Coexistence, dispersal and spatial structure in metacommunities: a stochastic model approach


We propose a stochastic model for interacting species in a metacommunity in order to study the factors affecting the intensity of the competition/colonization trade-off as a coexistence mechanism in metacommunities. We particularly focus on the role of the number of local communities and the number of refuges for the inferior competitor. The stochastic component is associated with the dispersal process and is represented by Poisson random measures. Thus, this stochastic model includes two dynamic scales: a continuous one, which refers to the interactions among species, and a low frequency one, referring to dispersal following a Poisson scheme. We show the well-posedness of the model and that it is possible to study its long-term behavior using Lyapunov exponents; the extinction of a species is associated with a negative slope in the time trajectory of the Lyapunov exponent, otherwise, it is equal to zero. We show that the competition/colonization trade-off is a function of the dispersal rate of the inferior competitor, and that it becomes less intense as the number of local communities increases, while the opposite is true with an increase in the number of refuges for the inferior competitor. We also show that under a priority effect type of scenario, dispersal can reverse priority effects and generate coexistence. Our results emphasize the importance of coexistence mechanisms related to the topology of the system of local communities, and its relationship with dispersal, in affecting the result of competition in local communities.

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We acknowledge funding from projects FONDECYT 1161023 and 1200925 and Project EcoDep. We also acknowledge funding from a postdoctoral fellowship to Cristóbal Quiñinao (FONDECYT 3170435), and support from projects CIMFAV-CIDI and Redes CONICYT 180018 to Rolando Rebolledo and AFB170008. We thank Sergio Navarrete for stimulating discussions on the role of dispersal and spatial structure in metacommunities and two anonymous reviewers for their insightful comments, suggestions, and criticisms.


FONDECYT 1161023, 1200925, FONDECYT 3170435), CIMFAV-CIDI, AFB170008, and Redes CONICYT 180018.

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Correspondence to Pablo A. Marquet.

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Appendix 1: Proofs

Proof of Proposition 1

Let ∥⋅∥ be the usual Euclidean norm in \(\mathbb {R}^{d}\), and for real matrices Q = (qlk)l,k= 1,..,d define the norm \(\parallel \cdot \parallel _{\max \limits }\) as \(\parallel Q\parallel _{\max \limits }=\underset {1\leq l,k\leq d}{\max \limits }\mid q_{lk}\mid \). Notice that for an \(\mathbb {R}^{d}\)-vector Y, we have \(\parallel QY\parallel \leq \sqrt {d}\parallel Q\parallel _{\max \limits }\parallel Y\parallel \). Let \(\boldsymbol {x}\in \mathbb {R}^{IJ}\). Then, we have the following inequalities: \(\parallel {\Lambda }(\boldsymbol {x})\boldsymbol {x}\parallel \leq \sqrt {IJ}\parallel {\Lambda }(\boldsymbol {x})\parallel _{\max \limits }\parallel \boldsymbol {x}\parallel \) and \(\parallel B(\boldsymbol {x},z)\parallel =\parallel \text {vec}(B(\boldsymbol {x},z))\parallel \leq \sqrt {IJ}\parallel B(\boldsymbol {x},z)\parallel _{\max \limits }\leq \sqrt {IJ}\parallel \beta (z)\parallel _{\max \limits }\parallel \boldsymbol {x}\parallel \), where

$$ \beta(z) = \left( \!\begin{array}{ccccc} \varepsilon_{11}1_{\left\{ z\leq b_{1}\right\} }\cdots\varepsilon_{1I}1_{\left\{ z\leq b_{1}\right\} } & 0 & \cdot & \cdot & 0\\ 0 & \cdot & & & \vdots\\ {\vdots} & & \cdot & & 0\\ 0 & \cdot & \cdot & 0 & \varepsilon_{I1}1_{\left\{ z\leq b_{J}\right\} }\cdots\varepsilon_{II}1_{\left\{ z\leq b_{J}\right\} } \end{array}\!\right). $$

