The potential for alternative stable states in nutrient-enriched invaded grasslands

Abstract

Nutrient enrichment of native grasslands can promote invasion by exotic plant species, leading to reduced biodiversity and altered ecosystem function. Empirical evidence suggests that positive feedbacks may make such transitions difficult to reverse. We developed a mathematical model of grassland dynamics in which one group of species (native) is a better competitor for nitrogen (N) and another group (exotic) is a better competitor for light. We parameterized the model for a grassland community and reproduced observed transitions from a native- to an exotic-dominated state under N loading. Within known bounds of parameter values, both smooth and hysteretic transitions are plausible. The model also predicts that N loading alone is insufficient to achieve a transition to an exotic-dominated state on a timescale relevant to grassland management (a few decades), and that therefore some other disturbance (e.g., fire suppression or heaving grazing) must be present to accelerate it. The model predicts that to restore a grassland to a native-dominated state after N inputs have been reduced, fire and carbon supplements would be most effective. Further field research in N-enriched invaded grasslands is required to establish the strengths of positive feedbacks and, in turn, the consequences of anthropogenic modification of grasslands worldwide.

Appendices

Appendix 1: Single-species equilibria

To obtain the native-dominated equilibrium, we set the left-hand sides of all three equations in (1) to zero and set ${\overline{B}}_e=0$, ${\overline{B}}_n>0$ to obtain

$$ \begin{array}{c}\hfill {\omega}_n{\nu}_n\overline{A}-{\mu}_n-{m}_n{g}_n\left({\overline{B}}_n,0\right)=0\hfill \\ {}\hfill I-k\overline{A}-{\overline{B}}_n\left({\nu}_n\overline{A}-\left({\mu}_n+{m}_n{g}_n\left({\overline{B}}_n,0\right)\right)\frac{\left(1-{\delta}_n\right)}{\omega_n}\right)=0\hfill \end{array} $$

(8)

From the first equation above (assuming that the maximum value of cover ${g}_n\left({\overline{B}}_n,0\right)=1$ can be approached asymptotically but not actually attained in biologically realistic settings), we obtain bounds on possible values of Ā:

$$ {\overline{A}}_{lo,n}\equiv \frac{\mu_n}{\omega_n{\nu}_n} < \overline{A}<\frac{\mu_n+{m}_n}{\omega_n{\nu}_n}\equiv {\overline{A}}_{hi,n} $$

(9)

In Eq. (8), Ā can be eliminated to give

$$ \left({\nu}_n{\delta}_n{\overline{B}}_n+k\right)\left({\mu}_n+{m}_n{g}_n\left({\overline{B}}_n,0\right)\right)=I{\nu}_n{\omega}_n $$

which shows that there is at most one solution for ${\overline{B}}_n$, because the left-hand side is an increasing function of ${\overline{B}}_n$. This means there is at most one native-dominated equilibrium. If I < I _min = kμ _n /ν _n ω _n (see main text), it is easily shown from the above equation that there is no native-dominated positive equilibrium (this makes sense, because this is the regime in which the trivial equilibrium is stable). Given that negative biomass and negative N make no sense, we hereafter use “equilibrium” and “positive equilibrium” synonymously. If I > I _min, there is exactly one native-dominated equilibrium.

Depending on the form of the function g _n , it may be possible to solve for ${\overline{B}}_n$ analytically but in general the solution can be found numerically. To calculate Ā, we then have

$$ \overline{A}=\frac{I}{k+{\overline{B}}_n{\nu}_n{\delta}_n} $$

The exotic-dominated equilibrium is symmetric with the native-dominated equilibrium. Thus, the exotic-dominated equilibrium exists when I > kμ _e /ν _e ω _e (by (2), this is a stricter condition than that required for the native-dominated equilibrium to exist).

Appendix 2: Stability analysis of trivial and single-species equilibria

The Jacobian for general B _n , B _e , and A is

$$ \begin{array}{c}\hfill M={\left[\begin{array}{ccc}\hfill {\omega}_n{\nu}_nA-{\mu}_n-{m}_n{f}_n\left({B}_n,{B}_e\right)-{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}\hfill & \hfill -{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_e}\hfill & \hfill {B}_n{\omega}_n{\nu}_n\hfill \\ {}\hfill -{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_n}\hfill & \hfill {\omega}_e{\nu}_eA-{\mu}_e-{m}_e{f}_e\left({B}_n,{B}_e\right)-{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_e}\hfill & \hfill {B}_e{\omega}_e{\nu}_e\hfill \\ {}\hfill x\hfill & \hfill y\hfill & \hfill -k-{B}_n{\nu}_n-{B}_e{\nu}_e\hfill \end{array}\right]}_{eq}\hfill \\ {}\hfill x=-{\nu}_nA+\frac{\left({\mu}_n+{m}_n{f}_n\left({B}_n,{B}_e\right)\right)\left(1-{\delta}_n\right)}{\omega_n}+{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}\frac{1-{\delta}_n}{\omega_n}+{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_n}\frac{1-{\delta}_e}{\omega_e}\hfill \\ {}\hfill y=-{\nu}_eA+\frac{\left({\mu}_e+{m}_e{f}_e\left({B}_n,{B}_e\right)\right)\left(1-{\delta}_e\right)}{\omega_e}+{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_e}\frac{1-{\delta}_n}{\omega_n}+{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_e}\frac{1-{\delta}_e}{\omega_e}\hfill \end{array} $$

where eq indicates that the matrix is evaluated at the equilibrium ${B}_n={\overline{B}}_n,{B}_e={\overline{B}}_e,A=\overline{A}$. We omit eq for brevity in much of the following.

