Modeling the Effects of Spatial Distribution on Dynamics of an Invading Melaleuca quinquenervia (Cav.) Blake Population

Abstract

To predict the potential success of an invading non-native species, it is important to understand its dynamics and interactions with native species in the early stages of its invasion. In spatially implicit models, mathematical stability criteria are commonly used to predict whether an invading population grows in number in an early time period. But spatial context is important for real invasions as an invading population may first occur as a small number of individuals scatter spatially. The invasion dynamics are therefore not describable in terms of population level state variables. A better approach is spatially explicit individual-based modeling (IBM). We use an established spatially explicit IBM to predict the invasion of the non-native tree, Melaleuca quinquenervia (Cav.) Blake, to a native community in southern Florida. We show that the initial spatial distribution, both the spatial density of individuals and the area they cover, affects its success in growing numerically and spreading. The formation of a cluster of a sufficient number and density of individuals may be needed for the invader to locally outcompete the native species and become established. Different initial densities, identical in number and density but differing in random positions of individuals, can produce very different trajectories of the invading population through time, even affecting invasion success and failure.

1. Introduction

The study of invasive species has been one of the major themes in ecology for the practical reason that invasive species pose a threat to native vegetation and to agricultural systems around the world. In attempting to estimate the risk of a non-native species invading a new environment, several factors need to be considered. One factor is how the climatic conditions of a non-native species compare with those of a potential new environment. Species distribution models, also called ecological niche models, are often used to estimate the suitability of the new environment (e.g., Elith and Leathwick 2009 [1]). However, predicting invasive success goes beyond determining suitable climate conditions and should take into consideration other traits of the potential invasive species as well as of the native community to which it has been introduced. These traits include its maximum growth rate, potential dispersal rate, competitive ability, and vulnerability to natural enemies.

Analytic models have been used to take into account some of these factors. Invasion criteria based on mathematical approaches are fundamental tools in quantitative ecology for predicting the dynamics of a species population entering an equilibrium community. Suppose the competitive interactions of populations of two species are described by the equations for biomasses of native species 1, N₁, and invasive species 2, N₂, where space can be considered one-dimensional and Lotka–Volterra competition occurs. The equations can be written as

$${\displaystyle \frac{d{N}_1}{dt}}=r_1\left(1-{c_{11}N}_1-c_{12}{N}_2\right)N_1+D_1{\displaystyle \frac{{\partial }^2{N}_1}{\partial x^2}}$$

(1a)

$${\displaystyle \frac{d{N}_2}{dt}}=r_2\left(1-{c_{21}N}_1-c_{22}{N}_2\right)N_2+D_2{\displaystyle \frac{{\partial }^2{N}_2}{\partial x^2}}$$

(1b)

where x is the spatial distance, r₁ and r₂ are the maximum growth rates, c₁₁ and c₂₂ are the intraspecific competition coefficients, c₁₂ and c₂₁ are the interspecific competition coefficients, and D₁ and D₂ are the dispersal rates. If the equilibrium $E(N_1^{*},0)$ exists where $N_1^{*}$ is the equilibrium size of species 1 and the invader biomass N₂ is initially very small, the criterion for the invasion of species 2 can be found by letting N₂ have a small value, which is represented by n₂:

$${\displaystyle \frac{d{n}_2}{dt}}=r_2\left(1-c_{21}{N}_1^{*}\right)n_2+D_2{\displaystyle \frac{{\partial }^2{n}_2}{\partial x^2}}$$

(2)

Suppose the small biomass of invading species 2 is spread uniformly over a small spatial interval of length L. Then, analytically, it can be shown that the population of species 2 can grow if

$$r_2\left(1-c_{21}{N}_1^{*}\right) \ge {\pi }^2{D}_2/L^2,$$

(3)

based on Kierstead and Slobodkin’s study (1953) [2]. More sophisticated analytic models can include such aspects as spatially heterogeneous environments [3], integro-difference equations [4,5], and the stochasticity of dispersal [6].