Using those inequalities, we have that:

$$ D(\boldsymbol{x})=\parallel F(\boldsymbol{x})\parallel^{2}+\parallel{\Lambda}(\boldsymbol{x})\boldsymbol{x}\parallel^{2}+\int\limits_{0}^{\infty}\parallel B(\boldsymbol{x},z)\parallel^{2}dz $$
$$ \begin{array}{@{}rcl@{}} &\leq&\parallel F(\boldsymbol{x})\parallel^{2}+IJ\parallel{\Lambda}(\boldsymbol{x})\parallel_{\max}^{2}\parallel\boldsymbol{x}\parallel^{2}\\&&+IJ\int\limits_{0}^{\infty}\parallel\beta(z)\parallel_{\max}^{2}dz\parallel\boldsymbol{x}\parallel^{2} \end{array} $$
$$ \leq K(\boldsymbol{x})(1+\parallel\boldsymbol{x}\parallel^{2}), $$

where \(K(\boldsymbol {x})=\max \limits \{\parallel F(\boldsymbol {x})\parallel ^{2},IJ\parallel {\Lambda }(\boldsymbol {x})\parallel _{\max \limits }^{2},IJ{\int \limits }_{0}^{\infty }\parallel \beta (z)\parallel _{\max \limits }^{2}dz\}\). By (A.1), ∥X(0) ∥ is bounded, so we can choose a large K0K(X(0)) such that D(x) ≤ K0(1+ ∥x2), for all x belonging to the set \(\{\boldsymbol {x}\in \mathbb {R}^{IJ}:K(\boldsymbol {x})\leq K_{0}\}\). This means that (4) has a local linear growth.

On the other hand, by (A.2) F(⋅) is a locally Lipschitz function. Let L(x) = Λ(x)x, for all \(\boldsymbol {x}\in \mathbb {R}^{IJ}\). Now, note that for any \(\boldsymbol {x},\boldsymbol {y}\in \mathbb {R}^{IJ}\) we have that L(x) − L(y) = [Λ(x) + Λ(y)][xy]/2 + [Λ(x) −Λ(y)][x + y]/2, and then, \(\parallel L(\boldsymbol {x})-L(\boldsymbol {y})\parallel \leq (\sqrt {IJ}/2)\parallel {\Lambda }(\boldsymbol {x})+{\Lambda }(\boldsymbol {y})\parallel _{\max \limits }\parallel \boldsymbol {x}-\boldsymbol {y}\parallel +(\sqrt {IJ}/2)\parallel {\Lambda }(\boldsymbol {x})-{\Lambda }(\boldsymbol {y})\parallel _{\max \limits }\parallel \boldsymbol {x}+\boldsymbol {y}\parallel \), where

$$ \begin{array}{@{}rcl@{}} \parallel{\Lambda}(\boldsymbol{x})-{\Lambda}(\boldsymbol{y})\parallel_{\max}&=&\underset{i,j}{\max}\mid{\sum}_{j^{\prime}=1}^{J}(x_{ij^{\prime}}-y_{ij^{\prime}})\lambda_{jj^{\prime}}\mid\\&\leq&{\sum}_{i^{\prime}=1}^{I}{\sum}_{j^{\prime}=1}^{J}\mid x_{ij^{\prime}}-y_{ij^{\prime}}\mid\mid\lambda_{jj^{\prime}}\mid \end{array} $$
$$ \leq\parallel\boldsymbol{x}-\boldsymbol{y}\parallel\sqrt{I\sum\limits_{j^{\prime}=1}^{J}\mid\lambda_{jj^{\prime}}\mid^{2}}, $$

by Hölder inequality. That is,

$$ \begin{array}{@{}rcl@{}} &&\parallel L(\boldsymbol{x}) - L(\boldsymbol{y})\parallel\!\leq\!\frac{\sqrt{IJ}}{2}\left( \parallel{\Lambda}(\boldsymbol{x})+{\Lambda}(\boldsymbol{y})\parallel_{\max}\vphantom{\sqrt{I{\sum}_{j^{\prime}=1}^{J}\mid\lambda_{jj^{\prime}}\mid^{2}}}\right.\\&&\left.\!+\parallel\boldsymbol{x} + \boldsymbol{y}\parallel\sqrt{I{\sum}_{j^{\prime}=1}^{J}\mid\lambda_{jj^{\prime}}\mid^{2}}\right)\parallel\boldsymbol{x} - \boldsymbol{y}\parallel, \end{array} $$

and thus for any \(\boldsymbol {x},\boldsymbol {y}\in \mathbb {R}^{IJ}\) such that ∥x ∥∨∥y ∥≤ M, where M is a constant such that M ≫∥X(0) ∥, there exists a constant HM such that ∥ L(x) − L(y) ∥≤ HMxy ∥. So, it means that L(⋅) is locally Lipschitz as well. Therefore, conditions of Theorem 9.1 in Chapter IV from Ikeda and Watanabe (2014) (linear growth and Lipschitz conditions) are locally satisfied, and hence, we have existence and uniqueness of a local solution. \(\square \)

