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     [13]Theoretical Ecology

     Article

               Plant-soil feedbacks and the coexistence of competing plants

     * Original Paper
     * [14]Open access
     * Published: 09 June 2012

     * Volume 6, pages 99-113, (2013)
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   [18]Theoretical Ecology [19]Aims and scope [20]Submit manuscript

   Plant-soil feedbacks and the coexistence of competing plants
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     * [22]Tomás A. Revilla^[23]1,[24]2,[25]7,
     * [26]G. F. (Ciska) Veen^[27]3,[28]4,[29]5,
     * [30]Maarten B. Eppinga^[31]6 &
     * ...
     * [32]Franz J. Weissing^[33]1

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     * 4100 Accesses
     * 48 Citations
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Abstract

   Plant-soil feedbacks can have important implications for the interactions among
   plants. Understanding these effects is a major challenge since it is inherently
   difficult to measure and manipulate highly diverse soil communities. Mathematical
   models may advance this understanding by making the interplay of the various
   processes affecting plant-soil interaction explicit and by quantifying the
   relative importance of the factors involved. The aim of this paper is to provide
   a complete analysis of a pioneering plant-soil feedback model developed by Bever
   and colleagues (J Ecol 85: 561-573, [35]1997; Ecol Lett 2: 52-62, [36]1999; New
   Phytol 157: 465-473, [37]2003) to fully understand the range of possible impacts
   of plant-soil feedbacks on plant communities within this framework. We analyze
   this model by means of a new graphical method that provides a complete
   classification of the potential effects of soil communities on plant competition.
   Due to the graphical character of the method, the results are relatively easy to
   obtain and understand. We show that plant diversity depends crucially on two key
   parameters that may be viewed as measures of the intensity of plant competition
   and the direction and strength of plant-soil feedback, respectively. Our analysis
   provides a formal underpinning of earlier claims that plant-soil feedbacks,
   especially when they are negative, may enhance the diversity of plant
   communities. In particular, negative plant-soil feedbacks can enhance the range
   of plant coexistence by inducing competitive oscillations. However, these
   oscillations can also destabilize plant coexistence, leading to low population
   densities and extinctions. In addition, positive feedbacks can allow locally
   stable forms of plant coexistence by inducing alternative stable states. Our
   findings highlight that the inclusion of plant-soil interactions may
   fundamentally alter the predictions on the structure and functioning of
   above-ground ecosystems. The scenarios presented in this study can be used to
   formulate hypotheses about the ways soil community effects may influence plant
   competition that can be tested with empirical studies. This will advance our
   understanding of the role of plant-soil feedback in ecological communities.

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Introduction

   Ecologists have long recognized that interactions between plants are mediated by
   many biotic (e.g., grazing, plant competition, and facilitation) and abiotic
   factors (e.g., soil texture, nutrient availability, and topography; Harper
   [42]1977; Tilman [43]1988). More recent research has stressed the influence of
   the soil community on competitive interactions between plants (Callaway et al.
   [44]2004; Klironomos [45]2002) by exerting positive or negative effects on the
   growth of specific plants (De Deyn et al. [46]2003; Gange et al. [47]1993;
   Klironomos [48]2003; Olff et al. [49]2000; van der Heijden et al. [50]1998a; van
   der Heijden et al. [51]2003; van der Putten and van der Stoel [52]1998; van der
   Putten et al. [53]1993). Therefore, plant-soil interactions can potentially be of
   crucial importance for species composition of plant communities and, more
   generally, the diversity of terrestrial ecosystems (van der Putten et al.
   [54]2001; Wardle et al. [55]2004).

   The interaction between a plant and the soil community, referred to as plant-soil
   feedback, is a two-step process: the presence of a specific plant changes the
   composition of the soil community, which in turn alters the growth rate of that
   specific plant (Bever [56]2003; Reynolds et al. [57]2003). Quantifying the effect
   of soil organisms on plant growth and vice versa is difficult due to the vast
   below-ground diversity, and the technical problems inherent to measuring and
   manipulating soil communities (Bever [58]2003; van der Putten et al. [59]2009).
   Therefore, there is a need for mathematical models that can help to generate
   insight into the potential implications of plant-soil feedbacks on species
   dynamics and diversity (van der Putten et al. [60]2009).

   Bever and colleagues ([61]1997, [62]1999, [63]2003) developed such a modeling
   framework. In a 2003 paper, Bever incorporated plant-soil interactions in the
   classical Lotka-Volterra competition model, an approach that motivated several
   subsequent modeling studies on similar topics (Bonanomi et al. [64]2005; Eppinga
   et al. [65]2006; Eppstein et al. [66]2006; Eppstein and Molofsky [67]2007;
   Umbanhowar and McCann [68]2005). Bever ([69]2003) highlighted two implications of
   plant-soil feedback for the coexistence of two competing plant species: (1)
   negative plant-soil feedback facilitates plant coexistence and (2) negative
   plant-soil feedback drives oscillations in plant abundances. These kinds of
   predictions are appealing to empirical plant-soil ecologists because they can be
   tested with relatively straightforward experiments (Bever [70]1994; Bever et al.
   [71]1997), without requiring specific knowledge on the composition of soil
   communities or the effects of individual soil-borne species on plant growth.

   Bever only performed a partial analysis of his model, thus leaving out a number
   of interesting predictions. A more complete analysis of the "Bever model" (i.e.,
   the model developed in Bever [72]2003) is required to fully understand the impact
   of plant-soil feedbacks on plant communities in this framework. The first goal of
   our paper is to develop intuitively appealing methods allowing such an analysis,
   thereby pointing out various routes by which soil communities can affect plant
   diversity. A second goal is to refine some of Bever's ([73]2003) conclusions,
   such as the usefulness of the "feedback parameter I [S]" (which was introduced on
   the basis of an earlier model) in the context of plant competition, and the
   characterization of the parameter range for which plant-soil feedback drives
   oscillations in plant abundances. Our third goal is to highlight that
   oscillations induced by negative feedbacks do not per se enhance plant diversity.
   In fact, we will show that such feedbacks can actually be detrimental for plant
   coexistence on a local scale.

   We start this paper by introducing the Bever model in "[74]The Bever model"
   section. In the "[75]Graphical analysis" section, we analyze and predict the
   outcomes of the model by means of a graphical method integrating plant-soil
   interactions and plant competition. In the "[76]Implications for plant
   coexistence" section, we derive the implications of our results for plant species
   coexistence, emphasizing some new results and scenarios not considered by Bever
   ([77]2003). Finally, we discuss the implications of our results in the
   "[78]Discussion" section.

