When does spatial diversification usefully maximise the durability of crop disease resistance?

Maximising the durability of crop disease resistance genes in the face of pathogen evolution is a major challenge in modern agricultural epidemiology. Spatial diversification in the deployment of resistance genes, where susceptible and resistant fields are more closely intermixed, is predicted to drive lower epidemic intensities over evolutionary timescales. This is due to an increase in the strength of dilution effects, caused by pathogen inoculum challenging host tissue to which it is not well-specialised. The factors that interact with and determine the magnitude of this spatial effect are not currently well understood however, leading to uncertainty over the pathosystems where such a strategy is most likely to be cost-effective. We model the effect on landscape scale disease dynamics of spatial heterogeneity in the arrangement of fields planted with either susceptible or resistant cultivars, and the way in which this effect depends on the parameters governing the pathosystem of interest. Our multi-season semi-discrete epidemiological model tracks spatial spread of wild-type and resistance breaking pathogen strains, and incorporates a localised reservoir of inoculum, as well as the effects of within and between field transmission. The pathogen dispersal characteristics, any fitness cost(s) of the resistance breaking trait, the efficacy of host resistance, and the length of the timeframe of interest, all influence the strength of the spatial diversification effect. These interactions, which are often complex and non-linear in nature, produce substantial variation in the predicted yield gain from the use of a spatial diversification strategy. This in turn allows us to make general predictions of the types of system for which spatial diversification is most likely to be cost-effective, paving the way for potential economic modelling and pathosystem specific evaluation. These results highlight the importance of studying the effect of genetics on landscape scale spatial dynamics within host-pathogen disease systems.

field experimental evolution study by Bousset et al. (2018) provides a degree of empirical support for this general theory, with results suggesting that higher 'genetic connectivity' within a host population 121 facilitates higher levels of infection. Arguably, a potential weakness of this study was the implicit 122 representation of 'genetic connectivity' by greater experimental inoculation of host variety patches 123 by their specialist pathogen strains, which means that it did not truly demonstrate a landscape scale 124 spatial effect. 125 A number of recent theoretical studies have begun comparing and contrasting the various avail-126 able strategies for the optimal deployment of resistance genes, such as using mosaics (between field 127 spatial diversification), mixtures, rotations and pyramids (Fabre et al., 2015, Djidjou-Demasse et al., 128 2017, Rimbaud et al., 2018). We believe however that the specific role of spatial diversification and 129 the dynamics which affect this strategy are not currently well understood. While existing studies 130 consistently point to smaller scales of spatial heterogeneity being optimal for durable and effec-131 tive disease control in multi-strain systems, there is generally little investigation of the factors that 132 influence the strength of this spatial effect. In order for such spatial strategies to be employed in 133 commercial agriculture, they will need to be cost effective, in that the benefit to resistance durability

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We extend the model used by Fabre et al. (2012) to explicitly include spatial structure and different 142 pathogen strains. Our SI (Susceptible, Infected) model tracks two pathogen strains, a 'wild-type' 143 (wt) and a 'resistance breaking' (rb) genotype (the principal variables and parameters used in this has reduced fitness (that may be zero) when infecting R fields. The rb strain has equal infective ability on both host genotypes, but may have a reproductive fitness cost (δ) (expressed on both host 153 genotypes) associated with its resistance breaking trait. The focus of our model on disease spread 154 at the landscape scale, driven by long range pathogen dispersal, makes it appropriate for application 155 to any gene-for-gene crop disease system with a foliar wind dispersed pathogen (such as many rusts 156 or powdery mildews). 157 The overall number of infected plants in each field is simulated for n d = 120 days over n y seasons 158 in a semi-discrete modelling approach. In this, continuous time dynamics in ordinary differential  Healthy plants can become infected through three alternative routes: from infected plants in the 171 same field at rate β F , from infected plants in other fields at rate β C , and from the reservoir of 172 inoculum at rates α wt,y or α rb,y in season y for the wt and rb pathogen strains respectively. The 173 pathogen population size in the reservoir is not explicitly modelled but is represented by these rate 174 parameters (α wt,y and α rb,y ) by scaling the baseline rate of infection from the reservoir component The values of the transmission parameters β F , β C and α E were optimised by following Fabre  and is described by the deterministic ODE system: in Eqn (2).

