Simulation model

Our study is based upon a stochastic individual-based model (IBM) representing a notional community in which two sessile species with overlapping generations occupy a uniform environment. The code can be accessed as a git repository hosted at https://github.com/jorgevc/IBM-ecology-simulator.git. Individuals occur at sites determined by a two-dimensional grid (x, y co-ordinates). When reproducing it is assumed that individuals are limited in their ability to disperse offspring. Similarly, competition among individuals for resources only takes place within a fixed radius. The spatial patterns that arise are therefore a direct consequence of the dispersal of individuals and their interactions rather than any external driver. The parameters of the model are defined in Table 1. To develop the system we used standard procedures employed in simulations of statistical mechanics [21].

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Table 1. Parameter definitions from the stochastic individual-based model used to develop the simulations.

https://doi.org/10.1371/journal.pone.0114979.t001

The arena was a two-dimensional square lattice of 150 units, giving a total 22,500 sites, with periodic boundaries forming a torus to prevent inward propagation of edge effects. Birth and death events took place in continuous time. The starting density of each species in the lattice was (the final output in the stationary state is not affected by this value). A relatively high starting density was chosen since invasibility was not used as the coexistence criterion within this study; an assessment of the implications of this distinction is reserved for the Discussion.

Initial individuals were distributed according to an homogeneous Poisson process (complete spatial randomness). In order to compensate for stochastic fluctuations due to the finite size of the system we present results based on ensemble averages. Each ensemble is composed of 20 realisations. The averages therefore have fluctuations equivalent to those of a single simulation performed on a lattice of 22,500×20 = 450,000 sites. The use of discrete space in our simulations is an optimisation to increase computational speed; results would not differ for ‘continuous’ space, which in modelling terms effectively means a lattice of higher resolution.

The frequency with which an individual belonging to species i makes an attempt to produce an offspring is given by the birth rate . The site occupied by the new offspring is chosen with equal probability among all sites that are closer or equal to the distance from its parent. If the randomly chosen site is already occupied then production of an offspring is prevented. Death of individuals can occur due to both intrinsic and extrinsic causes and therefore requires more than one parameter. The intrinsic death rate of species i, that which would occur in the absence of any competition, is given by . Inter-specific competition among individuals is defined as which is the rate at which an individual of species j acquires resources, potentially resulting in the death of an individual of species i. It is therefore an active process, expressing the ability of an individual to deplete resources and thereby kill (at least indirectly) its neighbours. This rate is uniform in space but with a maximum radius given by . This means that an individual of species j selects with equal probability a site within a distance with a rate . If the chosen site is occupied by an individual of species i, that individual dies. The intra-specific competition parameter is a special case equivalent to . It is the rate at which an individual of species i acquires resources, potentially resulting in the death of another individual of species i, uniform in space with a maximum radius . This treatment of interference competition as an active process differs from that more commonly used whereby a focal individual's likelihood of mortality is a function of the number of neighbours, in other words a passive process. The two treatments are mathematically equivalent and can be related by a similarity transformation. For a detailed discussion of the choice of active competition in this study and its implications see S1 Note.

An individual was chosen at random and the probability of a birth or death event calculated according to the corresponding rates. A computing time step was counted each time that on average all individuals of the system had been updated, and a generation was defined as being when on average all individuals had attempted to reproduce once. The simulation was automatically stopped when the change in total community density () fell below in 20 time steps. This arbitrary criterion was established during preliminary investigations to be a robust indicator of long-term outcomes, ensuring that we were close to the stationary state while reducing the risk of extinctions due to finite size fluctuations [22]. See S1 Fig. for an example of population density changes over time.

We obtained phase diagrams of the coexistence region for systems in which interactions occurred either over long or short distances, or over relatively longer distances for either the common or rare species. This was accomplished by running multiple simulations beginning with values of , and then with independent increments of 0.01 in each until , . All other parameters remained fixed (, , , ). This represents a total of 28,080 simulations for each composite phase diagram. Parameter values were chosen based on prior investigation which had determined that they encompassed the full range of potential outcomes within this system. We found the stationary densities of both species for each set of values and note where they do not fall to zero for either species, i.e. both species are maintained indefinitely. The phase diagrams obtained by keeping {c_2,1, d₂} fixed and altering other parameters have similar properties to those presented here, which are therefore used to illustrate a more general set of principles. Invasion analyses were conducted for a subset of parameter combinations and in all cases confirmed the outcome.

