Nik J Cunniffe

Christopher A Gilligan

A theoretical framework for biological control of soilborne plant pathogens: Identifying effective strategies

Keywords: Epidemiological model, invasion, persistence, basic reproductive number R 0, biocontrol

We develop and analyse a flexible compartmental model of the interaction between a plant host, a soil-borne pathogen and a microbial antagonist, for use in optimising biological control. By extracting invasion and persistence thresholds of host, pathogen and biological control agent, performing an equilibrium analysis, and numerical investigation of sensitivity to parameters and initial conditions, we determine criteria for successful biological control. We identify conditions for biological control (i) to prevent a pathogen entering a system, (ii) to eradicate a pathogen that is already present and, if that is not possible, (iii) to reduce the density of the pathogen. Control depends upon the epidemiology of the pathogen and how efficiently the antagonist can colonise particular habitats (i.e. healthy tissue, infected tissue and/or soil-borne inoculum). A sharp transition between totally effective control (i.e. eradication of the pathogen) and totally ineffective control can follow slight changes in biologically-interpretable parameters or to the initial amounts of pathogen and biological control agent present. Effective biological control requires careful matching of antagonists to pathosystems. For preventative/eradicative control, antagonists must colonise susceptible hosts. However for reduction in disease prevalence, the range of habitat is less important than the antagonist's bulking-up efficiency.

Introduction

Biological control uses a natural enemy (or antagonist) of a pathogen to effect a reduction in the level or prevalence of disease [START_REF] Cook | The Nature and Practice of Biological Control of Plant Pathogens[END_REF][START_REF] Baker | Evolving concepts of biological control of plant pathogens[END_REF]. There are obvious attractions.

However, biological control has all too often either failed to work or proved too unreliable to be a realistic proposition [START_REF] Thomashow | Biological control of plant root pathogens[END_REF][START_REF] Shtienberg | Incorporation of weather forecasting in integrated, biological-chemical management of Botrytis cinerea[END_REF][START_REF] Rosskopf | Field evaluation of Phomopsis amaranthicola, a biological control agent of Amaranthus spp[END_REF][START_REF] Guetsky | Combining biocontrol agents to reduce the variability of biological control[END_REF], despite successes in the controlled conditions of glasshouses and propagation systems [START_REF] Paulitz | Biological control in greenhouse systems[END_REF]. With chemical control ever more unattractive because of increasingly stringent legislative constraints [START_REF] Gullino | Social and political implications of managing plant diseases with restricted fungicides in Europe[END_REF][START_REF] Ragsdale | Social and political implications of managing plant diseases with decreased availability of fungicides in the United States[END_REF] and the economic and operational challenges posed by rapid evolution of resistant pathogens [START_REF] Parnell | Changes in fungicide sensitivity and relative species abundance: Oculimacula yallundae and O. acuformis populations (eyespot disease of cereals) in western Europe[END_REF][START_REF] Van Den Bosch | Models of fungicide resistance dynamics[END_REF], attention naturally reverts to explaining the hitherto disappointing failure of biological control in the field.

The physiological basis of biological control has attracted significant attention, and there is good understanding of a number of small-scale antagonistic mechanisms, including mycoparasitism [START_REF] Foley | Susceptibility of Pythium spp. and other fungi to antagonism by the mycoparasite Pythium oligandrum[END_REF][START_REF] Siwek | Mycoparasitism of Pythium ultimum by antagonistic binucleate Rhizoctonia isolates in agar media and on capsicum seeds[END_REF], antibiosis [START_REF] Raaijmakers | Antibiotic production by bacterial biocontrol agents[END_REF], induced resistance [START_REF] Vallad | Systemic acquired resistance and induced systemic resistance in conventional agriculture[END_REF] and hypovirulence [START_REF] Milgroom | Biological control of chestnut blight with hypovirulence: a critical analysis[END_REF]. However little is known at the population level, even though it is the coupled dynamics of the host, pathogen and antagonist at this larger scale that ultimately determine success. Disregarding purely statistical infection-dose responses that predict rather than explain [START_REF] Gilligan | Antagonistic interactions involving plant pathogens: fitting and analysis of models to non-monotonic curves for population and disease dynamics[END_REF][START_REF] Johnson | Dose-response relationships and inundative biological control[END_REF][START_REF] Raaijmakers | Dose-response relationships in biological control of fusarium wilt of radish by Pseudomonas spp[END_REF][START_REF] Montesinos | Dose-response models in biological control of plant pathogens: an empirical verification[END_REF][START_REF] Schisler | Effects of antagonist cell concentration and two-strain mixtures on biological control of fusarium dry rot of potatoes[END_REF][START_REF] Smith | Modeling dose response relationships in biological control: Partitioning host responses to the pathogen and biocontrol agent[END_REF][START_REF] Larkin | Mechanisms of action and dose response relationships governing biological control of fusarium wilt of tomato by nonpathogenic Fusarium spp[END_REF][START_REF] Cabrefiga | Analysis of aggressiveness of Erwinia amylovora using disease-dose and time relationships[END_REF], mathematical models and simulations have often concentrated on low-level mechanistic representations of the physiological responses detailed above [START_REF] Knudsen | Use of a computer simulation model to evaluate a plant disease biocontrol agent[END_REF][START_REF] Stack | A computer simulation model to predict the dispersal of biocontrol fungi in soil[END_REF][START_REF] Kessel | Competitive substrate colonization by Botrytis cinerea and Ulocladium atrum in relation to biological control of Botrytis cinerea in cyclamen[END_REF][START_REF] Knudsen | Individual-based approach to modeling hyphal growth of a biocontrol fungus in soil[END_REF].

Arguably a more illuminating approach, however, is to map these physiological responses to changes in one or more of a small set of epidemiologically-meaningful parameters, such as rates of infection and/or infectious periods, in a population-level model of disease [START_REF] Gilligan | An epidemiological framework for disease management[END_REF][START_REF] Gilligan | Epidemiological models for invasion and persistence of pathogens[END_REF]. Extensive theoretical work of this broad type has examined interactions between parasitoids and their insect hosts [START_REF] Mills | Modelling the biological control of insect pests: a review of host-parasitoid models[END_REF], and the ecology of these systems is now well-understood. However with certain exceptions [START_REF] White | Spatial heterogeneity in three-species, plantparasite-hyperparasite, systems[END_REF][START_REF] Jeger | A generic theoretical model for biological control of foliar plant diseases[END_REF][START_REF] Xu | Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent[END_REF], few generic studies have focussed on biological control of plant disease, and instead models have typically concentrated on specific host-pathogen-antagonist combinations. Partic-ularly well-studied are the interactions between Rhizoctonia solani and Trichoderma viride on radish [START_REF] Kleczkowski | Dynamically generated vari-ability in plant-pathogen systems with biological control[END_REF][START_REF] Bailey | Biological control of pathozone behaviour and disease dynamics of Rhizoctonia solani by Trichoderma viride[END_REF][START_REF] Gibson | Predicting variability in biological control of a plant-pathogen system using stochastic models[END_REF][START_REF] Bailey | Modeling and analysis of disease-induced host growth in the epidemiology of take-all[END_REF][START_REF] Gibson | A Bayesian analysis of botanical epidemics using stochastic compartmental models[END_REF][START_REF] Kleczkowski | Parameter estimation and prediction for the course of a single epidemic outbreak of a plant disease[END_REF], and between Sclerotina minor and Sporidesmium sclerotivorum on lettuce [START_REF] Gubbins | Population dynamics of a parasite and hyperparasite in a closed system: model analysis and parameter estimation[END_REF][START_REF] Gubbins | Biological control in a disturbed environment[END_REF][START_REF] Gubbins | Persistence of host-parasite interactions in a disturbed environment[END_REF][START_REF] Gubbins | A test of heterogeneous mixing as a mechanism for ecological persistence in a disturbed environment[END_REF]. Unfortunately, this narrow focus means that relatively few general messages have emerged. Here we have a broader ambition, and consider how microbial antagonists affect the spread of pathogens through host populations of plants in general.

