 { 0.0023) (0.0022 { 0.0033) 0.0037 0.017 (0.0024 { 0.0056) (0.007 { 0.041) 0.280 0.168 (0.238 { 0.329) (0.112 { 0.251) | | | 3 (1.9 { 4.4) Residual sum of squares 6.4 (11) 5.9 (10) Multiplication factors 14% 42% 8.26 2.23 2.11 0.8
22
0.0035 (0.031 { 0.039) 0.0091 (0.0060 { 0.0139) | 0.42 (0.37 { 0.48) | 6.6 (11) 80% 1 0