To convey the underlying spatial pattern of individuals we present the pair correlation function , a robust descriptor of spatial pattern structure [23], [24]. It is obtained from the first derivative of Ripley's function [25], which gives the expected number of points within a distance summed across all points in the pattern and divided by its average density . It can be estimated as

(1)

where r is the distance from each point i, is 1 for each j within r of i and otherwise 0, and n is the total number of points. This provides a cumulative function which can be converted to the pair correlation function . In ecological terms it describes the ‘plant's-eye perspective’ (sensu [5]) of neighbourhood density at increasing distance r. If densities are independent at a given distance, . When , pairs of individuals are more abundant than the spatial average, while indicates that they are less abundant.

The Mean-Field Approximation

We begin by defining when two species are expected to coexist if all individuals experience average conditions. These null expectations were obtained from a spatially-explicit extension of the classic Lotka-Volterra competition equations, referred to hereafter as the Mean Field Approximation (MFA), and using the same parameters as in Table 1. The dynamics of the mean density of species i can be described as:(2)

In this equation represents the ratio between the observed density of competitors around a given individual and the mean density of the whole system. When , individuals experience a greater density of competitors than would be expected based on average conditions, while the opposite is true for . Hence captures the difference from the mean density of competitors of species experienced by an individual of species when it attempts to reproduce. Likewise is the difference from the mean density of species experienced by species when it obtains resources. This is a useful construct as it quantifies how competition rates change as the result of spatial structure. It is related to the widely-used Ripley's function (equation 1; [25]) since for a given radius it is equivalent to the value of divided by the area of integration. Note that in our study is a state variable and not a parameter, as in some previous treatments. A generalised derivation is provided in S2 Note.

When the spatial structure of the system is not taken into account, the competition experienced by any single individual is the mean competition exerted by all others, irrespective of distance. It is equivalent to the behaviour of a community with homogeneous density equal to the mean of the entire system (). Equation (2) for a community of two species reduces to the well-known Lotka-Volterra model:

(3)

A stability analysis on the stationary solutions of (3) reveals that the following condition is necessary for coexistence to occur

(4)

This is a generalisation of the commonly-stated condition that intra-specific competition must exceed inter-specific [1] but taking into account the reduction of the birth rates due to overall density. When the left hand side of (4) is equal to the right the species are ecologically equivalent and the dynamics will be driven by ecological drift [26].

Based upon these equations, the region within which coexistence occurs for an arbitrary pair of parameters is illustrated in Fig. 1. Similar figures can be generated through the choice of any pair of parameters from Table 1 which include either of the inter-specific competition parameters (c_2,1 or c_1,2) and one with an independent direct effect on the population density of a single species (e.g. its rate of births b or death d). By varying two parameters, while keeping all others fixed, the model can lead to either competitive exclusion of one species, stable coexistence, or founder control whereby the species with an initial numerical advantage (determined by chance) comes to dominate.

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Figure 1. Phase diagram of competition outcomes over values of and predicted by the Mean Field Approximation (MFA; see Methods for details).

All other parameters fixed: , , , .

https://doi.org/10.1371/journal.pone.0114979.g001

Ecological pressure and local interactions

It is therefore apparent that whether a species has a greater inter-specific competition rate () is not a reliable criterion by which to infer its numerical dominance. Instead we propose an additional descriptor of the state of a population which we refer to as the ecological pressure acting upon it. Ecological pressure is defined as the sum of demographic forces that a particular population experiences. The change in density of a population is inversely related to the ecological pressure it experiences , and the units of ecological pressure can be chosen such that . The minus sign comes from the conceptual interpretation of ecological pressure as the forces that the environment exerts over the species (and not the species over its environment). This means that if the ecological pressure over a species in a community is positive the density of that species decreases, and vice versa if it is negative.