Proof of Proposition 2

Recalling the formulation Xij(t) of (3), we define the following quantity:

$$ Y_{j}(t) := \sum\limits_{i=1}^{I}X_{ij}(t), $$

which models the total biomass of species j over the whole system (i.e., archipelago) at time t. We recall that no biomass is transferred from outside the archipelago and all the emigrant biomass of any species in any community at most lands over some of the other communities. Therefore, the stochastic part of (3) must satisfy:

$$ \begin{array}{@{}rcl@{}} &&{\kern-9.5pt}\sum\limits_{i=1}^{I}\sum\limits_{i^{\prime}=1}^{I}\varepsilon_{ii^{\prime}}{{\int}_{0}^{t}}X_{i^{\prime}j}(s-){\int}_{0}^{\infty}1_{\{z\leq b_{j}\}}N_{i^{\prime}j}(dz,ds)\\ &&{} =\sum\limits_{i^{\prime}=1}^{I}\left( \sum\limits_{i=1}^{I}\varepsilon_{ii^{\prime}}\right){{\int}_{0}^{t}}X_{i^{\prime}j}(s-){\int}_{0}^{\infty}1_{\{z\leq b_{j}\}}N_{i^{\prime}j}(dz,ds)\! \leq\! 0,\!\!\\ \end{array} $$

almost surely, since \({\sum }_{i=1}^{I}\varepsilon _{ii^{\prime }}\leq 0\) for any \(i^{\prime }\). Let Y = (Y1,...,YJ)T. Now, in (6), let Ξ be the J × IJ matrix such that ΞX = Y. We have that:

$$ \boldsymbol{Y}(t)\leq\boldsymbol{Y}(0)+{\int\limits_{0}^{t}}{\Xi}{\Gamma}(\boldsymbol{X}(s))\boldsymbol{X}(s)ds, $$

by (16). Then we get:

$$ \begin{array}{@{}rcl@{}} \boldsymbol{Y}(t)&\leq&\boldsymbol{Y}(0)+{\int\limits_{0}^{t}}{\Xi}{\Gamma}(\boldsymbol{X}(s))\boldsymbol{X}(s)ds\leq\boldsymbol{Y}(0)\\&&+{\int\limits_{0}^{t}}\max_{i=1,..,I}\eta_{i}(\boldsymbol{X(s)}){\Xi}\boldsymbol{X}(s)ds \end{array} $$
$$ =\boldsymbol{Y}(0)+{\int\limits_{0}^{t}}\max_{i=1,..,I}\eta_{i}(\boldsymbol{X(s)})\boldsymbol{Y}(s)ds. $$

By (A.3), it is clear that \(\sup _{t\in \mathbb {R}_{+}}\max \limits _{j=1,..,J}Y_{j}(t)<\infty \), almost surely, and (7) follows from the fact that \(\max \limits _{j=1,..,J}Y_{j}(\cdot )\geq \max \limits _{i=1,..,I;j=1,..,J}X_{ij}(\cdot )\), almost surely.

Now, we shall prove that extinction occurs in the long term. First, notice that under structural assumptions made in (6), we can write in (3) fij(x) in the form \(\tilde {f}_{ij}(x)x\). Therefore, for any \(t\in \mathbb {R}_{+}\), i = 1,...,I and j = 1,...,J we have:

$$ \begin{array}{@{}rcl@{}} &&X_{ij}(t)\geq X_{ij}(0)+{{\int}_{0}^{t}}\tilde{f}_{ij}(X_{ij}(s))X_{ij}(s)ds\\ &&+{{\int}_{0}^{t}}{\sum}_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}X_{ij^{\prime}}(s)X_{ij}(s)ds\\&&-{{\int}_{0}^{t}}X_{ij}(s){\int}_{0}^{\infty}1_{\left\{z\leq b_{j}\right\}}N_{ij}(dz,ds) \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&=X_{ij}(0)+{{\int}_{0}^{t}}\tilde{f}_{ij}(X_{ij}(s))X_{ij}(s)ds\\&&+{{\int}_{0}^{t}}{\sum}_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}X_{ij^{\prime}}(s)X_{ij}(s)ds\\ &&-{{\int}_{0}^{t}}X_{ij}(s)b_{j}ds+{{\int}_{0}^{t}}X_{ij}(s)\mathcal{M}_{ij}([0,b_{j}],ds), \end{array} $$

