Biological Invasions

, Volume 11, Issue 6, pp 1247–1258

Initial conditions and their effect on invasion velocity across heterogeneous landscapes

Authors

    • Appalachian LaboratoryUniversity of Maryland Center for Environmental Science
  • Todd R. Lookingbill
    • Appalachian LaboratoryUniversity of Maryland Center for Environmental Science
Original Paper

DOI: 10.1007/s10530-008-9330-2

Cite this article as:
Ferrari, J.R. & Lookingbill, T.R. Biol Invasions (2009) 11: 1247. doi:10.1007/s10530-008-9330-2

Abstract

Accurate, time dependent control options are required to halt biological invasions prior to equilibrium establishment, beyond which control efforts are often impractical. Although invasions have been successfully modeled using diffusion theory, diffusion models are typically confined to providing simple range expansion estimates. In this work, we use a Susceptible/Infected cellular automaton (CA) to simulate diffusion. The CA model is coupled with a network model to track the speed and direction of simulated invasions across heterogeneous landscapes, allowing for identification of locations for targeted control in both time and space. We evaluated the role of the location of initial establishment insofar as it affected the pattern and rate of spread and how these are influenced by patch attributes such as size. Our results show that the location of initial establishment can significantly affect the temporal dynamics of an invasion. Traditional network metrics such as degree and measures of topological distance were insufficient for predicting the direction and speed of the invasion. Our coupled models allow the dynamic tracking of invasions across fragmented landscapes for both theoretical and practical applications.

Keywords

Biological invasionsNetwork theoryGraph theoryEpidemiologyDiffusionCellular automatonLandscape ecology

Introduction

Biological invasions are considered to be one of the greatest environmental problems of this century (Liebhold and Tobin 2006 and references therein) and are a serious threat to the economic and ecological sustainability of forested resources (Humble and Allen 2006). Pests and pathogens affecting dominant tree species can alter key successional and nutrient cycling processes that in turn affect the long-term ecological and evolutionary trajectories of forested ecosystems (Burdon et al. 2006; Hansen 1999). Accordingly, it is of great interest to the scientific, management and economic communities to understand how biotic invasions of non-indigenous and potentially harmful species expand their range through time (Mack et al. 2000). Theoretical biology is capable of contributing to the understanding of biological invasions (Hastings et al. 2005; Jeger et al. 2007), both in an academic and a practical sense. Theory-based models to accurately assess the spatial pattern of invasions, especially for heterogeneous landscapes, are a pressing research need of decision makers attempting to identify appropriate control measures (Hastings et al. 2005).

Various forms of diffusion models have been used to study biological invasions of mammals (e.g., Okubo 1980), plants (e.g., Skellam 1951), insects (e.g., Liebhold and Tobin 2006) and forest pathogens (e.g., Lee et al. 2007). Pure diffusion models (including reaction-diffusion) are often constrained by the assumption of habitat homogeneity, as incorporation of heterogeneity is computationally demanding (Seno and Koshiba 2005). Given that large portions of the world’s forests are becoming increasingly fragmented (Riitters et al. 2002), it is necessary to take landscape heterogeneity into account when modeling the spread of forest pests (Holdenreider et al. 2004). Diffusion processes are often simulated in heterogeneous landscapes using cellular automaton (CA) models, which can easily incorporate habitat heterogeneity (e.g., Hargrove et al. 2000) without the formal numerical solution of differential equations. In their simplest form, CA models are confined to allowing spread from an infected pixel (in a gridded land cover map) to the closest, or “adjacent”, neighbors of the same habitat type. Longer range jumps are accounted for using dispersal kernels and the concept of non-uniform adjacency (O’Sullivan 2001).

The concept of “adjacency” is also common in the literature of graph theory, which is a branch of discrete mathematics used to analyze networks (Gross and Yellen 2006). Network-adjacency is a binary assignment to all patch pairs in a landscape. If organism transfer is possible between patches A and B, the patches are considered connected and the network adjacency between them is given a value of 1, otherwise the value is set to zero (see Keitt et al. 1997). The entire ensemble of patch-patch adjacencies is represented by an N × N adjacency matrix, where N is the number of patches in the landscape. The adjacency matrix is the foundation of landscape-graphs which can be used to analyze broad-scale connectivity patterns, stepping-stone structures, etc. (Bunn et al. 2000; Urban and Keitt 2001).

