Oh the places they’ll go: improving species distribution modelling for invasive forest pests in an uncertain world

Abstract

Species distribution modelling (SDM) is a valuable tool for predicting the potential distribution of invasive species across space and time. Maximum entropy modelling (MaxEnt) is a popular choice for SDM, but questions have been raised about how these models are developed. Without biologically informed baseline assumptions, complex default SDM models could be selected, even though alternative settings may be more appropriate. Here we explored the effects of various SDM design strategies on distribution mapping of four forest invasive species (FIS) in Canada. We found that if we ignored the underlying FIS biology such as use of biologically relevant predictors, appropriate feature selection and inclusion of dispersal and biotic interactions when we developed our SDMs, we obtained complex SDMs (default) that provided an incomplete picture of the potential FIS invasion. We recommend simplifying SDM complexity and including biologically informed assumptions to achieve more accurate dispersal predictions, particularly when projecting FIS spread across time. We strongly encourage SDM users to perform species-specific tuning when modeling FIS distributions with MaxEnt to determine the best SDM design.

Appendices

Appendix 1

Evaluation summary of FIS models using TSS, correct classification rates, omission error and sensitivity metrics.

AGM
Evaluation metric	Sensitivity	TSS	CCR	OR
Default with all variables	0.500	0.393	0.696	0.500
Default with climate variables	0.589	0.482	0.741	0.411
Default with select variables	0.536	0.375	0.688	0.464
Default with select climatic variables	0.679	0.429	0.714	0.321
With tuned parameters	0.750	0.518	0.759	0.250
Without bias file	0.464	0.375	0.688	0.536
With sampling correction	0.736	0.528	0.764	0.264

ALB
Evaluation metric	Sensitivity	TSS	CCR	OR
Default with all variables	0.632	0.507	0.756	0.368
Default with climate variables	0.725	0.450	0.725	0.275
Default with select variables	0.684	0.484	0.744	0.316
Default with select climatic variables	0.750	0.475	0.738	0.250
With tuned parameters	0.737	0.537	0.769	0.263
Without bias file	0.526	0.501	0.756	0.474
With sampling correction	0.950	0.555	0.782	0.395

DED
Evaluation metric	Sensitivity	TSS	CCR	OR
Default with all variables	0.690	0.638	0.819	0.310
Default with climate variables	0.759	0.672	0.836	0.241
Default with select variables	0.621	0.569	0.784	0.379
Default with select climatic variables	0.690	0.569	0.784	0.310
With tuned parameters	0.707	0.586	0.793	0.293
Without bias file	0.759	0.724	0.862	0.241
With sampling correction	0.776	0.734	0.872	0.224

SOD
Evaluation metric	Sensitivity	TSS	CCR	OR
Default with all variables	0.789	0.684	0.842	0.211
Default with climate variables	0.789	0.684	0.842	0.211
Default with select variables	0.684	0.579	0.789	0.316
Default with select climatic variables	0.684	0.579	0.789	0.316
With tuned parameters	0.737	0.632	0.816	0.263
Without bias file	0.895	0.789	0.895	0.105
With sampling correction	0.947	0.842	0.921	0.053

Appendix 2

Potential distribution of selected FIS in current and future climate change scenarios along with MESS (multivariate environmental similarity surface) maps. For suitability maps, higher probability (red colors) represent areas suitable for FIS. Zero probability or lower probability (dark green) indicates areas less suitable. For MESS maps, increase in blue tone denotes increasing degree of extrapolation on at least one variable. Suitability predictions in those areas (blue) should be treated with high caution.

• (a) Asian gyspy moth

  

  

• (b) Asian longhoned beetle

  

  

• (c) Sudden oak death

  

  

• (d) Dutch elm disease

  

  

Appendix 3

See Fig. 8.

Fig. 8

Predicted potential distribution of selected FIS on a global scale. Higher probability (red colors) represent areas suitable for FIS. Zero probability or lower probability (dark blue) indicates areas less suitable

Appendix 4: environmental response curves

See Fig. 9.

Fig. 9

Relationships between environmental predictors and the probability of the presence of FIS: red curves show the mean response and blue margins are ± 1 SD calculated over 10 replicates

Appendix 5

Comparing dispersal limited to unlimited FIS dispersal projections under climate change conditions. Here numbers represents total number of cells colonized under each scenario.