These analytic results are interesting as broad indicators of invasion potential. But the parameters here are only gross population-level demographic characteristics of the species and do not describe how invading individuals interact with those of the native species. Simple analytic models also do not consider detailed site-specific aspects of the new environment such as spatial heterogeneity. For these reasons, individual-based models (IBMs), also called agent-based models (ABMs), have been employed to examine the effects of factors at the individual organism level. These allow models to make use of empirical data, such as those on giant hogweed (Heracleum mantegazzianum Sommier & Levier) [7], Russian knapweed (Acroptilon repens L.) [8], reed canary grass (Phalaris arundinacea L)) [9], paperbark tea tree (Melaleuca quinquenervia (Cav.) Blake) [10,11], and monocarpic thistle (Carduus nutans L.) [5]. Beyond their application to specific species, IBMs also allow the investigation of general types of mechanisms that might apply to a large number of potential invasive species, e.g., [12,13,14,15].

Among the general mechanisms that are known to be important in plant competition are the effects of ecological engineering, that is, one or more species having an effect on the environment that affects other species. One such component of ecological engineering is the litter produced by plants, which can affect competitive relationships, e.g., [9,16]. Leaf litter, depending on its nutrient contents and decomposition rate, can influence soil nutrient dynamics [17]. The effects of invasive Phalaris arundinacea on nutrient cycling in North American wetlands was modeled by [9]. Also, especially if litter accumulation can become great due to slow decomposition rates, litter can have a negative effect on seedling survival. This can be important in competition if the accumulation of litter of one species affects the survival of seedlings of the other species. The consequences of the litter suppression of seedlings were modeled by [17] for the invasion of stiltgrass (Microstegium vimineum (Trin.) A. Camus) in eastern and midwestern deciduous forests in the United States. High litter accumulation depends not only on litter decomposition rates but also on the number and local spatial densities of individuals of the species producing the litter. Keeping track of local densities requires a spatially explicit IBM, which can keep track of the spatial locations of all individuals. This is important as the invasion of a new species of an existing plant community can occur in the form of various initial spatial distributions of propagules.

These facts are relevant to our modeling of the invasion of southern Florida ecosystems by the paperbark tea tree (Melaleuca quiquenervia) [10,11] (Melaleuca hereafter). Our previous modeling has shown that the success of a simulated non-native Melaleuca invading a native community can depend not only its relative competitive abilities but also on the way the initial invaders are scattered in the native community, as well as their dispersal rates [10,11]. However, our earlier work did not systematically investigate the impact of different initial spatial starting points. Our objective here is to study the effect of the initial spatial configuration of the invading population systematically to obtain a better understanding of the factors contributing to invasion. This approach allows us to explore how variations in the initial spatial distribution influence invasion dynamics, providing a more detailed and realistic prediction of invasion outcomes.

We hypothesize that the initial spatial configuration of the invader is an important factor in determining its ability to spread, along with the above mathematical criteria, which can be tested with modeling. Our experience also shows that random differences in the initial spatial configuration of individuals with the same mean properties can lead to very different population trajectories. Thus, we also hypothesize that the outcomes for the initial conditions with the same average properties can vary dramatically.

To study both the hypothesis that different spatial configurations of the invading individuals lead to different invasion successes and the finding that random initial differences can also lead to different outcomes, we used a model called ManHam. ManHam was developed and tested for describing the invasion of the non-native tree species Melaleuca quinquenervia in a native community of southern Florida [10,11]. The ManHam model was developed to address the complex challenge of invasive species management and utilizes agent-based modeling to explore effective biocontrol strategies. Our approach was informed by insights into the dynamics of the invasive species, from Australia, Melaleuca, which is characterized by fast growth and high seed production in the environment of southern Florida due to its virtual lack of natural enemies in its new habitat [16,18]. Another advantage of Melaleuca is that its leaf litter has the capacity to suppress the emergence of native seedlings, which increases its competitive ability [19,20]. Melaleuca initially spread rapidly through natural communities in southern Florida. For the past two decades, biological control through a number of mostly arthropod agents has been shown to be able to control Melaleuca [21,22]. However, it is still important to model Melaleuca in southern Florida’s native communities to help predict what the long-term consequences of the control of the invader and the recovery of native communities will be.