The Bever model

   The "Bever model" (Bever [79]2003) studies the effect of two soil communities
   with densities S [A] and S [B] on the growth of two competing plant species with
   densities N [A] and N [B] (Fig. [80]1). The dynamics of the system are described
   by the following system of four differential equations:
   Fig. 1
   [81]figure 1

   Schematic representation of the interactions between two plants A and B and their
   associated soil communities S [A] and S [B]
   [82]Full size image
   $$ \frac{{d{N_A}}}{{dt}} = {r_A}{N_A}\left( {1 + {\alpha_A}{S_A} + {\beta_A}{S_B}
   - \frac{{{N_A} + {c_B}{N_B}}}{{{K_A}}}} \right) $$
   (1a)
   $$ \frac{{d{N_B}}}{{dt}} = {r_B}{N_B}\left( {1 + {\alpha_B}{S_A} + {\beta_B}{S_B}
   - \frac{{{N_B} + {c_A}{N_A}}}{{{K_B}}}} \right) $$
   (1b)
   $$ \frac{{d{S_A}}}{{dt}} = {S_A}{S_B}\frac{{{N_A} - v{N_B}}}{{{N_A} + {N_B}}} $$
   (1c)
   $$ \frac{{d{S_B}}}{{dt}} = {S_A}{S_B}\frac{{v{N_B} - {N_A}}}{{{N_A} + {N_B}}} $$
   (1d)

   The competition coefficients c [A] and c [B] are expressed as the per capita
   effects of each species on the growth rate of the competitor species
   (interspecific competition), relative to the per capita effect on the growth rate
   of its own population (intraspecific competition). r [A] and r [B] denote the
   intrinsic per capita growth rates of the plant species. K [A] and K [B] are the
   carrying capacities of the plant species when growing in isolation. S [A] and S
   [B] are the densities of the two soil communities, where S [A] is specifically
   associated with plant species A and S [B] with plant species B. The soil
   community is positively affected by the relative abundance of its associated
   plant species and negatively affected by the relative abundance of the other
   plant species. The parameter n is a scaling factor that quantifies the relative
   strength of the positive and negative effects of plants on soil community growth.
   The effect of the soil communities on plant growth is characterized by the
   parameters a [A], a [B], b [A], and b [B], respectively, which can be either
   positive, negative, or zero. The appearance of the product S [A] S [B] in
   ([83]1c) and ([84]1d) reflects the fact that Bever originally derived his model
   for the relative abundances of the two soil communities (Bever et al. [85]1997).
   In fact, adding ([86]1c) and ([87]1d) yields \( \frac{{d{S_{\text A}}}}{{dt}} +
   \frac{{d{S_{\text B}}}}{{dt}} = 0 \) and, as a consequence, S [A] + S [B] =
   const. Scaling the total abundance of the soil communities to one, we can express
   S [B] in terms of S [A] (i.e., S [B] = 1 - S [A]) and reduce system (1) to a
   system of three differential equations. This system can be written in the form:
   $$ \frac{{d{N_A}}}{{dt}} = {\rho_A}{N_A}\left( {1 - \frac{{{N_A} +
   {c_B}{N_B}}}{{{\kappa_A}}}} \right) $$
   (2a)
   $$ \frac{{d{N_B}}}{{dt}} = {\rho_B}{N_B}\left( {1 - \frac{{{N_B} +
   {c_A}{N_A}}}{{{\kappa_B}}}} \right) $$
   (2b)
   $$ \frac{{d{S_A}}}{{dt}} = {S_A}\left( {1 - {S_A}} \right)\left( {\frac{{{N_A} -
   v{N_B}}}{{{N_A} + {N_B}}}} \right) $$
   (2c)

   The intrinsic growth rates r [A ]and r [B ]and the carrying capacities k [A ]and
   k [B ]of the two plant species depend on the densities of the soil communities, S
   [A] and S [B] = 1 - S [A], and they are given by:
   $$ {\rho_A} = {\rho_A}\left( {{S_A}} \right) = {r_A} \cdot \left( {1 +
   {\alpha_A}{S_A} + {\beta_A}\left( {1 - {S_A}} \right)} \right) $$
   (3a)
   $$ {\rho_B} = {\rho_B}\left( {{S_A}} \right) = {r_B} \cdot \left( {1 +
   {\alpha_B}{S_A} + {\beta_B}\left( {1 - {S_A}} \right)} \right) $$
   (3b)
   $$ {\kappa_A} = {\kappa_A}\left( {{S_A}} \right) = {\kappa_A} \cdot \left( {1 +
   {\alpha_A}{S_A} + {\beta_A}\left( {1 - {S_A}} \right)} \right) $$
   (3c)
   $$ {\kappa_B} = {\kappa_B}\left( {{S_A}} \right) = {\kappa_B} \cdot \left( {1 +
   {\alpha_B}{S_A} + {\beta_B}\left( {1 - {S_A}} \right)} \right) $$
   (3d)

   For the rest of this paper, we assume that \(
   {\alpha_A},{\beta_A},{\alpha_B},{\beta_B} > - 1 \), thus preventing k [A] and k
   [B] from turning into negative equilibrium densities. All other model parameters
   are assumed to be positive.

Graphical analysis

   We analyze the Bever model in four steps. The first three steps do not show new
   results, but introduce concepts and a line of reasoning that is crucial for
   understanding our graphical methods in the final step. First, we discuss the
   effects of the soil dynamics on plants growing in monoculture, which will lead us
   to the concept of soil stability. Second, we consider the effects of a fixed soil
   composition on plant competition, which helps us to define the concept of
   competitive stability. Third, we take into account the net effects of the
   plant-soil feedbacks, and how to distinguish whether they are positive or
   negative. Fourth, we integrate our criteria for soil stability, competitive
   stability, and feedbacks in a graphical method for the analysis of the complete
   system (2).

Plant monocultures

   A monoculture of plant A is represented by Eqs. (2a, c) with N [B] = 0. This
   monoculture has two equilibrium states: a first one in which S [A] = 1 and \(
   {N_A} = {\kappa_A}(1) = {K_A}\left( {1 + {\alpha_A}} \right) \), and a second one
   in which S [A] = 0 and \( {N_A} = {\kappa_A}(0) = {K_A}\left( {1 + {\beta_A}}
   \right) \). When 0 < S [A] < 1, Eq. ([88]2c) implies that S [A] converges to 1
   while S [B] converges to 0, i.e., the soil community associated with plant A
   completely eliminates the soil community associated with plant B. In dynamical
   terms, the first equilibrium is stable with regards to perturbations in the soil
   community, and it will be classified as soil-stable. Accordingly, the second
   equilibrium is unstable with regards to perturbations in the soil community, and
   it will be classified as soil-unstable. Obviously, the same arguments apply to
   monocultures of plant species B. For later reference, we give the four
   monoculture equilibria a name: A [A ]: soil-stable monoculture of plant A, A [B
   ]: soil-unstable monoculture of plant A, B [B ]: soil-stable monoculture of plant
   B, and B [A ]: soil-unstable monoculture of plant B.

Effect of a static soil community on plant competition

   When a soil community remains static, i.e., at constant density, the coefficients
   r [A], r [B], k [A], and k [B] are constant and plant competition is described by
   a standard Lotka-Volterra model. The dynamics of this model is well-known (e.g.,
   Case [89]2000): the outcome of competition depends on the ability of each species
   to invade the monoculture of the other species when this species is at its
   carrying capacity (k [A] or k [B], respectively). From Eq. ([90]2a), it follows
   that species A can invade the monoculture of species B if c [B] k [B] < k [A],
   implying c [B] < k [A]/k [B]. Similarly, species B can invade in the monoculture
   of species A if c [A] < k [B]/k [A]. This leads to four possible outcomes of
   competition (Case [91]2000). If species A (resp. species B) is the only species
   able to invade the monoculture of the other species, species A (resp. species B)
   will win the competition. If both species are able to invade the monoculture of
   the other species, both species will stably coexist. If neither species is able
   to invade the monoculture of the other species, one of the species will win the
   competition, the winner depending on initial conditions.

