Effective competition determines the global stability of model ecosystems

Abstract

We investigate the stability of Lotka-Volterra (LV) models constituted by two groups of species such as plants and animals in terms of the intragroup effective competition matrix, which allows separating the equilibrium equations of the two groups. In matrix analysis, the effective competition matrix represents the Schur complement of the species interaction matrix. It has been previously shown that the main eigenvalue of this effective competition matrix strongly influences the structural stability of the model ecosystem. Here, we show that the spectral properties of the effective competition matrix also strongly influence the dynamical stability of the model ecosystem. In particular, a necessary condition for diagonal stability of the full system, which guarantees global stability, is that the effective competition matrix is diagonally stable, which means that intergroup interactions must be weaker than intra-group competition in appropriate units. For mutualistic or competitive interactions, diagonal stability of the effective competition is a sufficient condition for global stability if the inter-group interactions are suitably correlated, in the sense that the biomass that each species provides to (removes from) the other group must be proportional to the biomass that it receives from (is removed by) it. For a non-LV mutualistic system with saturating interactions, we show that the diagonal stability of the corresponding LV system close to the fixed point is a sufficient condition for global stability.

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Notes

  1. 1.

    We describe here the main notation used in this paper. A matrix M is said to be positive definite (the term Hermitian positive is more frequently used in the mathematical literature), which we write as M≻0, when its symmetric part \(M^{S}\equiv \frac {1}{2}\left (M+M^{T}\right )\) has all positive eigenvalues, where M T is the transposed matrix. This implies that \(\left \langle x,Mx \right \rangle \) is positive for all possible vectors x ≠ 0, where \(\left \langle x,y \right \rangle \) indicates the scalar product. We denote by \(M^{a}\equiv \frac {1}{2}\left (M-M^{T}\right )\) the antisymmetric part of matrix M. We also use the short notation \(M^{-T}\equiv \left (M^{T} \right )^{-1}\equiv \left (M^{-1}\right )^{T}\). We denote by \(\overline {M}\equiv MD^{-1}\) the matrix M multiplied by the right times the diagonal matrix \(\left (D \right )^{-1}\), which we interpret as the matrix M in the abundance units defined by D (see below).

  2. 2.

    In fact, it holds \(\left \langle x,D^{-1}QD^{-1}x \right \rangle =\left \langle (D^{-1}x),Q(D^{-1}x) \right \rangle \); therefore, \(Q^{\prime }\) is positive definite if and only if Q is such.

  3. 3.

    In fact, \(\left \langle x, \overline {E_{\mathrm {P}}}^{T} \left (\overline {B_{\mathrm {P}}}^{S} \right )^{-1}\overline {E_{\mathrm {P}}}x \right \rangle =\left \langle (\overline {E_{\mathrm {P}}}x), \left (\overline {B_{\mathrm {P}}}^{S} \right )^{-1}(\overline {E_{\mathrm {P}}}x) \right \rangle \geq 0\) by the hypothesis that \(\overline {B_{\mathrm {P}}}\) is positive definite.

  4. 4.

    In fact, Eq. 12 involves a symmetric matrix, and it can be rewritten as \(\left \langle x, \left (\overline {C_{\mathrm {A}}} \right )^{S}x \right \rangle >\left \langle x,\overline {E_{\mathrm {P}}}^{T} \left (\overline {B_{\mathrm {P}}}^{S} \right )^{-1}\overline {E_{\mathrm {P}}}x \right \rangle \geq 0\).

  5. 5.

    In fact, in the complex field the condition of positive definiteness can be written as x (M + M T)x>0∀x ≠ 0, where x denotes the conjugate transpose of x. In particular, if x is a right eigenvector with eigenvalue λ, it holds Mx = λxx M = λ x , thus the above condition implies that (λ + λ )(x x)>0 and (λ + λ )>0 since (x x) is positive by definition i.e. all eigenvalues of M must have positive real part.

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Acknowledgements

This work was supported by the Spanish Ministery of Economy through the grant BFU-40020 to UB and FPI grant BES-2009-013072 to APG. Research at the CBMSO is facilitated by the Fundación Ramón Areces. We thank the editor, Sebastian Schreiber, for useful comments that helped to improve the presentation, and an anonymous reviewer for useful suggestions and for finding a typo in a formula in the first version of this paper.

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Correspondence to Ugo Bastolla.