where \({\mathscr{M}}_{ij}([0,b_{j}],t)=-({\int \limits }_{0}^{\infty }1_{\left \{z\leq b_{j}\right \}}N_{ij}(dz,t)-tb_{j})=tb_{j}-N_{ij}([0,b_{j}],t)\) is a square-integrable martingale. This implies that:

$$ \begin{array}{@{}rcl@{}} X_{ij}(t)&\geq& X_{ij}(0)\exp\left\{{{\int}_{0}^{t}}\left[\tilde{f}_{ij}(X_{ij}(s))\vphantom{\sum\limits_{j^{\prime}\neq j}^{J}}\right.\right.\\&&\left.\left.+\sum\limits_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}X_{ij^{\prime}}(s)-b_{j}\right]ds+\mathcal{M}_{ij}([0,b_{j}],t)\right\} \end{array} $$
$$ \geq X_{ij}(0)\exp\left\{t\underset{\mathsf{x}\in R^{J}}{\inf}G_{ij}(\mathsf{x})+\mathcal{M}_{ij}([0,b_{j}],t)\right\}, $$


$$ \underset{\mathsf{x}\in R^{J}}{\inf}G_{ij}(\mathsf{x})\!:=\!\underset{(x_{1},...,x_{J})^{\mathrm{T}}\in R^{J}}{\inf}\left[\tilde{f}_{ij}(x_{j}) + \sum\limits_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}x_{j^{\prime}}-b_{j}\right], $$

and RJ is the bounded subset of \(\mathbb {R}^{J}_{+}\) which delimits the state space of (Xi1(⋅),...,XiJ(⋅))T for any i = 1,...,I. On the other hand, we have that:

$$ \mathbb{E}\left( \exp\left\{\mathcal{M}_{ij}([0,b_{j}],t)\right\}\right)=\exp\left\{b_{j}t\exp\{-1\}\right\}, $$

and thus,

$$ \mathbb{E}(X_{ij}(t))\!\geq\!\mathbb{E}(X_{ij}(0))\exp\left\{t(\underset{\mathsf{x}\in R^{J}}{\inf}G_{ij}(\mathsf{x})+b_{j}\exp\{-1\})\right\}. $$

Notice that it must satisfy \(b_{j}\exp \{-1\}+\inf _{\mathsf {x}\in R^{J}}G_{ij}(\mathsf {x})\leq 0\) if Xij(0) > 0, since otherwise \(\mathbb {E}(X_{ij}(t))\rightarrow \infty \) when \(t\rightarrow \infty \), which contradicts the fact that Xij(⋅) is bounded. So, if Xij(0) > 0, \(\exp \left \{t\inf _{\mathsf {x}\in R^{J}}G_{ij}(\mathsf {x})+{\mathscr{M}}_{ij}([0,b_{j}],t)\right \}\) never reaches zero in finite time, and then neither does Xij(⋅). \(\square \)

Proof of proposition 3

First notice that the vector b(x) can be decomposed into b(x) = βx, where \(\boldsymbol {\beta }={\int \limits }_{0}^{\infty }\beta (z)dz\) and β(z) is the IJ × IJ matrix defined in (14). Also, we have that \(\parallel B(\boldsymbol {x},z)\parallel \leq \sqrt {IJ}\parallel \beta (z)\parallel _{\max \limits }\parallel \boldsymbol {x}\parallel \).