2. Materials and Methods

Our modeling here uses a version of ManHam that is specially designed to study early invasion dynamics. In particular, a level of biocontrol on modeled Melaleuca growth rate and seedling production is imposed that makes the modeled native tree community roughly equal in competitive ability to the invading Melaleuca. The native community is modeled as a single ‘species’ to roughly represent a native hardwood hammock. Although these are somewhat artificial conditions, as biocontrol appears in reality to give an advantage to the native community, it is still useful to make this assumption to study how different initial spatial configurations of the invader affect its success. In our study, we first examine various initial spatial distributions of a relatively small number of invading pre-adult Melaleuca and examine how these distributions affect its early invasive dynamics. Secondly, we follow up on the results showing a high degree of variability in the simulation outcomes by determining whether this variability is simply due to the small initial number of invaders. We increase the number of invaders and examine whether this leads to more deterministic results.

In other words, the model simulates individual trees through their lifetimes from seedlings. Both age and growth in dbh are modeled. The model simulates competition between the invasive Melaleuca and native generic hardwood hammock species. Local competition occurs with both conspecific individuals and individuals of the other species. The litterfall and accumulation of the invader and its suppression of native seedlings are modeled.

2.1. Model

A spatially explicit, agent-based model (ABM), ManHam, is used, as in [10,11]. The purpose of the model is to test different initial spatial configurations of immature individuals of the invasive Melaleuca such that the same number of individuals at different spatial configurations leads to different probabilities of success in the early phase of an invasion. The model ManHam is described in detail in [10]. Here, we just mention the important aspects of each submodel.

The model is a Monte Carlo simulator of the competition between an invasive and a native tree species on a plot of 120 × 120 m. There are two types of entities: First, there are individual trees, which are termed agents, each of which is either an invader or a native tree. Individual trees have age, diameter at breast height (dbh) and spatial location. Canopy size is allometrically related to dbh. The individual trees are distributed in continuous space; that is, they can be located at any point within the 120 × 120 m plot. The second entities are the heterogenous litter accumulations across the plot, which are kept track of at the spatial resolution of 1 × 1 m.

2.1.1. Tree Growth

Individual trees grow from seedlings to senescence. At the start of the simulation, the trees that are initiated are assigned an initial tree diameter at breast height, dbh, as well as a location in continuous space. The code iterates over all trees and computes their growth in yearly increments in a way that is similar to earlier models developed for describing woody plant growth in agent-based simulations [23]. The growth and reproduction of the Melaleuca in the absence of biocontrol are both higher than the rates of the model native tree species. The growth of the invader is assumed affected by biocontrol. The growth of individual invader trees is sufficiently slowed by biocontrol by 80% to be one-third of the rate of growth of native trees. Slower growth over a period of time increases the likelihood of mortality.

2.1.2. Reproduction

The possibility of reproduction is assumed to occur for each mature tree each year and, in the absence of biocontrol, is greater for the invader than for the native species. The age of maturity is set arbitrarily at 20 years. The code specifies that each tree can produce an upper limit of N_{seedling,limit} viable seedlings each year, where each seedling has a probability, brate, to survive. Therefore, each mature tree can produce up to N_{seedling,limit} seedlings each year. A low value for N_{seedling,limit} is used here because of the low empirical probability of the survivorship of seedlings. The potential number of seedlings produced each year by invader trees under biocontrol is assumed to be half of that of native trees.

2.1.3. Seedling Dispersal

The distance from the parent from which an individual seedling is dispersed,

$$disper=-\mathrm{l}\mathrm{n}\left(rand\right)/c_3$$

(4)

is exponentially distributed, where rand is a number chosen randomly and uniformly on the interval (0, 1), and 1/c₃ is the mean dispersal distance. The dispersal can be in any direction (360°) with some seedlings landing outside the plot, where they do not survive. In the version of ManHam used here, the value of the mean dispersal distance, c₃, is the same for both species. This parameter, though important, is not varied here as our main interest is the initial spatial configurations of the invading individuals.