Write the characteristic equation of this matrix as

$$ -{\lambda}^3+{c}_1{\lambda}^2-{c}_2\lambda +{c}_3=0 $$

where

$$ {c}_1=\mathrm{T}\mathrm{r}(M),\ {c}_3=\mathrm{D}\mathrm{e}\mathrm{t}(M) $$

and c ₂ is the sum of principal minors of M. According to the three-dimensional Routh–Hurwitz criteria, an equilibrium will be stable if and only if c ₁ < 0, c ₃ < 0 and c ₁ c ₂ < c ₃. Alternatively, the eigenvalues can be checked directly (their real parts must all be negative for stability).

Trivial equilibrium

The Jacobian at the trivial equilibrium simplifies to

$$ M={\left[\begin{array}{ccc}\hfill {\omega}_n{\nu}_n\frac{I}{k}-{\mu}_n\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\omega}_e{\nu}_e\frac{I}{k}-{\mu}_e\hfill & \hfill 0\hfill \\ {}\hfill -{\nu}_n\frac{I}{k}+\frac{\mu_n\left(1-{\delta}_n\right)}{\omega_n}\hfill & \hfill -{\nu}_e\frac{I}{k}+\frac{\mu_e\left(1-{\delta}_e\right)}{\omega_e}\hfill & \hfill -k\hfill \end{array}\right]}_{eq} $$

The eigenvalues can be read off the diagonal, leading to the stability criterion

$$ I<k \min \left(\frac{\mu_n}{\omega_n{\nu}_n},\frac{\mu_e}{\omega_e{\nu}_e}\right)=k\frac{\mu_n}{\omega_n{\nu}_n}\equiv {I}_{\min } $$

where the min() is evaluated using constraint (2).

Native-dominated equilibrium

The Jacobian at the native-dominated equilibrium simplifies to

$$ \begin{array}{c}\hfill M={\left[\begin{array}{ccc}\hfill -{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}\hfill & \hfill -{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_e}\hfill & \hfill {B}_n{\omega}_n{\nu}_n\hfill \\ {}\hfill 0\hfill & \hfill {\omega}_e{\nu}_eA-{\mu}_e-{m}_e{f}_e\left({B}_n,0\right)\hfill & \hfill 0\hfill \\ {}\hfill x\hfill & \hfill y\hfill & \hfill -k-{B}_n{\nu}_n\hfill \end{array}\right]}_{eq}\hfill \\ {}\hfill x=-{\nu}_nA+\frac{\left({\mu}_n+{m}_n{f}_n\left({B}_n,0\right)\right)\left(1-{\delta}_n\right)}{\omega_n}+{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}\frac{1-{\delta}_n}{\omega_n}\hfill \\ {}\hfill y=-{\nu}_eA+\frac{\left({\mu}_e+{m}_e{f}_e\left({B}_n,0\right)\right)\left(1-{\delta}_e\right)}{\omega_e}+{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_e}\frac{1-{\delta}_n}{\omega_n}\hfill \end{array} $$

Note that because ${\overline{B}}_e=0$ at the equilibrium, we have $\frac{\partial {f}_n}{\partial {B}_n}>0$.

The matrix M has the following form, with a _ij  > 0 and the signs of w, x, y and z unknown:

$$ M=\left[\begin{array}{ccc}\hfill -{a}_{00}\hfill & \hfill w\hfill & \hfill {a}_{02}\hfill \\ {}\hfill 0\hfill & \hfill z\hfill & \hfill 0\hfill \\ {}\hfill x\hfill & \hfill y\hfill & \hfill -{a}_{22}\hfill \end{array}\right] $$

If we interchange the second and third rows of this matrix, and the second and third columns (this is equivalent to just reordering the variables) we obtain

$$ M\hbox{'}=\left[\begin{array}{ccc}\hfill -{a}_{00}\hfill & \hfill {a}_{02}\hfill & \hfill w\hfill \\ {}\hfill x\hfill & \hfill -{a}_{22}\hfill & \hfill y\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill z\hfill \end{array}\right] $$

Two eigenvalues of M′ are the eigenvalues of the upper two-by-two matrix, and these both have negative real part because the trace of this submatrix is negative and the determinant is

$$ {a}_{00}{a}_{22}-{a}_{02}y=k{m}_n\frac{\partial {f}_n}{\partial {B}_n}+{B}_n{\nu}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}{\delta}_n+{\delta}_n{\omega}_n{\nu}_n^2A $$

which is positive, because all terms on the right-hand side are positive. Hence, stability depends only on the sign of the remaining (real) eigenvalue z, i.e., the equilibrium is stable if and only if

$$ z={\omega}_e{\nu}_e\overline{A}-{\mu}_e-{m}_e{f}_e\left({\overline{B}}_n,0\right)<0 $$