We concentrate on soil-borne plant pathogens, which exemplify economicallyimportant systems for which biological control is considered to be a viable proposition [START_REF] Cook | Making greater use of introduced microorganisms for biological control of plant pathogens[END_REF][START_REF] Harman | Trichoderma species opportunistic, avirulent plant symbionts[END_REF][START_REF] Jacobsen | The role of bacillus-based biological control agents in integrated pest management systems: plant diseases[END_REF]. Our underlying methodology of analysing the likely efficacy of control by investigating its effect on epidemiologically-meaningful parameters has typically been cast in terms of effects on pathogen invasion and persistence [START_REF] Gilligan | Sustainable agriculture and plant disease: an epidemiological perspective[END_REF]. It has also been used to determine suitable controls for broad groups of pathogens, classified according to their epidemiology [START_REF] Hall | Invasion of drug and pesticide resistance is determined by a trade-off between biocide efficacy and relative fitness[END_REF][START_REF] Hall | Evaluating the performance of chemical control in the presence of resistant pathogens[END_REF][START_REF] Cunniffe | Invasion, persistence and control in models of soil-borne plant pathogens: the effect of host demography[END_REF], an arguably more challenging objective. However previous work has not specifically targetted biological control. In particular the effect(s) of control either remained fixed, or pulsed and decayed according to a simple schedule of treatments [START_REF] Hall | Evaluating the performance of chemical control in the presence of resistant pathogens[END_REF], and the more complex temporal variation corresponding to the three species interaction in biological control has not been considered, except with reference to the S. minor and S. sclerotivorum interaction [START_REF] Gubbins | Population dynamics of a parasite and hyperparasite in a closed system: model analysis and parameter estimation[END_REF][START_REF] Gubbins | Biological control in a disturbed environment[END_REF][START_REF] Gubbins | Persistence of host-parasite interactions in a disturbed environment[END_REF][START_REF] Gubbins | A test of heterogeneous mixing as a mechanism for ecological persistence in a disturbed environment[END_REF].

Here we extend an existing compartmental model of the interaction between a plant host and a soil-borne fungal pathogen [START_REF] Cunniffe | Invasion, persistence and control in models of soil-borne plant pathogens: the effect of host demography[END_REF] to include a bacterial or fungal antagonist. The effect(s) of each species upon the other is controlled by tunable parameters. In particular the antagonist can bulk-up and increase in density on three distinct habitats (healthy plant tissue and/or infected plant tissue and/or soil-borne inoculum), and can deleteriously affect any or all of the pathogen's epidemiological rates (e.g. rates of primary and secondary infection, rates of decay of infectious material). Any alteration to these rates depends on the density of the biological control agent, and so varies over time. As the interactions between host, pathogen and antagonist are controlled by parameters of the model, it retains sufficient flexibility to represent a range of systems.

We use the model to investigate how biological control is affected by (i) the properties of the host-pathogen interaction; (ii) the set of epidemiological rates the antagonist is capable of affecting; and (iii) the habitats the antagonist is capable of colonising. We examine:

1. preventative control, in which the antagonist prevents the pathogen from invading the system; 2. eradicative control, in which the antagonist eradicates the pathogen if it is already present;

3. reductive control, in which the antagonist reduces the density of the pathogen.

In the case of reductive control, we also characterise how the effectiveness of control (in terms of reduction in long-term pathogen density) depends on the antagonist's mode of action and population dynamics, and how suitable antagonists for particular pathogens are conditioned upon the pathogen's epidemiology. Finally we examine variations in the efficacy of control depending on the initial density of each species: host, pathogen and antagonist, and how under certain circumstances extreme changes in the efficacy of control can follow from only slight changes to either initial densities or to the parameters of the model.

Methods

Modelling

Host-pathogen interaction

The population of hosts is divided into two classes, susceptible (S) and infected (I). These variables may be defined in terms of the number or density of plants, or may be relative to smaller units such as roots, dependent upon the natural scale of the epidemic [START_REF] Gilligan | An epidemiological framework for disease management[END_REF]. Additionally we track the density of primary inoculum (X), which for fungal pathogens includes free-living infective stages such as spores and resting bodies including sclerotia and/or fragments of previously colonised host tissue:

dS dt = η (κ -(S + I)) -(β P X + β S I) S, ( 1
)
dI dt = (β P X + β S I) S -μI, ( 2
)
dX dt = νI -γX. ( 3
)
The model is a particular variant of a class of models introduced and analysed by Gubbins et al. [START_REF] Gubbins | Population dynamics of plant-parasite interactions: thresholds for invasion[END_REF]. It represents (at distinct rates β P and β S ) the dual pathways of primary and secondary infection characteristic of soil-borne plant pathogens [START_REF] Brassett | A model for primary and secondary infection in botanical epidemics[END_REF][START_REF] Gilligan | Population dynamics of botanical epidemics involving primary and secondary infection[END_REF][START_REF] Madden | A population-dynamic approach to assess the threat of plant pathogens as biological weapons against annual crops[END_REF][START_REF] Bailey | An epidemiological analysis of the role of disease-induced root growth in the differential response of two cultivars of winter wheat to infection by the take-all pathogen, Gaeumannomyces graminis. var. tritici[END_REF]. Infected hosts decay at per capita rate μ, corresponding to disease-induced mortality (this parameter could also represent a combination of natural and diseaseinduced mortality, or a rate of loss of infectiousness of infected host tissue). External inoculum loses infectiousness at rate γ, and is replenished by release from infectious hosts with efficiency ν, corresponding to infected hosts either producing or becoming sources of inoculum [START_REF] Gubbins | Biological control in a disturbed environment[END_REF][START_REF] Gilligan | Sustainable agriculture and plant disease: an epidemiological perspective[END_REF]. Replenishment of susceptible hosts is also included; without this the pathogen cannot persist in this class of model. Additionally for soilborne plant pathogens, growth/creation of host tissue typically occurs over timescales comparable to the epidemiological dynamics [START_REF] Bailey | Modeling and analysis of disease-induced host growth in the epidemiology of take-all[END_REF][START_REF] Bailey | Epidemiological analysis of take-all decline in winter wheat[END_REF][START_REF] Cunniffe | Invasion, persistence and control in models of soil-borne plant pathogens: the effect of host demography[END_REF], and so in contrast to models of aerial systems for which host demography is arguably less important [START_REF] Jeger | A generic theoretical model for biological control of foliar plant diseases[END_REF][START_REF] Xu | Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent[END_REF], a submodel for host growth is required. Host growth is linear, where both susceptible and infected hosts contribute to the carrying capacity (κ), and in which the dynamics are governed by rate parameter η [START_REF] Gilligan | Mathematical modeling and analysis of soilborne pathogens[END_REF]. The particular host growth function we have taken has been used in a number of previous investigations of soil-borne plant pathogens [START_REF] Jeger | The influence of root growth and inoculum density on the dynamics of root disease epidemics: theoretical analysis[END_REF][START_REF] Gilligan | Mathematical modeling and analysis of soilborne pathogens[END_REF][START_REF] Gilligan | Population dynamics of botanical epidemics involving primary and secondary infection[END_REF][START_REF] Truscott | Response of a deterministic epidemiological system to a stochastically varying environment[END_REF][START_REF] Stacey | A model for invasion and spread of rhizomania in the UK: implications for disease control strategies[END_REF], and additionally (when modelling at the scale with an individual plant as a single host) is applicable to the wide range of agricultural systems with continuous harvesting and replanting [START_REF] Madden | A theoretical assessment of the effects of vector-virus transmission mechanism on plant virus disease epidemics[END_REF][START_REF] Van Den Bosch | Disease control and its selection for damaging plant virus strains in vegetatively propagated staple food crops; a theoretical assessment[END_REF]. As our preliminary investigations indicated that other choices of the function, including logistic growth, appeared to have no effect on the qualitative results, we concentrated here on a relatively simple linear form in order to simplify both our analysis and the consequent presentation of our results.