2.1.4. Competition between Trees

Competition is described by the field of neighborhood (FON) model [24], which is used to describe the effect of close neighbors on the growth of each other. All neighbors within 5 m of each tree are determined at each point in time and put into a matrix called Neib_i so that their effects on annual diameter growth through FON interactions can be determined. This is carried out using the data on neighbors within 5 m of each tree that are stored in matrix Neib.

2.1.5. Litterfall and Its Accumulation

The leaf litterfall and accumulation of both the invader and native trees are simulated but only that of the invader is used in the simulations. Melaleuca leaf litter decomposes slowly and so that it can accumulate to depths such that it suppresses the emergence of native seedlings, though with less effect on Melaleuca’s own seedlings. Cumulative depth is calculated through time. The potential suppression of native seedlings by litter of the invader, Melaleuca, is based on the amount of leaf litter LF_accumulation, in the given 1 × 1 m area in which the seedling has landed. The probability of the seedling surviving is reduced according to the following:

$$Seedling survival suppression=e^{-L{F}_{accumulation}*s_{litter}},$$

(5)

where $s_{litter}$ is a parameter that is constant and that measures the effect of the litter of the invader on the seedlings of the native species. The value of $s_{litter}$ is not precisely known, and a plausible value is used here, which, in combination with biocontrol on the invader, leads to the two species being competitively similar. The parameters are estimated for Melaleuca based on [10].

2.1.6. Tree Mortality

There are three components of the net tree mortality. First, there is background mortality that is size-independent. Second, there is size-dependent morality that decreases with size, and third, there is density-dependent mortality. The last component is implemented by increasing the probability of mortality of a tree whose growth rate has decreased over time due to crowding. Parameters for the first two mortality sources are the same for the two species. However, based on the empirical data, it is assumed that the invader can tolerate greater crowding so that crowding-related mortality affects the native species within two meters of the stem, but it affects the invader only within one and one-half meters. Finally, all trees are assumed to have a maximum age. Details can be found in [10]. Biocontrol is assumed to affect only the invader. Insect herbivores were not simulated explicitly; a constant effect level of herbivory on each invasive tree was assumed.

2.2. Parameter Values

Most of the parameter values used are the same as those used for growth, reproduction, mortality, and the effects of biocontrol on growth and reproduction in [10]. Here, only the specific values noted in the model description above are given (see

Table 1. Parameter values for reproduction, seedling dispersal, and litter suppression of seedlings.

Parameter	Definition	Value	Units
brateinvader	Parameter for seedling establishment	0.07	dim’less
bratenative	“	0.75	“
Nseedling,limit	Upper limit on number of viable seedlings from a single tree	2	Number
c3,invader	Parameter related to dispersal	0.05	m−1
c3,native	“	0.05	m−1
slitter,invasive on invader	Parameter related to seedling suppression of invader	1	1/(kg m−2)
slitter,invasive on native	Parameter related to seedling suppression of native	10	1/(kg m−2)

). In [10], the parameters of seedling dispersal, $c_3$, and litter suppression $s_{litter}$ were varied for different simulations in that paper. Here, the specific values of these parameters were prescribed as those values were set such that the invasive and native trees had roughly similar competitive abilities so that the effects of spatial distribution of the invading Melaleuca could be examined.

The parameter values are in the ranges used by [10,11]. The non-native species was assumed to have high dispersal rates (c₃ = 0.15), and its litter had a negative suppressive effect on native seedlings (s_litter = 10). The advantages of the non-native species were balanced by the assumption that biocontrol reduced non-native tree growth by about two-thirds and reduced non-native seedling survival by 90%.

2.3. Simulations

Two sets of simulations are performed. In the first set of simulations, a relatively small number of invaders is established along one edge of the plot. The results of these simulations motivated a second set of simulations, in which the initial number of invaders was greatly increased.