This stability condition has a clear biological interpretation: it is simply the reverse of the invasion criterion for the exotic species (i.e., dB _e /dt > 0). The stability condition leads to a condition on Ā (by way of ${f}_e\left({\overline{B}}_n,0\right)={\alpha}_{en}{f}_n\left({\overline{B}}_n,0\right)={\alpha}_{en}\left(\left({\omega}_n{\nu}_n\overline{A}-{\mu}_n\right)/{m}_n\right)$:

$$ \left(\frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en}\right)\overline{A}<\frac{m_n}{\omega_n{\nu}_n}\frac{\mu_e}{\omega_e{\nu}_e}-\frac{m_e}{\omega_e{\nu}_e}\frac{\mu_n}{\omega_n{\nu}_n}{\alpha}_{en} $$

(10)

We know from relation (9) that if the equilibrium exists then we have

$$ {\overline{A}}_{lo,n}\equiv \frac{\mu_n}{\omega_n{\nu}_n}<\overline{A}<\frac{\mu_n}{\omega_n{\nu}_n}+\frac{m_n}{\omega_n{\nu}_n}\equiv {\overline{A}}_{hi,n} $$

and in fact we can write (from Eq. (1))

$$ \overline{A}=\frac{\mu_n}{\omega_n{\nu}_n}+{\overline{g}}_n\frac{m_n}{\omega_n{\nu}_n} $$

where ${\overline{g}}_n\in \left(0,1\right)$ is the cover of the native species at equilibrium. Substituting this expression for Ā into the stability condition and simplifying gives

$$ {\overline{g}}_n\left(\frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en}\right)<\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n} $$

To investigate this stability criterion further, we observe that from constraint (2), we know that the right-hand side is positive. A sufficient condition for stability is thus

$$ \frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en}\le \frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n} $$

(11)

(The quantity V defined in the main text is the left-hand side of this inequality minus the right-hand side.) If, however, condition (11) does not hold then the stability condition is satisfied only if

$$ {\overline{g}}_n<\frac{\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n}}{\frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en}}\equiv {\overline{g}}_{crit,n} $$

or, equivalently,

$$ \overline{A}<\frac{\frac{\mu_e}{m_e}-\frac{\mu_n}{m_n}\ {\alpha}_{en}}{\frac{\omega_e{\nu}_e}{m_e}-\frac{\omega_n{\nu}_n}{m_n}{\alpha}_{en}}\equiv {\overline{A}}_{crit,n} $$

Corresponding critical values of ${\overline{B}}_n$ and I can be calculated as follows. First, note that

$$ {\overline{B}}_n={\left.{g}_n^{-1}\right|}_{{\overline{B}}_e=0}\left({\overline{g}}_n\right)={\left.{g}_n^{-1}\right|}_{{\overline{B}}_e=0}\left(\frac{\omega_n{\nu}_n\overline{A}-{\mu}_n}{m_n}\right) $$

(12)

Substituting in the critical value ${\overline{g}}_{crit,n}$ gives

$$ {\overline{B}}_{crit,n}={\left.{g}_n^{-1}\right|}_{{\overline{B}}_e=0}\left(\frac{\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n}}{\frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en}}\right) $$

We can also derive a critical value of the control parameter I:

$$ \begin{array}{c}\hfill I=\left(k+{\overline{B}}_n{\delta}_n{\nu}_n\right)\overline{A}\hfill \\ {}\hfill {I}_{crit,n}=\left(k+{\overline{B}}_{crit,n}{\delta}_n{\nu}_n\right){\overline{A}}_{crit,n}\hfill \end{array} $$

Because ${\overline{B}}_n$ is monotonically and positively related to Ā (Eq. (12) with relation (9)), and I is monotonically and positively related to both ${\overline{B}}_n$ and Ā, we have stability for I < I _crit,n . It is easy to show that we have I _crit,n  > I _min. So the equilibrium exists and is stable for I _min < I < I _crit,n and exists and is unstable for I > I _crit,n .

In summary, there will always be at least some values of the control parameter I (nutrient input) for which there is a stable native-dominated equilibrium. The equilibrium always exists for nutrient input above a threshold I _min , but it will be unstable above a higher threshold I _crit,n if the native species cannot draw down nitrogen to sufficiently low levels (i.e., if condition (11) does not hold). At this point, the exotic can invade.

Exotic-dominated equilibrium

Bounds on Ā at the exotic-dominated equilibrium can be derived that are symmetric with the bounds at the native-dominated equilibrium:

$$ {\overline{A}}_{lo,e}\equiv \frac{\mu_e}{\omega_e{\nu}_e} < \overline{A}<\frac{\mu_e+{m}_e}{\omega_e{\nu}_e}\equiv {\overline{A}}_{hi,e} $$

(13)

And we can write

$$ \overline{A}=\frac{\mu_e}{\omega_e{\nu}_e}+{\overline{g}}_e\frac{m_e}{\omega_e{\nu}_e} $$

where ${\overline{g}}_e\in \left(0,1\right)$ is the cover of the exotic species at equilibrium. The results of the stability analysis of the exotic-dominated equilibrium are also symmetric with that of the native-dominated equilibrium, up to the point where we derive the general stability criterion:

$$ \left(\frac{m_e}{\omega_e{\nu}_e}-\frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}\right)\overline{A}<\frac{m_e}{\omega_e{\nu}_e}\frac{\mu_n}{\omega_n{\nu}_n}-\frac{m_n}{\omega_n{\nu}_n}\frac{\mu_e}{\omega_e{\nu}_e}{\alpha}_{ne} $$

(14)

or, in terms of equilibrium cover of the exotic species:

$$ {\overline{g}}_e\left(\frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}-\frac{m_e}{\omega_e{\nu}_e}\right)>\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n} $$