The parallel definitions of CA and network adjacency represent two scales of organism movement. At the CA level, adjacencies are most often local and represent organism population expansion through diffusive processes generally within patches. At the graph level, network adjacencies represent broader, landscape-scale connections among patches. In this work, we combine the two concepts of adjacency to form coupled CA-network models to simulate the temporal progress of biological invasions at both the patch and landscape scales simultaneously. Given the pressing need for more time-series analysis of biological invasions (Mack et al. 2000), we believe the development of these types of theoretical models are a valuable complement to empirical field studies. Epidemiological models are often used to evaluate the temporal dynamics of population growth and spread of diseases (Watts and Strogatz 1998; Mack et al. 2000; Huang et al. 2005; Vasquez 2006). Because many of the same factors influencing disease epidemiology are common to all biotic invasions (Mack 1985), we chose a well known Susceptible/Infected cellular automaton as the basis for diffusive simulations.

The consideration of both local and long range spread is extremely important for initial condition problems in which the invader is observed in a localized region and it is necessary to understand how it may invade the entire landscape–e.g., for the purpose of targeted treatment/control to stop or slow the spread. In fragmented landscapes, where the direction and speed of an invasion is reliant on the spatial structure of suitable habitat, we hypothesize that these dynamics may influence the routing and speed of the invasion across a network of patches. Our work applies a spatially explicit coupled CA-network model to a hypothetical landscape with varying patch size and initial conditions to explore the sensitivity of invasion velocity to these factors. Then we apply our modeling approach to forest patches in Manassas National Battlefield Park to demonstrate how invasion velocity, and ultimately control efforts, are tied to the location of initial establishment.

The simulations are not intended to represent any particular organism, but to demonstrate the sensitivity of the widely accepted diffusion modeling paradigm to initial conditions by explaining invasion behavior using a network approach. By observing the routing taken by an invasion as it spreads across a heterogeneous landscape, the exercise offers an approach for identification of potential control points and provides insight into the utility of network models and commonly used network metrics and concepts for this purpose.

Methods

We performed two sets of simulations across heterogeneous landscapes using a Susceptible/Infected (SI) epidemiological cellular automaton model (e.g., Cruickshank et al. 1999; Park et al. 2002; Turner et al. 2002) as the basis of diffusive spread simulations. The first application was designed to investigate the role of patch size and the location of initial establishment on invasion dynamics using a deterministic model. The second applies a stochastic model to a real landscape.

Experiment 1

The first application was designed to test the sensitivity of the invasion process to initial conditions for a landscape in which inter-patch adjacencies are established a priori, which is typical for landscape applications of graph theory. Local diffusion from infected (status I) pixels to susceptible (status S) pixels was restricted to the nearest neighbors in the 4 cardinal directions (for gridded digital land cover maps). Pixels in separate patches within 5 pixels of each other were considered “adjacent”, above and beyond the simple nearest neighbor CA adjacency. This longer range adjacency defines patch-patch (network) adjacencies required to construct the landscape graph. These rules were applied to a hypothetical 80 × 80 pixel landscape with 10 forest patches surrounded by nonforest “matrix” (Fig. 1). The corresponding graph structure is also shown (Fig. 2). This hypothetical landscape was constructed so that all inter-patch distances between network adjacent patches were equivalent (5 pixels), so that no single connection was more heavily weighted (by distance). Our choice of a dispersal kernel maximum distance (5 pixels) was arbitrary, but the approach is similar to defining network adjacency based on Euclidean distance (e.g. D’Eon et al. 2001; Urban and Keitt 2001; Rudd et al. 2002).
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Fig. 1

Hypothetical landscape, 80 × 80 pixels. Susceptible sites, or “habitat”, colored dark gray. Patches are referred to by number. Initial condition starting points for individual simulations denoted by capital letters

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Fig. 2

Graph representation of the landscape shown in Fig. 1. Each patch is the equivalent of a “node”, indicated by black circles scaled with diameter proportional to area. Inter-patch adjacency is indicated by “edges”, or dark lines connecting adjacent nodes

Eight simulations were performed, each with different “release points” which identify the initial establishment location of an invasive organism (points A–H, Fig. 1). We limited the number of release points so we could present the routing of the invasion through time for each simulation. Release points were chosen to represent invasions starting from small and large patches with varying levels of network adjacencies, and are therefore representative of the range of initial conditions possible on the simulated landscape. Simulations were performed until at least one pixel in all patches was infected. The computational time step at which transfer was made from the patch in which the initial infection occurred to each previously uninfected patch was recorded for each simulation. The time at which all patches had at least one pixel infected was designated the Transfer Time, T(t).