Antagonist

Our extension to the model introduces the density of an antagonist species (A):

dS dt = η (κ -(S + I)) - β P X 1 + α P A + β S I 1 + α S A S, ( 4
)
dI dt = β P X 1 + α P A + β S I 1 + α S A S -μ (1 + ω μ A) I, ( 5
)
dX dt = νI -γ 1 + ω γ A X, ( 6
)
dA dt = (ρ S S + ρ I I + ρ X X -σ -ξA)A. (7)
The antagonist affects the pathogen and acts as an agent of biological control by (potentially) decreasing the rate(s) of infection and/or by increasing the rate(s) of decay of infectious material. The per capita parameters α P , α S , ω μ , ω γ characterise the pathogen-antagonist interaction. The antagonist is able to bulk-up upon susceptible hosts, infected hosts and/or soil-borne inoculum: we define each of these as a habitat.

Antagonist bulking-up depends upon the habitat-specific parameters ρ S , ρ I and ρ X , providing a mechanism to represent habitat-generalists (ρ S = ρ I = ρ X ), habitat-specialists (only one of ρ S , ρ I , ρ X non-zero), or anywhere between these extremes. The antagonist density decays at per capita rate σ, corresponding to inter-specific competition from other soil-borne organisms and the natural death of the antagonist. There is densitydependence acting upon the antagonist population, controlled by the parameter ξ, and which prevents unbounded increase of antagonist density.

Non-dimensionalisation

To simplify the analysis we introduce the dimensionless variables

Ŝ = Sκ -1 , Î = Iκ -1 , X = ηXν -1 κ -1 , Â = ξAη -1 , t = ηt, (8)
and parameters

βP = β P νκη -2 , βS = β S κη -1 , μ = μη -1 , γ = γη -1 , αP = α P ηξ -1 , αS = α S ηξ -1 , ωμ = ω P ηξ -1 , ωγ = ω S ηξ -1 , ρS = ρ S κη -1 , ρI = ρ I κη -1 , ρX = ρ X νκη -2 , σ = ση -1 . ( 9
)
The model is transformed to (Table 1)

d Ŝ d t = 1 -Ŝ + Î - βP X 1 + αP Â + βS Î 1 + αS Â Ŝ, (10) d
Î d t = βP X 1 + αP Â + βS Î 1 + αS Â Ŝ -μ 1 + ωμ Â Î, ( 11
)
d X d t = Î -γ 1 + ωγ Â X, ( 12
)
d  d t = ρS Ŝ + ρI Î + ρX X -σ - Â. ( 13
)
Scaling according to Equations ( 8) and ( 9) leads to a dimensionless system parameterised in terms of the three key interactions which we focus upon: the effect of the pathogen on its plant host; the effect of the antagonist on the pathogen; and the response of the antagonist to its habitat.

*** INSERT TABLE ONE NEAR HERE ***

Numerical methods

Our analysis of Equations ( 10)-( 13) is supplemented by numerical solution. We take as an example the control of a particular pathogen, with (unless otherwise stated) βP = 0.5, βS = 0.375, μ = 0.25, γ = 0.8, (

and

Ŝ0 = 1.0, Î0 = 0, X0 = 0.1, Â0 = 0.1, (15)
corresponding to the simultaneous introduction of a small density of inoculum and antagonist to a host population at its carrying capacity. We focus upon three key numerical scenarios (Table 2).

*** INSERT TABLE TWO NEAR HERE ***

Results

Equilibrium analysis

Without antagonism

The basic reproductive number of the pathogen in the absence of the antagonist

(Appendix A.1) is R 0 = R P 0 + R S 0 = 1 μ βP γ + βS , ( 16
)
where this key threshold may be partitioned into distinct components R P 0 and R S 0 corresponding to primary and secondary infection. If R 0 < 1 then the pathogen cannot invade, and the host density stabilises at its carrying capacity, with

Ŝ∞ , Î∞ , X∞ = (1, 0, 0). ( 17
)
However if R 0 > 1 then the pathogen invades the host population, and

Ŝ∞ , Î∞ , X∞ = 1 R 0 , 1 1 + μ 1 - 1 R 0 , 1 γ (1 + μ) 1 - 1 R 0 . ( 18
)
Furthermore it can be shown that the pathogen always persists at this level if R 0 > 1 (Appendix A.2).

Including antagonism

The full model with  = 0 introduces two equilibria in addition to analogues of Equations ( 17) and ( 18) with Â∞ = 0. The first corresponds to host and antagonist coexisting, with the pathogen absent:

Ŝ∞ , Î∞ , X∞ , Â∞ = (1, 0, 0, ρS -σ) . ( 19
)
For Equation [START_REF] Foley | Susceptibility of Pythium spp. and other fungi to antagonism by the mycoparasite Pythium oligandrum[END_REF] to predict biologically plausible densities, the rate at which the antagonist bulks-up on susceptible hosts ( ρS ) must be greater than its per capita rate of decay ( σ).

If we define

R( Â) = 1 μ 1 + ωμ Â βP γ 1 + αP Â 1 + ωγ Â + βS 1 + αS Â , ( 20
)
where R( Â) is a criterion for invasion, and R( Â = 0) = R(0) = R 0 of the underlying model, the other additional equilibrium is given implicitly by

Ŝ∞ = 1 R( Â∞ ) , ( 21
) Î∞ = 1 1 + μ 1 + ωμ Â∞ 1 - 1 R( Â∞ ) , ( 22
) X∞ = 1 γ 1 + ωγ Â∞ 1 + μ 1 + ωμ Â∞ 1 - 1 R( Â∞ ) , ( 23
) Â∞ = ρS Ŝ∞ + ρI Î∞ + ρX X∞ -σ. ( 24
)
The equilibrium specified by Equations ( 20)-( 24) corresponds to all three species coexisting. For the densities of infected hosts and inoculum to be biologically plausible, R( Â∞ ) > 1 is required, and since R(•) is a decreasing function and Â∞ must be greater than zero, a precondition is that the pathogen can invade the antagonist-free system (i.e. R(0) = R 0 > 1). In principle the expressions in Equations ( 20)-( 23) could be substituted into Equation ( 24) to give a sixth order polynomial fixing Â∞ , but the complex expression that results adds little insight. The four equilibria of the full model, together with existence criteria, are summarised in Table 3.

*** INSERT TABLE THREE NEAR HERE ***

Invasion criteria

We examine invasion criteria for all three species: host, pathogen and antagonist.