The right-hand side is positive here by constraint (2). Because $0<{\overline{g}}_e<1$, a necessary condition for stability is

$$ \frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}-\frac{m_e}{\omega_e{\nu}_e}>\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n} $$

(15)

(The quantity U defined in the main text is the left-hand side of this minus the right-hand side). We can then see that necessary and sufficient conditions for stability of the exotic-dominated equilibrium are condition (15) together with

$$ {\overline{g}}_e>\frac{\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n}}{\frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}-\frac{m_e}{\omega_e{\nu}_e}}\equiv {\overline{g}}_{crit,e} $$

or, equivalently,

$$ \overline{A}>{\overline{A}}_{crit,e}\equiv \frac{\frac{\mu_n}{m_n}-\frac{\mu_e}{m_e}\ {\alpha}_{ne}}{\frac{\omega_n{\nu}_n}{m_n}-\frac{\omega_e{\nu}_e}{m_e}{\alpha}_{ne}} $$

A corresponding critical value of ${\overline{B}}_e$ can then be derived:

$$ {\overline{B}}_e={\left.{g}_e^{-1}\right|}_{{\overline{B}}_n=0}\left(\frac{\omega_e{\nu}_e\overline{A}-{\mu}_e}{m_e}\right) $$

(16)

This is always real as guaranteed by (13). Then we have

$$ {\overline{B}}_{crit,e}={\left.{g}_e^{-1}\right|}_{{\overline{B}}_n=0}\left(\frac{\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n}}{\frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}-\frac{m_e}{\omega_e{\nu}_e}}\right) $$

and

$$ \begin{array}{c}\hfill I=\left(k+{\overline{B}}_e{\delta}_e{\nu}_e\right)\overline{A}\hfill \\ {}\hfill {I}_{\mathrm{crit},e}=\left(k+{\overline{B}}_{\mathrm{crit},e}{\delta}_e{\nu}_e\right){\overline{A}}_{\mathrm{crit},e}\hfill \end{array} $$

Because ${\overline{B}}_e$ increases monotonically with Ā, and I increases monotonically with ${\overline{B}}_e$ and Ā, we have stability for I > I _crit,e . It is easy to show that we have I _crit,e  > kμ _e /ν _e ω _e , which is the threshold above which the exotic-dominated equilibrium exists.

In summary, there will be a stable exotic-dominated equilibrium above some threshold nutrient input I _crit,e if and only if condition (15) holds.

Bistability

We are also interested to know when bistability between the native-dominated equilibrium and the exotic-dominated equilibrium is possible. The two equilibria both exist when I > kμ _e /ν _e ω _e . We get bistability in two cases: (i) if conditions (2) and (6) hold, then we have bistability for I > I _crit,e ; (ii) if condition (2) fails but (6) holds and I _crit,e  < I _crit,n , then we have bistability for I _crit,e  < I < I _crit,n . The difference between these two cases is biologically significant. In the first case, the native-dominated equilibrium never destabilizes, even for very high nutrient inputs. In the second case, the native-dominated equilibrium destabilizes for high nutrient inputs, in which case the only stable state is dominated by the exotic.

Appendix 3: Coexistence equilibria

Here, we conduct a general investigation of the coexistence equilibria and their stability without specifying a functional form of the light competition functions g _i . Observe that at a coexistence equilibrium (straight from the model definition given by Eq. (1)) we have

$$ \begin{array}{c}\hfill \overline{A} = \frac{\mu_n+{m}_n{f}_n\left({\overline{B}}_n,{\overline{B}}_e\right)}{\omega_n{\nu}_n}\hfill \\ {}\hfill \overline{A}=\frac{\mu_e+{m}_e{f}_e\left({\overline{B}}_n,{\overline{B}}_e\right)}{\omega_e{\nu}_e}\hfill \end{array} $$

(17)

We can obtain expressions relating the cover of each species ${\overline{g}}_i$ to the available nitrogen pool Ā at the coexistence equilibrium by substituting in the expression for the f _i functions into (8) and solving:

$$ \begin{array}{c}\hfill {\overline{g}}_n=\frac{1}{1-{\alpha}_{ne}{\alpha}_{en}}\left\{\frac{\omega_n{\nu}_n\overline{A}-{\mu}_n}{m_n}-{\alpha}_{ne}\left(\frac{\omega_e{\nu}_e\overline{A}-{\mu}_e}{m_e}\right)\right\}\hfill \\ {}\hfill {\overline{g}}_e=\frac{1}{1-{\alpha}_{ne}{\alpha}_{en}}\left\{\frac{\omega_e{\nu}_e\overline{A}-{\mu}_e}{m_e}-{\alpha}_{en}\left(\frac{\omega_n{\nu}_n\overline{A}-{\mu}_n}{m_n}\right)\right\}\hfill \end{array} $$

(18)

The ${\overline{g}}_i$ will clearly be real if Ā is real. For a coexistence equilibrium to be biologically sensible, we require ${\overline{g}}_n>0$, ${\overline{g}}_e>0$, and ${\overline{g}}_n+{\overline{g}}_e<1$ (we also require ${\overline{B}}_i$ real and positive; for the specific form of g _i introduced in Eq. (7), we can always get a real solution with ${\overline{B}}_i>0$ if these conditions on ${\overline{g}}_i$ are satisfied). We now use the conditions on ${\overline{g}}_i$ to establish equilibrium levels of the available nitrogen pool Ā at which coexistence equilibria exist.