For the network representation of the landscape (Fig. 2) we calculated the degree of each patch, where degree equals the number of connections from that patch to any others (see Gross and Yellen 2006). Topological distance was also calculated. Topological distance is the integer number of jumps between patches required to move from one patch to another (Pascual-Hortal and Saura 2006). If the patches are adjacent, the topological distance is one. If the patches are separated by an intermediate patch, the topological distance is two. If multiple pathways exist between 2 patches, the shortest is chosen. In real landscapes, the Euclidean distances among patches are non-topological, however we constructed the landscape in Fig. 1 such that all inter-patch distances are equivalent and no single connection is weighted more heavily than others due to differences in Euclidean distance. In other words, the Euclidean distance is simply the topological distance scaled by a factor of 5.

Experiment 2

The second application used a stochastic cellular automaton based on the SI model in which probabilistic long-range transfers were possible beyond the nearest neighbors of infected pixels. This type of approach can be considered a stratified diffusion model (Shigesada et al. 1995; Shigesada and Kawasaki 1997). We used the model to track inter-patch (network) adjacencies as they developed through time over a real landscape defined as forest patches within Manassas National Battlefield Park (MNBP), Virginia (Fig. 3).
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Fig. 3

Location map for Manassas National Battlefield Park (MNBP)

Battlefield parks such as MNBP provide valuable case studies of heterogeneous landscapes because of their long-history of fragmentation. MNBP was established to preserve the scene of two significant Civil War battles, and by management mandate much of the park has retained its battlefield character within the rapidly developing Washington, D.C. metropolitan region. Though designated primarily for its cultural resources, the park now serves as an important biological refuge in the region and maintains approximately 50% of its 2,056 ha in forest (Gardner et al. 2008). External stressors such as new species introductions dominate park natural resource management concerns. Land cover for MNBP was taken from the 2001 National Landcover Dataset (NLCD) in grid format (Homer et al. 2004). Forest classes (NLCD class 41, 42 and 43) were re-classified to “forest”. All other pixels, the majority of which were agriculture and/or pasture, were classified as “non-habitat”. The resultant MNBP landscape had 34 forest patches (Fig. 4).
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Fig. 4

Manassas National Battlefield Park. Dark lines represent administrative boundaries. Shaded areas are forest patches. Light shaded areas represent the two patches from which initial conditions (IC1, IC2, IC3, and IC4) were applied for Experiment 2

The stochastic, stratified diffusion model was based on the SI cellular automaton with the following features. Similar to the first application, at any time step an infected pixel (status I) spread via local diffusion to susceptible (status S) pixels in the 4 cardinal directions. In addition, each pixel with status I had non-zero probability, PLDD = 0.1, of contributing a long-distance dispersal (LDD) event. If an LDD event occurred, the distance of the jump was calculated using a negative exponential decay (e.g., Bunn et al. 2000).
$$ P = - e^{\theta d} $$
(1)
The decay coefficient θ was determined using an approach similar to Bunn et al. (2000) and Urban and Keitt (2001); the probability P in Eq. 1 was set to 0.05 for a tail distance of 270 meters and the decay coefficient θ calculated. The tail distance of 270 m was determined using the LANDGRAPHS software package (Urban 2003) and was found to be the smallest edge-edge Euclidean distance for which all 34 patches coalesced into a single cluster of inter-connected patches. Smaller tail distance resulted in isolated patches.

If an LDD event occurred, a random direction was assigned and the distance of the jump calculated by solving for d in Eq. 1. If the location of the destination pixel for the LDD event corresponded to a pixel suitable for establishment (habitat) and had status S (susceptible), that pixel status was changed from S to I. If the origin pixel and the destination pixel were in different habitat patches, and if the destination patch was previously pristine (all pixels status S), then the destination patch was considered infected.