In particular we determine whether or not these species can invade, increasing in density when introduced to a system otherwise at equilibrium, and characterise how this depends upon the rates controlling infection and/or reproduction. The host can always invade (and in fact persist at non-zero density), as its birth rate at low densities is independent of its own population size, and so there is a constant influx of hosts into the system whenever the host density is small. We therefore focus upon invasion of pathogen and antagonist, firstly in the absence of the other, but thereafter when the other species is present (Table 4).

*** INSERT TABLE FOUR NEAR HERE ***

If the antagonist is absent (and so only the host is present), the pathogen can invade only if

R(0) = R 0 > 1, (25)
using the results for the underlying model. When the pathogen is absent, the antagonist can invade the host population if

ρS σ > 1, (26)
i.e. if it is able to bulk-up more quickly on susceptible hosts than it decays.

Invasion of each species in the presence of the other is more complex. If the antagonist is present at equilibrium with the host, then the pathogen can only invade if (cf.

Equation ( 20))

R( ρS -σ) > 1. ( 27
)
Note that since R(•) is decreasing, and because ρSσ must be greater than zero for the antagonist to be present in the absence of the pathogen (Equation ( 26)), R 0 > 1 is a necessary precondition for invasion of the pathogen when the antagonist is present (this is a consequence of the antagonist's deleterious effect on the pathogen). However, if the pathogen is present, the antagonist can only invade if

ρS σ Ŝ∞ + ρI σ Î∞ + ρX σ X∞ > 1, (28)
where the values of Ŝ∞ , Î∞ and X∞ follow from the antagonist-free equilibrium in Equation [START_REF] Van Den Driessche | Reproduction numbers and subthreshold epidemic equilibria for compartmental models of disease transmission[END_REF], i.e. when ρS σR 0

+ ρI σ (1 + μ) 1 - 1 R 0 + ρX σγ (1 + μ) 1 - 1 R 0 > 1. ( 29
)
Depending on the preferred habitat of the antagonist (i.e. to what extent it can bulkup on susceptible hosts, infected hosts and pathogen inoculum), invasion can become more or less likely. For example a habitat-specialist antagonist which can only bulk-up on susceptible hosts (i.e. ρS > 0, ρI = ρX = 0) is less likely to invade in the presence of the pathogen, whereas a similarly-specialised antagonist with a preference for infected hosts (i.e. ρI > 0, ρS = ρX = 0) requires the pathogen to be present to have any chance of invading.

Control without antagonism

Biologically-plausible control strategies lead to reductions in the dimensionless rates of transmission ( βP and/or βS ), and/or increases in the dimensionless rates of decay of infected hosts and inoculum (μ and/or ω). Changes to these dimensionless parameters depend upon the intensity of control and the host-pathogen system in question. The efficacy of control may be conveniently characterised according to its effect on R 0 , and in particular we distinguish: (i) eradication, in which the pathogen is excluded in the long term (R 0 < 1); and (ii) reduction, in which the pathogen persists at a smaller density (R 0 > 1). Certain control strategies can never lead to eradication in systems which have R P 0 > 1 or R S 0 > 1 in the absence of control, no matter how intensively applied (Table 5). This emphasises the need to match any control strategy with the host-pathogen interaction in question.

*** INSERT TABLE FIVE NEAR HERE ***

Control including antagonism

We initially assume that the antagonist is able to bulk-up very quickly, and so that it is able to persist in the system at a very large density, thereby identifying lower bounds for the endemic equilibrium pathogen density ( Î∞ ) when the antagonist is present.

Thereafter we extend this by numerical examination of several scenarios (Table 2), progressively investigating the effects upon Î∞ of smaller rates of antagonist bulkingup (and so lower antagonist density); the habitats that the antagonist is able to colonise;

and the initial densities of antagonist, pathogen and host.

Maximum reductive control (i.e. minimum Î∞ )

If the population dynamics of the antagonist allow it to persist in the system, the best that it can achieve in reducing Î∞ may be inferred directly from the antagonist-free behaviour. The maximum effect of a particular class of antagonist depends upon R P 0 and R S 0 for the host-pathogen system, and on the rate(s) that the antagonist is capable of affecting (Table 5). These lower bounds on Î∞ ignore the antagonist's per capita effect and/or its density, and therefore may not be attained in practice (however, see below).

We note, however, that according to this analysis only single-mode antagonists able to affect the rate at which infected hosts decay (i.e. to shorten the infectious period of infected hosts, ωμ > 0) are capable of eradicating all classes of pathogen.

Antagonist density (Scenario A)

We first assume a habitat-generalist antagonist which bulks-up at equal rate λ on all habitats. Numerical analysis of the endemic level of infection (Figure 1), then shows the effect on Î∞ of any decrease in a per capita rate of antagonism may be compensated for by a suitably-sized increase in the antagonist's ability to bulk-up (as this leads to a larger equilibrium antagonist density and so an equal force of antagonism overall). Additionally whenever the antagonist has a large enough per capita effect on the pathogen and/or is able to bulk-up sufficiently, the limiting lower bounds upon the minimum infected density from Section 3.

Habitat-specificity (Scenario B)

We examine habitat-specificity by fixing the per capita effect of the antagonist while allowing a pair of habitat-specific bulking-up parameters (i.e. two of ρS , ρI , ρX ) to vary simultaneously (Figure 2). The contours of infected density are linear; this is because

Â∞ = ρS Ŝ∞ + ρI Î∞ + ρX X∞ -σ, (30)
and so, all other things being equal, any increase in (say) ρS can be exactly offset by a suitably-sized decrease in (say) ρI to give an equally-sized antagonist population. We note there is no requirement for the antagonist to be able to bulk-up on all classes of habitat in order to attain the maximal control outlined in Section 3.3.1 (Figure 2b).

However, if the antagonist is theoretically able to eradicate the pathogen, this is only actually possible when ρS > 0, i.e. when the antagonist can bulk-up on susceptible hosts. Any antagonist that was not able to bulk-up upon healthy tissue but that eradicated the pathogen would destroy its own only habitat by exerting its antagonistic effect to the maximum possible extent.

*** INSERT FIGURE TWO NEAR HERE ***

Bistability, eradication and feedback (Scenario C)

It is possible that neither the pathogen nor the antagonist can invade when the other is present at equilibrium, when neither invasion criterion according to Equations [START_REF] Gilligan | Population dynamics of botanical epidemics involving primary and secondary infection[END_REF] and ( 29) is satisfied. Accordingly the model is bistable for certain sets of parameters, with eradication of either pathogen or antagonist dependent on initial conditions. If the initial conditions are held fixed, bistability manifests itself with a sharp transition in the endemic infected density, as a small change in a parameter such as ρS leads to a sudden switch from no control to eradication (Figure 3b). Examining the dynamics on either side of this transition illustrates the mechanism by which alternate equilibria are attained (Figure 4). For values of ρS (ability of the antagonist to bulk-up on healthy host tissue) either side of the transition point marked by a green dot in Figure 3b, the antagonist is able to invade initially and to bulk-up quickly to an intermediate plateau. Nevertheless for the smaller value of ρS , the effective reproductive number of the pathogen remains above one, and the antagonist is eradicated as the pathogen establishes itself and the antagonist's habitat is removed. For the slightly larger value of ρS , however, the effective reproductive number drops below one at the intermediate plateau, and so the pathogen density begins to fall. Since any decrease in pathogen density leads to a corresponding increase in antagonist density as the latter has more habitat, and because this leads to a larger force of antagonism and so a further decrease in pathogen density, the pathogen is eradicated via a feedback mechanism.