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In Eqns (2) and (3), x indicates variables pertaining to a particular field, K[z, x] is the dispersal 208 kernel coupling field z to field x, and z = x. Here I wt,x,y is the number of plants infected by the 209 wt strain while I rb,x,y is the number infected by the rb strain (both being in field x and in season 210 y). The cost of the resistance breaking trait is δ, which is assumed act on the rate of sporulation 211 on both the primary and reservoir host, while γ is the susceptibility of the R host to the wt strain.

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If resistance to the wt strain is complete γ = 0, meaning that the wt epidemic in an R field (Eqn

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The terms α wt,x,y and α rb,x,y represent the specific infection rates of each pathogen genotype 215 from the local reservoir, where the reservoir host is assumed to be selectively neutral to both pathogen in season y are given by: Here, A wt,x,y−1 = n d 0 I wt,x,y−1 (t) dt and A rb,x,y−1 = n d 0 I rb,x,y−1 (t) dt, the AUDPCs for the epidemics    edge/area (E/A) ratio of the landscape is increased (Fig. 3). This is due to the larger dilution caused 237 by greater mixing of the two host genotype field types at smaller scales of spatial heterogeneity.

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Epidemic intensity is initially measured here over a 40 season time period, to balance short term The wt epidemic is suppressed at higher E/A ratios (Fig. 3) due to the smaller spatial grain 245 causing an increase in the proportion of dispersed wt inoculum from S fields that is wasted as it 246 lands on, but is unable to infect, R fields. The rb epidemic in R fields meanwhile is suppressed by 247 a corresponding process in which there is an increase in the proportion of rb inoculum that lands 248 on S fields. The rb inoculum is in direct competition with wt inoculum for uninfected host tissue 249 within S fields, resulting in a lower intensity rb epidemic than would take place in an R field. The 250 consequent rb genotype dispersal from these S fields back onto the nearby R fields therefore has a 251 lower force of infection than in R field to R field transmission over the same distance. The reduction 252 in the intensity of the rb epidemic on R fields is compensated to a certain extent by the increase 253 in the frequency of the rb genotype on S fields as the two field genotypes are more closely mixed 254 together in space. Whether this compensation ultimately increases or decreases the intensity of the 255 overall landscape rb epidemic, as the E/A ratio is increased, depends upon the genetic parameters 256 (Fig. 4). The gradient of the spatial suppressive effect (the rate of change in the response of average seasonal 259 epidemic intensity to E/A ratio) decreases as the E/A ratio is increased (Fig. 3). The rate of 260 gradient change at low E/A ratios is faster however with a higher mean dispersal distance (flatter 261 dispersal kernel). This washes out and limits the strength of the spatial effect at smaller scales of 262 spatial heterogeneity, where the high mean dispersal distance of the pathogen limits the impact of 263 any further decrease in the scale of spatial heterogeneity. In general, this means that in the reverse direction, as the mean dispersal distance decreases, the spatial suppressive effect is relevant over a 265 larger range of E/A ratios. The overall size of the spatial effect increases as the cost of the rb trait δ is increased from 0 to 268 0.4 (Fig. 4). Furthermore, the extent to which the pathogen genotype frequencies on the S host 269 change, as we move from larger to smaller scales of spatial heterogeneity, also depends on the cost 270 of the rb trait. When δ = 0 (Fig. 4a) the rb genotype is able to take advantage of the greater 271 proportion of host fields that it can infect, and the close proximity of the two field types at smaller 272 spatial scales of heterogeneity, allowing it to outcompete the wt genotype. This replacement of 273 pathogen genotypes at different E/A ratios is seen to a lesser extent when δ = 0.2 (Fig. 4b), and is 274 almost absent when δ = 0.4 (Fig. 4c). As the fitness cost δ increases the rb genotype is unable to 275 compete as effectively with the wt genotype on the S host at small scales of spatial heterogeneity.