First, we consider the conditions ${\overline{g}}_n>0$ and ${\overline{g}}_e>0$. If 1 − α _ne α _en  > 0, the condition ${\overline{g}}_e>0$ reduces to the reverse of stability criterion (10) for the native-dominated equilibrium, ${\overline{g}}_n>0$ reduces to the reverse of the stability criterion (5) for the exotic-dominated equilibrium, and we have a potential coexistence equilibrium only if Ā _crit,n  < Ā < Ā _crit,e . So potential coexistence equilibria occur only for values of the available nitrogen pool Ā at which there do not exist stable native- and exotic-dominated equilibria, and thresholds for destabilization of boundary equilibria are also thresholds for the appearance of coexistence equilibria. A coexistence equilibrium is guaranteed to be well-defined infinitesimally close to one of these thresholds, because it interpolates continuously with the boundary equilibrium, but it may or may not be stable. If 1 − α _ne α _en  < 0, then the conditions just discussed reverse and we have a potential coexistence equilibrium only if Ā _crit,e  < Ā < A _crit,n , i.e., for values of Ā where there is both a stable native-dominated and a stable exotic-dominated equilibrium, and thresholds for stabilization of boundary equilibria are thresholds for the appearance of coexistence equilibria.

So, regardless of the sign of 1 − α _ne α _en , we have potential coexistence equilibria only over a single interval of Ā, and as seen from Eq. (18), ${\overline{g}}_i$ linearly interpolates from ${\overline{g}}_{crit,i}$ to 0 over an interval for Ā. Thus, we must have ${\overline{g}}_{\mathrm{crit},i}>0$ for both species, if each species’ cover is to be defined sensibly somewhere along an interval. Also, to have potential coexistence over some range of Ā, at least one of the two species must have ${\overline{g}}_{\mathrm{crit},i}<1$, because otherwise we would have ${\overline{g}}_{\mathrm{crit},n}+{\overline{g}}_{\mathrm{crit},e}>1$ over the whole interval for Ā.

We now establish the exact ranges of Ā over which a coexistence equilibrium will be present (not necessarily a stable one) by investigating when the condition ${\overline{g}}_n+{\overline{g}}_e<1$ is satisfied. We will have a potential critical value of Ā when ${\overline{g}}_n+{\overline{g}}_e=1$, i.e., when (from Eq. (18)):

$$ \overline{A}={\overline{A}}_{\mathrm{crit},c}\equiv \frac{1-{\alpha}_{ne}{\alpha}_{en}+\frac{\mu_n}{m_n}\left(1-{\alpha}_{en}\right)+\frac{\mu_e}{m_e}\left(1-{\alpha}_{ne}\right)}{\frac{\omega_n{\nu}_n}{m_n}\left(1-{\alpha}_{en}\right)+\frac{\omega_e{\nu}_e}{m_e}\left(1-{\alpha}_{ne}\right)} $$

Because ${\overline{g}}_n+{\overline{g}}_e$ is linear in Ā, a coexistence equilibrium can be well-defined only for Ā < Ā _crit,c or for Ā > Ā _crit,c but not both. We now divide possible parameterizations of the model into the same four cases (A–D) used for assessing stability of the boundary equilibria in the main text.

In Case A (U < 0, V < 0), it can be shown by manipulating the conditions U < 0 and V < 0 that ${\overline{g}}_{\mathrm{crit},i}\notin \left[0,1\right]$ for both species, so there is no way to get coexistence.

In Case B (U > 0, V < 0), U > 0 guarantees $0<{\overline{g}}_{\mathrm{crit},e}<1$. And from the conditions U > 0 and V < 0 we can deduce that

$$ \frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en} $$

has the same sign as 1 − α _ne α _en . Using this together with condition V < 0, we can see that if 1 − α _ne α _en  > 0 we must have ${\overline{g}}_{\mathrm{crit},n}>1$ and thus have a coexistence equilibrium for Ā _crit,c  < Ā < Ā _crit,e , and that if 1 − α _ne α _en  < 0 we must have ${\overline{g}}_{\mathrm{crit},n}<0$ and thus have coexistence equilibrium for Ā _crit,e  < Ā < Ā _crit,c .

In Case C (U < 0, V > 0), V > 0 guarantees $0<{\overline{g}}_{\mathrm{crit},n}<1$. Also, we can use U < 0, V > 0 and α _ne  > α _en to show that 1 − α _ne α _en  > 0 and then that

$$ \frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}-\frac{m_e}{\omega_e{\nu}_e}<0 $$

implying ${\overline{g}}_{\mathrm{crit},e}<0$. So from these constraints on ${\overline{g}}_{\mathrm{crit},e}$ and ${\overline{g}}_{\mathrm{crit},n}$ we see that we get a coexistence equilibrium for Ā _crit,n  < Ā < Ā _crit,c .

In Case D (U > 0, V > 0), we are guaranteed $0<{\overline{g}}_{\mathrm{crit},i}<1$ for both species, and we thus get a coexistence equilibrium for every value of Ā in the interval Ā _crit,n  < Ā < Ā _crit,e .