The exact value of the per capita rate of antagonist bulking-up on susceptible habitat, ρS , (with all other parameters fixed) at which there is a sharp transition depends on the initial conditions (Figures 3c and3d). We note that, although this value depends upon the density of antagonist and pathogen at the initial plateau, the critical value of ρS is relatively irresponsive to Â0 and Ŝ0 (since the dynamics of antagonist and host are fast, and the initial condition is soon "washed out" of the system). However the initial pathogen density (shown in Figures 3c and3d via the proxy of initial inoculum density) has a large effect on the value of ρS . This counter-intuitive result can be attributed to the following dynamics (Figure 4): the initial pathogen density exerts a large influence on the density of susceptible hosts that corresponds to the primary infection plateau, via the Ŝ + Î term in the host population's carrying capacity. This, in turn, leads to changes in the value of the bulk-up parameter required for the sharp transition, via the feedback described above.

Discussion

We have extended a well-studied and generic model of soil-borne plant pathogens to encompass biological control, by including the dynamics of an antagonist population. The antagonist can increase in density on a range of habitats, including susceptible hosts, infected hosts and soil-borne inoculum. The rate of increase on each habitat depends on a parameter, and so is configurable depending on the antagonist in question.

The antagonist acts as an agent of biological control by affecting the epidemiologi-cal processes that underpin the host-pathogen interaction; these effects are translated via effects on selected epidemiological parameters including rates of primary and secondary infection, and infectious periods of infected hosts and inoculum. Reduction(s) in these rates/periods depend(s) jointly on the antagonist's density and on a per capita parameter for the effectiveness of the antagonist. By allowing the control effect to depend on antagonist density, the complex temporal variation corresponding to the three species interaction in biological control is reflected. By decoupling the range of habitat(s) the antagonist is capable of colonising from its action(s) on the epidemiology of the pathogen, and by allowing both these aspects of its biology to be be controlled by tunable parameters, the model can target diverse pathogen-antagonist interactions. As the underlying epidemiological model is equally flexible, and can in principle represent any host-pathogen combination, the full model is therefore applicable to a wide range of host-pathogen-antagonist triplets.

It is instructive to show this flexibility in practice. Using take-all on wheat, caused by the fungus Gaeumannomyces graminis var. tritici, as an illustrative example, a number of studies have used a variant of our underlying model to investigate the hostpathogen dynamics [START_REF] Bailey | Modeling and analysis of disease-induced host growth in the epidemiology of take-all[END_REF][START_REF] Bailey | Epidemiology and chemical control of take-all on seminal and adventitious roots of wheat[END_REF][START_REF] Bailey | An epidemiological analysis of the role of disease-induced root growth in the differential response of two cultivars of winter wheat to infection by the take-all pathogen, Gaeumannomyces graminis. var. tritici[END_REF][START_REF] Bailey | Epidemiological analysis of take-all decline in winter wheat[END_REF]. Both primary and secondary infection and host growth were shown to have an important role, and in particular it is necessary to take β P , β S > 0 in the epidemiological model. Turning to the biological control agent, a range of mechanisms for the antagonistic effect of Pseudomonas spp. bacteria have been proposed.

However the current consensus [START_REF] Weller | Microbial populations responsible for specific soil suppressiveness to plant pathogens[END_REF] emphasises the role of antibiotic production (either 2,4-diacetylphloroglucinol [START_REF] Pierson | Use of mixtures of fluorescent pseudomonads to suppress take-all and improve the growth of wheat[END_REF] or phenazine-1-carboxylic acid [START_REF] Thomashow | Role of a phenazine antibiotic from Pseudomonas fluorescens in biological control of Gaeumannomyces graminis var. tritici[END_REF][START_REF] Raaijmakers | Changes in populations of rhizosphere bacteria associated with take-all disease of wheat[END_REF]). Antibiotics reduce the rates of both initial primary infection [START_REF] Bull | Relationship between root colonization and suppression of Gaeumannomyces graminis var. tritici by Pseudomonas fluorescens strain 2-79[END_REF] and of the growth of lesions and secondary spread of the pathogen [START_REF] Cook | Making greater use of introduced microorganisms for biological control of plant pathogens[END_REF][START_REF] Gardner | Changes in populations of rhizosphere bacteria associated with take-all disease of wheat[END_REF]. This would correspond to the bacterium reducing the effective rates of primary (β P ), and secondary (β S ) infection, respectively, and so to α P , α S > 0 in the model. Finally, although the bacteria are acknowledged to colonise healthy roots [START_REF] Thomashow | Role of a phenazine antibiotic from Pseudomonas fluorescens in biological control of Gaeumannomyces graminis var. tritici[END_REF], populations are much larger on diseased roots [START_REF] Gardner | Changes in populations of rhizosphere bacteria associated with take-all disease of wheat[END_REF], and so we would take ρ I > ρ S > 0. In principle a similar characterisation of a plausible set of non-zero parameters could easily be outlined for any host-pathogen-antagonist interaction.

Biological control can conveniently be divided into (i) preventative, in which preemptive application aims to stop the pathogen from entering the system; and (ii) reactive, with control applied after the pathogen has already invaded the population of plant hosts. Reactive control can be further subdivided into (a) eradicative, where the pathogen is driven out of the system by the antagonist; and (b) reductive, with the more modest aim of reducing the density of the pathogen. The model allows us to understand each of these types of control.

For preventative control, the pathogen's invasion criterion in the presence of the antagonist (Equations ( 20) and ( 27)) illustrates the importance of both the antagonist's density when the pathogen is absent, and its effect(s) on the epidemiology of the pathogen. Clearly to be able to prevent pathogen invasion, the antagonist must be able to bulk-up on susceptible host tissue (i.e. has ρS > 0). However, depending on the division of the pathogen's basic reproductive number into distinct components corresponding to primary and secondary infection, R 0 = R P 0 + R S 0 , even a high density of antagonist may not be sufficient to stop invasion. In particular, a pathogen with R P 0 > 1 can only be prevented from invading by an antagonist that is able to alter at least one of the rates associated with primary infection (i.e. that affects the effective rate of primary infection, βP ; of decay of inoculum, γ; or of infected hosts, μ; and so has at least one of αP , ωγ or ωμ greater than zero). There is an analogous result for secondary infection. Note that an antagonist which affects the rate of decay of infectious hosts (i.e. has ωμ > 0) is, in principle at least, theoretically capable of preventing invasion irrespective of the pathogen's balance between primary and secondary infection. This is because the infectious period of infected hosts is implicated in both infection pathways. However, whether or not such an antagonist does indeed prevent the pathogen from entering the system depends not only on the antagonist's value of ωμ , but also on its density in the absence of the pathogen (i.e. on the balance between its bulk-up rate on susceptible hosts, ρS , and its natural decay rate, σ).

The partitioning of R 0 is equally critical for eradicative reactive control. The maximum possible effect of any control which independently affects a single epidemiological mechanism is shown in Table 5. Again the result is driven by the partitioning Even if the antagonist is able to bulk up to a large extent on the available habitat, it will not be able to eradicate the pathogen unless a correct combination of epidemiological mechanism(s) are targetted.

R 0 = R P 0 + R S 0 . For example, if both R P 0 , R S 0 > 1,
The significance of the antagonist's population dynamics for reactive control, however, is twofold. Firstly the antagonist must be able to invade when the pathogen is present (cf. Equation ( 29)). This depends on a complex balance of the available density of susceptible and infected hosts and soil-borne inoculum, and which of these habitats the antagonist is capable of colonising. Secondly, and arguably more importantly, useful reductive control is possible even if the pathogen is not eradicated. As shown in Figure 1, broadly-speaking, the better the antagonist is at bulking-up on available habitat, the more effective it will be at controlling the pathogen, given a fixed per capita efficiency of antagonism. This is unsurprising. However less obvious (Figure 2)

is that any increase in (say) the rate of increase on susceptible hosts, ρS , can be exactly offset by a suitably-sized decrease in (say) the rate of increase on infected hosts, ρI .