   A community equilibrium exists if either both species or none of the species can
   invade the monoculture of the other species, i.e., if c [A] - k [B]/k [A] and c
   [B] - k [A]/k [B] have the same sign. The community equilibrium is stable if the
   two species can mutually invade each other. Multiplying both sides of the
   invasion criteria c [A] < k [B]/k [A] and c [B] < k [A]/k [B] yields the
   stability condition c [A] c [B] < 1, which is often interpreted as "interspecific
   competition is on average weaker than intraspecific competition". Summarizing:
   $$ existence\,of\,community\,equilibrium:\left( {{c_A} - {\kappa_B}/{\kappa_A}}
   \right)\left( {{c_B} - {\kappa_A}/{\kappa_B}} \right) > 0 $$
   (4a)
   $$ stability\,of\,community\,equilibrium:{c_A}{c_B} < 1 $$
   (4b)

   Figure [92]2 illustrates the conditions for equilibrium and stability in a plot
   where the co-ordinate axes correspond to the ratios of (soil dependent) carrying
   capacities: x = k [B]/k [A] and y = k [A]/k [B] (this approach is similar in
   spirit as the "recovery plane" analysis of Eppinga et al. ([93]2006) where k
   [B]/k [A] - c [A] and k [A]/k [B] - c [B] are plotted against each other). The
   relation between x and c [A] determines whether B can invade the monoculture of
   A, while the relation between y and c [B] determines whether A can invade the
   monoculture of B. The parameters x and y are not independent but constrained by
   \( xy = \left( {{{{{\kappa_B}}} \left/ {{{\kappa_A}}} \right.}} \right)\left(
   {{{{{\kappa_A}}} \left/ {{{\kappa_B}}} \right.}} \right) = 1 \). In other words,
   all parameter combinations describing a competitive plant system actually lie on
   the hyperbola xy = 1. If c [A] c [B] < 1 (Fig. [94]2a), this hyperbola intersects
   the coexistence region III, which means that stable coexistence is possible for
   certain values of k [B]/k [A]. In the case of c [A] c [B] > 1 (Fig. [95]2b), the
   hyperbola intersects the mutual exclusion region IV and stable coexistence is not
   possible at all.
   Fig. 2
   [96]figure 2

   Outcomes of the Lotka-Volterra system. The axes correspond to the ratios of the
   carrying capacities: \( x = {\kappa_B}/{\kappa_A},y = {\kappa_A}/{\kappa_B} \).
   Since y = 1/x, all feasible systems are constrained to the hyperbola xy = 1.
   Plant B invades the monoculture of plant B if x > c [A ]and plant A invades the
   monoculture of B if y > c [B ]. As a result the plane can be divided into four
   invasibility zones I, II, III, and IV. a When c [A] c [B] < 1, the hyperbola xy =
   1 intersects region III, but not region IV. Hence, the community equilibrium is
   stable (if it exists). b When c [A] c [B] > 1, the hyperbola xy = 1 intersects
   region IV but not region III. Now the community equilibrium is unstable (if it
   exists), and depending on the initial conditions, the system will either converge
   to a monoculture of plant A or to a monoculture of plant B
   [97]Full size image

   For future reference, we will use the term competitive stability to refer to
   systems allowing a stable plant community equilibrium (c [A] c [B] < 1) and the
   term competitive instability for systems where only monoculture equilibria can be
   stable (c [A] c [B] > 1). As we shall see later on, the soil community effects
   (indirectly) affect the balance between intra- and interspecific plant
   competition among the plant species. Therefore, such soil community effects have
   the ability to shift the plant system from one competitive regime to another.

Positive and negative plant-soil feedback

   Given a pair of values of k [A] and k [B] determined by S [A] (Eqs. [98]3c, d)
   the corresponding competitive system can be represented as a point on the
   hyperbola xy = 1 in Fig. [99]2. Since S [A] can vary from 0 to 1, all feasible
   competitive systems can be mapped as a continuous set of points, i.e., a
   feasibility arc. The two endpoints of the feasibility arc have the coordinates:
   $$ {S_A} = 1:x = {x_A} = \frac{{{\kappa_B}(1)}}{{{\kappa_A}(1)}} =
   \frac{{{\kappa_B}\left( {1 + {\alpha_A}} \right)}}{{{\kappa_A}\left( {1 +
   {\alpha_B}} \right)}},y = {y_A} = \frac{1}{{{x_A}}} = \frac{{{\kappa_A}\left( {1
   + {\alpha_A}} \right)}}{{{\kappa_B}\left( {1 + {\alpha_B}} \right)}} $$
   (5a)
   $$ {S_A} = 0:x = {x_B} = \frac{{{\kappa_B}(0)}}{{{\kappa_A}(0)}} =
   \frac{{{\kappa_B}\left( {1 + {\beta_B}} \right)}}{{{\kappa_A}\left( {1 +
   {\beta_A}} \right)}},y = {y_B} = \frac{1}{{{x_B}}} = \frac{{{\kappa_A}\left( {1 +
   {\beta_A}} \right)}}{{{\kappa_B}\left( {1 + {\beta_B}} \right)}} $$
   (5b)

   Figure [100]3 pictures the feasibility arc as an arrow with its head at S [A ]= 1
   (Eq. [101]5a) and its tail at S [A ]= 0 (Eq. [102]5b). With respect to the y-axis
   the arc can have an upward or a downward orientation. If the orientation is
   upwards (Fig. [103]3a), large values of S [A] correspond to systems close to (or
   in) parameter regime I of Fig. [104]2a, where plant species A is competitively
   dominant. Small values of S [A] correspond to systems close to (or in) regime II
   where plant B is dominant. In this situation there is a positive plant-soil
   feedback in the sense that the dominance of one type of soil community (S [A] or
   S [B]) favors the competitive dominance of the associated plant species (A or B).
   Similarly, there is negative plant-soil feedback if the arc points downwards
   (Fig. [105]3b) because the dominance of one type of soil community favors the
   competitive dominance of the plant species not associated with the dominant soil
   community.
   Fig. 3
   [106]figure 3

   Effect of positive versus negative plant-soil feedback. The range of possible
   soil communities (S [A] ranging from 0 to 1) determines a feasibility arc,
   corresponding to a segment on the hyperbola xy = 1. The direction of increasing S
   [A] is indicated by an arrow head. a If the arrow points upwards (J [S ]> 0),
   increasing S [A] shifts the system towards region I where plant species A wins
   the competition (see Fig. [107]2). Hence, the soil community of A has a positive
   effect on the competitive position of A (positive soil-plant feedback). b If the
   arrow points downwards (J [S] < 0), increasing S [A] shifts the system towards
   region II where species B wins. Hence, the soil community of A has a negative
   effect on A (negative soil-plant feedback)
   [108]Full size image

