Spatial Heterogeneity in Soil Microbes Alters Outcomes of Plant Competition

Plant species vary greatly in their responsiveness to nutritional soil mutualists, such as mycorrhizal fungi and rhizobia, and this responsiveness is associated with a trade-off in allocation to root structures for resource uptake. As a result, the outcome of plant competition can change with the density of mutualists, with microbe-responsive plant species having high competitive ability when mutualists are abundant and non-responsive plants having high competitive ability with low densities of mutualists. When responsive plant species also allow mutualists to grow to greater densities, changes in mutualist density can generate a positive feedback, reinforcing an initial advantage to either plant type. We study a model of mutualist-mediated competition to understand outcomes of plant-plant interactions within a patchy environment. We find that a microbe-responsive plant can exclude a non-responsive plant from some initial conditions, but it must do so across the landscape including in the microbe-free areas where it is a poorer competitor. Otherwise, the non-responsive plant will persist in both mutualist-free and mutualist-rich regions. We apply our general findings to two different biological scenarios: invasion of a non-responsive plant into an established microbe-responsive native population, and successional replacement of non-responders by microbe-responsive species. We find that resistance to invasion is greatest when seed dispersal by the native plant is modest and dispersal by the invader is greater. Nonetheless, a native plant that relies on microbial mutualists for competitive dominance may be particularly vulnerable to invasion because any disturbance that temporarily reduces its density or that of the mutualist creates a window for a non-responsive invader to establish dominance. We further find that the positive feedbacks from associations with beneficial soil microbes create resistance to successional turnover. Our theoretical results constitute an important first step toward developing a general understanding of the interplay between mutualism and competition in patchy landscapes, and generate qualitative predictions that may be tested in future empirical studies.

Basic model for microbe-mediated plant competition: the D R = D I = 0 case The absence of the microbe in patch x means that with no dispersal, patch x is governed by the well-known Lotka-Volterra competition model. In this model, Notice that both of these equilibria are stable (i.e. we have alternative stable states) when c R > 1 and c I > 1.
With no dispersal, patch m also has two equilibria at which one plant excludes the other: R * m = ,Ĩ * m = 0 is, , (S1.1) which has eigenvalues − r R (bR * 2 m +k) k+R * m and r I (1 − c RR * m ). The equilibrium is stable if both of these eigenvalues are negative. The first eigenvalue is always negative and the second is The Jacobian matrix evaluated at the second equilibrium, Thus, the equilibrium [R * m = 0,Ĩ * m = 1] is stable whenever c I > k a . Because k a < 1 (see main text), notice that any c I > 1 will satisfy this condition.
We seek parameters that allow the microbe-independent plant to exclude the responsive plant in patch x ([R * x = 0,Ĩ * x = 1] stable: c R > 1) and the responsive plant to exclude the independent plant in patch m . These conditions define our parameter range of interest (main text inequalities (6)).
Notice that within this range, we always have alternative stable states in patch m ([R * m = 0,Ĩ * m = 1] is always stable since c I > 1 guarantees c I > k a ). We also may have alternative stable states in patch x, if c R > 1. There is, however, a range The D R , D I = ∞ case To study the dynamics with infinitely high dispersal rates, we follow the approach of Hastings (1982) and define new new variables: R s =R m +R x , R d =R m −R x , I s =Ĩ m +Ĩ x , and I d =Ĩ m −Ĩ x . As D R and D I → ∞, differences between patches vanish and R d and I d → 0, respectively. Dynamics of the remaining variables, R s and I s , are given by,

