Correction to: A general stochastic model shows that plant-soil feedbacks can buffer plant species from extinction risks in unpredictable environments

Correction to : Plant and Soil

https://doi.org/10.1007/s11104-022-05698-6

Main text:

The original version of the article unfortunately contained two inaccuracies in two equations, which missed one bracket and that for one variable used a lower case “a” rather than an upper case “A”. Although these writing inaccuracies have no effect on the results, also because models were implemented correctly with the form of equations used in Bever 2003 (New Phytol 157:465–473. https://doi.org/10.1046/j.1469-8137.2003.00714.x), and the equations remain understandable, we decided to publish this erratum to remove any ambiguity. The correct system of eqs. 1 should read

$$\left\{\begin{array}{c}\frac{dA}{dt}=A ({r}_{A}-{m}_{A}A-{c}_{BA }B+{\alpha }_{A}{S}_{A}+{\beta }_{A}\left(1-{S}_{A}\right))\\ \frac{dB}{dt}=B ({r}_{B}-{m}_{B}B-{c}_{AB }A+{\alpha }_{B}{S}_{A}+{\beta }_{B}\left(1-{S}_{A}\right))\\ \frac{d{S}_{A}}{dt}={S}_{A}\left(1-{S}_{A}\right)\left[\frac{A}{A+B}-v\frac{B}{A+B}\right]\end{array}\right.$$

In which we have added a bracket at the end of the first two equations, and changed the Sa with SA in the first two equations.

Also, in the original text, just after reporting eqs. 1, we said [quote]:

“Note that the parameters αa, αb, βa and βb are either positive or negative. The two alphas parameters (αa and αb) describe the net effect of soil of A on plant A and B, while the beta parameters (βa and βb)…”.

That is all good but in the parameters alphas and betas we should have used upper case A and B rather than lower case “a” and “b”, just for consistency with eqs. 1. And the same should have applied whenever, throughout the text, we used the same parameters alphas and betas. For example, we defined the soil feedback as I_s = α_a—α_b—β_a+β_b. There, again, for consistency with eqs. 1, the lower case “a” and “b” should have been upper case A and B, and so the equation should better read:

$${I}_{s}={\alpha }_{A}-{\alpha }_{B}-{\beta }_{A}+{\beta }_{B}.$$

Basically, we inconsistently used lower case “a” and “b” and upper case A and B to simply mean plant A and B. We should have always consistently use either lower case “a” and “b” or, better, upper case “A” and “B” for plant A and B.

Exactly the same changes should be applied to the stochastic version of the system, that should read:

$$\left\{\begin{array}{c}dA=[A \left({r}_{A}-{m}_{A}A-{c}_{BA }B+{\alpha }_{A}{S}_{A}+{\beta }_{A}\left(1-{S}_{A}\right))\right]dt+{\sigma }_{A}d{W}_{1} \\ dB=[B \left({r}_{B}-{m}_{B}B-{c}_{AB }A+{\alpha }_{B}{S}_{A}+{\beta }_{B}\left(1-{S}_{A}\right))\right]dt+{\sigma }_{B}d{W}_{2} \\ d{S}_{A}=\left[{S}_{A}\left(1-{S}_{A}\right)\left[\frac{A}{A+B}-v\frac{B}{A+B}\right]\right]dt+ {\sigma }_{S}d{W}_{3}\end{array}\right.$$

The original article has been corrected.