Numerically, we observe that a point in parameter space where a coexistence equilibrium (internal equilibrium) appears and a boundary equilibrium destabilizes corresponds to a transcritical bifurcation where the two equilibria exchange stability. At one of these bifurcations, there can be a smooth transition between a boundary equilibrium and the internal equilibrium (e.g., transitions between the native-dominated and coexistence equilibria in Fig. 1, right panels, and in Fig. A2). But if the internal equilibrium is not biologically well-defined on one side of the bifurcation (${\overline{B}}_i<0$), the system can exhibit a flip from one boundary equilibrium to the other (e.g., Fig. 1, left panels starting at the exotic-dominated equilibrium for high I and then decreasing I below I _crit,e ). In the next subsection, we investigate the stability of the coexistence equilibrium mathematically.

Stability of coexistence equilibria

For brevity, we omit the horizontal bars indicating equilibrium values in much of the following. Taking derivatives of Eq. (8) with respect to B _n gives

$$ \begin{array}{c}\hfill \frac{dA}{d{B}_n}=\frac{m_n}{\omega_n{\nu}_n}\left(\frac{\partial {f}_n}{\partial {B}_n}+\frac{\partial {f}_n}{\partial {B}_e}\frac{d{B}_e}{d{B}_n}\right)\hfill \\ {}\hfill \frac{dA}{d{B}_e}=\frac{m_e}{\omega_e{\nu}_e}\left(\frac{\partial {f}_e}{\partial {B}_e}+\frac{\partial {f}_e}{\partial {B}_n}\frac{d{B}_n}{d{B}_e}\right)\hfill \end{array} $$

(19)

Define

$$ \begin{array}{c}\hfill {\xi}_1\equiv \frac{m_n}{\omega_n{\nu}_n}\frac{\partial {f}_n}{\partial {B}_e}-\frac{m_e}{\omega_e{\nu}_e}\frac{\partial {f}_e}{\partial {B}_e}=\left(\frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en}\right)\frac{\partial {g}_n}{\partial {B}_e}+\left(\frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}-\frac{m_e}{\omega_e{\nu}_e}\right)\frac{\partial {g}_e}{\partial {B}_e}\hfill \\ {}\hfill {\xi}_2\equiv -\frac{m_n}{\omega_n{\nu}_n}\frac{\partial {f}_n}{\partial {B}_n}+\frac{m_e}{\omega_e{\nu}_e}\frac{\partial {f}_e}{\partial {B}_n}=\left(\frac{m_n}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}{\alpha}_{en}\right)\frac{\partial {g}_n}{\partial {B}_n}+\left(\frac{m_n}{\omega_n{\nu}_n}{\alpha}_{ne}-\frac{m_e}{\omega_e{\nu}_e}\right)\frac{\partial {g}_e}{\partial {B}_n}\hfill \\ {}\hfill {\xi}_3\equiv -\frac{\partial {f}_n}{\partial {B}_n}\frac{\partial {f}_e}{\partial {B}_e}+\frac{\partial {f}_n}{\partial {B}_e}\frac{\partial {f}_e}{\partial {B}_n}=-\left(1-{\alpha}_{ne}{\alpha}_{en}\right)\left(\frac{\partial {g}_n}{\partial {B}_n}\frac{\partial {g}_e}{\partial {B}_e}-\frac{\partial {g}_e}{\partial {B}_n}\frac{\partial {g}_n}{\partial {B}_e}\right)\hfill \end{array} $$

Solving Eq. (19) for dA/dB _n and dB _e /dB _n then gives

$$ \begin{array}{c}\hfill \frac{d{B}_e}{d{B}_n}=\frac{\xi_2}{\xi_1}\hfill \\ {}\hfill \frac{dA}{d{B}_n}=\frac{m_n}{\omega_n{\nu}_n}\frac{m_e}{\omega_e{\nu}_e}\frac{\xi_3}{\xi_1}\hfill \\ {}\hfill \frac{dA}{d{B}_e}=\frac{m_n}{\omega_n{\nu}_n}\frac{m_e}{\omega_e{\nu}_e}\frac{\xi_3}{\xi_2}\hfill \end{array} $$

The Jacobian at a coexistence equilibrium simplifies to

$$ \begin{array}{c}\hfill M=\left[\begin{array}{ccc}\hfill -{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}\hfill & \hfill -{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_e}\hfill & \hfill {B}_n{\omega}_n{\nu}_n\hfill \\ {}\hfill -{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_n}\hfill & \hfill -{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_e}\hfill & \hfill {B}_e{\omega}_e{\nu}_e\hfill \\ {}\hfill x\hfill & \hfill y\hfill & \hfill -k-{B}_n{\nu}_n-{B}_e{\nu}_e\hfill \end{array}\right]\hfill \\ {}\hfill x=-{\delta}_n{\nu}_nA+{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}\frac{1-{\delta}_n}{\omega_n}+{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_n}\frac{1-{\delta}_e}{\omega_e}\hfill \\ {}\hfill y=-{\delta}_e{\nu}_eA+{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_e}\frac{1-{\delta}_n}{\omega_n}+{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_e}\frac{1-{\delta}_e}{\omega_e}\hfill \end{array} $$

Define the following function:

$$ \eta \left({B}_n,{B}_e,A\right) = kA+{B}_n\left({\nu}_nA-\left({\mu}_n+{m}_n{f}_n\left({B}_n,{B}_e\right)\right)\frac{\left(1-{\delta}_n\right)}{\omega_n}\right)+{B}_e\left({\nu}_eA-\left({\mu}_e+{m}_e{f}_e\left({B}_n,{B}_e\right)\right)\frac{\left(1-{\delta}_e\right)}{\omega_e}\right) $$

At equilibrium, we have I = η(B _n , B _e , A)|_eq, where the subscript eq. indicates that the function is evaluated at ${B}_n={\overline{B}}_n$, ${B}_e={\overline{B}}_e$, A = Ā.