This indicates that, if the antagonist is able to persist in the system, the range of habitats that it is capable of colonising is less important than the rate at which it is able to bulk-up on those habitats that it can use. In particular there is no requirement for the antagonist to be able to bulk-up on all classes of habitat in order to exert the maximal reductive control it is capable of as per Table 5, so long as it is sufficiently able to utilise those habitats it can colonise. Of course there is the important proviso in the limiting case of eradication that the antagonist must be able to bulk up on susceptible hosts (as otherwise it destroys its own habitat in exerting its antagonist effect). Finally we note that the maximum effect of reductive control again follows from a combination of R P 0 and R S 0 for the underlying host-pathogen interaction and the set of epidemiological mechanisms that the antagonist can affect.

We used the equilibrium density of infected hosts, Î∞ , to assess the quality of biological control. This approach is fairly standard for models of this type [START_REF] Gilligan | An epidemiological framework for disease management[END_REF][START_REF] Cunniffe | Invasion, persistence and control in models of soil-borne plant pathogens: the effect of host demography[END_REF] and certainly has the dual advantages of simplicity and lack of ambiguity. However in certain circumstances it is possible that either (i) the equilibrium may not be reached within the timescale of interest for a particular application of the model (eg. within a single growing season); or (ii) the approach to equilibrium is oscillatory, and so the final density of infected hosts understates the impact of the pathogen on the quantity of practical interest (eg. the yield of a crop plant). Other approaches are possible, often based on some variant of the area under the disease progress curve (AUDPC) [START_REF] Madden | The Study of Plant Disease Epidemics[END_REF]. A particularly useful metric which concentrates on the yield within a single growing season of length T max and which addresses both of these potential problems was proposed by Hall et al. [START_REF] Hall | Invasion of drug and pesticide resistance is determined by a trade-off between biocide efficacy and relative fitness[END_REF]

y = T max t=0 w(t)S(t)dt, (31)
where w(t) gives an appropriate weighting to any growth stages that have a disproportionate effect on yield. However this approach (i) targets the particular case of withinseason growth of a crop, and so is inappropriate for a generic framework such as that we present here; (ii) can only be calculated for any particular set of parameters using simulation; and (iii) requires the weighting function w(t) to be defined. Additionally, we note that this type of metric would be most useful if the approach to equilibrium were strongly oscillatory; this does not appear to be the case for our model, at least for the parameter sets we have examined. This is perhaps in part a consequence of the linear function we used to model host growth, which has recently been shown to be associated with a smooth approach to equilibrium in the underlying epidemiological model (in contrast to non-linear host growth functions such as logistic, which promote cycling of the state variables [START_REF] Cunniffe | Invasion, persistence and control in models of soil-borne plant pathogens: the effect of host demography[END_REF]). Finally we note that a rise then fall in the number of infected roots appears to be rare for the soil-borne systems we are most focussed upon [44, 20,

The above analyses of invasion have depended on either the antagonist or pathogen being well-established, and so one species or the other being initially present in the system at its equilibrium density. However this is not necessarily the case. While the above analyses remain broadly correct, it is possible that both the pathogen-free and antagonist-free equilibria are locally stable. In this bistable case, either pathogen or antagonist can eventually be eradicated. The final outcome of attempted control then depends critically on the initial conditions at the time of deployment. Interestingly for a fixed initial condition there is a sharp jump from totally effective eradicative control (i.e. eradication of the pathogen) to totally ineffective control (i.e. the pathogen persists at its antagonist-free equilibrium) as the parameters of the model are slightly altered. As changes in parameters can be driven by changes in environmental or other conditions [START_REF] Kleczkowski | Dynamically generated vari-ability in plant-pathogen systems with biological control[END_REF], this mechanism arguably offers a plausible explanation for the wideranging outcomes of biological control in practice, and for spatial differences in the effectiveness of biological control in response to small-scale environmental changes.

We have chosen not to explicitly model responses to environmental variables such as temperature and moisture levels, in the interests of parsimony and to avoid obscuring the messages of this introduction to the model framework. However we note that a flexible technique based on rewriting the model as a stochastic differential equation and coupling it to a simple Markov-chain weather-generating model was presented by Truscott and Gilligan [START_REF] Truscott | Response of a deterministic epidemiological system to a stochastically varying environment[END_REF], and our result illustrating very large effects of small parameter changes indicates that this may be a fruitful area for our future work.

The generic nature of our work distinguishes it from previous models of biological control of soil-borne pathogens [START_REF] Gubbins | Population dynamics of a parasite and hyperparasite in a closed system: model analysis and parameter estimation[END_REF][START_REF] Kleczkowski | Dynamically generated vari-ability in plant-pathogen systems with biological control[END_REF][START_REF] Bailey | Biological control of pathozone behaviour and disease dynamics of Rhizoctonia solani by Trichoderma viride[END_REF][START_REF] Gibson | Predicting variability in biological control of a plant-pathogen system using stochastic models[END_REF][START_REF] Gubbins | Biological control in a disturbed environment[END_REF][START_REF] Gubbins | Persistence of host-parasite interactions in a disturbed environment[END_REF][START_REF] Gubbins | A test of heterogeneous mixing as a mechanism for ecological persistence in a disturbed environment[END_REF][START_REF] Bailey | Modeling and analysis of disease-induced host growth in the epidemiology of take-all[END_REF][START_REF] Gibson | A Bayesian analysis of botanical epidemics using stochastic compartmental models[END_REF][START_REF] Kleczkowski | Parameter estimation and prediction for the course of a single epidemic outbreak of a plant disease[END_REF]. However a flexible model of the biological control of airborne pathogens was recently introduced by Jeger et al. [START_REF] Jeger | A generic theoretical model for biological control of foliar plant diseases[END_REF], and further investigated by Xu et al. [START_REF] Xu | Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent[END_REF]. Our model is more closely targetted to the soil-borne systems we consider, and in particular includes the distinct pathways of primary and secondary infection that have been shown to control epidemics of soil-borne disease [START_REF] Brassett | A model for primary and secondary infection in botanical epidemics[END_REF][START_REF] Gilligan | Population dynamics of botanical epidemics involving primary and secondary infection[END_REF][START_REF] Madden | A population-dynamic approach to assess the threat of plant pathogens as biological weapons against annual crops[END_REF][START_REF] Bailey | An epidemiological analysis of the role of disease-induced root growth in the differential response of two cultivars of winter wheat to infection by the take-all pathogen, Gaeumannomyces graminis. var. tritici[END_REF]. Furthermore our model includes the growth of the host, which is now well-acknowledged to be a crucial driver of the dynamics of soil-borne pathogens [START_REF] Bailey | Modeling and analysis of disease-induced host growth in the epidemiology of take-all[END_REF][START_REF] Bailey | Epidemiological analysis of take-all decline in winter wheat[END_REF]. Host growth was excluded from the models of Jeger et al. [START_REF] Jeger | A generic theoretical model for biological control of foliar plant diseases[END_REF] and Xu et al. [START_REF] Xu | Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent[END_REF] on the grounds of expediency in simplifying analytic solution. Instead those authors allowed a proportion of tissue colonised by the biological control agent (their class H b ) to continuously become removed (R) or to revert to susceptible (H s ). The latter transition allows the pathogen to persist in the system. The former transition (i.e. H b → R) was removed in the updated version of the model due to Jeger et al. [START_REF] Xu | Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent[END_REF] to ameliorate the unrealistic immediate removal of a large proportion of host tissue following a large one-time application of biological control. As a consequence of our focus on soil-borne pathogens, it is the more extensive treatment of host growth and primary and secondary infection that distinguishes our work from the models of Jeger et al. [START_REF] Jeger | A generic theoretical model for biological control of foliar plant diseases[END_REF] and Xu et al. [START_REF] Xu | Numerical studies of biocontrol efficacies of foliar plant pathogens in relation to the characteristics of a biocontrol agent[END_REF].

