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We now derive the metapopulation model used in this paper. We start by deriving a general metapopulation model that is based on the seminal work of Levin50. Assuming that the inter-patch migrations are detailed-balanced, we make use of the formulation in Eq. (8) to derive a balanced metapopulation model. We then show that the balanced model admits a unique coexistence equilibrium that is asymptotically stable if the dispersal network is heterogeneous, whereas the same equilibrium is neutrally stable in the case of a homogeneous network.General metapopulation modelMathematical models based on traditional metapopulation theory usually assume that the metapopulation is made up of many neighboring spatially homogeneous habitat patches connected via dispersal. Consider an interconnected network of m discrete patches each being inhabited by the same n species. In addition, assume that species can migrate from one patch to some or all of the other patches. The rate of migration of each species between two patches is directly proportional to the proportion of the particular species in the originating patch, with a (nonnegative) constant of proportionality being the same across species. This constant of proportionality will be referred to as the rate constant associated with the migration. It is assumed that if there is migration between two given patches, then it is bidirectional, i.e., the rate constant of migration from j to k is strictly positive if and only the same holds for the migration from k to j. Just like in the case of a reversible single-species chemical reaction network, inter-patch migrations may be described by a weighted symmetric directed graph (G_2=(V_2,E_2)) where (V_2={1,ldots ,m}) is the set of patches (vertices) and an edge ((j,k)in E_2) means that every species can migrate from patch j to patch k. Finally, it is also assumed that the graph (G_2) corresponding to the inter-patch migration is connected, i.e., there is a path between every two distinct vertices of the graph.The flow of species between the patches can be summarized in a weighted (mtimes m) adjacency matrix ({mathbf {A}}) with entry (A_{jk}) being equal to the rate constant of migration of species from the (j{ {text {th}}}) to the (k{ {text {th}}}) patch. The diagonal elements of ({mathbf {A}}) are hence equal to 0. Due to the bidirectional nature of migration, it holds that (A_{jk} >0 Leftrightarrow A_{kj} >0) and (A_{jk}=0 Leftrightarrow A_{kj}=0), for any (jne k). Let (Delta =text {diag}(delta _1,ldots ,delta _m)) denote the m-dimensional diagonal matrix whose (j{ {text {th}}}) entry is given by$$begin{aligned} delta _{j}=sum _{k=1}^{m} A_{jk}. end{aligned}$$Define ({mathbf {L}}:=Delta -{mathbf {A}}^top ). Note that$$begin{aligned} (mathbb {1}^m)^{top }{mathbf {L}}=(mathbb {1}^m)^{top }Delta -big ({mathbf {A}}mathbb {1}^mbig )^{top }=({mathbf {0}}^m)^top . end{aligned}$$Let ({mathbf {x}}in S^{mn}), with (x_{i,j}) the proportion of species i in patch j across the entire metapopulation, then the net migration rate (psi _{i,j}) of species i from other patches to patch j is given by$$begin{aligned} psi _{i,j}=sum _{k=1}^{m}A_{kj}x_{i,k}-sum _{k=1}^{m}A_{jk}x_{i,j}=sum _{k=1}^{m}A_{kj}x_{i,k}-delta _{j}x_{i,j}=-sum _{k=1}^{m}L_{jk}x_{i,k}. end{aligned}$$Let us denote (Psi _i:=left( psi _{i,1},psi _{i,2},ldots ,psi _{i,m}right) ^{top }) and ({mathbf {r}}_i:=left( x_{i,1},x_{i,2},ldots ,x_{i,m}right) ^{top }), then$$begin{aligned} Psi _{i}=-{mathbf {L}}{mathbf {r}}_{i}. end{aligned}$$
(9)
Within each patch, the proportions of species are affected by other patches only via migration. Let (phi _{i,j}) denote the rate of change of the proportion of species i in patch j in the absence of migration. Since the dominance relationships among the species (described by a tournament matrix ({mathbf {T}})) are assumed to be the same for all patches and since the habitat patches are spatially homogeneous, the expression for (phi _{i,j}) is given by the right-hand side of System (1):$$begin{aligned} phi _{i,j}=x_{i,j}left( {mathbf {T}}{mathbf {p}}_{j}right) _{i}, end{aligned}$$
(10)
where ({mathbf {p}}_j:=left( x_{1,j},x_{2,j},ldots , x_{n,j}right) ^{top }), (i=1,ldots ,n) and (j=1,ldots ,m). Assuming migration among the patches, the proportion of a species within a patch is influenced by two factors: the first is the interaction with other species within the patch and the second is the migration of that particular species to or from other patches. Thus, the metapopulation model describing the dynamics of the n species in the m-patch network is described by the system of mn differential equations;$$begin{aligned} {dot{x}}_{i,j}=phi _{i,j}+psi _{i,j}=x_{i,j}left( {mathbf {T}}{mathbf {p}}_{j}right) _{i}-left( {mathbf {L}}{mathbf {r}}_{i}right) _{j},qquad i=1,ldots ,n,quad j=1,ldots ,m . end{aligned}$$
(11)
This system evolves on the unit simplex (S^{mn}).
Proposition 2

