Cluster formation in a heterogeneous metapopulation model

Abstract

A spatially explicit heterogeneous metapopulation model with two different patch types is analyzed. Some network topologies support a partially synchronized dynamics, a state where two different clusters of patches are formed. Within each cluster the dynamics of all patches are synchronized. The linearized asymptotic stability of the partially synchronized attractor is studied. The transversal stability is analyzed and a simple expression for the transversal Lyapunov number of partially synchronized attractors is obtained.

Appendix

Lemma 1

$S^{\bot }$ is C-invariant.

Proof

Since the set $B_{S^{\bot }}$ is a basis for $S^{\bot }$, it will be enough to show that $Cv\in S^{\bot }$ for all $v\in S^{\bot }$. Clearly for $i=1,2\ldots ,k-1$ we have

$$\begin{aligned} Cu_i \in S^{\bot }\Leftrightarrow Cu_i =a_{1,i} u_1 +\cdots +a_{k-1,i} u_{k-1} +a_{k,i} w_1 +\cdots +a_{n-2,i} w_{n-k-1},\nonumber \\ \end{aligned}$$

(37)

for some coefficients $a_{mi} \in \mathfrak {R}$, $m=1,2,\ldots ,n-2$. Taking in account the definitions of the basis vectors $u_i$’s and $w_i$’s given in (16), solving for the $a_{mi}$’s we can write the right hand side of (37) in the equivalent form

$$\begin{aligned} \left\{ {\begin{array}{ll} -\sum \limits _{m=1}^{k-1} {a_{mi} =-c_{11} +c_{1,i+1} } &{}\\ a_{mi} =-c_{m+1,1} +c_{m+1,i+1},&{}\quad \quad m=1,2,\ldots ,k-1 \\ -\sum \limits _{m=k}^{n-2} {a_{mi} =-c_{k+1,1} +c_{k+1,i+1} } &{}\\ a_{mi} =-c_{m+2,1} +c_{m+2,i+1},&{}\quad \quad m=k,k+1,\ldots ,n-2. \\ \end{array}} \right. \end{aligned}$$

(38)

Substituting the $a_{mi}$’s by the corresponding expressions depending on the coefficients of the matrix C, we can restate the above condition using only the coefficients of the connectivity matrix to obtain

$$\begin{aligned} \left\{ {\begin{array}{l} \sum \limits _{m=1}^{k-1} {(c_{m+1,1} -c_{m+1,i+1} )} =-c_{11} +c_{1,i+1} \\ \sum \limits _{m=k}^{n-2} {(c_{m+2,1} -c_{m+2,i+1} )} =-c_{k+1,1} +c_{k+1,i+1}. \\ \end{array}} \right. \end{aligned}$$

(39)

Rearranging the sums in (39) we can finally write

$$\begin{aligned} Cu_i \in S^{\bot }\Leftrightarrow \left\{ {\begin{array}{l} \sum \limits _{m=1}^k {c_{m1} =\sum \limits _{m=1}^k {c_{m,i+1} } } \\ \sum \limits _{m=k+1}^n {c_{m1} =\sum \limits _{m=k+1}^n {c_{m,i+1} } } \\ \end{array}} \right. \end{aligned}$$

(40)

But using the hyppothesis that the connectivity matrix can be partitioned into four submatrices of equal column sum given in Eq. (15) we can verify that $\sum \nolimits _{m=1}^k {c_{m1} =\sum \nolimits _{m=1}^k {c_{m,i+1} ={\alpha }^{\prime }}}$ and that $\sum \nolimits _{m=k+1}^n {c_{m1} =\sum \nolimits _{m=k+1}^n {c_{m,i+1} ={\gamma }^{\prime }}}$ for all $i=1,2,\ldots ,k-1$. Therefore, (40) implies that $Cu_i \in S^{\bot }$ for all $i=1,2\ldots ,k-1$.