For MNBP, four initial conditions were tested. The four initial conditions, IC1–IC4, were chosen to demonstrate the influence of initial conditions within large patches, each being >3 km in length (Fig. 4). One hundred simulations were performed for each initial condition, and each simulation was allowed to progress until all patches were infected. For each initial condition and for each time step, the number of patches infected was averaged over the 100 iterations and 95% confidence intervals calculated. Transfer time T(t) was calculated for each initial condition as the mean time (averaged over 100 iterations) at which all patches became infected. T-tests were used to test for statistical difference between transfer times.

An assumption in experiments 1 and 2 was that the time taken to make a long-range transfer is identical to that required for a short-range, within-patch expansion to nearest neighbor pixels. This assumption is commonly used in network-based studies that explore non-uniform adjacencies (e.g., Ball et al. 1997), CA models that use long-range, probability-based dispersal kernels (e.g., Hargrove et al. 2000; Flather and Bevers 2002) and stratified-diffusion models (e.g., Shigesada et al. 1995).

Results

Experiment 1

Infection times outside of the patch in which initial establishment occurred varied from 4 to 61 time steps (Table 1). The time sequence of patch infections (Table 1) indicated patches first infected were those directly network-adjacent with topological distance of 1 (Table 2). However, in cases where an infected patch had >1 patch connected to it (degree > 1), the invasion had multiple directions it could go. Topological distance did not inform which would be the first “jump” taken. For example, patch 2 had topological distance of 1 to patches 1, 4, 7 and 9, which were the first patches to be infected for simulated release points in patch 2 (Table 1, points A, B and C). But patch 2 was also separated by a topological distance of 1 from patch 3 (Table 2). For release from point A (Fig. 1), patch 3 was the third patch to be infected. Patches 4 and 6 were infected before patch 3, but patch 6 was located 2 units of topological distance from the source patch (patch 2). Similarly, patch 3 was the third and sixth patch to be infected for release points B and C, even though it was topologically “closer” to patch 2 than patches infected before it.
Table 1

Time to infection to any patch outside source for all patches for the eight simulated release points in Experiment 1

Release point

A

B

C

D

E

F

G

H

In Patch

2

2

2

6

9

9

4

4

Time to infect

Patch 1

28

4

28

44

42

55

31

60

Patch 2

na

na

na

5

4

23

13

38

Patch 3

14

27

24

8

29

48

39

61

Patch 4

13

15

11

27

24

23

na

na

Patch 5

18

35

19

3

24

43

42

56

Patch 6

13

37

13

na

18

37

36

50

Patch 7

12

36

12

4

18

37

35

50

Patch 8

22

46

22

8

25

46

45

57

Patch 9

23

35

11

18

na

na

34

37

Patch 10

45

47

33

40

23

4

25

22

Numerical values of transfer time, T(t), indicated by bold type

Table 2

Topological distance matrix—i.e., the number of jumps required to move among patches

Patch

1

2

3

4

5

6

7

8

9

10

1

1

2

2

3

3

2

4

2

3

2

 

1

1

2

2

1

3

1

2

3

  

2

1

2

3

3

2

3

4

   

3

3

2

4

2

1

5

    

1

2

2

3

4

6

     

1

1

3

4

7

      

2

2

3

8

       

4

5

9

        

1

10

         

Matrix is symmetric so values only shown for upper triangular

Transfer time, T(t), or the time for all patches to become at least partially infected, varied from 33 to 61 time steps (Table 1). In all simulations except from release point G, the patch located the greatest topological distance from the source (Table 2) was infected prior to T(t). Thus the patch located the furthest topological distance was not necessarily and in fact frequently not the last infected.