The unit simplex (S^{mn}) is positively invariant for System (11).

Proof
To show the invariance of the unit simplex (S^{mn}) under the flow of System (11), it suffices to show that each of the faces of the simplex cannot be crossed, i.e., the vector field points inward from the faces of (S^{mn}).
On the one hand, if (x_{i,j}=0) for some i, j, then$$begin{aligned}{dot{x}}_{i,j}=sum _{k=1}^{m}A_{kj}x_{i,k}ge 0,end{aligned}$$which implies that (x_{i,j}=0) cannot be crossed from positive to negative. In an ecological context, this condition simply states the obvious fact that an extinct species is in no danger of declining. On the other hand, if (x_{i,j}=1) for some i, j, then obviously (x_{l,k}=0) for any (lne i) or (kne j) and$$begin{aligned}{dot{x}}_{i,j}=-delta _{j}< 0.end{aligned}$$Hence, the vector field associated with System (11) points inward from the faces of (S^{mn}). So, (S^{mn}) is positively invariant under the flow of System (11). (square ) Note that Proposition 2 does not exclude the solution trajectories of System (11) from approaching the boundary equilibria of the system as (trightarrow infty ). We call metapopulation model (11) persistent if for every ({mathbf {x}}_0in S^{mn}_{+}), the (omega )-limit set (omega ({mathbf {x}}_0)) does not intersect the boundary of (S^{mn}). In other words, a metapopulation model is persistent if the initial existence of all the species implies that none of the species goes extinct with the passage of time.Balanced homogeneous and heterogeneous metapopulation modelsWe say that the inter-patch migration of a metapopulation model is detailed balanced if the overall migration rate of any species between any two patches is zero for a certain positive set of proportions of that species in the different patches. From the theory of detailed-balanced reaction networks described in “Detailed-balanced single species mass action reaction networks” section, it follows that a detailed-balanced inter-patch migration network corresponds to a detailed-balanced single species mass action reaction network. Let B denote the incidence matrix corresponding to the directed graph (G_2) describing the inter-patch migrations and let r denote the number of edges in (G_2). Comparing Eqs. (8) and (9), it follows that if the inter-patch migration is detailed balanced, then there exist diagonal matrices ({mathcal {K}}in {mathbb {R}}^{rtimes r}) and ({mathbf {Z}}^*in {mathbb {R}}^{mtimes m}) with positive diagonal entries such that ((mathbb {1}^m)^{top }{mathbf {Z}}^*mathbb {1}^m=1) and$$begin{aligned} {mathbf {L}}={mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }({mathbf {Z}}^*)^{-1}. end{aligned}$$Let ({mathbf {Z}}^*=text { diag}({mathbf {z}}^*)). Equation (9) can now be rewritten as$$begin{aligned} Psi _{i}=-{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) . end{aligned}$$ (12) Henceforth in this manuscript, we restrict our analysis to metapopulation models of type (11) for which the interactions within each patch correspond to a tournament with a completely mixed optimal strategy and whose inter-patch migration is detailed balanced. Such metapopulation models will be referred to as balanced metapopulation models.We have seen earlier in “Species interactions and tournament matrices” section that if the interactions within every patch correspond to a tournament with a completely mixed optimal strategy, then the corresponding mean-field model admits a unique coexistence equilibrium ({mathbf {y}}^*in S^{n}_{+}) with ({mathbf {T}}{mathbf {y}}^*={mathbf {0}}^n). Thus, for a balanced metapopulation model, System (10) can be rewritten as$$begin{aligned} phi _{i,j}=x_{i,j}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i, end{aligned}$$ (13) where ({mathbf {Y}}^*:=) diag(({mathbf {y}}^*)). Consequently, from Eqs. (11)–(13), it follows that the dynamics of a balanced metapopulation model containing n species and m patches can be described by mn differential equations$$begin{aligned} {dot{x}}_{i,j}=x_{i,j}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) right) _{j} ,qquad i=1,ldots ,n,quad j=1,ldots ,m . end{aligned}$$ (14) If all the elements of ({mathbf {z}}^*) in the above