The spatial Lotka-Volterra model (equation (2)) can be obtained from the following form for the ecological pressure on species i:

(5)

Ecological pressure may change over time for various reasons. These include fluctuations in the densities of one or more species, variation in external abiotic factors, or even shifts in the spatial organisation of the community. The relationship between and allows inference of the actual ecological pressure in natural systems from measurements of the change in population density over time. This is important because it provides a potential means to assess the performance of a theoretical model by comparing the predicted ecological pressure with the observed demographic change.

Over a finite period of time , the actual change in population density can be found by the series expansion(6)where denotes the total time derivative of P_i. The concept of ecological pressure is useful in this study because calculating how it changes as a result of variation in the range or strength of interactions allows us to account for changes in population densities or outcomes of competition. When the system is in a stationary state, i.e. both populations are at equilibrium, the total ecological pressure is zero. By altering the parameters of one or more species we can identify how ecological pressure will change as a result. For example, in a two species system the ecological pressure over species j (equation (5)) will be diminished by reducing the inter-specific competition range of the other species (), which in turn enables an increase in the population density of species j.

Model outputs

When compared to the results of spatially-explicit simulations of the individual-based model, the mean-field approximation fails to accurately predict the region of parameter space within which coexistence occurs. This effect is particularly pronounced when interactions occur over short ranges, as expected in nature when individuals are most influenced by their nearest neighbours.

With long range dispersal, mean-field predictions are almost recovered, regardless of the values of the competition range (Fig. 2a). This is expected since offspring are able to escape from regions of high density, removing the constraints on recruitment imposed by spatial population structure. The parameter space within which stable coexistence occurs becomes smaller when the dispersal range is short, making coexistence less likely (Fig. 2b). This is the result of a reduction in population-level birth rates due to a higher density of individuals near their parents, i.e. the number of successful offspring decreases. Localised dispersal and short range interactions for both species lead to a reduction in the parameter space for coexistence in comparison with long range interactions (Fig. 2c). This is in agreement with a previous mathematical result [27] and occurs because the area from which the parents obtain resources is reduced, making competition in the local neighbourhood more intense, thus reducing even further the number of successful offspring near their parents.

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Figure 2. Coexistence diagrams for a system of two species obtained by changing the values of the inter-specific competition rate of species 1 () and the intrinsic death rate of species 2 ().

All other parameters fixed at , , , . Each point represents the outcome of 20 realisations of the model (see Methods). Blue circles: only species 1 survives; red squares: only species 2 survives; brown rhombus: coexistence. Solid black lines represent predictions based on the mean-field approximation. The table above each diagram shows the values of the range of dispersal , inter-specific and intra-specific competition (, ). a) long range dispersal; b) localised dispersal; c) short-range interactions with equal values for both species; d) variation in intra- and inter-specific competition ranges among species. Definition of parameters in Table 1; specimen patterns based on c) and d) are shown in S3 Fig.

https://doi.org/10.1371/journal.pone.0114979.g002

Only when dispersal is highly localised is the region of coexistence modified by the ranges of competitive interactions. Some parameter combinations decrease the potential for coexistence (Fig. 2b, 2c), while others allow coexistence in regions where it was not predicted by the MFA (Fig. 2d). These results are robust to alternative parameter values (e.g. S2 Fig.).

The transition from coexistence to competitive exclusion in the simulations occurs due to continuous changes in the densities () of each species, rather than a sharp boundary (Fig. 3). Fig. 3a demonstrates that there is an important region of parameter space in which despite species 2 being competitively weaker () it is numerically dominant (i.e. the red line is above the blue). From Fig. 3b we can see that the susceptibility of a species' population density to changes in its intrinsic death rate is greater (i.e. a steeper slope) when another species is present, even as a minority element. These patterns reinforce our view that complementarity is central to understanding coexistence, since the density of each species responds to the presence of the other; it is not merely the case that an inferior competitor fits around the pattern generated by the stronger species.

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Figure 3. Typical density of two species obtained from linear transects through the coexistence diagram in Fig. 2c for the simulated communities.