The degree, or number of network connections for each patch, ranged from 1 to 5, with the largest being for patch 2 (Table 3). Transfer time (Table 1) for two of the release points in patch 2 (A and B) were approximately equal to values obtained for initial releases in other patches (D, E, G) with smaller degree (Table 3). Plots of the number of patches infected versus time (Fig. 5) were expected to yield a sharp increase beyond the infection of patch 2, indicating rapid transmission across the network due to the high degree of patch 2. Our results showed this to be true for initial conditions A, C and possibly D, but not for the other initial conditions.
Table 3

Degree of each patch as indicated in Fig. 2

Patch number

Degree

1

1

2

5

3

2

4

2

5

2

6

3

7

2

8

1

9

2

10

2

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Fig. 5

Number of patches infected versus computational time for Experiment 1. Labels a-h refer to initial conditions for each of the eight simulations. Circles indicate the time of infection of patch 2. The oval in 5F occurred because patches 2 and 4 were infected at the same time step

Experiment 2

For the tail distance of 270 m (used to define the decay coefficient θ in Eq. 1) the mean transfer time, T(t), was 921 and 950 for initial conditions IC1 and IC2 in the large patch in the northeast portion of MNBP, and 910 and 1022 for initial conditions IC3 and IC4 in the large patch in the southern portion of MNBP. Differences in T(t) were not significant between either IC1 and IC2 (t = 0.403, P = 0.679) or IC3 and IC4 (t = 1.704, P = 0.090), indicating that the location of initial condition within a patch did not influence transfer time. Initial conditions in different patches also had negligible influence on T(t). Mean transfer time for IC1 and IC2 (936 time steps) was not significantly different than for IC3 and IC4 (966; t = 0.605, P = 0.545).

The dynamics of invasion did differ, however, during the early phase of the invasions for all 4 initial conditions (Fig. 6). The number of patches infected versus time varied for different initial conditions within patches (IC1 vs. IC2; IC3 vs. IC4) as well as for release points in different patches (IC1 and IC2 vs. IC3 and IC4). For example, at time = 100, the mean number of patches infected (8.61, 13.86, 6.38 and 4.37) for release from initial conditions IC1 through IC4 were all significantly different (< 0.001).

Discussion

Understanding how invading organisms propagate across landscapes is key to designing effective management prescriptions for forest invasions. Mathematical models provide a tool to inform managers on the best use of treatment options. Because forests are often fragmented to some extent, these tools must be appropriate for heterogeneous landscapes.

Recent studies have advocated the use of network theory for the study of invasions across fragmented forested landscapes (Holdenreider et al. 2004; Jeger et al. 2007). Network, or graph models are computationally efficient and easy to implement (Calabrese and Fagan 2004). Sensitivity analyses can be performed to determine which network elements (e.g. forest patches or potential connections between them) are most important for overall connectivity (e.g., Urban and Keitt 2001). Often the emphasis is on preserving these elements to maintain connectivity, but from the perspective of managing biological invasions, these elements represent key control points to disrupt transmission. Network models typically treat nodes (people, computers, forest patches) as being instantaneous transmitters of information (Huang et al. 2005), in which case the connection patterns and distances between nodes drive processes across the network. Missing from these analyses is consideration of the time required for internal transmission of an organism/pathogen within nodes/patches.

Patch size and the concept of “Hubs”

To account for within-patch dynamics as an inherent feature of a network, we combined a Susceptible/Infected (SI) cellular automaton with network representation of a landscape through the use of non-uniform adjacency, which allowed us to track the time-dependent routing of the simulated pathogen among and within patches. In our study of the hypothetical landscape, we found that information on patch degree was only marginally predictive of the dynamics of spread. Patches with large degree, which have been referred to as “hubs” (e.g., Brooks 2006; Minor and Urban 2008) or “spiders” (Cantwell and Forman 1993) in landscape studies, did not necessarily accelerate the invasion. This usage of the term hub is not in strict agreement with the formal definition of hub as a node with orders of magnitude greater degree than the mean (e.g., Barabasi 2002). The relaxed definition in the landscape literature can be attributed to the fact that studies of habitat networks typically have far fewer nodes relative to networks analyzed in the classical network literature, in which 106 or more nodes is common. For example, Acosta et al. (2003) and Pascual-Hortal and Saura (2006) analyzed landscape networks with <12 nodes; Bunn et al. (2000); Ricotta et al. (2000); Rudd et al. (2002); Rothley and Rae (2005) and Neel (2008) analyzed habitat networks with <100 nodes; and Keitt et al. (1997); Urban and Keitt (2001); D’Eon et al. (2002); Brooks (2006) and Bodin and Norberg (2007) investigated networks with <1,000 nodes. Because “hubs” as defined for landscape applications do not necessarily have degree orders of magnitude greater than the mean, it is questionable whether these patches will have the properties expected of traditionally defined hubs. In fact, recent work in the sociology literature suggests that across artificial small-world networks, minor variations in node degree have no influence on transfer time (Huang et al. 2005). Our findings raise additional concerns about the applicability of this term to many landscape analyses.