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This is despite the close proximity of large numbers of R fields, which act as a major source of rb 277 inoculum, to the S fields. allows these contrasting spatial scenarios to be compared (Fig. 5). For both E/A ratios, epidemic 281 intensities decrease as δ is increased, up to δ = 0.6 where the rb trait is too expensive for that 282 pathogen genotype to invade and there is no further effect of increasing δ (Fig. 5a). The variability 283 and strength of the spatial suppressive effect can be ascertained by plotting the difference between 284 the epidemic intensities for the two E/A ratio values (Fig. 5b). The strength of the suppressive 285 effect increases from δ = 0 to 0.3, but decreases from δ = 0.4 to 0.6. A small suppressive effect 286 is still seen at δ = 0 due to the small scale of spatial heterogeneity disrupting the transient wt 287 epidemic, before the wt strain is outcompeted by the rb and reaches its near zero evolutionary 288 equilibrium frequency.

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The initial increase in the strength of the suppressive effect is due to a steeper gradient of change, 290 in the response of overall epidemic intensity to changes in δ, at small scales of spatial heterogeneity 291 (E/A ratio = 1.4) (Fig. 5a). This steeper change with δ is in turn caused by rapidly changing rb 292 dynamics on the S host, which are a larger driver of system sensitivity with greater field mixing 293 (Supporting Information Fig. S2). Here, any increase in δ reduces the competitive ability of the 294 rb genotype against the wt on susceptible hosts, which consequentially increases the amount of rb 295 inoculum that is 'wasted' as it disperses onto these S hosts, thereby increasing the spatial effect 296 strength.

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The subsequent fall in the strength of the suppressive effect is correspondingly due to a steeper 298 gradient of change with δ at large scales of spatial heterogeneity (E/A ratio = 0.1) (Fig. 5a). In 299 this range of δ values there is an increased sensitivity to changes in δ of the rb epidemic on the R 300 host, relative to that on the S host (Supporting Information Fig. S2). This is primarily because 301 the rb genotype is already unable to compete effectively with the wt on the susceptible host in this 302 range, and therefore does not respond to further changes in the fitness cost. These rapidly changing 303 R rb dynamics are a larger driver of system sensitivity with less field mixing.

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The value of δ for which the gradient of overall epidemic intensity change with δ is equal at both

Partial resistance 310
Here, we relax the assumption that the wt strain cannot infect resistant hosts (Fig. 6). As γ is 311 increased above 0, the wt genotype becomes able to infect the R host, and at higher frequencies 312 with smaller scales of spatial heterogeneity (high E/A ratios) (Fig. 6a,b). The reduced efficacy of 313 the resistance gene allows the wt strain to compete more effectively with the rb strain on the R 314 host, particularly at high E/A ratios where the field types are more greatly mixed in space. The 315 strength of the spatial suppressive effect, from using a high rather than a low E/A ratio, is shown in effect decreases to zero as γ is increased from 0 to 0.6. This effect is due to a reduction in the 318 proportion of wt inoculum that is 'wasted' in its increased dispersal onto R fields at smaller scales 319 of spatial heterogeneity. When γ = 1 there is no effect of the scale of spatial heterogeneity, as the 320 landscape is then made up of entirely susceptible hosts, and therefore the wt genotype is able to 321 fully outcompete the rb genotype.