   The feedback is positive if \( y({S_A}) = {{{{\kappa_A}({S_A})}} \left/
   {{{\kappa_B}({S_A})}} \right.} \) is increasing with S [A] (as in Fig. [109]3a)
   or, in other words, if dy/dS [A] > 0. This derivative is given by \( dy/d{S_A} =
   \left( {{K_A}{K_B}/\kappa_B^2} \right) \cdot {J_S} \), where
   $$ {{J}_{{\text{S}}}} = \left( {1 + {{\alpha }_{{\text{A}}}}} \right)\left( {1 +
   {{\beta }_{{\text{B}}}}} \right) - \left( {1 + {{\alpha }_{{\text{B}}}}}
   \right)\left( {1 + {{\beta }_{{\text{A}}}}} \right) $$
   (6)

   Since \( {K_A}{K_B}/K_B^2 \) is always positive, the plant-soil feedback is
   positive if J [s] > 0 and it is negative if J [s] < 0. In the absence of
   plant-soil feedback (J [s] = 0), the feasibility arc collapses into a single
   point (see the "[110]Effect of a static soil community on plant competition"
   section).

   The importance of the sign of the plant-soil feedback was pointed out by Bever
   and colleagues (Bever et al. [111]1997; Bever [112]1999) who argued that positive
   feedback tends to favor competitive dominance and, hence, species-poor plant
   communities, while negative feedback tends to favor plant coexistence. To
   quantify plant-soil feedbacks, Bever introduced an interaction coefficient I [S]
   that is defined by
   $$ {I_S} = {\alpha_A} + {\beta_B} - {\alpha_B} - {\beta_A} $$
   (7)

   There is a simple relationship between the coefficient I [S] introduced by Bever,
   and the coefficient J [S] resulting from our analysis
   $$ {J_S} = {I_S} + {\alpha_A}{\beta_B} - {\alpha_B}{\beta_A} $$
   (8)

   Hence, in the case of parameter symmetry (i.e., \( {\alpha_A}{\beta_B} =
   {\alpha_B}{\beta_A} \)) Bever's interaction coefficient I [S] correctly predicts
   plant-soil feedback also in the case of plant-plant competition. J [S]
   generalizes I [S] to systems without parameter symmetry. In asymmetric scenarios,
   the difference between J [S] and I [S] is often small. But it is easy to
   construct examples where J [S] and I [S] differ in sign, that is, where Bever's
   coefficient I [S] does not correctly indicate the sign of the feedback.

   In analogy to Lotka-Volterra competition coefficients we also define a net
   plant-soil feedback effect, which can be derived from the ratio between effects
   on the host plant (analogous to intraspecific competition) and cross-effects on
   the other plant (analogous to interspecific competition):
   $$ {H_S} = {x_B}{y_A} = \frac{{\left( {1 + {\alpha_A}} \right)\left( {1 +
   {\beta_B}} \right)}}{{\left( {1 + {\alpha_B}} \right)\left( {1 + {\beta_A}}
   \right)}} $$
   (9)

   In view of ([113]6), H [S] > 1 implies that host-plant effects of a soil
   community are more favorable than cross-effects, meaning that plant-soil feedback
   is positive (J [S] > 0). H [S] < 1 implies that cross-effects are more favorable
   than host plant effects, meaning that plant-soil feedback is negative. We also
   want to remark that the natural logarithm of H [S], which has the literal signs
   (+ or -), as well as comparable scales for positive and negative feedbacks, can
   be approximated by Bever's I [S] for small values of alphas en betas (i.e., first
   order Taylor series).

Combining plant and soil dynamics

   Combining the plots in Figs. [114]2 and [115]3 provides us with a graphical
   method that is often sufficient for a complete characterization of the dynamics
   of the coupled plant-soil community described by Eq. ([116]2). There are 20
   different ways in which the feasibility arc can intersect the four parameter
   domains corresponding to the plant competition scenarios I to IV (Fig. [117]4),
   which are characterized by the monoculture invasion criteria. A detailed overview
   of the scenarios is given in [118]Appendix A.
   Fig. 4
   [119]figure 4

   Intersection of the feasibility arc of Fig. [120]3 with the invasion zones of
   Fig. [121]2. The arc is represented as an arrow which indicates the direction of
   increasing S [A]. There are 20 scenarios, differing in the relative position and
   orientation of the feasibility arc with respect to the invasion zones
   [122]Full size image

   Figure [123]5a depicts case 12, where the feasibility arc lies completely within
   region II where only species B can invade. We can therefore conclude that
   irrespective of the state of the soil only plant species B can grow in this
   scenario. Thus the system converges to the monoculture equilibrium B [B ]in which
   S [A] = 0. In fact, in all cases where the feasibility arc lies within regions I
   or II (cases 1, 2, 11, and 12) one of the plants will always win, irrespective of
   the initial conditions. When the arc lies in region III (cases 3 and 13), each
   plant species can invade the monoculture of the other species, implying that the
   two species will stably coexist at equilibrium. When the arc lies within region
   IV (cases 4 and 14) neither plant species can invade when rare. Both plant
   monocultures are stable, and the winner depends on the initial conditions
   (founder control; Bolker et al. [124]2003). Summarizing, we can conclude that in
   the eight scenarios where the feasibility arc lays completely inside one of the
   four competition regions the qualitative outcome of plant competition does not
   depend on the sign of the plant-soil feedback.
   Fig. 5
   [125]figure 5

   Graphical analysis of two scenarios discussed in the text. a Case 12 in
   Fig. [126]1: the whole feasibility arc is lying in region II. In this case,
   species B will win the competition, and the soil composition will converge to S
   [A] = 0 (i.e., the "tail" of the arrow). b Case 15 in Fig. [127]1: the tail of
   the arc is in region I, whereas the head is in region II. As long as the system
   is in region I, S [A] will increase. Eventually region III will be reached,
   corresponding to the stable coexistence of the two plant species
   [128]Full size image

   Figure [129]5b depicts case 15, which is more complex because the feasibility arc
   spans two invasion zones. However, the analysis is still straightforward. A
   monoculture of plant B can never be stable, since the whole arc lies in the
   parameter region where A can invade such a monoculture. In parameter regime I, a
   monoculture of plant A is to be expected in the absence of plant-soil feedback.
   However, as long as plant A is in monoculture, the associated soil community S
   [A]will increase. Hence the system will be shifted along the feasibility arc in
   the direction of the arrow, until the plant coexistence regime III is reached. We
   conclude that in this case the (negative) plant-soil feedback enables coexistence
   of the two plant species.