We start by showing the existence of a Lyapunov exponent for (6). Consider its representation given by (11). According to Theorem 6.8.2 in Applebaum (2009), we have to show that the alternative condition to the Assumption 6.8.1, given in Equation (6.42), is satisfied for our system (11) for all xR. That is, the existence of a positive constant \(L^{\prime }\) such that \(\boldsymbol {x}^{\mathrm {T}}{\Gamma }(\boldsymbol {x})\boldsymbol {x}+\boldsymbol {x}^{\mathrm {T}}\boldsymbol {b}(\boldsymbol {x})+{\int \limits }_{0}^{\infty }\parallel B(\boldsymbol {x},z)\parallel ^{2}dz\leq L^{\prime }(1+\parallel \boldsymbol {x}\parallel ^{2})\). In fact, by the above, we have that:

$$ \begin{array}{@{}rcl@{}} \boldsymbol{x}^{\mathrm{T}}{\Gamma}(\boldsymbol{x})\boldsymbol{x}+\boldsymbol{x}^{\mathrm{T}}\boldsymbol{b}(\boldsymbol{x})+{\int}_{0}^{\infty}\parallel B(\boldsymbol{x},z)\parallel^{2}dz\leq\boldsymbol{x}^{\mathrm{T}}{\Gamma}(\boldsymbol{x})\boldsymbol{x}\\+\boldsymbol{x}^{\mathrm{T}}\boldsymbol{\beta}\boldsymbol{x}+IJ\parallel\boldsymbol{x}\parallel^{2}{\int}_{0}^{\infty}\parallel\beta(z)\parallel_{\max}^{2}dz\leq L^{\prime\prime}\parallel\boldsymbol{x}\parallel^{2}, \end{array} $$

where \(L^{\prime \prime }=\max \limits \{\sup _{\boldsymbol {x}\in \boldsymbol {R}}{\Gamma }({\boldsymbol {x}}),\parallel \boldsymbol {\beta }\parallel ,IJ{\int \limits }_{0}^{\infty }\parallel \beta (z)\parallel _{\max \limits }^{2}dz\}\). Clearly it also implies that the process (12) has a Lyapunov exponent.

Now, consider that there exists a constant c > 0 such that \({\mathscr{L}}\langle u_{IJ},\boldsymbol {x}\rangle \leq -c\langle u_{IJ},\boldsymbol {x}\rangle \) for all xR. Define \(\mathbb {E}_{\boldsymbol {x}}(\langle u_{IJ},\boldsymbol {X}(t)\rangle ):=\mathbb {E}(\langle u_{IJ},\boldsymbol {X}(t)\rangle \mid \boldsymbol {X}(0)=\boldsymbol {x})\), for all \(t\in \mathbb {R}_{+}\). Applying it to (12) we obtain:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},\boldsymbol{X}(t)\rangle)&=&\langle u_{IJ},\boldsymbol{x}\rangle+{\int\limits_{0}^{t}}\mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},{\Gamma}(\boldsymbol{X}(s))\boldsymbol{X}(s)\\&&+\boldsymbol{b}(\boldsymbol{X}(s))\rangle) ds \end{array} $$
$$ \leq \langle u_{IJ},\boldsymbol{x}\rangle-c{\int\limits_{0}^{t}}\mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},\boldsymbol{X}(s)\rangle)ds. $$

Then, by Gronwall’s inequality we get:

$$ \mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},\boldsymbol{X}(t)\rangle)\leq \langle u_{IJ},\boldsymbol{x}\rangle\exp\{-ct\}, $$

which implies that 〈uIJ,X(⋅)〉 goes exponentially fast to 0. \(\square \)

Proof of Lemma 4

For a chosen uIJ, there exists \(i^{\prime }=1,...,I\) and \(j^{\prime }=1,...,J\) such that \(\mathbb {E}_{\boldsymbol {x}}(X_{i^{\prime }j^{\prime }}(\cdot ))\leq \mathbb {E}_{\boldsymbol {x}}(\langle u_{IJ},\boldsymbol {X}(\cdot )\rangle )\), where \(\mathbb {E}_{\boldsymbol {x}}(X_{i^{\prime }j^{\prime }}(\cdot ))\) satisfies (17) for \(i=i^{\prime }\) and \(j=j^{\prime }\) given \(X_{i^{\prime }j^{\prime }}(0)=\boldsymbol {x}\). Therefore, extinction can only occur in an exponentially fast way. \(\square \)