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There is a range of γ values, from 0.6 to 0.9 for δ = 0.3 (Fig. 6d), for which the strength of generally increases as the number of seasons is increased (Fig. 8a,b,c). This is due to the increasing 343 frequency of the rb genotype, which facilitates greater infection of R fields as the system approaches 344 its long term evolutionary equilibrium. In the case where the fitness cost of the rb trait δ = 0, there 345 is no significant spatial suppressive effect on epidemic intensities over 80 seasons (i.e. the red and 346 blue curves converge in Fig. 8a). Here, the rb strain is able to completely outcompete the wt on 347 both host genotypes at the long term evolutionary equilibrium, meaning that the scale of spatial 348 heterogeneity has no effect. A transient spatial effect does however occur over a low to medium 349 number of seasons, as the system has not yet reached its equilibrium state and the wt strain is still 350 present at significant frequencies. There is a specific number of seasons, for a given fitness cost of the rb trait (δ), over which 352 the greatest spatial suppressive effect on epidemic intensities can be achieved (Fig. 8d,e,f). For 353 δ = 0 and δ = 0.2 (Fig. 8d,e) this peak effect strength occurs over a relatively short time period, 354 approximately 4 and 11 seasons respectively, whereas with δ = 0.4 the greatest spatial suppressive 355 effect is observed as the system approaches its long term evolutionary equilibrium.

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The initial increase in the strength of the spatial suppressive effect, from the first season to the 357 intermediate peak, is due to a faster increase in epidemic intensities at a low E/A ratio compared 358 with a high E/A ratio. This is because the newly emergent rb genotype is able to propagate rapidly in the highly aggregated R fields. The rb genotype initially spreads less quickly in S fields, and 360 therefore at high E/A ratios, due to the high level of competition with the coexisting wt pathogen 361 genotype. Beyond the peak in spatial suppression, if it is present, the strength of the suppressive 362 effect declines as epidemic intensities begin to increase faster at high compared to low E/A ratios.

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This is due to the increased importance of the rb strain spread on S hosts to the sensitivity of the S host (Fig. 4c).

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The variation in epidemic intensities due to the stochastic placement of fields in the landscape 379 (size of the error bars), is highest at the points where the strength of the spatial suppressive effect 380 is greatest (Fig. 8). This implies that the effect of the specific stochastic arrangement of fields, 381 and therefore the precise degree of spatial heterogeneity, is greatest when the general sensitivity to 382 spatial dynamics is maximised. focussed on the qualitative differences between local short ranged dispersal through diffusion, and 418 stratified dispersal that also included a separate long distance component, rather than looking at 419 spatial heterogeneity as a continuous scale.

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The spatial suppressive effect of cropping pattern on epidemic spread is maximised at an inter-421 mediate value for the fitness cost associated with the resistance breaking trait (Fig. 5) cost values. The exact fitness cost value for this peak in the spatial suppressive effect is lower with 429 a less effective resistance gene (Fig. 7).

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A less effective resistance gene lowers the strength of the spatial effect, as long as the resistance 431 breaking strain has a high enough fitness to be able to invade the agricultural landscape (Figs 6, 7). context, as long as the resistance breaking pathogen strain is fit enough to invade and persist within 445 the landscape (Fig. 7). The fact that spatial diversification can actually worsen epidemics when 446 only a wild-type strain is present, and a partially effective resistance gene is used, highlights the 447 necessity of understanding the state of the pathogen community and the genetic nature of the 448 system before implementing such control strategies. A spatial strategy will be less effective when 449 there are no fitness costs for the resistance breaking strain, however an effect is still observed due to 450 the time required for this strain to fully take over the pathogen population. This naturally becomes 451 particularly apparent when looking over a lower number of seasons.

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A critical factor that is generally neglected within the study of resistance durability is the length that a spatial strategy is most likely to be effective over short timescales for resistance breaking 469 strains that carry little or no fitness costs, and over longer timescales for more costly traits.

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In the current study we have restricted the cropping ratio of the susceptible and resistant cultivars 471 to 50 : 50, in order to avoid having to consider potential interactions between the effects of the scale 472 of spatial heterogeneity and the amount of resistant crop deployed. The potential ways in which the 473 patterns we have described might be influenced by different cropping ratios is a valid area for further 474 study however, as is the way that optimal cropping ratios might in turn be influenced by spatial