   As indicated by Fig. [130]5, the position of the end point of the feasibility arc
   with respect to the invasion zones tells us whether a plant monoculture can be
   invaded or not, and hence how the plant-soil interactions may determine the plant
   dynamics. For a monoculture of plant A, the "head" of the feasibility arc (S [A]
   = 1), which corresponds to the equilibrium state A [A ], is relevant. This state
   can be invaded by plant B if and only if c [A] < x [A]. By symmetry, a
   monoculture of plant B corresponds to the state B [B ], which is located at the
   "tail" of the feasibility arc (S [A] = 0) and can be invaded by plant A, if and
   only if c [B] < x [B]. Multiplying these two inequalities and noticing that \(
   {{x}_{{\text{A}}}}{{y}_{{\text{B}}}} = {{\left(
   {{{x}_{{\text{B}}}}{{y}_{{\text{A}}}}} \right)}^{{ - 1}}} = H_{{\text{S}}}^{{ -
   1}} \), we obtain a necessary condition for mutual invasion of the two
   monoculture equilibria:
   $$ {{c}_{{\text{A}}}}{{c}_{{\text{B}}}} < H_{{\text{S}}}^{{ - 1}} $$
   (10)

   In the absence of plant-soil feedback (J [S] = 0 or H [S] = 1) condition (10)
   turns into condition (4b) for the stable coexistence of the two plant species in
   the Lotka-Volterra competition model. In line with the predictions of Bever and
   colleagues (Bever [131]1999; Bever et al. [132]1997), condition (10) leads to the
   conclusion that the conditions for plant coexistence are more stringent in the
   case of a positive feedback (where \( H_{\text{S}}^{{ - 1}} < 1 \)), while they
   are more relaxed in the case of a negative plant-soil feedback.

   Indeed we found that positive feedback generally enhances plant monocultures
   (cases 7-10), while net negative feedback enhances plant coexistence (cases 15,
   16, 19, and 20). However, we found a few exceptions where plants can coexist
   under positive feedback (cases 5 and 6), and where negative feedback drove the
   system towards plant monocultures (cases 17 and 18). Therefore, in contrast to
   standard Lotka-Volterra theory, monoculture invasion conditions as those
   considered above do no longer provide a complete picture of the outcome of plant
   competition. In the "[133]Implications for plant coexistence" section, we will
   demonstrate in more detail that the two plant species can stably coexist even in
   cases where condition (10) for the mutual invasibility of the two plant
   monocultures is not satisfied.

Implications for plant coexistence

Can plant-soil feedback drive community oscillations?

   One of the most important conclusions from Bever ([134]2003), resulting from a
   numerical example, is that negative plant-soil feedback can enhance coexistence
   by inducing competitive oscillations. However, a mathematical analysis of the
   same example (Revilla [135]2009) reveals that these oscillations rapidly dampen
   out and that the system converges to a stable coexistence equilibrium. (Only,
   when numerically integrating the model with a large integration step size we
   obtained the oscillations described in Bever's Fig. [136]1c.). Therefore, we
   first explore if and under what conditions oscillation can arise in the Bever
   model.

   Population cycles are frequently associated with predator-prey dynamics, but they
   are also a common feature in Lotka-Volterra competition models (Gilpin [137]1975;
   May and Leonard [138]1975) and in resource competition models (Huisman and
   Weissing [139]2001; Revilla and Weissing [140]2008). In these models,
   oscillations require at least three competitors such that species R outcompetes
   species S, S outcompetes P, and P outcompetes R, as in the Rock-Scissors-Paper
   game. Mathematically, the monocultures of R, P, and S are connected by
   heteroclinic orbits, i.e., a sequence of paths R -> P -> S -> R that forms a
   cycle. Although the Bever model involves only two plant competitors, it has in
   fact four monoculture states (A [A ], A [B ], B [B ], B [A ]). This allows us to
   construct a heteroclinic cycle as follows.

   Consider the configuration corresponding to case 20 in Fig. [141]6a. From the
   positioning of the end points of the feasibility arc in competition regimes I and
   II, we can conclude that the following inequalities are satisfied:
   Fig. 6
   [142]figure 6

   Graphical analysis (a) and dynamical analysis (b) of case 20 in Fig. [143]1. a As
   long as the system is in region I, S [A] will increase, driving the monoculture
   of A from an uninvasible state into an invasible state. Similarly, as long as the
   system is in region III, S [A] will decrease, driving the monoculture of B toward
   an invasible state. In both directions, the system passes through the zone of
   unstable community equilibria (IV). b The parameter space is mapped into a phase
   space where circles represent equilibria (white, unstable; gray, saddle) at the
   corresponding invasion zones (see [144]Appendix A for details on mapping into the
   phase space). The diagonal line represents the A-B nullcline (species A grows
   towards the right B grows towards the left), and the dashed line is the soil
   nullcine (S [A] increases in the right and decreases in the left); their
   intersection corresponds to the internal equilibrium predicted in (a).
   Monocultures are either competitively stable or soil stable, but not both: they
   are saddle points forming a heteroclinic cycle around the internal equilibrium AB
   which is competitively unstable. The system converges to a heteroclinic orbit
   (the border of the phase space)
   [145]Full size image
   $$ {x_B} < {c_A} < {x_A}{ }{\text{and}}{ }{y_A} < {c_B} < {y_B} $$
   (11)

   As discussed above, the inequalities c [A] < x [A] and c [B] < y [B] imply that
   both plant monocultures can be invaded by their plant competitor when their
   associated soil communities (i.e., soil community A is associated to plant
   monoculture A, and soil community B to plant monoculture B) are dominant. In
   other words, A [A ]and B [B ]are competitively unstable. The inequalities x [B] <
   c [A] and y [A] < c [B] imply that the two monocultures cannot be invaded when
   the invader's soil biota is dominant. In other words, A [B ]and B [A ]are
   competitively stable. However, as shown in the "[146]Plant monocultures" section,
   these two equilibria are not soil stable. In such a situation, we expect a cyclic
   sequence A [A ]-> B [A ]-> B [B ]-> A [B ]-> A [A ]of successions or at least
   oscillations following this sequence (Fig. [147]6b).

   This is confirmed by Fig. [148]7 which depicts the time course of the system
   under the above scenario. For Bever's parameterization (Fig. [149]7a;
   corresponding to Bever [150]2003, Fig. [151]1c), the system exhibits damped
   oscillations that converge to a coexistence equilibrium. By increasing the
   intensity of competition, c [A ]c [B ], this equilibrium becomes unstable and
   gives rise to a limit cycle (Fig. [152]7b). With a further increase in
   competition intensity the cycle becomes a heteroclinic orbit, i.e., the system
   "visits" the four monoculture equilibria in a cyclic fashion, remaining in the
   vicinity of each monoculture state for increasingly longer times (Fig. [153]7c).
   Fig. 7
   [154]figure 7

   Competitive oscillations due to a cyclic succession of competitive instability
   and soil instability. The top panels show the densities of plant A (solid line)
   and B (dashed line); the bottom panels depict the density of the soil biota S
   [A]. The parameter values K [A] = 100, K [B] = 120, r [A] = 0.7, r [B] = 0.5, n =
   0.8, a [A] = -0.03, a [B] = b [A] = 0.10, b [B] = -0.20, and c [B] = 0.980 are
   kept fixed, while the competition coefficient c [A] is increased from c [A] =
   0.885 in (a) via c [A] = 1.005 in (b), to c [A] = 1.050 in (c). Scenario (a)
   corresponds to Bever's ([155]2003) Fig. 4c with increasing intensity of
   competition, the asymptotic behavior of the system changes from convergence to a
   stable coexistence equilibrium in (a), via convergence to a limit cycle in (b) to
   convergence to a heteroclinic orbit in (c). In this example, soil-plant feedback
   is negative (J [s] = -0.434). Because of x [A] ~= 1.36, y [A] ~= 0.73, x [B] ~=
   0.87, y [B] ~= 1.15, the conditions (13) for the cyclical succession A [A ]-> B
   [A ]-> B [B ]-> A [B ]-> A [A ]are satisfied
   [156]Full size image