We can follow the same steps of the above argument to show that $Cw_i\in S^{\bot }$ for all $i=1,2,\ldots ,n-k-1$. It is important to notice that similar to the previous case, the vectors $Cw_i$’s can be written as a linear combination of the elements of the basis $B_{S^{\bot }}$. Thus, $i=1,2,\ldots ,n-k-1$ we can have

$$\begin{aligned} Cw_i= & {} a_{1,k-1+i} u_1 +\cdots +a_{k-1,k-1+i} u_{k-1} +a_{k,k-1+i}w_1\nonumber \\&+\cdots +a_{n-2,k-1+i} w_{n-k-1}, \end{aligned}$$

(41)

where the coefficients are given by

$$\begin{aligned} \left\{ {\begin{array}{ll} a_{m,k-1+i} =-c_{m+1,k+1} +c_{m+1,k+i+1},&{}\quad m=1,2,\ldots ,k-1 \\ a_{m,k-1+i} =-c_{m+2,k+1} +c_{m+2,k+i+1},&{}\quad m=k,k+1,\ldots ,n-2. \\ \end{array}} \right. \end{aligned}$$

(42)

Lemma 2

The entries of the matrix $\tilde{C}$ are given in (23).

Proof

The columns of the matrix $J_{B_{S\bot }}(s_t)$ are the vectors $J(s_t)u_i$, $i=1,2,\ldots ,k-1$ and $J(s_t)w_i$, $i=1,2,\ldots ,n-k-1$. Using (12) we can write,

$$\begin{aligned} J(s_t ) u_i =(I-\mu (I-C))D_t u_i,\quad i=1,2,\ldots ,k-1. \end{aligned}$$

(43)

Observe that as a consequence of (13) we have $D_tu_i={f}^{\prime }(x_t)u_i$, thus

$$\begin{aligned} J(s_t)u_i=(1-\mu ){f}^{\prime }(x_t)u_i+\mu {f}^{\prime }(x_t)Cu_i. \end{aligned}$$

(44)

But, from Lemma 1, $S^{\bot }$ is C-invariant, thus $Cu_i$ is a linear combination of the vectors in $B_{S^{\bot }}$ and consequently

$$\begin{aligned} J(s_t)u_i= & {} (1-\mu ){f}^{\prime }(x_t )u_i \nonumber \\&+\;\mu {f}^{\prime }(x_t ) \left( {\sum \limits _{m=1}^{k-1} {a_{mi} u_m +\sum \limits _{m=1}^{n-k-1} {a_{k-1+m,i}w_m}}}\right) ,\quad i=1,2,\ldots ,k-1.\nonumber \\ \end{aligned}$$

(45)

Regrouping the terms we may write,

$$\begin{aligned} J(s_t)u_i= & {} b_{1,i} u_1 +\cdots +b_{k-1,i} u_{k-1} +b_{k,i} w_1 +\cdots +b_{n-2,i} w_{n-k-1},\nonumber \\&i=1,2,\ldots ,k-1, \end{aligned}$$

(46)

where the coefficients of the above linear combination are given by

$$\begin{aligned} b_{mi} =\left\{ {\begin{array}{ll} \mu a_{mi} {f}^{\prime }(x_t ),&{}\quad m\ne i \\ (1-\mu (1-a_{mm} )){f}^{\prime }(x_t ),&{}\quad m=i \\ \end{array}} \right. \end{aligned}$$

(47)

$m=1,2,\ldots ,n-2,\quad i=1,2,\ldots ,k-1$. Similar calculations lead to

$$\begin{aligned} J(s_t )w_i= & {} b_{1,k-1+i} u_1 +\cdots +b_{k-1,k-1+i} u_{k-1} +b_{k,k-1+i} w_1 \nonumber \\&+\cdots +b_{n-2,k-1+i} w_{n-k-1}, \end{aligned}$$

(48)

where

$$\begin{aligned} b_{m,k-1+i}=\left\{ {\begin{array}{ll} \mu a_{m,k-1+i} {g}^{\prime }(y_t ),&{}\quad m\ne k-1+i \\ (1-\mu (1-a_{mm} )){g}^{\prime }(y_t ),&{}\quad m=k-1+i \\ \end{array}} \right. \end{aligned}$$

(49)

$m=1,2,\ldots ,n-2,\quad i=1,2,\ldots ,n-k-1$. The coefficients on the left hand side of (47) and (49) are the entries of the matrix $J_{B_{S\bot }}(s_t)$. Define the n $-$ 2 $\times$ n $-$ 2 matrix A with the entries given in (38) and (42). As a consequence of (47) and (49) we can write

$$\begin{aligned} J_{B_{S\bot } } (s_t )=(I-\mu (I-A))\tilde{D}_t. \end{aligned}$$

(50)

It follows from a comparison of (50) with (21) that $\tilde{C}=A$. The formulas in (23) are obtained by reindexing the entries in (38) and (42).