Appendix 2. Some results of a classical competitive Lotka-Volterra system

In the present appendix we recall some results of a classical competitive Lotka-Volterra system and its corresponding parameter space, in order to enlighten the long-term solutions and the reader realize how the presence of dispersal can transform such parameter space and bifurcation diagrams. Notice that as we will start treating the classic case, in a local community without dispersal, first we do not consider the subscript i. Consider then the equations:

$$ \begin{array}{@{}rcl@{}} {x}_{1}^{\prime}(t) & = x_{1}(t)\left( r_{1}+\alpha_{11}x_{1}(t)+\alpha_{12}x_{2}(t)\right), \\ {x}_{2}^{\prime}(t) & = x_{2}(t)\left( r_{2}+\alpha_{21}x_{1}(t)+\alpha_{22}x_{2}(t)\right). \end{array} $$

For any steady state (x1,x2), the eigenvalues of the Jacobian matrix are given by:

$$ \frac12\left( \text{tr}(J)\pm\sqrt{\text{tr}(J)^{2}-4\det(J)}\right), $$

where tr(J) and \(\det (J)\) stand for the trace and determinant of the matrix

$$ J = \left( \begin{array}{cc} r_{1}+2\alpha_{11}x_{1}+\alpha_{12}x_{2} & \alpha_{12}x_{1} \\ \alpha_{21}x_{2} & r_{2}+\alpha_{21}x_{1}+2\alpha_{22}x_{2} \end{array}\right). $$

In the following we simply write T = tr(J) and \(D=\det (J)\). The standard trace-determinant analysis tells us that

  • Case \(4\det (J)>\text {tr}(J)^{2}\): the system has two complex eigenvalues, If tr(J) < 0 then the steady state is stable. On the other hand, if tr(J) > 0 then the steady state is unstable.

  • Case \(\det (J)<0\): in this scenario, both eigenvalues are positive and have different sign, thus the steady state is unstable.

  • Case \(0<4\det (J)<\text {tr}(J)^{2}\): if tr(J) > 0 then the two eigenvalues are positive and the steady state is unstable. Otherwise, tr(J) < 0 then the configuration is asymptotically stable.

Recall also that steady states are either (0,0), (0,−r2/ α22), (−r1/α11, 0) or

$$ \bar x_{1} = \frac{-r_{1}\alpha_{22}+r_{2}\alpha_{12}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}},\quad \bar x_{2} = \frac{-r_{2}\alpha_{11}+r_{1}\alpha_{21}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}. $$

Then we have:

  • (0,0): it follows that T = r1 + r2 and D = r1r2, in which case the extinction scenario is unconditionally unstable.

  • (0,−r2/α22): we have

    $$ T=r_{1}-r_{2}\frac{\alpha_{12}}{\alpha_{22}}-r_{2}\text{ and }D=-r_{2}\left( r_{1}-r_{2}\frac{\alpha_{12}}{\alpha_{22}}\right). $$

    Stability of this fixed point is reduced to the condition

    $$ r_{1}<r_{2}\frac{\alpha_{12}}{\alpha_{22}}. $$
  • (−r1/α11, 0): similarly,

    $$ T=-r_{1}+r_{2}-r_{1}\frac{\alpha_{22}}{\alpha_{11}}\text{ and }D=-r_{1}\left( r_{2}-r_{1}\frac{\alpha_{21}}{\alpha_{11}}\right), $$

    by consequence, the stability of this fixed point holds true if and only if

    $$ r_{2}<r_{1}\frac{\alpha_{21}}{\alpha_{11}}. $$
  • \((\bar x_{1},\bar x_{2})\): it follows that

    $$ T=\alpha_{11}\bar x_{1}+\alpha_{22}\bar x_{2}\text{ and }D=\big(\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}\big)\bar x_{1}\bar x_{2}, $$

    the existence and stability of this final fixed point is only under the conditions

    $$ \alpha_{11}\alpha_{22}>\alpha_{12}\alpha_{21},\quad r_{1}\alpha_{22}<r_{2}\alpha_{12}\quad\text{and}\quad r_{2}\alpha_{11}<r_{1}\alpha_{21}. $$

    Notice that, since αii < 0 then second and third conditions are equivalent to

    $$ r_{1}>r_{2}\frac{\alpha_{12}}{\alpha_{22}}\quad\text{and}\quad r_{2}>\frac{r_{1}\alpha_{21}}{\alpha_{11}}, $$

    thus we have the instability of all previous scenarios.