equation are equal, i.e., if (z_j^*=frac{1}{m}) for (j=1,ldots ,m), then we say that the balanced metapopulation model is homogeneous, otherwise we call it heterogeneous. Whether a balanced metapopulation model is homogeneous or not can be checked from the adjacency matrix ({mathbf {A}}) corresponding to its inter-patch migration graph (G_2). If ({mathbf {A}}) is symmetric, then the model is homogeneous, otherwise it is heterogeneous. Remark 3 In35, the authors assume that migrations from one patch to other patches are random with a probability of migration (or migration constant) equal to the reciprocal of the number of dispersal links from a patch to other patches. They thus define a dispersal graph to be homogeneous if all nodes have the same degree (number of links), otherwise the graph is heterogeneous. With this definition, homogeneity, in general, is equivalent to the existence of cycles in the dispersal graph, whereas heterogeneity is equivalent to their absence. However, with our new definition, it is clear that this is not necessary. An example of such a case is shown in Fig. 2.Figure 2Left: A heterogeneous dispersal graph according to35. Right: A homogeneous dispersal graph according to our definition.Full size image Coexistence equilibrium and its uniquenessIn this section, we present a theorem that gives an expression for a coexistence equilibrium of a balanced metapopulation model. Before we state our main theorem in this section, we need the following lemma. Lemma 4 Let ({mathbf {B}}in {mathbb {R}}^{mtimes r}) denote the incidence matrix of a finite connected directed graph (G_2) and let ({mathcal {K}}in {mathbb {R}}^{rtimes r}) denote a diagonal matrix with positive diagonal entries. For any ({mathbf {w}}in {mathbb {R}}_+^m), it holds that (-{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) ge 0). Moreover (-{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) = 0) if and only if ({mathbf {w}}=qmathbb {1}^m), where (qin {mathbb {R}}_+). Proof Assume that the (p{ {text {th}}}) edge of the graph (G_2) is directed from vertex (i_p) to vertex (j_p). Hence, (B_{i_pp}=-1), (B_{j_pp}=1) and (B_{kp}=0) for (i_pne kne j_p). Thus,$$begin{aligned} -{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) =sum _{p=1}^m(w_{j_p}-w_{i_p})kappa _pleft( frac{1}{w_{i_p}}-frac{1}{w_{j_p}}right) =sum _{p=1}^mfrac{kappa _p}{w_{i_p}w_{j_p}}left( w_{j_p}-w_{i_p}right) ^2ge 0. end{aligned}$$Moreover, (-{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) =0) if and only if (w_{j_p}=w_{i_p}) for (p=1,ldots ,m), which is equivalent with ({mathbf {B}}^{top }{mathbf {w}}={mathbf {0}}^r). Since the graph (G_2) is connected, we recall from48 that (text {rank}({mathbf {B}})=m-1) and furthermore (text {ker}({mathbf {B}}^{top })=mathbb {1}^m). Therefore ({mathbf {B}}^{top }{mathbf {w}}={mathbf {0}}^r) if and only if ({mathbf {w}}=qmathbb {1}^m), where (qin {mathbb {R}}_+). This completes the proof. (square ) We now state the main theorem of this section. Theorem 5 A balanced metapopulation model described by System (14) admits a unique coexistence equilibrium ({mathbf {x}}^*in S^{mn}_{+}). The proportion (x_{i,j}^{*}) of species i in patch j at the unique coexistence equilibrium is given by$$begin{aligned} x_{i,j}^*=y^{*}_iz^{*}_j. end{aligned}$$ (15) for (i=1,ldots ,n) and (j=1,ldots ,m). Proof We divide the proof into two parts. In the first part we prove that System (15) indeed yields an equilibrium for the model. In the second part, we prove that this coexistence equilibrium is unique. Let us define$$begin{aligned} {mathbf {p}}_{j}^*:=left( x_{1,j}^*, x_{2,j}^*, ldots , x_{n,j}^*right) ^top =z_j^*{mathbf {y}}^*; quad {mathbf {r}}_{i}^{*}:=left( x_{i,1}^{*}, x_{i,2}^{*}, ldots , x_{i,m}^{*}right) ^top =y_i^*{mathbf {z}}^*. end{aligned}$$For ({mathbf {x}}^*) to be an equilibrium of System (14), it should render the right-hand side equal to zero. Note that$$begin{aligned} mathbf {TY}^*left( frac{{mathbf {p}}_{j}^*}{{mathbf {y}}^*}right) =z_j^*mathbf {TY}^*mathbb {1}^n=z_j^*{mathbf {T}}{mathbf {y}}^{*}={mathbf {0}}^n end{aligned}$$and$$begin{aligned} {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^*}{{mathbf {z}}^{*}}right) =y_i^*{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }mathbb {1}^m={mathbf {0}}^m. end{aligned}$$In addition,$$begin{aligned} (mathbb {1}^{mn})^{top }{mathbf {x}}^*=sum _{i=1}^nsum _{j=1}^mx_{i,j}^*=sum _{i=1}^{n}y_i^*sum _{j=1}^mz_j^{*}=1. end{aligned}$$Thus, ({mathbf {x}}^*) is a coexistence equilibrium of System (14). Assume that there exists another coexistence equilibrium ({mathbf {x}}^{**}in , S^{mn}_{+}). Let (x_{i,j}^{**}) denote the corresponding proportion of species i in patch j and define$$begin{aligned} {mathbf {p}}_{j}^{**}:=left( x_{1,j}^{**}, x_{2,j}^{**}, ldots , x_{n,j}^{**}right) ^top ; qquad {mathbf {r}}_{i}^{**}:=left( x_{i,1}^{**}, x_{i,2}^{**}, ldots , x_{i,m}^{**}right) ^top . end{aligned}$$It follows that for any i, j it holds that$$begin{aligned} x_{i,j}^{**}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=0. end{aligned}$$ (16) Multiplying both sides of this equality with (frac{x_{i,j}^*}{x_{i,j}^{**}}), we get$$begin{aligned} x_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i- frac{x_{i,j}^*}{x_{i,j}^{**}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=0. end{aligned}$$Summing the left-hand side of the above expression over the different species and patches, we get$$begin{aligned} sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i- sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}^{**}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=0. end{aligned}$$ (17) Now consider the two terms in the left-hand side of the above equality separately. For the first term, note that for any j it holds that$$begin{aligned} sum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i= & {} sum _{i=1}^nx_{i,j}^*left( {mathbf {T}}{mathbf {p}}_{j}^{**}right) _{i} = sum _{i=1}^{n}x_{i,j}^{*}left( sum _{l=1}^{n}T_{il}x_{l,j}^{**}right) =-sum _{l=1}^{n}x_{l,j}^{**}left( sum _{i=1}^{n}T_{li}x_{i,j}^{*}right) \= & {} -sum _{l=1}^{n}x_{l,j}^{**}left( sum _{i=1}^nT_{li}y_i^{*}z_j^*right) =-z_j^*sum _{l=1}^{n}x_{l,j}^{**}({mathbf {T}}{mathbf {y}}^*)_l=0. end{aligned}$$Hence,$$begin{aligned} sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i=0. end{aligned}$$For the second term, we find$$begin{aligned} -sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}^{**}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=-sum _{i=1}^ny_i^*sum _{j=1}^mfrac{z_j^*}{x_{i,j}^{**}}left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=-sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}^{**}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) . end{aligned}$$Thus, Eq. (17) can be simplified as$$begin{aligned} -sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}^{**}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) =0. end{aligned}$$Since (y_i^* >0) for (i=1,ldots ,n), it holds for any (i=1,ldots ,n) that$$begin{aligned} -left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}^{**}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) =0. end{aligned}$$
(18)
From Eq. (18) and Lemma 4, it follows that ({mathbf {r}}_{i}^{**}=q_i{mathbf {z}}^*) with (q_iin {mathbb {R}}_+) for (i=1,ldots ,n). Thus, (x_{i,j}^{**}=q_iz_{j}^*) and ({mathbf {p}}_{j}^{**}=z_j^*{mathbf {q}}) for (i=1,ldots ,n) and (j=1,ldots ,m). Substituting the latter in the left-hand side of Eq. (16), we get$$begin{aligned} x_{i,j}^{**}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=q_i{z_j^*}^2left( {mathbf {T}}{mathbf {Y}}^*left( frac{{mathbf {q}}}{{mathbf {y}}^*}right) right) _i-q_ileft( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }mathbb {1}^mright) _j=q_i{z_j^*}^2(mathbf {Tq})_i. end{aligned}$$Since (q_i >0) for (i=1,ldots ,n), for Eq. (16) to hold, we should have (mathbf {Tq}={mathbf {0}}^n). Also note that$$begin{aligned} (mathbb {1}^{mn})^{top }{mathbf {x}}^{**}=sum _{i=1}^nsum _{j=1}^{m}x_{i,j}^{**}=sum _{i=1}^nq_isum _{j=1}^mz_j^* =sum _{i=1}^nq_i=1. end{aligned}$$Since the metapopulation model is balanced, it follows that ({mathbf {q}}={mathbf {y}}^*). Thus, (x_{i,j}^{**}=y_i^*z_j^*=x_{i,j}^*) for (i=1,ldots ,n) and (j=1,ldots ,m). This proves the uniqueness of the coexistence equilibrium ({mathbf {x}}^*). (square )
We now give examples of two balanced metapopulation models.