In summary our results highlight the importance of both population dynamics and the mechanism(s) of antagonism for effective biological control of soil-borne plant pathogens. We illustrate how successful biological control depends crucially on the epidemiology of the host-pathogen interaction and the habitats that the antagonist is able to colonise. While we acknowledge our underlying modelling framework is rather simple, by restricting ourselves to a non-spatial, autonomous, deterministic variant of the SIRX framework, we have avoided the proliferation of state variables and parameters which would have been associated with more complex models. Additionally models of this ostensibly simple type have been extensively and successfully confronted with data [START_REF] Gilligan | Population dynamics of botanical epidemics involving primary and secondary infection[END_REF][START_REF] Gubbins | Biological control in a disturbed environment[END_REF][START_REF] Gubbins | Persistence of host-parasite interactions in a disturbed environment[END_REF][START_REF] Gubbins | A test of heterogeneous mixing as a mechanism for ecological persistence in a disturbed environment[END_REF][START_REF] Bailey | Modeling and analysis of disease-induced host growth in the epidemiology of take-all[END_REF][START_REF] Bailey | Epidemiology and chemical control of take-all on seminal and adventitious roots of wheat[END_REF][START_REF] Bailey | An epidemiological analysis of the role of disease-induced root growth in the differential response of two cultivars of winter wheat to infection by the take-all pathogen, Gaeumannomyces graminis. var. tritici[END_REF][START_REF] Bailey | Epidemiological analysis of take-all decline in winter wheat[END_REF]. However, our future work will concentrate on extending the framework to include stochasticity [START_REF] Gibson | A Bayesian analysis of botanical epidemics using stochastic compartmental models[END_REF][START_REF] Gilligan | Epidemiological models for invasion and persistence of pathogens[END_REF]; spatial effects [START_REF] Park | Invasion and persistence of disease in a spatially structured metapopulation[END_REF][START_REF] Stacey | A model for invasion and spread of rhizomania in the UK: implications for disease control strategies[END_REF]; environmental variation [START_REF] Truscott | Response of a deterministic epidemiological system to a stochastically varying environment[END_REF] and the periodic removal of hosts associated with commercial cropping in agricultural systems [START_REF] Gubbins | Persistence of host-parasite interactions in a disturbed environment[END_REF][START_REF] Madden | A population-dynamic approach to assess the threat of plant pathogens as biological weapons against annual crops[END_REF].

Invasion of Already present Precondition

Invasion condition 5 are attained in practice, as λ and αP or ωμ → ∞, but the actual density of infected hosts for particular parameters depends upon a combination of the bulk-up behaviour and the per capita effect of antagonism. 3: Sharp transitions: response to antagonist habitat, antagonist mode of action and the initial density of host, pathogen and antagonist. Bistability (here promoted by large antagonist death rate) leads to large changes in behaviour with small changes to the parameters. Figures 2(a) and 2(b) are replicated for an antagonist with large death rate ( σ = 5.0). When the antagonist is capable of eradicating the pathogen (i.e. Figure (b)) there can be a sharp transition on the ρS axis near the value ρS = 8.3 (marked by a green dot), where the density of infected hosts abruptly decreases from its maximum value to zero (with initial conditions Ŝ0 = 1.0, Î0 = 0, X0 = 0.1, Â0 = 0.1). Figures (c) and(d) show the response of the location of the sharp transition in parameter space to the initial conditions. The value of ρS (the rate at which antagonist bulks-up on susceptible hosts) at which there is a phase transition between eradication and totally ineffective control is shown for different values of ( Ŝ0 , X0 ) (c) and ( Â0 , X0 ) (d). Here ρI = ρS = 0, and so there is no bulking-up on infected hosts, although the qualitative behaviour generalises to antagonists that are also able to bulk-up on these habitats.

 Ŝ None ρS σ > 1 Ŝ, Î, X R(0) = R 0 > 1 ρS σ Ŝ∞ + ρI σ Î∞ + ρX σ X∞ > 1 Î Ŝ None R(0) = R 0 > 1 Ŝ,  ρS -σ > 0 R( ρS -σ) > 1

  3.1 are attained. *** INSERT FIGURE ONE NEAR HERE ***

*

  ** INSERT FIGURE THREE NEAR HERE *** ** INSERT FIGURE FOUR NEAR HERE ***

Figures 2 Figures 3c,d and 4 Table 2 :

 242 Figures 2 and 3a,b

5 :

 5 Dependence of the minimum long-term infected density, Î∞ , on R P 0 , R S 0 and the epidemiological mechanism affected by control.

Figure 1 :

 1 Figure 1: Habitat-generalist antagonism. Figure (a) shows the dependence of the endemic density of infected hosts upon the per capita antagonistic effect on the rate of primary infection ( αP ) and the rate at which the antagonist can bulk-up ( λ = ρS = ρI = ρX ). Figure (b) shows the dependence on antagonist bulking-up and the rate of decay of infected hosts ( ωμ ). The lower bounds in Table5are attained in practice, as λ and αP or ωμ → ∞, but the actual density of infected hosts for particular parameters depends upon a combination of the bulk-up behaviour and the per capita effect of antagonism.

Figure 2 :

 2 Figure 2: Habitat-specialist antagonism. All figures show the endemic density of infected hosts as individual components of bulking-up behaviour are altered. Figures (a) and (b) demonstrate the variation in ( ρS , ρI ) space (with ρX = 0), whereas Figures (c) and (d) demonstrate the variation in ( ρI , ρX ) space (with ρS = 0). The antagonist is able to affect only the rate of primary infection in Figures (a) and (c) ( αP = 1.0, αS = ωμ = ωγ = 0), whereas it affects only the rate of decay of infected hosts in Figures (b) and (d) ( ωμ = 1.0, αP = αS = ωγ = 0). Note the maximum reduction in infected density is achievable for antagonists that do not bulk-up on all classes of host, the linear contours of infected density, and the necessity for ρS > 0 for the pathogen to be eradicated (i.e. the varying gradient of the contours in Figure (b)).

Figure

  Figure3: Sharp transitions: response to antagonist habitat, antagonist mode of action and the initial density of host, pathogen and antagonist. Bistability (here promoted by large antagonist death rate) leads to large changes in behaviour with small changes to the parameters. Figures2(a) and 2(b) are replicated for an antagonist with large death rate ( σ = 5.0). When the antagonist is capable of eradicating the pathogen (i.e. Figure(b)) there can be a sharp transition on the ρS axis near the value ρS = 8.3 (marked by a green dot), where the density of infected hosts abruptly decreases from its maximum value to zero (with initial conditions Ŝ0 = 1.0, Î0 = 0, X0 = 0.1, Â0 = 0.1). Figures(c) and (d)show the response of the location of the sharp transition in parameter space to the initial conditions. The value of ρS (the rate at which antagonist bulks-up on susceptible hosts) at which there is a phase transition between eradication and totally ineffective control is shown for different values of ( Ŝ0 , X0 ) (c) and ( Â0 , X0 ) (d). Here ρI = ρS = 0, and so there is no bulking-up on infected hosts, although the qualitative behaviour generalises to antagonists that are also able to bulk-up on these habitats.