Analysis of the phase plane

If we come back to the numerical example with 2 species in a local community without dispersal, the system of equations are:

$$ \begin{array}{@{}rcl@{}} {x}_{1}^{\prime}(t) & = x_{1}(t)\left( r_{1}-\frac{r_{1}}{K_{1}}x_{1}(t)+\lambda_{12}x_{2}(t)\right), \\ {x}_{2}^{\prime}(t) & = x_{2}(t)\left( r_{2}+\lambda_{21}x_{1}(t)-\frac{r_{1}}{K_{2}}x_{2}(t)\right), \end{array} $$

thus defining the interacting matrix

$$ \left( \begin{array}{cc} -{r_{1}}/{K_{1}} &\lambda_{12} \\ \lambda_{21} & -{r_{1}}/{K_{2}} \end{array}\right). $$

Parameters rj’s are intrinsic for each species, thus we can consider them as fixed. Therefore, we can make an analysis of the stability of the fixed points under the two nonnegative quantities:

$$ p_{1} = \frac{\alpha_{21}}{\alpha_{11}},\qquad p_{2}=\frac{\alpha_{12}}{\alpha_{22}}, $$

where α21 = λ21, α12 = λ12, α11 = −r1/K1 and α22 = −r2/K2. It follows that the coexistence of the two population can happen only under the restriction

$$ p_{1}p_{2}<1,\quad \frac{r_{1}}{r_{2}}>p_{2},\quad \frac{r_{2}}{r_{1}}>p_{1}, $$

and similar conditions appear for any of the other fixed points. We summarize this parameter dependence of fixed points, in Fig. 8, where without loss of generality we have assumed that r1r2. Long-term behavior correspond to the classical Lotka-Volterra competition: a) coexistence, b) species 1 wins, c) species 2 wins, and d) strong competition (where the species with the largest initial biomass wins), that correspond to each of the zones in Fig. 8. The conditions for each scenario are:

  1. (a)

    p1 < r2/r1 and p2 < r1/r2. Since in this case p1p2 < 1, coexistence of species is expected.

  2. (b)

    p1 > r2/r1 and p2 < r1/r2. It follows that species 1 kicks in and at large times it is expected to converge asymptotically towards (−r1/α11, 0) = (K1, 0).

  3. (c)

    p1 < r2/r1 and p2 > r1/r2. Species 2 dominates and the solution is expected to converge asymptotically towards (0,−r2/α22, 0) = (0,K2). Moreover, since p1 < r2/r1 and p1p2 > 1 we get that

    $$ \bar x_{1} = \frac{-r_{1}\alpha_{22}+r_{2}\alpha_{12}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}} = \frac{1}{\alpha_{11}}\frac{r_{2}p_{2}-r_{1}}{1-p_{1}p_{2}}>0, $$


    $$ \bar x_{2} = \frac{-r_{2}\alpha_{11}+r_{1}\alpha_{21}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}=\frac{1}{\alpha_{22}}\frac{r_{1}p_{1}-r_{2}}{1-p_{1}p_{2}}<0, $$

    i.e., the second coordinate of the coexistence scenario vanishes and

  4. d)

    p1 > r2/r1 and p2 > r1/r2. In this last zone conditions imply that both winner-takes-all scenario are possible, and that the coexistence fixed point \((\bar x_{1},\bar x_{2})\) exists but it is unstable. Therefore, according to initial condition, we expect to converge at large times towards either (0,K2) or (K1, 0).

Fig. 8

Outcomes of the model for a single community and two populations when r1 > r2 and on the parameter space p1 = −λ21K1/r1 v/s p2 = −λ12K2/r2. Long-term behavior on each zone: (a) coexistence, (b) species 1 wins, (c) species 2 wins and (d) strong competition. Points 1 and 3 are the coordinates for the analyzes carried out in “Coexistence and the competition/colonization trade-off” and “Effect of the metacommunity spatial structure” and Appendix C, respectively

Appendix 3. Additional results for two species competition with priority effects

Results for the dynamics of two species in a central community within a metacommunity architecture with three local communities

For each pair (b1,b2) with bi taking values from 0.1 to 0.9 with incremental step 0.1, we performed 150 simulations. We report the empirical average of biomass of both species on the central community.