The Jacobian matrix can be written as

$$ M={\left[\begin{array}{ccc}\hfill -{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_n}\hfill & \hfill -{B}_n{m}_n\frac{\partial {f}_n}{\partial {B}_e}\hfill & \hfill {B}_n{\omega}_n{\nu}_n\hfill \\ {}\hfill -{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_n}\hfill & \hfill -{B}_e{m}_e\frac{\partial {f}_e}{\partial {B}_e}\hfill & \hfill {B}_e{\omega}_e{\nu}_e\hfill \\ {}\hfill -\frac{\partial \eta }{\partial {B}_n}\hfill & \hfill -\frac{\partial \eta }{\partial {B}_e}\hfill & \hfill -\frac{\partial \eta }{\partial A}\hfill \end{array}\right]}_{eq} $$

The condition on the determinant for stability is

$$ \mathrm{D}\mathrm{e}\mathrm{t}(M)={B}_n{B}_e\left[{\omega}_n{\nu}_n{\omega}_e{\nu}_e\left(\frac{\partial \eta }{\partial {B}_n}\left\{\frac{m_n}{\omega_n{\nu}_n}\frac{\partial {f}_n}{\partial {B}_e}-\frac{m_e}{\omega_e{\nu}_e}\frac{\partial {f}_e}{\partial {B}_e}\right\}+\frac{\partial \eta }{\partial {B}_e}\left\{-\frac{m_n}{\omega_n{\nu}_n}\frac{\partial {f}_n}{\partial {B}_n}+\frac{m_e}{\omega_e{\nu}_e}\frac{\partial {f}_e}{\partial {B}_n}\right\}\right)+\frac{\partial \eta }{\partial A}{m}_n{m}_e\left\{-\frac{\partial {f}_n}{\partial {B}_n}\frac{\partial {f}_e}{\partial {B}_e}+\frac{\partial {f}_n}{\partial {B}_e}\frac{\partial {f}_e}{\partial {B}_n}\right\}\right]<0 $$

(20)

Using the definitions of ξ _i from earlier, the stability criterion (11) can be written as

$$ {\xi}_3{B}_n{B}_e\left[\frac{\omega_n{\nu}_n{\omega}_e{\nu}_e}{m_n{m}_e}\left(\frac{\partial \eta }{\partial {B}_n}\frac{\xi_n}{\xi_3}+\frac{\partial \eta }{\partial {B}_e}\frac{\xi_2}{\xi_3}\right)+\frac{\partial \eta }{\partial A}\right]<0 $$

(21)

From (12) the necessary condition for stability then can be written as:

$$ \left(1-{\alpha}_{ne}{\alpha}_{en}\right){\left[\frac{\partial \eta }{\partial {B}_n}\frac{d{B}_n}{dA}+\frac{\partial \eta }{\partial {B}_e}\frac{d{B}_e}{dA}+\frac{\partial \eta }{\partial A}\right]}_{eq}>0 $$

But the bracketed term on the left-hand side is just dη/dA| _eq . Because I = η(B _n , B _e , A)| _eq , this tells us that a necessary condition for stability (corresponding to the second Routh–Hurwitz criterion) is that

$$ \left(1-{\alpha}_{ne}{\alpha}_{en}\right)\frac{dI}{d\overline{A}}>0 $$

(22)

An equivalent criterion that is easier to interpret biologically is

$$ \left(1-{\alpha}_{ne}{\alpha}_{en}\right)\frac{d\overline{A}}{dI}>0 $$

For stability of a coexistence equilibrium, there are two other necessary conditions (the other two Routh–Hurwitz criteria):

$$ \begin{array}{c}\hfill \begin{array}{c}\hfill {c}_1=\mathrm{T}\mathrm{r}(M)<0\hfill \\ {}\hfill {c}_1{c}_2<{c}_3\hfill \end{array}\hfill \\ {}\hfill {c}_1=\mathrm{T}\mathrm{r}(M)<0\hfill \\ {}\hfill {c}_1{c}_2<{c}_3\hfill \end{array} $$

Numerical investigations indicate that for most biologically plausible functional forms of the g _i functions and in most regions of parameter space where (13) holds, these two conditions also hold, although it is possible to find exceptions (e.g., Fig. 2d, although the bifurcation occurs only slightly to the left of the maximum value of I).

Appendix 4: Coexistence equilibria for specific form of biomass–cover functions

We now use the specific form of the light competition functions g _i given by Eq. (7) and derive implicit equations for the coexistence equilibria in terms of the control parameter I. We also take the stability analysis from Appendix III further. We assume that the model is parameterized such that native-dominated and exotic-dominated equilibria both exist over some range of the control parameter (i.e., that we have bistability between these two states). This means that condition (15) holds, and we will make use of this below. We will also make use of condition (15), which says that the native species is a better competitor for N.