   In Fig. [157]7a, b, c [A] c [B] < 1 and the parameter configuration corresponds
   to case 19 in Fig. [158]1. Now the feasibility arc intersects parameter region
   III for stable competitive coexistence. In the absence of soil community effects,
   there exists a stable coexistence equilibrium, but this equilibrium can be
   destabilized when soil community effects are incorporated. This is indeed the
   case if the competition intensity c [A] c [B] is just below 1 (as in
   Fig. [159]7b). In fact, it can be shown analytically (Revilla [160]2009) that
   stable equilibrium coexistence (as in Fig. [161]7a) is only possible if c [A] c
   [B] < 1 - d, where d is a positive quantity that can be calculated from the
   remaining system parameters. In Fig. [162]7c, c [A] c [B] > 1, as a consequence,
   the parameter configuration corresponds to case 20 in Fig. [163]1, where the
   feasibility arc intersects parameter region IV for mutual competitive exclusion
   (see also Fig. [164]6a). In this region, there exists a coexistence equilibrium,
   but this equilibrium is unstable. As a consequence, the sequence of events
   described unfolds, and the system converges to the heteroclinic cycle A [A ]-> B
   [A ]-> B [B ]-> A [B ]-> A [A ](see Fig. [165]6b).

   Notice that the conditions (11) imply that \( {H_S} =
   {x_{\text{B}}}{y_{\text{A}}} < {x_{\text{A}}}{y_{\text{B}}} = H_{\text{S}}^{{ -
   1}} \), which is only possible for H [S] < 1 or, equivalently, J [S] < 0. As a
   consequence, the above scenario for the development of oscillations can only
   occur in case of negative plant-soil feedback. Since the analytical results of
   Revilla ([166]2009) suggest that this is the only route to oscillations in the
   Bever model, we conclude that competitive oscillations will only occur if two
   requirements are met: (a) the plant-soil feedback is negative and (b) both
   soil-stable monoculture equilibria are susceptible to invasion.

Does negative plant-soil feedback enhance coexistence?

   Coexistence between plant species is favored when each plant species can invade
   the monoculture of the other species. This is possible in many scenarios
   (Fig. [167]1 scenarios 1, 2, 3, 11, 12, 13, 15, 16, 19, and 20). When we exclude
   the scenarios where feedback does not determine invasion (1, 2, 3, 11, 12, and
   13) we can see that only negative plant-soil feedback allows mutual invasion.
   Although this generally enhances plant coexistence, it does not necessarily lead
   to equilibrium coexistence. Instead, as shown in the previous paragraph
   oscillations can occur. If these oscillations take the form of a limit cycle, the
   two plant species will still stably persist, although not in equilibrium. But the
   oscillations can also take the form of a heteroclinic cycle, where the
   populations are repeatedly driven to very low densities. In the mathematical
   model, they still survive, but in the real world local extinction can occur.
   Thus, in case of a heteroclinic cycle mutual invasibility of the two monocultures
   may not result in long-term coexistence.

   To assess the relative importance of limit cycles versus heteroclinic cycles, we
   investigated the pattern in Fig. [168]7 numerically. Using
   continuation-bifurcation analysis software (program: XPPAUT, Ermentrout 2002), we
   found that as c [A] c [B] increases, regular oscillations start at c [A] c [B] =
   0.977 (where a Hopf bifurcation occurs). Already at c [A] c [B] = 0.998, these
   oscillations turn into a heteroclinic orbit. Switching between the four
   monoculture states persists until c [A] c [B] = 1.559 (at this point, the
   requirement for mutual invasion (12) is no longer satisfied, and the chain of
   heteroclinic orbits connecting the four monocultures is broken). Thus, while the
   system does not converge to equilibrium for 0.977 < c [A] c [B] < 1.559, a limit
   cycle only occurs for the much smaller range 0.977 < c [A] c [B] < 0.998. In
   other words, more than 95 % of the parameter range corresponding to
   non-equilibrium dynamics corresponds to the occurrence of heteroclinic cycles,
   and hence, to eventual extinction.

   Accordingly, negative feedback has a much smaller potential for the facilitation
   of plant coexistence than Bever's analysis seems to suggest. One should notice,
   however, that in a spatial context, the risk of extinction via the
   large-amplitude oscillations associated with a heteroclinic cycle may be
   counteracted by the repeated re-immigration of the locally extinct species. Under
   such conditions, also a heteroclinic cycle can allow the regional coexistence of
   the plant species.

Coexistence as an alternative stable state

   The previous example highlights that plant-soil feedbacks can generate
   competitive dynamics that are not observed in standard competition models. In
   this section, we will also show an example of how soil community effects can
   fundamentally alter the stability of equilibria. This has important implications
   for the utility of invasion criteria to analyze stability. When properly used,
   invasion criteria are very useful to predict the range of dynamics that a
   dynamical system can possibly display. In the present context, we can list all
   possible dynamics of Bever's model (Table [169]1) by just considering whether A
   [A ]and B [B ]can be invaded or not (i.e., whether c [A] < x [A], c [B] < y [B]
   hold or not), and the direction of the feedback (the sign of J [S ]). However, as
   illustrated by the following example an invasion analysis can also have its
   limitations.
   Table 1 Plant community composition for different scenarios of plant invasion and
   net plant-soil feedback
   [170]Full size table

   Consider the scenario in Fig. [171]8a (case 5) where the feedback is positive (J
   [S] > 0) and where c [A] > x [A] and c [B] < y [B]. In this case, condition (10)
   for mutual invasibility does not hold: while plant A is able to invade the
   monoculture of plant B (since c [B] < y [B]), plant B cannot invade the
   monoculture of plant A (since c [A] < x [A] does not hold). However, stable
   coexistence may still be possible. As indicated in Fig. [172]8a, the "tail" of
   the feasibility arc (S [A] = 0) lies in zone III, which means that for S [A] = 0
   there is a competitively stable community equilibrium. Following our previous
   notation, this equilibrium will be denoted by AB [B ]since the soil is dominated
   by B's soil biota, as seen in Fig. [173]8b. Provided that N [A] < vN [B], this
   community equilibrium is a stable attractor: according to Eq. ([174]1c), this
   condition ensures dS [A]/dt < 0; and hence, the soil stability of AB [B ](in
   addition to the competitive stability of this equilibrium). Interestingly, AB [B
   ]is not the only attractor: the "head" of the feasibility arc (S [A] = 1) lies in
   zone I, where plant coexistence is not possible and plant A attains a
   competitively stable and soil stable monoculture A [A ].
   Fig. 8
   [175]figure 8