Densities of the responsive plants in both patches converge to Rs
Hereafter, we refer to Rs 2 simply as R. Likewise, the microbe-independent plant converges to a density that we will call I, equal to Is 2 , in both patches. We can then rewrite equations (S1.3) to give the dynamics in each patch under infinite dispersal, The Jacobian matrix for equations (S1.4) is, . (S1.5) At the trivial equilibrium, [R * = I * = 0], the eigenvalues of (S1.5) are r R 2 and r I , and so this equilibrium is never stable. The equilibrium where the microbe-independent plant excludes the responsive plant in both patches, [R * = 0, I * = 1], has eigenvalues r R 2 1 − c I a k − c I and −r I . The former is negative as long as c I > k k+a , which is automatically satisfied in our parameter range of interest, where c I > 1. Thus, the microbe-independent plant can always stably exclude the responsive plant.
Model (S1.4) potentially has 2 additional types of equilibria: exclusion by the responsive plant and coexistence. Exclusion of the microbe-independent plant by the responsive plant occurs where I * = 0 and R * is a solution to, (S1.6) Because equation (S1.6) is concave up (b + 1 > 0) with a negative y-intercept (−2k < 0), it has only 1 positive root. This root is always real. Plugging the positive solutions to equation (S1.6) into the Jacobian (S1.5) yielded no stable solutions for over 1 million randomly selected parameter sets from our range of interest; we therefore believe that this equilibrium is never stable under infinite dispersal, although we have not proven it analytically. Finally, there is a coexistence equilibrium where R * is a solution to, and I * = 1 − c R R * . From the same > 10 6 randomly drawn parameter sets mentioned above, none found a positive real root of (S1.7) that was < 1 c R (which is needed for I * > 0). Therefore, we conclude that coexistence is never feasible under infinite dispersal.
Realistic levels of plant dispersal (0 < D R , D I < ∞) The Jacobian of the full 2-patch model (equations (5) in the main text) is . The dashed lines in the Jacobian matrix (S1.8) divide the matrix into four 2×2 blocks, which is convenient for analysis because when the Jacobian is evaluated at several of the equilibria, the off-diagonal blocks (upper-right and/or lower-left) contain only zeros. In those cases, the eigenvalues of the entire Jacobian are the combined set of eigenvalues for the diagonal (upper-left and lower-right) blocks. Taking advantage of this and considering the diagonal blocks one at a time, rather than working with the entire Jacobian, is convenient because the Routh-Hurwitz criteria for eigenvalues < 0 (indicating a stable equilibrium point) are particularly simple when applied to 2×2 matrices.
Our model has four equilibria: 1. The trivial equilibrium:R * m =R * x =Ĩ * m =Ĩ * x = 0 2. Exclusion of the microbe-responsive plant in both patches:R * m =R * x = 0, I * m =Ĩ * x = 1 3. Exclusion of the microbe-resistant plant in both patches:R * m ,R * x > 0,Ĩ * m =Ĩ * x = 0 4. Coexistence in both patches:R * m ,R * x ,Ĩ * m ,Ĩ * x > 0 At the trivial equilibrium, both off-diagonal blocks of the Jacobian matrix contain only zeros so the eigenvalues of the entire Jacobian are the eigenvalues of the two diagonal blocks. These blocks both have the form, where s = R in the upper-left block and I in the lower-right, and have the eigenvalues r s and r s − 2D s . The eigenvalues of the Jacobian thus include both r R and r I , which are necessarily positive given that our parameters all have positive values. Therefore, the trivial equilibrium is always unstable. When the microbe-responsive plant is excluded in both patches, the upper right block contains only zeros and again we can get all the eigenvalues from the diagonal blocks, (S1.10) The second has eigenvalues −r I and −r I − 2D I , which are always negative. The eigenvalues of the first block are more complicated expressions, but we can apply the Routh-Hurwitz criteria to show that both are always negative as well. The Routh-Hurwitz criteria guarantee that the eigenvalues of a 2×2 matrix are negative if the determinant of the matrix is positive and the trace is negative. Here, the determinant is, As explained in the main text, the definitions of our rescaled parameters necessarily makes a k > 1. In addition, under the conditions for microbe-mediated competition, c I > 1. Thus, (1 − c I ), 1 − c I a k , and 2 − c I 1 + a k are all negative. The first term of (S1.11) is then positive and when we subtract the negative second term, the result will be positive and the first Routh-Hurwitz stability condition is satisfied. The trace of this block is, which is clearly always negative, satisfying the second stability condition. Therefore, the equilibrium wherein the microbe-responsive plant is excluded from both patches is always stable under the conditions of microbe-mediated competition.
At the equilibrium where the responsive plant excludes the microbe-independent plant in both patches,Ĩ * m =Ĩ * x = 0 andR * m is given by the implicit equation, (S1.14) At the fourth equilibrium, where both species persist in both patches, it is not possible to write even an implicit equation for any of the equilibrium densities, and so we found the population densities at this final equilibrium by simulation.
The fact that we cannot explicitly solve forR * m ,R * x > 0 limits our ability to derive useful stability criteria for the remaining two equilibria. We found the stability regions for these equilibria by finding the equilibrium densities for a given parameter combination using equations (S1.13)-(S1.14) (when the responsive plant excludes the microbe-independent plant) or by simulation (for coexistence), then plugging those densities into the full Jacobian matrix (S1.8) and finding the eigenvalues in Matlab.