Example 1

It is easy to verify that the network shown in Fig. 3 corresponds to a balanced metapopulation model governed by System (14) with$$begin{aligned} {mathbf {T}} = left[ begin{array}{rrr} 0 &{}quad 1 &{}quad -1\ -1 &{}quad 0 &{}quad 1\ 1 &{}quad -1 &{}quad 0 end{array}right] ; quad {mathbf {B}} = left[ begin{array}{rrr} -1 &{}quad 0 &{}quad 1\ 1 &{}quad -1 &{}quad 0\ 0 &{}quad 1 &{}quad -1 end{array}right] ; end{aligned}$$({mathbf {y}}^*=left( frac{1}{3}, frac{1}{3}, frac{1}{3} right) ^{top }), ({mathbf {z}}^*=left( frac{1}{5}, frac{2}{5}, frac{2}{5} right) ^{top }) and ({mathcal {K}}=text { diag}left( frac{1}{10},frac{3}{10},frac{1}{10}right) ). Note that this metapopulation model is heterogeneous. From Theorem 5, it follows that the species proportions at the unique coexistence equilibrium for this model are given by (x_{i,1}^*=frac{1}{15}) and (x_{i,2}^*=x_{i,3}^*= frac{2}{15}).Figure 3A metapopulation network composed of three patches. Each patch contains a local population composed of three species (1, 2 and 3), in cyclic competition, as shown by the black arrows. The red arrows denote migrations among the patches in the directions shown.Full size image

Example 2

It is easy to verify that the network shown in Fig. 4 corresponds to a balanced metapopulation model governed by System (14) with$$begin{aligned} {mathbf {T}} = left[ begin{array}{rrr} 0 &{}quad 1 &{} quad -1\ -1 &{}quad 0 &{} quad 1\ 1 &{}quad -1 &{}quad 0 end{array}right] ; quad {mathbf {B}} = left[ begin{array}{rrr} 1 &{}quad -1\ 0 &{}quad 1\ -1 &{}quad 0 end{array}right] ; end{aligned}$$({mathbf {y}}^{*}={mathbf {z}}^*=left( frac{1}{3}, frac{1}{3}, frac{1}{3} right) ^{top }) and ({mathcal {K}}=frac{1}{3}text { diag}(mathbb {1}_2)). Note that this metapopulation model is homogeneous. From Theorem 5, it follows that the species proportions at the unique coexistence equilibrium in this case are all given by (x_{i,j}^*=frac{1}{9}) for (i,j= 1,2,3).Figure 4A metapopulation network composed of three patches. Species can migrate from patch 1 to the other two patches and vice versa. However, there exists no migrations between patches 2 and 3.Full size image
StabilityWe now prove the local stability of the unique coexistence equilibrium corresponding to the balanced metapopulation model (14). For the proof, we make use of the same Lyapunov function as in “Neutral stability” section, coupled with LaSalle’s invariance principle51, (52, Section 4.2), (53, pp. 188–189).

Theorem 6

Consider the balanced metapopulation model (14) with coexistence equilibrium ({mathbf {x}}^*in , S^{mn}_{+}).

1.

If the model is heterogeneous, then ({mathbf {x}}^*) is locally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}) in the neighbourhood of ({mathbf {x}}^*). Furthermore, if the model is persistent, then ({mathbf {x}}^*) is globally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}).

2.

If the model is homogeneous and persistent, then as (trightarrow infty ), the solution trajectories converge to a limit cycle satisfying the equation ({dot{x}}_{i,j}=x_{i,j}({mathbf {T}}{mathbf {p}}_{j})_i) with (x_{i,j}=x_{i,k}), for (i=1,ldots ,n) and (j,k=1,ldots ,m).