Blue line and circles: species 1; red line and squares: species 2. a) for variation in with , b) for variation in with . All other parameters as Fig. 1.

https://doi.org/10.1371/journal.pone.0114979.g003

In Fig. 4 we illustrate three cases where spatial coexistence can be achieved through localised interactions in regions where the MFA based upon the expectations of the Lotka-Volterra equations fails to predict it. Each phase diagram represents a particular combination of parameters, while the summary statistics illustrate the intra- and inter-specific pair correlation functions calculated at the point marked on the corresponding phase diagram. By comparing the empirical correlation function with that expected under the MFA we can observe the change in the dimensions of clusters caused by switching from long-range interactions to localised. The correlation functions reveal that each case exhibits a unique spatial pattern, thereby indicating different underlying coexistence mechanisms.

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Figure 4. Three examples of spatial coexistence beyond the predictions of the Mean Field Approximation (MFA) achieved through localised interactions.

Left column: phase diagrams obtained as in Fig. 2 with predictions of the MFA (solid black lines), though for clarity we omit individual points representing each set of simulations. Circles indicate the point at which corresponding spatial pattern statistics were calculated. Middle column: the intra-specific pair correlation function of species 1, , indicating the deviation from average density of individuals of the same species at distance r from any single individual. Thin solid line is the identical function but with long-range interactions (), representing the MFA (as in Fig. 2a). Right column: cross-pair correlation function of species 1 to species 2, . The deviation from the average density of species 2 at a distance r as experienced by an individual of species 1 is proportional to the value of this function at r. Thin solid line is the cross-pair correlation in the case of long range interactions.

https://doi.org/10.1371/journal.pone.0114979.g004

Fig. 4a shows that short-range interactions (localised intra- and inter-specific competition) promote coexistence outside the MFA predictions. Coexistence occurs as a consequence of increased ecological pressure on the numerically-dominant species due to intense intra-specific competition, following from reduction of the range over which this occurs . In conjunction there is a reduction in ecological pressure on the rarer species due to shortening of the inter-specific competition range of species 1 (). This can be verified by describing the change in ecological pressure for each species from equation 5:(7)

The phenomenon holds at least up to the first order in a time expansion of the ecological pressure (equation (6)), given that individuals within species are aggregated () and segregated between (). This mechanism lies behind the spatial segregation hypothesis [12], but note that enhanced coexistence is only exhibited near the point of ecologically-equivalent species.

In Fig. 4b the numerically-dominant species 1 has shorter-range inter-specific than intra-specific competition (), generating an increase in the coexistence region consistent with heteromyopia [18]. The reduction in ecological pressure on the rarer species 2 arises because of the relatively shorter range of inter-specific competition from the numerically-dominant species:(8)

The effect holds at least up to the first order in time (equation (6)) with segregation between species (). The same process is also present in spatial segregation but the lack of intensified intra-specific competition identifies it as a distinct phenomenon.

Finally, Fig. 4c shows coexistence via an additional mechanism which has not previously been identified. The effect of changing from Fig. 4b to Fig. 4c is to transform species 1 from being numerically dominant to rare. At the lower bound of the coexistence region, short-range inter-specific competition by the rare species reduces the ecological pressure acting upon it. This can be seen from the decrease in height of in Fig. 4c (middle column) as a result of shortening . From the perspective of individuals of the rarer species there is a reduction of conspecifics in its neighbourhood (i.e. reduced clustering), thereby reducing intra-specific competition. This behaviour can only be predicted via the second order term in equation (6), meaning it is observed only once a second generation is born (grandchildren). For this reason we refer to it as a triadic mechanism; its elucidation depends on third order spatial moments, i.e. a minimum of three individuals.

This can be seen by demonstrating that there is no immediate change in the ecological pressure on species 1 as a result of changing (see equation (8)). This means that the simple linear term in the time expansion of ecological pressure cannot account for the coexistence region in Fig. 4c. Only the second order in time of equation (6) can account for this effect:(9)

In general and . For a further demonstration based on spatial moments see S3 Note. To summarise, in the triadic mechanism it is the ability of grandchildren to escape the competition of their grandparents which enables the persistence of the rarer species. Note that it is the characteristics of the minority species which dictate whether coexistence occurs.