Figure 4 :

 4 Figure 4: Sharp transitions: mechanism by which alternate equilibria are attained. The density of (a) susceptible hosts, (b) infected hosts, (c) inoculum and (d) antagonists for ρ S = 8.3 (blue solid line) and ρS = 8.4 (red dotted line). The extreme difference in behaviour for slightly different values of ρS is due to bistability in the model and the feedback described in the main text.

  an antagonist is only able to eradicate the pathogen (i.e. drive Î∞ to zero) if it is able to interfere with both infection pathways simultaneously. This can either be because the antagonist is capable of affecting both primary and secondary infection (eg. αP , αS > 0, although other combinations are possible), or because it can reduce the infectious period of infected hosts (i.e. ωμ > 0).

Table 1 :

 1 Dimensionless variables and parameters (with illustrative parameter values and initial conditions, where appropriate).

	Variable or Definition	Description	Default
	parameter					value
	Ŝ	Sκ -1	Density of susceptible hosts	-
	Î	Iκ -1	Density of infected hosts	-
	X	ηXν -1 κ -1	Density of soil-borne inoculum	-
	Â	ξAη -1	Density of antagonist	-
	t	ηt			Time	-
	Ŝ0	S 0 κ -1	Initial density of susceptible hosts	1.0
	Î0	I 0 κ -1	Initial density of infected hosts	0
	X0	ηX 0 ν -1 κ -1	Initial density of soil-borne inoculum	0.1
	Â0	ξA 0 η -1	Initial density of antagonist	0.1
	βP	β P νκη -2	Rate of primary infection	0.5
	βS	β S κη -1	Rate of secondary infection	0.375
	μ	μη -1	Death rate of infected hosts	0.25
	γ	γη -1	Decay rate of soil-borne inoculum	0.8
	R 0	βP	ĉ-1 + βS	μ-1 Pathogen's basic reproductive number (no Â) 4.0
	R P 0	βP	ĉ-1 μ-1	Component of R 0 due to primary infection	2.5
	R S 0	βS	μ-1	Component of R 0 due to secondary infection 1.5
	αP	α P ηξ -1	Controls reduction in βP by antagonist	0-5
	αS	α S ηξ -1	Controls reduction in βS by antagonist	0-5
	ωμ	ω P ηξ -1	Controls increase in μ by antagonist	0-5
	ωγ	ω S ηξ -1	Controls increase in γ by antagonist	0-5
	ρS	ρ S κη -1	Bulk-up rate upon susceptible hosts	0-10
	ρI	ρ I κη -1	Bulk-up rate upon infected hosts	0-10
	ρX	ρ X νκη -2	Bulk-up rate upon soil-borne inoculum	0-10
	σ	ση -1	Rate of decay of antagonist	1 or 5

Table 4 :

 4 Invasion criteria.

	when control or antagonist reduces 1
	Minimum long-term infected density Î∞
	Pathogen

Note that lower bounds on infected density do not take into account the population dynamics of the antagonist, but are attained for reasonable values of the antagonist bulking-up parameters (see main text and Figures1 and 2). Note also that here we restrict attention to single-mode controls/antagonists which are capable of affecting the effective size of a single epidemiological parameter.

Acknowledgements

We thank the BBSRC for funding in the form of a PhD studentship (NJC) and a Professorial Fellowship (CAG). NJC would also like to thank Douglas Bailey for extensive discussions concerning biological control, and Stephen Parnell and Matt Castle for their critical reading of early drafts of this manuscript.

Equilibrium

Interpretation

Existence criterion Ŝ∞ , Î∞ , X∞ , Â∞ ( , ✗, ✗, ✗)

Only host persists None ( , , , ✗) Host and pathogen persist

Host and antagonist persist ρSσ > 0 ( , , , ) Coexistence of all three species ∃ Â∞ > 0 in Equation [START_REF] Gilligan | An epidemiological framework for disease management[END_REF] with R 0 = R( Â∞ ) > 1

Appendix A. Stability analysis 636

Stability of the underlying epidemiological model (i.e. Equations ( 10)- [START_REF] Bull | Relationship between root colonization and suppression of Gaeumannomyces graminis var. tritici by Pseudomonas fluorescens strain 2-79[END_REF] with  fixed at zero) is determined by the Eigenvalues of its Jacobian, which at a general point

Appendix A.1. Pathogen-free equilibrium 637 At the pathogen-free equilibrium (Equation ( 17))

The Eigenvalues λ satisfy

Clearly one Eigenvalue is always -1, and so the Routh-Hurwitz (R-H) criteria [START_REF] Britton | Essential Mathematical Biology[END_REF] applied to the inner quadratic factor indicates that all three Eigenvalues have negative real parts if and only if

which in combination form a condition for the pathogen-free equilibrium to be stable

Since all parameters are positive this may be reduced to

which forms a final stability condition for the pathogen-free equilibrium of the underlying model. As the R-H criterion is a two-way implication, whenever Equation (A.7)

is not satisfied the pathogen-free equilibrium is unstable, and the pathogen can invade a population of hosts at its carrying capacity in the absence of antagonism, allowing us to identify the basic reproductive number as

That the particular expression given in Equation (A.8) is the basic reproductive number of the pathogen (rather than for example a related quantity with similar threshold behaviour, such as its square root [START_REF] Heffernan | Perspectives on the basic reproductive ratio[END_REF]) may be confirmed by either a retrospective biological interpretation of its components [START_REF] Van Den Bosch | The basic reproduction number of plant pathogens: matrix approaches to complex dynamics[END_REF] or more formally by the Next Generation method [START_REF] Diekmann | On the definition and the computation of the basic reproductive ratio R 0 in models for infectious diseases[END_REF][START_REF] Van Den Driessche | Reproduction numbers and subthreshold epidemic equilibria for compartmental models of disease transmission[END_REF]. Full details for a similar model are given in [START_REF] Cunniffe | Invasion, persistence and control in models of soil-borne plant pathogens: the effect of host demography[END_REF].

Appendix A.2. Pathogen-present equilibrium

At the pathogen-present equilibrium of Equation ( 18), noting that

the Jacobian reduces to

The characteristic equation is given by

where

The R-H criteria for cubic equations indicate that all three Eigenvalues have negative real part if and only if

)

Clearly (A.15) is satisfied only if R 0 > 1, and so the biological existence criterion R 0 > 1 is also necessary for the equilibrium to be stable. Rewriting

indicates that

and hence that (A. [START_REF] Cunniffe | Invasion, persistence and control in models of soil-borne plant pathogens: the effect of host demography[END_REF]) is true whenever R 0 > 1 (note that this is a sufficient rather than necessary condition for (A.16) to hold). The product in (A.17) can then be rearranged

which is definitely positive if R 0 > 1 (again this is a sufficient rather than necessary condition). Overall a necessary and sufficient condition for all three R-H criteria to be satisfied, and therefore for the pathogen-present equilibrium of the underlying model to be stable, and for the pathogen to be able to persist at its non-zero equilibrium density in the host population in the absence of antagonism, is just R 0 > 1.

4. Figure 4. Figure 4. Figure