In the first set of simulations, each simulation is started with 3500 native trees assumingly randomly distributed in the 120 × 120 m plot, with a uniform random distribution of ages between 0 and 100. A typical initial distribution is shown in Figure 1. There are an initial 200 invading Melaleuca saplings between 0 and 20 years. Because the age of maturity is 20, all invading individuals are initially immature.

The invading Melaleuca saplings are initiated according to six different cases. The trees are randomly located in each case between 0 and 120 along the y-axis but between (case 1) 0 and 5 m, (case 2) 0 and 10 m, (case 3) 0 and 15, (case 4) 0 and 20 m, (case 5) 0 and 30 m, and (case 6) 0 and 40 m along the x-axis for the respective cases (Figure 2). That is, the invading species is assumed to be invading along the front from the left-hand side of the plot.

In the second set of simulations, the invading species is assumed to initially cover the whole left half of the plot. They are assumed to have been artificially kept from the possibility of invading the right-hand side of the plot until the start of the simulation. All simulations were performed using Matlab R(2022)a, (MathWorks).

3. Results

3.1. Small Initial Invasions

The resident native trees have typical distributions, as shown in Figure 1. The invading Melaleuca trees have six initial distributions similar to those shown in Figure 2. For each of the six initial configurations, or cases, seven simulations were performed with different random number initiators so that the specific spatial locations and pre-mature ages differed. The results of Monte Carlo simulations are shown in Figure 3. Within each case, with each simulation performed with different random number initiators, the trajectories of the invading species differ. In fact, in five of the six cases, at least one extinction occurred. But there are clear differences between the six cases. In case 3 (0 to 15 m), the trajectories are all upward and tightly packed. In case 5 (0 to 30 m) and case 6 (0 to 40 m), the trajectories are more scattered, and four simulations lead to extinction in case 6. Cases 1, 2, and 4 are similar in that each has one extinction, and the trajectories show only a moderate degree of cohesion.

By examining some of the spatial patterns of individual invaders along these trajectories, one can obtain an idea of how the spatial dynamics can lead to successful invasions. Two examples are shown, one trajectory from case 1, in which the invaders were all initially within 5 m of the left edge of the plot, and one trajectory from case 6, in which all invaders were distributed within 40 m of the left edge of the plot. Ten snapshots are shown for each case, from year 10 to year 1600.

In case 1, although the individuals are first configured along a narrow line, by year 600, they start to rearrange into a tight semi-circular cluster, which then advances towards the center of the plot (Figure 4). In case 6, the initially diffused distribution of 200 individuals over 40 × 120 = 4400 m² begins to form two round clusters. These join together to form a single cluster by year 1400, which by then is rapidly advancing (Figure 5).

Each of the six panels of Figure 3 contains seven trajectories that have the same parameter values but different random number initiators. In only one of the cases (case 3 for individuals initially between 0 and 15 m) are all trajectories relatively similar, all showing clear invasion success. In the other five cases, one or more failures to invade occurred and the trajectories of the successful invasions were generally scattered. These results show that random differences in the initial configuration of individuals can lead to divergent results for the early stages of an invasive process but that a particular initial condition, case 3, leads to fairly coherent set of trajectories for different starting conditions. These results motivated the second set of simulations below.

3.2. Invader Has Equal Initial Numbers to Those of the Native Population

The high degree of randomness in trajectories within each of the six cases shown above in Figure 3, which results only from the random rearrangements of individuals within the given initial areas of the invaders, is interesting as it suggests a degree of uncertainty in making predictions. However, one might argue that the small size (200 individuals) of the initial number of invaders made possible substantial differences in the initial configurations. Therefore, we repeated these invasion experiments starting with much larger numbers of invaders. Typical initial conditions are shown in Figure 6, in which 2000 individuals each of the invader and native populations initially occupy opposite sides of the plot and where the invading species is prevented from invading the right-hand side of the plot until the start of the simulation. Differences in circle size, representing different ages, are not considered in Figure 6.