(a) Asian gypsy moth

GCM/scenario	Infestation point-Vancouver port			Infestation point-Toronto port			Unlimited dispersal
	ccsm4	hadgem2es	miroc5	ccsm4	hadgem2es	miroc5	ccsm4	hadgem2es	miroc5
rcp26	3416	4143	5074	5393	5260	5499	19,142	19,509	19,265
rcp45	5262	4608	5115	5479	5532	5726	19,383	19,592	19,445
rcp85	5318	5473	5939	5619	5788	5843	19,548	19,770	19,599

(b) Asian longhorned beetle

GCM/scenario	Infestation point-Vancouver port			Infestation point-Toronto port			Unlimited dispersal
	ccsm4	hadgem2es	miroc5	ccsm4	hadgem2es	miroc5	ccsm4	hadgem2es	miroc5
rcp26	71	76	68	2482	2497	2446	7701	7366	7647
rcp45	75	82	72	2528	2539	2613	8117	7904	8116
rcp85	76	91	79	2567	2587	2663	8716	8510	8334

(c) Dutch elm disease

GCM/scenario	Infestation point-Toronto port			Unlimited dispersal
	ccsm4	hadgem2es	miroc5	ccsm4	hadgem2es	miroc5
rcp26	1438	1379	1443	2051	2045	1992
rcp45	1538	1555	1524	2026	2061	2013
rcp85	1577	1616	1622	2039	2059	2040

(d) Sudden oak death

GCM/scenario	Infestation point-Vancouver port			Unlimited dispersal
	ccsm4	hadgem2es	miroc5	ccsm4	hadgem2es	miroc5
rcp26	879	625	893	1166	977	1061
rcp45	606	650	848	1063	947	1097
rcp85	433	500	759	1033	911	1036

Appendix 6

FIS dispersal limited distributions under different climate change scenarios and two hypothesized infestation points-

AGM (Infestation point-Vancouver port)

Dispersal restricted future distribution of AGM under GCM-CCM4 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of AGM under GCM-HADGEM2ES and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of AGM under GCM-MIROC5 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

AGM (Introduction point-Toronto port)

Dispersal restricted future distribution of AGM under GCM-CCSM4 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of AGM under GCM-HADGEM2ES and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of AGM under GCM-MIROC5 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

ALB (Introduction point-Vancouver port)

Dispersal restricted future distribution of ALB under GCM-CCSM4 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of ALB under GCM-HADGEM2ES and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of ALB under GCM-MIROC5 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

ALB (Introduction point-Toronto port)

Dispersal restricted future distribution of ALB under GCM-CCSM4 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of ALB under GCM-HADGEM2ES and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of ALB under GCM-MIROC5 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

DED (Introduction point-Toronto port)

Dispersal restricted future distribution of DED under GCM-CCM4 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of DED under GCM-HADGEM2ES and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of DED under GCM-MIROC5 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

SOD (Introduction point-Vancouver port)

Dispersal restricted future distribution of SOD under GCM-CCM4 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of SOD under GCM-HADGEM2ES and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Dispersal restricted future distribution of SOD under GCM-MIROC5 and RCP 2.6, 4.5 and 8.5 climate change scenarios. Color gradient from blue to grey represents the first 10 years of the simulation time frame when colonization first occurred, the light grey to light yellow color gradient represents the next 10 years followed by orange and rose color gradients (years 2031–2050). Pink colored pixel indicates the hypothesized point of DED introduction (port of Toronto) while the green pixels represent suitable areas that were not colonized due to dispersal limitations and not reached by FIS during the time of simulation.

Appendix 7

AGM life history parameters and associated references

AGM is a potent invader with more than 600 known hosts. AGM females are capable of flight and can lay eggs on human-made objects.

• Generations per year

  • Univoltine—one generation per year (Elkinton and Liebhold 1990)

• Dispersal

  • Adult females disperse and spread their population naturally by sustained flight and wind-borne dispersal of first instars (Keena et al. 2008).

  • Attracted to lights at night (Montgomery and Wallner 1988; Schaefer and Strothkamp 2014)

• Dispersal distance

  • Frequent long distance dispersal flights (average less than 1 km to max range of 20–40 km) (Iwaizumi et al. 2010; Keena et al. 2008)

  • Russian females may fly distances up to 100 km and eastern Siberian females seen crossing mountain ranges in large groups during outbreaks (Rozhkov and Vasilyeva 1982)

  • Egg masses in Japanese cities found within 1 km of forests(Liebhold et al. 2008).

  • Average flight distance of 1 day old Chinese females in 8 h on flight mills was 5.65 km and maximum was 10.67 km (Yang et al. 2017)

• Reproductive capacity

  • Producing an average of 600–1000 eggs per egg mass (USDA)

• Distribution

  • Found throughout temperate Asia. Usually east of the Ural Mountains into Far East Russia, through most of Japan, China and Korea. It is not found east of the Himalayan range in India (USDA)

• Critical temp.

  • AGM populations may struggle in regions experiencing longer periods of temperatures ≥ 30 °C and survival rate is highest between 15and 25 °C (Limbu et al. 2017).