   (a) Graphical analysis and (b) phase space analysis of case 5 in Fig. [176]1. (a)
   In this configuration a monoculture of species A is not invasible by B (and hence
   stable), while a monoculture of species B is invasible by A. As explained in the
   text, the community equilibrium corresponding to the tail of the feasibility arc
   can also be a stable equilibrium. Hence, the system allows two alternative stable
   states. (b) The phase space shows the equilibria (white, unstable; black, stable;
   gray, saddle) and the invasion zones. The diagonal line representing the A-B
   nullcline (both species grow towards it) intersects S [A] = 0 giving rise to the
   coexistence equilibrium AB [B ]predicted in part (a). If the vertical dashed line
   representing the soil-nullcline (S [A] increases to the right and decreases to
   the left) intersects the A-B nullcline there will be an internal equilibrium AB
   that is a saddle point; and depending on the initial conditions the system moves
   towards the monoculture of A [A ]dominated by A's soil biota or towards the
   coexistence equilibrium AB [B ]dominated by B's soil biota
   [177]Full size image

   This example illustrates three points. First, a locally stable community
   equilibrium is not necessarily globally stable. In addition to the community
   equilibrium (here AB [B ]), there can be other stable attractors (here A [A ]),
   and the competitive outcome depends on the initial conditions. Second, invasion
   analysis is a very powerful technique, but it has its limitations. In the above
   example, AB [B ]can be stable even in the absence of mutual invasibility (the
   monoculture equilibrium A [A ]cannot be invaded by plant species B). Third, since
   our graphical analysis reflects an invasion analysis, it also has its
   limitations. In the above example, the stability of AB [B ]depends on the
   magnitude of the parameter n, while n does not play a significant role in the
   graphical analysis.

   Our conclusion that mutual invasibility is not required for stable plant
   coexistence is not only of theoretical interest, but also of empirical relevance.
   In fact, many experimental studies on plant-soil feedback use a set-up that is
   quite comparable to a mathematical invasion analysis (i.e., introduction of a
   plant species in a community dominated by other species). The above example
   demonstrates that--in contrast to classical competition theory--predictions on
   plant coexistence that are solely based on the outcome of invasion experiments
   have to be treated with care.

Discussion

   In this paper, we aimed at a better understanding of plant species dynamics and
   coexistence in the presence of plant-soil feedback. To this end, we developed
   mathematical and graphical techniques allowing a rather complete analysis of
   Bever's ([178]2003) model. We found that plant species coexistence is possible if
   the intensity of interspecies competition (c [A] c [B]) is smaller than \(
   H_{\text{S}}^{{ - 1}} \), a parameter closely related to the net plant-soil
   feedback parameter J [S]. When plant-soil feedback is positive, plant coexistence
   is more difficult to achieve than in the pure competition model. In contrast, a
   negative plant-soil feedback allows plant coexistence under more relaxed
   conditions and thus has the potential to enhance plant community diversity (Bever
   et al. [179]1997; van der Heijden et al. [180]2008). Recent empirical studies, as
   well as a meta-analysis of more than 300 plant-soil feedback experiments, indeed
   show that negative feedback favors coexistence and diversity (Bever [181]1994;
   Bradley et al. [182]2008; De Deyn et al. [183]2003; Kulmatiski et al. [184]2008;
   Mills and Bever [185]1998).

   Most of the results from our graphical analysis have an intuitive explanation
   because plant-soil feedback contributes to the competitive strength of the plant
   species. For example, in the case of negative plant-soil feedback (J [S] > 0),
   the competitive position of a plant is weakened when its associated soil
   community increases (i.e., the system shifts towards the region where the other
   plant species wins, Figs. [186]2 and [187]3). However, plant-soil interactions do
   not affect the competition intensities c [A] and c [B], and hence, also not the
   stability of an internal equilibrium if it exists (Fig. [188]2).

   While our analysis supports Bever's conclusion that negative feedback favors
   coexistence, we also demonstrated that the standard criterion for plant
   coexistence (the mutual invasibility of monocultures) is not a necessary
   condition for coexistence. As a consequence, plant coexistence under positive
   feedback is more likely than Bever's ([189]2003) analysis suggests. Surely, in
   natural situations positive plant-soil feedback, for example, mediated by the
   presence of arbuscular mycorrhizal fungi is often associated with an increase in
   the dominance of a particular plant species and a reduction in diversity
   (Hartnett and Wilson [190]1999). Moreover, exotic plant species that escaped from
   species-specific soil pathogens may still profit from mutualistic root symbionts
   in their new range which favors their dominance (Callaway et al. [191]2004;
   Klironomos [192]2002). However, it has also been shown that positive plant-soil
   feedbacks can enhance plant coexistence. For example, the presence of arbuscular
   mycorrhizal fungi may enhance plant diversity in grasslands (Grime et al.
   [193]1987; van der Heijden et al. [194]1998b) by promoting seedling establishment
   and enhancing the competitive ability of subordinate plant species (Grime et al.
   [195]1987; van der Heijden et al. [196]2008).

   Our results provide a mechanistic understanding of how negative plant-soil
   feedback can drive oscillations in plant species abundances. Oscillations could
   occur in situations where plants would otherwise competitively exclude each
   other, meaning that oscillations can enable coexistence. We also found, however,
   that oscillations along a heteroclinic orbit could occur in situations where
   plants would otherwise stably coexist, meaning that oscillations can also enable
   stochastic extinction. Therefore, negative plant-soil feedback does not
   necessarily enhance coexistence in all situations. Although many studies
   indicated that negative plant-soil feedback can be important driving plant
   dynamics (Olff et al. [197]2000) and ecological succession (De Deyn et al.
   [198]2003; van der Putten et al. [199]1993), there are no studies directly
   testing its impact on competitive oscillations. Yet, some of these empirical
   studies show that soil-borne organisms have the potential to decline the
   competitive strength of plants to such an extent that can be replaced by others
   (van der Putten and van der Stoel [200]1998; van der Putten et al. [201]1993).

   We introduced two alternative interaction coefficients J [S] and H [S]. These
   coefficients are more generally applicable to the current model, than Bever's I
   [S]. In the original plant-soil feedback model of Bever et al. ([202]1997), plant
   abundances as well as soil communities are accounted in terms of proportions (not
   densities) and plant competition is "apparent" (sensu Holt [203]1977), being a
   by-product of the feedbacks. As a consequence, the original Bever et al. model
   ([204]1997) has only two outcomes that can be predicted in terms of plant-soil
   feedback effects, i.e., the sign of I [S]. In contrast, the version of the Bever
   ([205]2003) model that we studied here is a modification of the Lotka-Volterra
   model. This model has more degrees of freedom because plant densities can vary
   independently, and because this model contains additional parameters related to
   resource competition. Consequently, equilibrium stability and global dynamics
   cannot be fully accounted for by solely considering combinations of feedback
   effects (I [S], J [S], H [S]), but must consider resource competition (parameters
   c [A] and c [B]) as well. In this study, we have shown how Bever's coefficient I
   [S] can be generalized to capture situations involving plant competition as well
   (J [S] or H [S]).