Appendices

Appendix A: Conditions for diagonal stability of A

Theorem

the interaction matrix \(\overline {A}=AD\) is positive definite; hence, the matrix A describes a globally stable system, if and only if the matrix

$$\begin{array}{@{}rcl@{}} H = \left( \begin{array}{ll} \left( \overline{B_{\mathrm{P}}} \right)^{S} & \overline{E_{\mathrm{P}}} \\ \overline{E_{\mathrm{P}}}^{T} & \left( \overline{C_{\mathrm{A}}} \right)^{S} \end{array} \right) \end{array} $$
(26)

is positive definite, where \(\overline {C_{\mathrm {A}}}=\overline {B_{\mathrm {A}}}- \overline {{\Gamma }_{\mathrm {A}}}\left (\overline {B_{\mathrm {P}}} \right )^{-1}\overline {{\Gamma }_{\mathrm {P}}}\) , and \(\overline {E_{\mathrm {P}}}=\frac {1}{2} \overline {B_{\mathrm {P}}}^{T}\left (\left (\overline {B_{\mathrm {P}}} \right )^{-1}\overline {{\Gamma }_{\mathrm {P}}}-\right .\left . \left (\overline {B_{\mathrm {P}}}\right )^{-T}\overline {{\Gamma }_{\mathrm {A}}}\right )\) . We denote the symmetric part of a matrix by the superscript s, \(\left (\overline {C_{\mathrm {A}}} \right )^{S}= \frac {1}{2}\left (\overline {C_{\mathrm {A}}}+\overline {C_{\mathrm {A}}}^{T}\right )\) , and denote B −T as a shortcome of \(\left (B^{T} \right )^{-1}\) or, equivalently, \(\left (B^{-1}\right )^{T}\).

Proof

The block effective competition matrix C A corresponds to the Schur complement of B P in the matrix A. One can go from A to a matrix containing C A in a diagonal block by means of the Aitken’s block diagonalization formula (Zhang 2005). This corresponds to

$$\begin{array}{@{}rcl@{}} A &=& \left( \begin{array}{ll} B_{\mathrm{P}} & -{\Gamma}_{\mathrm{P}} \\ -{\Gamma}_{\mathrm{A}} & B_{\mathrm{A}} \end{array}\right) \\ &=& \left( \begin{array}{ll} I_{\mathrm{P}} & 0 \\ -{\Gamma}_{\mathrm{A}} \left( B_{\mathrm{P}} \right)^{-1} & I_{\mathrm{A}} \end{array}\right) \left( \begin{array}{ll} B_{\mathrm{P}} & 0 \\ 0 & C_{\mathrm{A}} \end{array}\right) \left( \begin{array}{ll} I_{\mathrm{P}} & -\left( B_{\mathrm{P}} \right)^{-1}{\Gamma}_{\mathrm{P}} \\ 0 & I_{\mathrm{A}} \end{array}\right) \\ &\equiv& L \mathcal{C} U\, , \end{array} $$
(27)

where L and U correspond to the lower and upper triangular matrices of the Aitken’s transform, respectively, and \(\mathcal {C}\) is the matrix with diagonal blocks equal to B P and C A. Next, we substitute this expression for A in the condition for diagonal stability, obtaining

$$ Q^{\prime} =\frac{1}{2}\left( A\!D + D\,A^{T}\right)= \frac{1}{2}\left[ L\,\mathcal{C}\, U\!D + D\,U^{T}\! \mathcal{C}^{T}\, L^{T} \right]\succ 0. $$
(28)

We now apply a congruence by U −1 to get the matrix H

$$\begin{array}{@{}rcl@{}} H&=&U^{-T}\! Q^{\prime}\! U^{-1}= \frac{1}{2}\left[ \!\left( U^{-T}L\right)\mathcal{C}D+\!D\mathcal{C}^{T}\left( U^{-T}L\right)^{T} \right]\\ &=& \left( \begin{array}{ll} \overline{B_{\mathrm{P}}}^{S} & \overline{E_{\mathrm{P}}} \\ \overline{E_{\mathrm{P}}}^{T} & \overline{C_{\mathrm{A}}}^{S} \end{array}\right), \end{array} $$
(29)

with \(\overline {E_{\mathrm {P}}}\) given by Eq. 13 in the main text. Thus, A will be diagonally stable if and only if there is a positive diagonal matrix D such that H in Eq. 29 is positive definite. □

From this calculation and from Theorem 7.7.6 in Horn and Johnson (1985), we see that the interaction matrix \(\overline {A}\) will be positive definite if and only if the following holds:

$$ \overline{C_{\mathrm{A}}}^{S} - \overline{E_{\mathrm{P}}}^{T} \left( \overline{B_{\mathrm{P}}}^{S} \right)^{-1}\overline{E_{\mathrm{P}}} \succ 0\, . $$
(30)