ALB life history parameters and associated references

• Sex ratio

  • 1-!:14 male–female (Bancroft and Smith 2005)

  • 1:1 male–female (Trotter et al. 2019)

  • Other papers use only the females to model spread since she drives the establishment of new infestations

• Generations per year

  • Temperature dependent

    • (Haack et al. 2010)

    • (Keena and Moore 2010)

    • (Faccoli and Gatto 2016)

    • (Favaro et al. 2015)

  • Not strictly univoltine (1 year); may take multiple years to develop

    • (Keena and Moore 2010; Trotter and Keena 2016) (in Finland may take 10+ years)

    • (Straw et al. 2015)

      • 3 years for Paddock Woods

    • (Kappel et al. 2017)

      • Do not use the Newtonian Cooling model to estimate within tree temps—may not accurately reflect temps within tree

      • But estimated that in northern states will take minimum 2–3 years to complete development, some areas up to 5–6 years

• Dispersal distance

  • Frequent short distance dispersal flights (< 1.5 km)

    • (Javal et al. 2018)

  • Tendency to remain on and reinfest natal tree

    • (Haack et al. 2010)

  • Dispersal occurs when tree host quality deteriorates

    • (Sawyer 2011)

  • Rare long distance flights (< 1.5 km)

    • Human mediated transport likely more significant at farther distances

      • (Fournier and Turgeon 2017)

  • ~ 10 km (modeled and based on graph)

    • (Trotter et al. 2019)

  • Longest single sustained flight on flight mill = 4006 m; median = 247.6 m

    • (Javal et al. 2018)

  • Lifetime dispersal for a female = 14,060 m; median = 3964 m

    • (Javal et al. 2018)

• Spread rates

  • in England in one stand, mean rate of population spread 29.3 m/year

    • (Straw et al. 2016)

  • Jersey City spread 50 m/year

    • (Sawyer et al. 2011)

  • New Jersey spread 2.4–3.2 km in 5–6years

    • (Sawyer et al. 2011)

  • Italy spread 2 × 2 km in 5 years

    • Faccoli et al. (2015)

• Probability to disperse

  • 55% of tethered test flights = no flight

    • Javal et al. 2018

  • < 50% took flight in a number of laboratory experiments, esp females

    • (Keena 2018)

• Critical temps

  • 10.2C-egg hatch

  • Temperature developmental model-(Trotter and Keena 2016)

  • Adult emergence in spring after 400-degree-days (10C threshold)

    • (Smith et al. 2004)

  • Dispersal ceases below 15 °C

    • (Keena 2018)

• Habitat preferences

  • Edge preference

    • (Williams et al. 2004)

    • (Shatz et al. 2013)

DED biology and vector life history

DED is vectored by several species of bark beetle: Hylurgopinus rufipes (native), Scolytus multistriatus (introduced-Europe), and Scolytpus schevyrewi (introduced-Asia).

• Surprising lack of dispersal information for above three species

  • (Harwood et al. 2011)

• Dispersal kernel

  • (Harwood et al., 2011)

    • DED vectors

      • Negative exponential kernel of 20 km (15–40 km)

      • Experts estimate max dispersal = 12.88 km

      • Most dispersal within 500 m of host

      • Median dispersal distance of 150 m for a negative square power law function for incorporating radial dispersal

      • Probability of 0.002 for dispersal > 12.88 km

      • Combined beetle and firewood kernel of 3:1 beetle:firewood movement gives a reasonable pattern of spread in early stages of epidemic

• History of DED in UK

  • (Tomlinson and Potter 2010)

• Review of factors influencing flight in bark beetles (Jones et al. 2019)

• Bark beetles (= Scolytinae) contain vectors of DED

  • Flight capacity versus dispersal—distinct

    • Capacity = physiological ability to fly

    • Dispersal = capacity + imapct of external factors (e.g. environment)

    • Long distance dispercal characterized by above canopy flight carried by wind (e.g. Mountain pine beetle dispersal over Rocky Mountains; 30–100 km/day via wind)

• Dispersal distance

  • mean for beetles ranges from 500 m to 6 km; max distances can be > 25 km, but this is a long thin tail, bulk of the pop is short distance

• Fat-tailed dispersal kernel needed to capture potential for bark beetles to disperse long distances

• Min temp for flight initiation in bark beetles range from 10.6C–21C; mean = 15.6C

• Dispersal distance

  • Mark recapture-38% pop close to release site, 52% within 400–600 m from release site (Strobel and Lanier 1981)

  • 5–6 km dispersal (Wolfenbarger and Jones 1943)

  • Mark recapture-1 km (Pines and Westwood 2008)

  • 400–600 m dispersal (Wollerman 1979)

SOD biology and vector life history parameters and associated references

There are no known vectors of SOD other than humans but any organism that can move soil is potentially a vector of SOD.(Grünwald et al. 2012, 2019; Kliejunas 2010; Rizzo et al. 2005)

• Dispersal

  • Long range spread of disease through sporangia and chlamydospores, chlamydospores can survive for a week at a constant temperature of 55 °C.