We need to solve for Ā, ${\overline{B}}_n$, and ${\overline{B}}_e$ from Eq. (17) and the following equation (again from the model definition (1)):

$$ \overline{A}=\frac{I}{k+{\overline{B}}_n{\delta}_n{\nu}_n+{\overline{B}}_e{\delta}_e{\nu}_e} $$

(23)

Equation (17) can be used to eliminate Ā and to express ${\overline{B}}_e$ in terms of ${\overline{B}}_n$. Then equating the right-hand sides of (23) and the first equation in (17) and rearranging gives an implicit equation for I in terms of ${\overline{B}}_n$:

$$ I=\frac{\left\{{m}_n\left(c-b-{\alpha}_{ne}\left(a-b\right)\right)-{\mu}_n\left(a-c\right)\right\}{\overline{B}}_n^p+\left({m}_n{\alpha}_{ne}b+{\mu}_nc\right){h}^p}{\omega_n{\nu}_n\left(-\left(a-c\right){B}_n^p+c{h}^p\right)}\left(k+{\delta}_n{\nu}_n{\overline{B}}_n+{\delta}_e{\nu}_e{\overline{B}}_e\right) $$

(24)

where

$$ {\overline{B}}_e={\left(\frac{-\left(a-b\right){\overline{B}}_n^p+b{h}^p}{c-b}\right)}^{\frac{1}{p}} $$

(25)

and

$$ \begin{array}{c}\hfill a=\frac{m_n}{\omega_n{\nu}_n}-\frac{m_e{\alpha}_{en}}{\omega_e{\nu}_e}\hfill \\ {}\hfill b=\frac{\mu_e}{\omega_e{\nu}_e}-\frac{\mu_n}{\omega_n{\nu}_n}>0\hfill \\ {}\hfill c=\frac{m_n{\alpha}_{ne}}{\omega_n{\nu}_n}-\frac{m_e}{\omega_e{\nu}_e}>0\ \hfill \end{array} $$

Note that we have c > b > 0 from inequalities (2) and (15), but that α may be positive or negative, depending on the parameterization.

An expression relating I directly to ${\overline{B}}_e$ can be written by switching the n and e subscripts in Eq. (24). An expression relating I to Ā can be found by substituting

$$ {\overline{B}}_n=h{\left(\frac{c\overline{A}+{m}_e{\mu}_n-{m}_n{\alpha}_{ne}{\mu}_e}{\left(a-c\right)\overline{A}-\left({m}_n{m}_e\left(1-{\alpha}_{ne}{\alpha}_{en}\right)+{m}_n{\mu}_e\left(1-{\alpha}_{ne}\right)+{m}_e{\mu}_n\left(1-{\alpha}_{en}\right)\right)}\right)}^{\frac{1}{p}} $$

in Eq. (24).

Note that I is a single-valued function of ${\overline{B}}_n$ (Eq. (24)). We also know that I is a single-valued function of Ā and of ${\overline{B}}_e$, because Ā and ${\overline{B}}_e$ are monotonically decreasing functions of ${\overline{B}}_n$ (see the calculations above). So for any given value of ${\overline{B}}_n$ (or ${\overline{B}}_e$ or Ā) there is at most one possible value of the control parameter I.

To establish the range of values of ${\overline{B}}_n$ that correspond to real and non-negative values of the control parameter I (Eq. (15)), first observe that if ${\overline{B}}_n$ and ${\overline{B}}_e$ are both real and non-negative, then $\overline{B}$ and I will both be real and non-negative too, by (17) and (23). To determine which real and non-negative values of ${\overline{B}}_n$ correspond to real and non-negative values of ${\overline{B}}_e$, we must consider the sign of a − b. If a − b > 0, then real and non-negative values of ${\overline{B}}_n$ that correspond to real and non-negative values of ${\overline{B}}_e$ (Eq. (25)) are those in the closed interval $\left[0,{\left(\frac{b}{a-b}\right)}^{\frac{1}{p}}h\right]$. This means that the points corresponding to coexistence equilibria form a single arc in $\left\{I,\overline{A},{\overline{B}}_n,{\overline{B}}_e\right\}$ space. The upper limit is at ${\overline{B}}_n={\left(\frac{b}{a-b}\right)}^{\frac{1}{p}}h$ = B _crit,n , ${\overline{B}}_e=0$ and I = I _crit,n ; this is the point at which the native-dominant equilibrium changes from stable to unstable. At the lower limit we have ${\overline{B}}_n=0$, ${\overline{B}}_e={B}_{\mathrm{crit},e}$ and I = I _crit,e ; this is the point at which the exotic-dominant equilibrium changes from stable to unstable. Together, these results show that if a − b > 0, the points corresponding to coexistence equilibria form a single arc whose two endpoints are the points where the native- and exotic-dominant equilibria change stability.

If, on the other hand, a − b < 0, then any real and non-negative value of ${\overline{B}}_n$ corresponds to a real and non-negative value of ${\overline{B}}_e$ (Eq. (25)) and hence a real and non-negative value of the control parameter I. In this case, there is a semi-infinite arc of coexistence equilibria in $\left\{I,\overline{A},{\overline{B}}_n,{\overline{B}}_e\right\}$ space that extends from the point at which the exotic-dominant equilibrium changes from stable to unstable (${\overline{B}}_n=0,\ {\overline{B}}_e={B}_{crit,e}$) to ${\overline{B}}_n\to \infty$, with exactly one coexistence equilibrium for each value of ${\overline{B}}_n$.