Proof
Let (x_{i,j}) denote the proportion of species i in patch j. Assuming that ({mathbf {x}}in S^{mn}_{+}), consider the Lyapunov function$$begin{aligned} V({mathbf {x}})=-(mathbf {x^{*}})^{top }text {Ln}left( frac{{mathbf {x}}}{{mathbf {x}}^*}right) . end{aligned}$$
(19)
By Gibbs inequality, V(x) is positive on (S^{mn}_{+}) and is equal to zero only if ({mathbf {x}}={mathbf {x}}^*). Taking the time derivative of V, we have$$begin{aligned} {dot{V}}({mathbf {x}})=-sum _{j=1}^msum _{i=1}^{n}left( frac{x_{i,j}^{*}}{x_{i,j}}right) {dot{x}}_{i,j}. end{aligned}$$From Eq. (14), it follows that$$begin{aligned} {dot{V}}({mathbf {x}})= -sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i+ sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) right) _{j}. end{aligned}$$As in the proof of Theorem 5, it can be verified that$$begin{aligned} sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i=0 end{aligned}$$and$$begin{aligned} sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) right) _{j}=sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) . end{aligned}$$Thus,$$begin{aligned} {dot{V}}({mathbf {x}})=sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) . end{aligned}$$Since (y_i^* >0) for (i=1,ldots ,n), it follows from Lemma 4 that ({dot{V}}({mathbf {x}})le 0) and ({dot{V}}({mathbf {x}})=0) if and only if ({mathbf {r}}_i=q_i{mathbf {z}}^*) with (q_iin {mathbb {R}}_+), for (i=1,ldots ,n). Thus,$$begin{aligned} x_{i,j}=q_iz_j^*, end{aligned}$$
(20)
for (i= 1,ldots ,n) and (j=1,ldots ,m). Since ((mathbb {1}^{mn})^{top }{mathbf {x}}=1), we obtain$$begin{aligned} sum _{i=1}^nsum _{j=1}^{m}x_{i,j}=sum _{i=1}^nq_isum _{j=1}^mz_j^* =sum _{i=1}^nq_i=1. end{aligned}$$Let ({mathcal {E}}subset S^{mn}_{+}) be the set of all vectors ({mathbf {x}}) for which condition (20) is satisfied with ((mathbb {1}^n)^{top }{mathbf {q}}=1). We now determine the largest subset of ({mathcal {E}}) that is positively invariant w.r.t. System (14). Assume that ({mathbf {x}}) continuously takes values from ({mathcal {E}}) and satisfies System (14). Since ({mathbf {x}}) takes values from ({mathcal {E}}), we have ({dot{x}}_{i,j}=z_j^*{dot{q}}_i). Since ({mathbf {x}}) also satisfies System (14), we have$$begin{aligned} {dot{x}}_{i,j}=x_{i,j}left( {mathbf {T}}{mathbf {p}}_{j}right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{*}}{{mathbf {z}}^{*}}right) right) _{j}=q_i{z_j^*}^2(mathbf {Tq})_i-q_ileft( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }mathbb {1}^mright) _j=q_i{z_j^*}^2(mathbf {Tq})_i. end{aligned}$$Thus, (z_j^*{dot{q}}_i=q_i{z_j^*}^2(mathbf {Tq})_i) which implies that$$begin{aligned} {dot{q}}_i=z_j^*q_i(mathbf {Tq})_i, end{aligned}$$
(21)
for (i=1,ldots ,n) and (j=1,ldots ,m). We now consider two cases.
Case 1: The model is heterogeneous, i.e., the vector ({mathbf {z}}^*) is not parallel to (mathbb {1}^m).
In this case, Eq. (21) will be satisfied only if (q_i(mathbf {Tq})_i=0) for (i=1,ldots ,n). Since (q_iin {mathbb {R}}_+) for (i=1,ldots ,n), it follows that (mathbf {Tq}={mathbf {0}}^n). Since ((mathbb {1}^n)^{top }{mathbf {q}}=1), we have ({mathbf {q}}={mathbf {y}}^*). This implies that (x_{i,j}=y_i^*z_j^*=x_{i,j}^*) for (i=1,ldots ,n) and (j= 1,ldots ,m). Thus, the largest subset of ({mathcal {E}}) that is positively invariant w.r.t. System (14) consists of just the unique equilibrium ({mathbf {x}}^*in S^{mn}_{+}). By LaSalle’s invariance principle, it follows that the equilibrium ({mathbf {x}}^*) is locally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}) in the neighbourhood of ({mathbf {x}}^*), and globally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}) provided that System (14) is persistent.