  • Natural dispersal of SOD is by movement of plant material, waterborne and soilborne chlamydospores, and by waterborne, soilborne and wind-blown rain containing sporangia (Rizzo and Garbelotto 2003; Rizzo et al. 2005; Grünwald et al. 2019)

• Dispersal distance

  • Splash dispersal-propagules can travel up to 60 cm above infested surfaces (Kuske 1983).

  • Local spread < 1 km (ecological (Condeso et al. 2007; Ellis et al. 2010) and genetic (Mascheretti et al. 2008, 2009))

  • Most inoculum remains within 10 m of the host (Davidson et al. 2005)

  • Maximum dispersal distance < 8 km during rare storm events (apsnet.org).

  • Number of trees infected was higher on public lands that were open to recreation than on adjacent lands lacking public access and higher human population densities within 50 km increased chances of fungal infection (Cushman et al. 2008).

• Effects of temperature and moisture on growth and sporulation

  • Fungal growth occurs 10–31 °C (Tooley et al. 2009)

  • Exposure to temperatures over 30 °C decreases survival and a few minutes at 40 °C kills the fungus (Browning et al. 2008)

  • Sporangia production occurs over the temperature range of 16–22 °C (Englander et al. 2006)

  • A dew period of as little as 1 h was enough for fungal development but moisture for 24–48 h is required for maximal disease development in the laboratory (Tooley et al. 2009)

  • Most clonal hyphal colonies can survive 24 h exposure to − 5 C and some can withstand − 25 C for 24 h. (Browning et al. 2008).

• Distribution

  • SOD is distributed only in Europe and parts of North America, with three identified clonal lineages (EU1, NA1 and NA2), named for the continent where they were first found, followed by a number indicating the order of discovery (Grünwald et al. 2009)

• Habitat

  • Coastal forest types (Rizzo and Garbelotto 2003; Rizzo et al. 2002), moist and moderate climates (Rizzo et al. 2005).

Appendix 8 (MigClim: Calibration of P_Disp)-based on source (Engler and Guisan 2009)

P_Disp, is the colonization probability of a pixel given its distance from a source pixel. To calibrate P_Disp we defined a dispersal kernel used to model regular seed dispersal. The kernel was based on the following negative exponential seed dispersal probability distribution function (Eq. 1).

$${\text{P}}_{seed} \left( {\text{x}} \right) = {\text{e}}^{{\left( {{\text{x}} - {\text{pixelsize}}} \right) \left( {\frac{{ln\left( {1 - k} \right)}}{DispDist}} \right)x.\left( {\frac{{ln\left( {1 - k} \right)}}{DispDist}} \right)}}$$

(1)

Further simplified in more conventional simple negative exponential form (Eq. 2)

$${\text{P}}_{seed} \left( x \right) = \left( {\left( {1 - k} \right)^{{ - \frac{pixelsize}{DispDist}}} - 1} \right) .{\text{e}}^{{ x.\left( {\frac{{ln\left( {1 - k} \right)}}{DispDist}} \right)}}$$

(2)

where Pseed is the probability of a seed reaching distance x ≥ pixelsize, pixelsize is the one-dimensional size of a pixel, DispDist is the dispersal distance reached by the proportion k of the seeds.

Since the surface composed of pixels located at distance j from a source cell increases with distance from that source cell, the probability of a pixel to receive a seed is computed as (Eq. 3)

$${\text{P}}_{seed} \left( {Pixel_{j} } \right) = {\raise0.7ex\hbox{${{\text{P}}_{{seed^{\left( x \right)} }} }$} \!\mathord{\left/ {\vphantom {{{\text{P}}_{{seed^{\left( x \right)} }} } {Surface_{j} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${Surface_{j} }$}}$$

(3)

where Surfacej is the number of pixels covered by all pixels belonging to a same distance class. Assuming that the distribution of successful seeds (i.e. seeds leading a pixel to become colonized) is proportional to the overall distribution of seeds (P_seed), P_Disp is computed as (Eq. 4):

$${\text{P}}_{Disp} \left( {Pixel_{j} } \right) = 1 - (1 - {\text{P}}_{seed} \left( {Pixel_{j} } \right))^{successful seeds}$$

(4)

where P_Disp is the probability of colonisation for a target pixel with distance j from a source pixel and Successful Seeds the number of successful seeds produced by a fully mature source pixel.