As in the above simulations for each of the six cases in Figure 3, the simulations based on the situation in Figure 6 all employ the same parameter values but start with different random number initiators for setting initial spatial locations and ages. The simulations were carried out for different periods of time, with only a few being carried out long enough to reach extinctions (Figure 7). The intention was not to show which species eventually won but to show the vast variability in trajectories in the early stages of competition. The simulations show the extreme variability of trajectories in which only the initial spatial configurations of the individuals of the two species varied.

4. Discussion

A basic question regarding the introduction of non-native species to an environment is what the risk is that they will become invasive. Many factors have been considered in mathematical and simulation models of invasive species. Of particular importance in assessing the ultimate success of an invader is whether or not the population is able to grow in population size and spread from initially low numbers.

4.1. Invasion Success as Function of Initial Spatial Distribution of Invading Individuals

Based on the results of the simulations shown in Figure 3, an initial conclusion can be drawn that the initial configuration of individuals of an invasive tree species can influence its early success in growing and spreading. In particular, for a fixed number of immature individuals, the combination of population density and area occupied by the invading species influences the reliability of a successful invasion. For the set of starting conditions compared in Figure 3, case 3 (0 to 15 m) appears to be the best combination as a fairly tight bundle of increasing trajectories is formed. Case 6 (0 to 40) had the most (four) unsuccessful invasion attempts.

Some indication of the explanation of what leads to a successful invasion can be inferred from the examples of the case 1 simulation in Figure 4 and the case 6 simulation in Figure 5. In both cases, after some time, the invader manages to form a large dense cluster of individuals that can then grow in size at a substantial rate. The reason for this appears to be a version of the Allee effect. The Allee effect was already reflected in the classical mathematical result of [2] in that the initial size of the area occupied by the invader, L, must be large enough for inequality (3) to hold for the population to grow. In our case, the number of invading individuals is fixed, so the Allee effect depends on the initial density of individuals. Because an important mechanism for the success of invading Melaleuca is that it can produce litter that suppresses native seedlings, it is important that the invader exists in a high enough density that it can produce a sufficient litter layer across the space. The invader must also occupy enough space that it can gain a foothold based on its ability to produce seedlings suppressing litter. Of course, the area that is occupied by individuals and the density of individuals are tradeoffs when the number of individuals is fixed.

It appears that there is a particular tradeoff that is most effective. Case 1 (0 to 5 m) and case 2 (0 to 10 m) are generally successful but are less reliably successful than case 3 as they are slower to increase land coverage. At the opposite extreme, in case 5 (0 to 30 m) and case 6 (0 and 40 m), the initial distribution of invading Melaleuca was dilute. For those cases, it was more difficult for the Melaleuca to develop high litter densities. Nevertheless, in some simulations, even in case 6, Melaleuca was able to be successful in the early invasive stage (Figure 5). In these cases, although it took some time, eventually, the initially dilute density of the invaders was by chance able to form local concentrations that were sufficiently dense, occupying enough area to eventually start to grow.

4.2. Sensitivity of Invading Population Trajectories and Success to Initial Spatial Distribution

The second general feature of the simulations is that even within each of the six cases of the initial conditions shown in Figure 2, there is a large amount of variation in the trajectories. Because the initial number of invaders (200) is relatively small, to reduce the possible effect of small initial numbers on sensitivity to the initial conditions, simulations were performed with the starting conditions shown in Figure 6, that is, with 2000 individuals each of the invader and native species populations occupying opposite halves of the plot. The random number initiator was changed for each of the 10 different simulations. The trajectories show remarkable sensitivity to the initial conditions. What is most striking is the variability in the direction of population change within some of the simulations, in which the apparent dominance of one species is abruptly reversed.