In our study, the major reason the node with highest degree in experiment 1 did not accelerate the invasion was because the patch with largest degree also contained the most habitat. The large size of this patch introduced a delay mechanism as the patch itself became populated and then infected nearby patches. A management implication of this finding is that patch degree alone should not be used to target control points without accounting for within-patch dynamics, which are inherently a function of patch size and shape. At a minimum, patch size should be considered, as the goal would typically be to minimize the amount of habitat infected, not necessarily the number of patches. In our example, for initial condition D in patch 6 (Fig. 1), the simulated infection spread (in chronological sequence) to patch 5, 7, 2, and then to patches 3 and 8, etc. (see Table 1). Patch 2 is the largest in the network, and transmission to patch 2 occurs first across patches 5 and 7. Given a known establishment in patch 6, patches 5 and 7 could be targeted for control efforts to stop or slow the invasion from reaching patch 2.

Utility of topological distance measures

Control options identified by tracking simulated invasion velocity and direction require knowledge of the location of the initial establishment (initial conditions), which are typically not accounted for in network representations of landscape connectivity. More commonly, landscape connectivity studies define potential networks, which can be useful for identifying global control points under equilibrium conditions. However, biological invasions are by definition not at equilibrium with their environment (Meentemeyer et al. 2008); therefore time-dependent simulations would be more appropriate for identifying control/eradication efforts.

Graph-based epidemic models (e.g., Watts and Strogratz 1998) often assign higher pathogen spread rates (and shorter T(t)) to shorter topological distances among nodes. Topological distance measures include the mean value of shortest path distances among patches, called the characteristic path length L(p), and the largest shortest path length, called the graph diameter d(G). Small d(G), for example, has been associated with more easily traversable landscapes (Bunn et al. 2000; Urban and Keitt 2001; Minor and Urban 2008). However, network distance measures such as L(p) and d(G) are based on inter-patch distances and do not take patch attributes into account (Ferrari et al. 2007). Because the network in our first application was static, L(p) and d(G) were constant. However, transfer time varied from 33 to 61 (Table 1, Fig. 3), indicating a degree of variability in this metric for a “fixed” network that is highly dependent on initial conditions. In addition, in only one of the eight simulations was the patch topologically farthest from the source patch populated last. This finding implies that graph-theoretic measures based on topological distances, such as L(p), d(G), the Harary Index (Ricotta et al. 2000) or integral index of connectivity (Pascual-Hortal and Saura 2006) may not be useful for inferring rates of spread across landscape networks, specifically when the network is composed of forest patches of varying size and shape and the process being modeled is not at equilibrium.

Initial conditions and the stochastic model

Simulations across Manassas National Battlefield Park (MNBP) using the stochastic model further illustrated the effect of initial conditions on invasion dynamics. The lack of difference in T(t) for the different initial conditions was due to the fact that as the simulated invasions occupied greater amounts of habitat there were greater numbers of long-distance dispersal events, leading to an accelerated invasion which quickly made up for any differences early in the invasion process. This would be expected for a stratified diffusion process in which neighborhood diffusion is accompanied by dispersers that move medium to long distances from occupied portions of the landscape, similar to the “Type 2” and “Type 3” range expansion dynamics described by Shigesada et al. (1995).