Case 2: The model is homogeneous, i.e.
({mathbf {z}}^*=frac{1}{m}mathbb {1}^m)

In this case, Eq. (21) takes the form ({dot{q}}_i=frac{q_i}{m}(mathbf {Tq})_i). We have (x_{i,j}=q_iz_j^*=frac{q_i}{m}) and$$begin{aligned} {dot{x}}_{i,j}=frac{{dot{q}}_i}{m}=frac{q_i}{m^2}(mathbf {Tq})_i=x_{i,j}({mathbf {T}}{mathbf {p}}_{j})_i. end{aligned}$$Consequently, the largest subset of ({mathcal {E}}) that is positively invariant w.r.t. System (14) consists of all vectors ({mathbf {x}}(t)in , S^{mn}_{+}) satisfying ({dot{x}}_{i,j}=x_{i,j}({mathbf {T}}{mathbf {p}}_{j})_i) with (x_{i,j}=x_{i,k}) for (i=1,ldots ,n) and (j,k=1,ldots ,m). The proof for Case 2 again follows from LaSalle’s invariance principle. (square )
The above results can be illustrated by simulating System (14) for the metapopulation models shown in Fig. 3 and 4 in Examples 1 and 2, respectively. The results of the simulations are shown in Figs. 5 and 6, respectively.Figure 5Left: Dynamics of the metapopulation model in Fig. 3 for patches 1 and 3 showing asymptotic stability of the coexistence equilibrium. Right: The time evolution of the proportion of species 1 in the three patches.Full size imageFigure 6Left: Dynamics of the metapopulation model in Fig. 4 for patches 1 and 3 showing a limit cycle arising from the neutral stability of the coexistence equilibrium. Right: Time evolution of the proportion of species 1 in the three patches. Note that the dynamics in all patches are the same and thus the three graphs overlap.Full size image More