Appendix B: Conditions for positivity of C

We prove here that the effective competition matrix C A is positive definite if and only if it holds

$$\begin{array}{@{}rcl@{}} \left( {B_{\mathrm{A}}} \right)^{S} \,-\,{\left( {\Gamma}\right)^{S}_{\mathrm{A}}}\left( {B_{\mathrm{P}}}^{S} \right)^{-1}{\!\left( {\Gamma}\right)^{S}_{\mathrm{P}}} \!& + &\! {E_{\mathrm{A}}}^{T} \left( {B_{\mathrm{P}}}^{S} \right)^{-1}{\!E_{\mathrm{A}}} \\ \,=\, {S_{\mathrm{A}}} \!& + &\! {E_{\mathrm{A}}}^{T}\! \left( {B_{\mathrm{P}}}^{S} \right)^{-1}{E_{\mathrm{A}}} \succ 0 \end{array} $$
(31)

where S A is given by Eq. 11 and E A is given by Eq. 13 in the main text. We denote by superscripts S and a the symmetric and antisymmetric part, respectively, and we use the block notation where B is the matrix whose diagonal blocks are B P and B A, and Γ is a matrix whose off-diagonal blocks are ΓP and ΓA, respectively. With this notation, we can compactly write C = B−ΓB −1Γ. The computation is based on expressing the matrix B −1 as the sum of its symmetric part, \(\left (B^{-1} \right )^{S}=B^{-T} B^{S} B^{-1}=\left (B^{S} \right )^{-1}+ \left (B^{-T}B^{a}\right )\left (B^{S} \right )^{-1}\left (B^{a} B^{-1}\right )\) and antisymmetric part \(\left (B^{-1} \right )^{a}=-B^{-T} B^{a} B^{-1}\).

$$\begin{array}{@{}rcl@{}} -\left( C \right)^{S}+\left( B \right)^{S} & = & \frac{1}{2}\left[ {\Gamma}\left( B \right)^{-1}{\Gamma} + {\Gamma}^{T}\left( B\right)^{-T}{\Gamma}^{T} \right] \\ & = & \frac{1}{2} \left[ \left( {\Gamma}^{S}+{\Gamma}^{a}\right) \left( \left( B^{-1} \right)^{S}+\left( B^{-1} \right)^{a}\right)\right.\\ &&\left.\left( {\Gamma}^{S}+{\Gamma}^{a}\right) \right. \\ & & + \left. \left( {\Gamma}^{S}-{\Gamma}^{a}\right) \left( \left( B^{-1} \right)^{S}-\left( B^{-1} \right)^{a}\right)\right.\\ &&\left.\left( {\Gamma}^{S}-{\Gamma}^{a}\right) \right] \\ & \sim & {\Gamma}^{S} \left( B^{-1} \right)^{S} {\Gamma}^{S}+ {\Gamma}^{a} \left( B^{-1} \right)^{S} {\Gamma}^{a}\!+ \!2 {\Gamma}^{a}\\ &&\left( B^{-1} \right)^{a} {\Gamma}^{S} \\ & = & {\Gamma}^{S} \left( B^{S} \right)^{-1}{\Gamma}^{S}+ {\Gamma}^{S}\\ &&\left( B^{-T}B^{a}\left( B^{S} \right)^{-1}B^{a}B^{-1}\right) {\Gamma}^{S} \\ & & + {\Gamma}^{a} \left( B^{-T}B^{S}B^{-1}\right){\Gamma}^{a} - 2 {\Gamma}^{a}\\ &&\left( B^{-T}B^{a}B^{-1}\right){\Gamma}^{S} \\ & = & {\Gamma}^{S} \left( B^{S} \right)^{-1}{\Gamma}^{S} \\ & & - \left( B^{a}B^{-1}{\Gamma}^{S}-B^{S}B^{-1}{\Gamma}^{a}\right)^{T}\left( B^{S} \right)^{-1}\\ &&\left( B^{a}B^{-1}{\Gamma}^{S}-B^{S}B^{-1}{\Gamma}^{a}\right) \\ & = &{\Gamma}^{S} \left( B^{S} \right)^{-1}{\Gamma}^{S} + E^{T} \left( B^{S} \right)^{-1}E \end{array} $$
(32)

where MN indicates that \(\left \langle x,Mx \right \rangle =\left \langle x,Nx \right \rangle \) (i.e. we eliminate asymmetric components, such as ΓS B T B a B −1ΓS or \({\Gamma }^{a} \left (B^{S} \right )^{-1}{\Gamma }^{a}\)). This proves Eq. 31 above.

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Ferrera, A., Pascual-García, A. & Bastolla, U. Effective competition determines the global stability of model ecosystems. Theor Ecol 10, 195–205 (2017). https://doi.org/10.1007/s12080-016-0322-z

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Keywords

  • Ecological models
  • Population dynamics
  • Global stability
  • Structural stability