It is tempting to compare the sensitivity to the initial condition exhibited here with the well-known sensitivity to the initial conditions in some deterministic mathematical models called ‘deterministic chaos’ (e.g., [25]). A difference, of course, is that the model here uses Monte Carlo simulations. However, the numbers in the Figure 7 simulations are large, so one might expect greater similarity in the trajectories. It is clear that a great deal remains to be learned about what drives these dynamics. Collective dynamics over spatial scales occurs within the populations. By this we mean that a locally dense cluster of invading individuals, which may occur by chance, can be self-reinforcing because they can collectively produce enough litter to exclude native seedlings. Thus, there is a small chance that the cluster can grow and that we can determine the outcome of the competitive interaction. Another factor that may help explain some of the sudden changes in the direction of population trajectories in Figure 7 is the upper limit on the lifetime of trees. Rapid die-offs of an age cohort of individuals in a large part of the landscape can lead to a change in the relative population growth rates of species populations. As demonstrated, random differences in the initial spatial configurations can produce varied results, highlighting the importance of considering spatial dynamics in invasion biology.

The third general feature of the simulations is that the results involved with some novel edge effects. The ecological dynamics at the edge of species’ range limits are complex and involve multiple interacting factors, including dispersal, competition, and environmental variability. Turchin and Ellner [26] demonstrated that species at the edge of their range often exhibit different ecological behaviors compared to those in the core areas. For instance, increased dispersal ability and higher reproductive rates are often observed at range edges, which can facilitate range expansion despite suboptimal conditions [27,28]. These adaptive strategies highlight the importance of spatial dynamics and local adaptation in shaping species distributions. Edges can act as barriers or filters, affecting the dispersal of organisms and their propagules, which in turn can influence the spatial distribution of populations and community composition. For example, edges can impede the movement of pollinators, thereby reducing gene flow among plant populations and altering reproductive success and plant diversity [29].

4.3. Novelty of Results

The results here are new. Although, as discussed in the Introduction, IBMs have been used in studying invasions of specific plant species, we know of no study, systematic or otherwise, that focuses on the effects of the initial spatial distribution of invaders. It is well known that the success of invaders in highly variable. There may be many introductions of a novel species to an environment before the species becomes invasive. A consideration of not just the number but also the initial spatial configuration of individuals of the species may help explain why some invasions are successful and others are not. Also, our results, showing the high degree of variability in population trajectories, even when introduced in large numbers, are new. We showed a high degree of sensitivity to spatial configurations of 2000 invading species, which resembles the ‘chaotic’ behavior of some mathematical models but has a quite different origin.

4.4. Limits of Model

Our model simulates a specific invasive tree species, Melaleuca quinquenervia, in the specific environment of southern Florida. However, we believe that the nature of the interaction of the invasive and native species in our model has features that are general. Our spatially explicit IBM follows the general format of other models of forest dynamics, e.g., [24]. Also, the inhibitory effect of the invasive species, Melaleuca, on the native species through mechanisms, such as the litter suppression of native seedlings, is not uncommon. In our simulations, this allowed the invader to often be successful even though relatively strong biocontrol effects on growth and reproduction were imposed. Mechanisms of inhibition, such as through characteristics of litter or allelopathy, are common. Also, invading plant species affect the local soil microbial community, which can favor the invader [30]. Therefore, we believe that our model and results have implications for other systems.

The cases modeled here are an extremely limited slice of the vast range of initial spatial conditions that could be studied. We looked at only a few cases of invasions from one side of a plot carried out by a certain number of invaders. The parametrization was chosen such that, with a level of biocontrol imposed on the invader, the two species had very similar competitive fitness, although they differed in types of advantages. Further studies will be needed to provide stronger conclusions concerning how the initial spatial configuration influences invasive success.

5. Conclusions

We used a tested spatially explicit IBM of the competition of Meleleuca quinquenervia, invasive to southern Florida, with a generic native species. The model was used to explore the effects of different spatial distributions of the invasive population. The results showed that the initial density of individuals of the invader had an effect on its success. This was due to there likely being a critical threshold density for the sufficient accumulation of the invader’s litter to have a strong suppressive effect on the native species. The simulations also showed extreme sensitivity of the competitor population trajectories to the initial spatial conditions, which resembled the ‘chaotic’ behavior seen in some mathematical models. Future work will entail studying a greater variety of initial distributions and understanding the causal mechanisms that underlie the results.