The finding of significant differences in invasion rates early on in the invasion for different release points within the same patch (Fig. 6) reinforces the results from experiment 1 that the detailed location of a release point influences the amount of time it takes for an invasion to spread in a network That is, our results show that the location of initial establishment within a patch may influence the routing/trajectory of the invasion as much as the location of the patch in which initial establishment occurs. Analysis using epidemiological models has potential to identify time-dependent and spatially explicit locations for targeted control efforts that account for both patch and sub-patch variability in initial detection of an invader.
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Fig. 6

Number of patches infected versus computational time for four simulations using the stochastic model applied to Manassas National Battlefield Park (MNBP). Plots shown for the initial conditions in the northeastern patch in MNBP (IC1 and IC2) and for the initial conditions in the southern patch (IC3 and IC4). Dashed lines represent 95% confidence intervals from the 100 replicate runs

Direction for future work

Our work made use of simple binary landscapes, which is ideal for exploring the behavior of underlying models. We have shown important features of commonly used epidemic/diffusion models using network theory as well as demonstrated some limitations of network theory for the analysis of biotic invasions. However, for applications of these types of models to real invasions we feel that several issues would need to be addressed. First, identifying “habitat” for invading organisms is typically more complex than simply locating “forest” (or other habitat type) pixels in a land cover map. Multiple environmental factors (soil attributes, climate, vegetation type/structure/age class, etc.), stochastic factors and Allee effects can play important roles in the successful (or not so successful) spread of biotic invasions (Mack et al. 2000). Often the accelerated range expansion of biological invasions is attributed to long-distance dispersal events, which are addressed by stratified diffusion models. However, stratified diffusion models typically assume dispersal kernels are constant in space and time, which is more than likely not realistic (Urban et al. 2008). Certain organisms have been observed to evolve in their new locations, possibly altering their reproductive and dispersal characteristics relative to studies of populations where they are considered native (Mack et al. 2000; Hastings et al. 2005; Urban et al. 2008). Without accounting for variations in habitat quality, climate, etc., models may over-predict range expansion through time (Meentemeyer et al. 2008).

Cellular automaton models are well-suited for integration of this type of variability. For example, while not applied to biological invasions, the EMBYR fire spread model (Hargrove et al. 2000) is essentially a form of SI stratified diffusion model applied to gridded land cover maps. The model accounts for fuel class (vegetation type), fuel moisture content and wind speed to determine the direction and the extent of fire spread. For biological invasions, similar process variables would need to be defined for the organism and the landscape of interest. Prior to integration of these factors, however, it is necessary to fully understand the underlying mechanisms of the diffusion and network models, which have inherent features independent of any specific application. Combining diffusion modeling and network theory, as demonstrated here, can simplify analysis and provide insight that is not otherwise available. For example, results of diffusion models are often presented as the amount of habitat invaded as a function of time, but do not identify temporal routing trajectories or account for the number of nodes/patches infected, features that can be incorporated in network models.

In addition, our results are most definitely specific to model parameters and the scales of the landscapes and dispersal distances analyzed. However, the results can provide insight towards invasion processes in general. In both of our experiments, the maximum dispersal distance was smaller than the length scale of the largest patches. This type of situation would be most similar to invasive pathogens such as sudden oak death (Condeso and Meentemeyer 2007) or other organisms which have limited dispersal ability relative to the size of the forested tracts they invade. As the length scales of dispersal approach and exceed the length scales of the patches within a given landscape, it is expected that within-patch variation would exert diminishing influence.

Conclusions

When biological invasions or epidemics saturate a population (of habitat patches or people), the size of the invasion/epidemic often makes treatment or control options at best costly and at worst impractical. Empirical determination of time dependent range expansion (of biological invasions) is considered a research priority (Mack et al. 2000), and we feel that development of theoretical models should parallel this effort. This type of parallel development, and control strategies designed to prevent saturation, is under development in the epidemiology literature (e.g. Eubank et al 2004; Toraczkai and Guclu 2007). Our work is the first demonstration of this strategy that we are aware of, to biological invasions across heterogeneous landscapes, using a novel coupling of a Susceptible/Infected cellular automaton with a network representation of fragmented landscapes. Although our study was general in nature, it demonstrated that landscape control efforts are dependent on the initial conditions as well as patch or node attributes that are typically not accounted for in network epidemic models (Huang et al. 2005). We showed that traditional network metrics related to degree or topological distances are not as informative as could be hoped for in terms of predicting the direction and speed of invasion processes across heterogeneous landscapes. Future work involving analysis of biological invasions from a network perspective, as advocated in the literature (e.g. Holdenreider et al 2004; Jeger et al 2007), should account for the effect of patch size and shape on population expansion, and not just network attributes such as inter-patch distance and patch degree.

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© Springer Science+Business Media B.V. 2008