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SoilContents of risk metals in soilsLands of localities from which soil and plant samples were taken belong to agricultural lands.Soil reaction is one of the factors that most affects the behaviour of heavy metals in soil. Low pH values pose a risk of reduced nutrient intake and increase the availability of heavy metals for plants29,30.The presence of risk elements in the soil was evaluated based on their contents in bioavailable form (mobile forms), determined in soil extracts NH4NO3, and the total contents of risk elements were determined in soil extract by aqua regia (Table 1).Table 1 The contents of risk elements (Cu, Ni, Pb, Cd, Hg, Mn) in soil (mg/kg).Full size tableAccessible heavy metals for plants are those which are present in the soil solution as soluble components or those which are easily dissolved by root exudates31. The highest Cu contents determined in soil extract by NH4NO3, were in the cadastre of Gemerská Poloma (max. 0.390 mg/kg) (Table 1). However, even the highest determined concentration of Cu in its bioavailable form did not exceeded the determined critical value for this element18. Nickel is a beneficial element for plants. Elevated Ni concentrations in soils have a potential negative effect on plants32. Content of bioavailable forms of nickel is lower than the determined critical value in all analysed samples. Cadmium and lead present a risk to agricultural activity in this area. Cadmium in soil is highly bioavailable and has higher mobility in plants compared to other heavy metals. It is easily transported by roots to shoots. In contrast, lead is one of the least mobile heavy metals. It is naturally concentrated in the upper layers of the soil33. The contents of the available forms of cadmium and lead exceed the critical values for these elements. In case of lead, the determined contents are from 0.257 Henckovce to 0.676 Gemerská Poloma. Takáč et al.34 determined in 20 soil samples from the Central Spiš region 7.2–257.6 mg Cu/kg soil and 1.0–84.8 mg Pb/kg in their potentially mobilizable form and 0.4–1.4 mg Cu/kg soil and 4.3–7.1 mg Pb/kg in their mobile form. In comparison with our results, Vilček et al.35 determined a lower content of Cd (0.04), Pb (0.17), Ni (0.15) and higher Cu content (0.48) mg/kg in forms accessible to plants in 16 soil samples from locality Nižná Slaná in the years 2006–2008. However, high concentrations of metals in soil do not necessarily mean the availability of metals for plants36. As a result, extractable Mn is often a better indicator of Mn availability. Mn2+ is generally considered to be bioavailable22. The highest concentration of Mn was measured in soil samples from the cadastre of Nižná Slaná. On the contrary, the lowest concentrations were detected in samples from Gemerská Poloma cadastre, which is the furthest cadastre from the source. No critical limit is set up for manganese according to Slovak legislation, it is not possible to classify these soils as contaminated/uncontaminated. For comparison, the EDTA-extractable content of Mn ranged from 22.7 to 127 mg/kg dry soil (China)29; the mobile concentrations between 0.32 and 202.0 mg/kg and the available concentrations from 5.4 to 126.3 mg/kg (Egypt)37.Based on results of statistical analysis, significant higher content of Cu, Pb and Cd can be stated in samples from Gemerská Poloma cadastre. These soils are classified as gley fluvisols, soils from the other two localities are cambisols (from medium heavy to light) and acid cambisols (Henckovce), cambisols from medium heavy to light and typically acid cambisols (Nižná Slaná). The soil profile of fluvisols is repeatedly disrupted by floods, which often enriches them with a new layer of sludge sediments2.Another method for determination of metal content in soil is mineralisation using aqua regia, which dissolves most of the soil constituents except those strongly bound in silicate minerals. This content is sometimes referred to as pseudototal (determined in aqua regia). In this way, all elements that are likely to become bioavailable in the long term are determined38.Pseudototal contents of risk metals (Table 1) determined in soil extract using aqua regia were higher than their limit value in case of Cu (Gemerská Poloma cadastre), Cd (all cadastres) and Hg (cadastre of Henckovce and Gemerská Poloma).Due to the fact that the hygienic condition of agricultural soils is assessed according to the exceeding of the limit values of at least one risk substance, the monitored plots can be classified as contaminated (Cu  > 60.0, Cd  > 0.7, Hg  > 0.5 mg/kg soil).Manganese is not classified as risk element in Slovak legislation.Tóth et al.39 classified European soils into four categories: (1) no detectable content of HM, (2) the concentration of the investigated element is above the threshold value (Hg 0.5, Cd 1, Cu 100, Pb 60 and Ni 50 mg/kg), but below the lower guideline value (Hg 2, Cd 10, Cu 150, Pb 200 and Ni 100 mg/kg), (3) concentration of one or more element exceeds the lower guideline value but is below the higher guideline value (Hg 5, Cd 20, Cu 200, Pb 750 and Ni 150 mg/kg), (4) samples having concentrations above the higher guideline value.In comparison with the threshold and guideline values, soils in cadastres of Gemerská Poloma (Cu), Henckovce, Nižná Slaná, Gemerská Poloma (Cd, Hg) represent the ecological risk. Threshold and guideline values for Mn were not defined.The Spiš region and the northern part of the Gemer region belong to the most polluted areas in Slovakia in terms of soil contamination due to mining and metallurgical activities that have been carried out here in the past. Soils near the sludge in Nižná Slaná contain 3.17–53.3 (14.2–301, 0.71–20.6, 3.33–177, 12.9–223 and 675–11,510, respectively) mg Cd (Cu, Hg, Ni, Pb and Mn, respectively)/kg of soil14. In loaded area of Dongchuan, (China), contained Cd (Cu, Hg, Ni and Pb, resp.) 0.20–3.57 (45.38–2026, 0.02–0.23, 24.06–95.9 and 6.83–146.6, resp.) mg/kg40. In contrast, in the agricultural area of Punjab of the India, the soil contamination was caused by an excessive use of agrochemicals and polluted irrigation sources. Increased Cu (Pb and Cd) contents were determined in the soil samples: 9.0–48.5 (5.5–9.67 and 0.516–1.58, resp.) mg/kg41.However, in most cases, a large portion of the total element content is not available for immediate uptake by plants. Available forms represent a small proportion of this total content which is potentially available to plants. Availability is affected by many factors, including pH, redox state, macronutrient levels, available water content and temperature29,33,36,38.Indicators of soil contaminationContamination factors and degree of contaminationThe contamination character may be described in a uniform, adequate and standardised way by means of the contamination factor and the degree of contamination. Hakanson24 reported four Contamination degrees of individual metal (({mathrm{C}}_{mathrm{f}}^{mathrm{i}})) – low (({mathrm{C}}_{mathrm{f}}^{mathrm{i}}) < 1), moderate (1 ≤ ({mathrm{C}}_{mathrm{f}}^{mathrm{i}})   More

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