   A major strength of the plant-soil feedback modeling approach introduced by Bever
   and colleagues is its close connection to experimental plant-soil feedback pot
   experiments (Bever [206]1994; Bever [207]1999; Bever et al. [208]1997). However,
   the assumption of exponential plant growth, made in the original version of the
   model, is problematic, even in shorter-term pot experiments. This may be one
   reason why the original model does not predict the outcome of such experiments
   very well (Kulmatiski et al. [209]2011). By linking plant-soil feedback to the
   classical Lotka-Volterra competition model, Bever ([210]2003) included plant
   competition in his model. By providing the mathematical and graphical tools for
   analyzing this more complicated model, we hope to facilitate the extension of the
   plant-soil feedback approach to longer-term experiments and field situations.

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Acknowledgments

   We thank Wim van der Putten, Han Olff, Johan van de Koppel, Peter de Ruiter, and
   one anonymous reviewer for comments on previous versions of the manuscript. TR
   was supported by a grant from the Netherlands Organization for Scientific
   Research (NWO).

Open Access

   This article is distributed under the terms of the Creative Commons Attribution
   License which permits any use, distribution, and reproduction in any medium,
   provided the original author(s) and the source are credited.

Author information

Authors and Affiliations

    1. Theoretical Biology Group, Centre for Ecological and Evolutionary Studies,
       University of Groningen, P.O. Box 11103, 9700 CC, Groningen, The Netherlands
       Tomás A. Revilla & Franz J. Weissing
    2. Station d'Ecologie Experimentale du CNRS a Moulis, USR 2936, Moulis, 09200,
       Saint-Girons, France
       Tomás A. Revilla
    3. Community and Conservation Ecology Group, Centre for Ecological and
       Evolutionary Studies, University of Groningen, P.O. Box 11103, 9700 CC,
       Groningen, The Netherlands
       G. F. (Ciska) Veen
    4. Department of Terrestrial Ecology, Netherlands Institute of Ecology, P.O. Box
       50, 6700 AB, Wageningen, The Netherlands
       G. F. (Ciska) Veen
    5. Department of Forest Ecology and Management, Swedish University of
       Agricultural Sciences, 901 83, Umeå, Sweden
       G. F. (Ciska) Veen
    6. Department of Innovation and Environmental Sciences, Utrecht University, P.O.
       Box 80155, 3508 TC, Utrecht, The Netherlands
       Maarten B. Eppinga
    7. Instituto de Zoologia y Ecologia Tropical, Universidad Central de Venezuela,
       Av. Paseo Los Ilustres, Los Chaguaramos, Caracas, 1041-A, Venezuela
       Tomás A. Revilla

   Authors
    1. Tomás A. Revilla
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Corresponding author

   Correspondence to [344]Franz J. Weissing.

Additional information

   Tomás A. Revilla and G. F. (Ciska) Veen contributed equally to this paper.

Appendix A: Graphical analysis

Appendix A: Graphical analysis

   Figure [345]4 is useful for classifying and describing the main features of the
   dynamics in terms of the invasion conditions. For example in cases 1, 2, 11, and
   12, it is easy to see that only one species A or B always wins because only one
   of them is able to grow for any soil composition. In cases 4 and 14, any resident
   species will be protected against invasion, and any equilibrium that may exist
   will be competitively unstable, for any soil composition.

   However, cases 3, 5-10, 13, and 15-20 are more complicated. In these situations,
   it is sometimes useful to have a graphical representation in the familiar form of
   a phase space and nullclines. Although this is possible for a three-dimensional
   system like the Bever model, the following two-dimensional representation is more
   convenient:
   figure a

   The phase space is constructed according to the following rules:
     * The horizontal axis indicates the plant composition: plant A is dominant on
       the right, plant B on the left. The vertical axis indicates soil composition:
       A's soil biota dominates on the top, B's soil biota dominates on the bottom.
     * The corners represent the plant monocultures, with A [B ], B [B
       ]corresponding to S [A] = 0, and A [A ], B [A ]corresponding to S [A] = 1.
       Depending on the competitive stability conditions (7, 8, 11, 12) and their
       soil stability, the corners are classified as stable, unstable, or saddle
       points.
     * The invasion zones intersected by the feasibility arc become the domains of
       attraction in the phase plot, placed in the same order as they are
       encountered by traversing the arc from S [A] = 0 to 1 (and using the same
       fill patterns).
     * If the arc intersects zones III or IV, the corresponding domain of attraction
       in the phase plot is divided by a diagonal line. This line, representing
       coexistence equilibria, is the plant nullcline, i.e., a nullcline for the
       plant composition, not the plant densities. In the case of intersecting zone
       III, the plant composition moves towards the line (communities are
       competitively stable). In the case of intersection with zone IV, the plant
       composition moves away from the line (communities are competitively
       unstable).
     * The plane is divided by vertical that represents the non-trivial soil
       nullcline: S [A] increases at the right of the line (N [A] > vN [B] in Eq.
       2), and decreases at the left (N [A] < vN [B] in Eq. 2). The smaller the v
       the bigger the portion of the plane where S [A] increases, and vice versa.
       The top (S [A] = 1) and the bottom (S [A] = 0) sides of the plane are trivial
       soil nullclines.
     * A coexistence equilibrium corresponds to the intersection of the plant
       nullcline with a soil nullcline, trivial or not. For this reason, there can
       be border equilibria where S [A] = 0, S [A] = 1, or an internal equilibrium
       where S [A] is intermediate. Depending on its location with respect to the
       attraction domains and the non-trivial plant nullcline, an equilibrium is
       declared stable, unstable or a saddle point.

   Because of symmetry, we do not show cases 6, 8, 16, and 18 because they are
   qualitatively equivalent to cases 5, 7, 15, and 17 (by swapping the "A" and "B"
   labels). Cases 3 and 13 are very similar in the stability of their monocultures,
   and because of having border equilibria. However, they display qualitatively
   different dynamics. Under net positive feedbacks (case 3) the system can display
   alternative stable states: coexistence with dominance of plant A and its soil
   community or coexistence with dominance by plant B and its soil community. On the
   other hand, under net negative feedbacks (case 13) there cannot be alternative
   stable states, and oscillations may develop (though we suspect they dampen out
   given the geometry of the nullclines).

   The majority of cases under net positive feedback result in competitive
   exclusion. However, some can display alternative stable states, and coexistence
   depending on the initial conditions (5 and 6). On the other hand, the majority of
   scenarios under net negative feedback promote mutual invasion and coexistence (17
   and 18 are the exceptions), including coexistence through oscillations (19 and
   20).

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   Attribution 2.0 International License
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   use, distribution, and reproduction in any medium, provided the original work is
   properly cited.

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Cite this article

   Revilla, T.A., Veen, G.F., Eppinga, M.B. et al. Plant-soil feedbacks and the
   coexistence of competing plants. Theor Ecol 6, 99-113 (2013).
   https://doi.org/10.1007/s12080-012-0163-3

   [348]Download citation
     * Received: 26 April 2011
     * Accepted: 10 May 2012
     * Published: 09 June 2012
     * Issue Date: May 2013
     * DOI: https://doi.org/10.1007/s12080-012-0163-3

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Keywords

     * [349]Graphical analysis
     * [350]Oscillations
     * [351]Diversity
     * [352]Plant community dynamics
     * [353]Bever model

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