The results of the model are shown in the upper subplots of Fig. 9. One observes that the species with larger dispersal rates dominates the long-term solutions in the central community. In the bottom panel of Fig. 9 we plot the probability that species 1 wins as a function of the difference b1b2. We find a shape very similar to the ones described in literature (see, e.g., Calcagno et al. 2006).

Fig. 9

Simulations made for parameters r1 = r2 = 2, Ki1 = Ki2 = 10 (for all i = 1,2,3). All εij are 0 except for ε21 = ε23 = −ε11 = −ε33 = 0.1 and λ21 = λ12 = − 0.21. (top) We show the empirical mean calculated for species 1 and 2 in a central community under a strong competition scenario where species with higher initial biomass will win according to the classical Lotka-Volterra model of competition without stochastic dispersal. (down) We show the probability that species 1 wins as a function of the difference between colonization rates b1b2. As expected, under a symmetric dynamics the species with larger bi dominates at long times. Dots represents the empirical mean of 150 simulations of the model, for the three-community architecture where the dispersion parameter of species 1 varies. The solid line corresponds to a nonlinear statistical fit

Results for two species competition under priority effects in a metacommunity with two local communities

The first case is a symmetrical one, in which the outcome is the expected one for the classical Lotka-Volterra model (i.e., for each community, the species with the larger initial biomass will win). In our example, species 1 wins in community 2 due to its larger initial biomass condition and species 2 wins in community 2 for analogous reasons (Fig. 10 upper panel). Here, species 1 cannot disperse and species 2 dispersal rate b2 is low. In the second scenario, a small increase in b2 and no dispersal of species 1, will reverse the competence in community 1, and species 2 will win in both communities (Fig. 10, middle panels). And in the third scenario, species 1 only lives in community 1, and a stronger dispersal intensity of species 2 and no dispersal of species 1, will cause coexistence in community 1 (Fig. 10, bottom panel). This last outcome is due to the rapid and massive exchange of biomass of species 2 from one community to another, which can maintain its biomass levels in both communities, specifically causing in community 1 that the biomass condition for winning frequently changes: sometimes species 1 locally dominates due to a temporary larger biomass and sometimes species 2 locally dominates due to the same reason. In Fig. 11 we can visualize the final outcome of the dynamics of both species in community 1 and under each scenario, through the slope (in the long term) of the graph \(\log (X_{ij}(t))\) vs t, which corresponds to the (estimated) Lyapunov exponent.

Fig. 10

Here we show the biomass dynamics of two species in two communities under a priority effect scenario whereby the species that wins in competition is the one with higher initial biomass. The three panels represents three cases of the dynamics of the two species in the two communities, where the dispersal parameter of the inferior competitor species (the one with the lowest initial biomass) varies in each of these columns. The simulations where made over a mean of 50 trajectories and we considered the common parameters r1 = 1.6, r2 = 1, Ki1 = 12, Ki2 = 10 (for all i = 1.2), ε12 = ε21 = −ε11 = −ε22 = 0.5, λ12 = − 0.168 and λ21 = − 0.0875 in (8) for the three cases, with initial conditions given by X11(0) = 10, X12(0) = 1, X21(0) = 1 and X22(0) = 10. The first case is one where the dispersal rate of species 2 is low, b2 = 0.003, and no dispersal of species 1. In the second case we consider only a small increase in the dispersal rate of species 2 (b2 = 0.05). Notice that this subtle change makes this species to increase in biomass in community 1 reversing the priority effect. Finally, in the third case, in local community 2 only species 2 exists but as we increase its dispersal rate (b2 = 1) the priority effect coexistence in local community 1 is now possible

Fig. 11

Convergence of Lyapunov exponents of species 1 and species 2 in community 1 for the corresponding cases of Fig. 10, from top to bottom. In continuous line is depicted one typical trajectory and in dashed line the mean of 50 trajectories, where we can see a softened version of \(\log (X_{ij}(t))\) vs t. The slopes of the long-term tendencies tell us the destination of the species: slope ≈ 0 will mean persistence and negative slope indicates extinction

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Tejo, M., Quiñinao, C., Rebolledo, R. et al. Coexistence, dispersal and spatial structure in metacommunities: a stochastic model approach. Theor Ecol (2021).

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  • Metacommunity
  • Island biogeography
  • Lotka-Volterra model
  • Poisson measure
  • Lyapunov exponent
  • Trade-off
  • Priority effect